Note
Go to the end to download the full example code or to run this example in your browser via Binder.
Bayesian analysis with nested sampling#
A demonstration of a Bayesian analysis using the nested sampling technique.
Context#
1. Bayesian analysis#
Bayesian inference uses prior knowledge, in the form of a prior distribution, in order to estimate posterior probabilities which we traditionally visualise in the form of corner plots. These distributions contain more information than a maximum likelihood fit as they reveal not only the “best model” but provide a more accurate representation of errors and correlation between parameters. In particular, non-Gaussian degeneracies are complex to estimate with a maximum likelihood approach.
2. Limitations of the Markov Chain Monte Carlo approach#
A well-known approach to estimate this posterior distribution is the Markov Chain Monte Carlo (MCMC). This uses an ensemble of walkers to produce a chain of samples that after a convergence period will reach a stationary state. Once convergence is reached, the successive elements of the chain are samples of the target posterior distribution. However, the weakness of the MCMC approach lies in the “Once convergence” part. If the walkers are started far from the best likelihood region, the convergence time can be long or never reached if the walkers fall in a local minima. The choice of the initialisation point can become critical for complex models with a high number of dimensions and the ability of these walkers to escape a local minimum or to accurately describe a complex likelihood space is not guaranteed.
3. Nested sampling approach#
To overcome these issues, the nested sampling (NS) algorithm has gained traction in physics and astronomy. It is a Monte Carlo algorithm for computing an integral of the likelihood function over the prior model parameter space introduced in Skilling, 2004. The method performs this integral by evolving a collection of points through the parameter space (see recent reviews from Ashton et al., 2022, and Buchner, 2023). Without going into too many details, one important specificity of the NS method is that it starts from the entire parameter space and evolves a collection of live points to map all minima (including multiple modes if any), whereas Markov Chain Monte Carlo methods require an initialisation point and the walkers will explore the local likelihood. The ability of these walkers to escape a local minimum or to accurately describe a complex likelihood space is not guaranteed. This is a fundamental difference with MCMC or Minuit which will only ever probe the vicinity along their minimisation paths and do not have an overview of the global likelihood landscape. The analysis using the NS framework is more CPU time consuming than a standard classical fit, but it provides the full posterior distribution for all parameters, which is out of reach with traditional fitting techniques (N*(N-1)/2 contour plots to generate). In addition, it is more robust to the choice of initialisation, requires less human intervention and is therefore readily integrated in pipeline analysis. In Gammapy, we used the NS implementation of the UltraNest package (see here for more information), one of the leading package in Astronomy (already used in Cosmology and in X-rays). For a nice visualisation of the NS method see here : sampling visualisation. And for a tutorial of UltraNest applied to X-ray fitting with concrete examples and questions see : BXA Tutorial.
Note: please cite UltraNest if used for a paper
If you are using the “UltraNest” library for a paper, please follow its citation scheme: Cite UltraNest.
Proposed approach#
In this example, we will perform a Bayesian analysis with multiple 1D spectra of the Crab nebula data and investigate their posterior distributions.
Setup#
As usual, we’ll start with some setup …
import matplotlib.pyplot as plt
import numpy as np
import astropy.units as u
from gammapy.datasets import Datasets
from gammapy.datasets import SpectrumDatasetOnOff
from gammapy.modeling.models import (
SkyModel,
UniformPrior,
LogUniformPrior,
)
from gammapy.modeling.sampler import Sampler
Loading the spectral datasets#
Here we will load a few Crab 1D spectral data for which we will do a fit.
Model definition#
Now we want to define the spectral model that will be fitted to the data. The Crab spectra will be fitted here with a simple powerlaw for simplicity.
model = SkyModel.create(spectral_model="pl", name="crab")
Warning
Priors definition: Unlike a traditional fit where priors on the parameters are optional, here it is inherent to the Bayesian approach and are therefore mandatory.
In this case we will set (min,max) prior that will define the
boundaries in which the sampling will be performed.
Note that it is usually recommended to use a LogUniformPrior for
the parameters that have a large amplitude range like the
amplitude parameter.
A UniformPrior means that the samples will be drawn with uniform
probability between the (min,max) values in the linear or log space
in the case of a LogUniformPrior.
model.spectral_model.amplitude.prior = LogUniformPrior(min=1e-12, max=1e-10)
model.spectral_model.index.prior = UniformPrior(min=1, max=5)
datasets.models = [model]
print(datasets.models)
DatasetModels
Component 0: SkyModel
Name : crab
Datasets names : None
Spectral model type : PowerLawSpectralModel
Spatial model type :
Temporal model type :
Parameters:
index : 2.000 +/- 0.00
amplitude : 1.00e-12 +/- 0.0e+00 1 / (TeV s cm2)
reference (frozen): 1.000 TeV
Defining the sampler and options#
As for the Fit object, the Sampler object can receive
different backend (although just one is available for now).
The Sampler comes with “reasonable” default parameters, but you can
change them via the sampler_opts dictionary.
Here is a short description of the most relevant parameters that you
could change :
live_points: minimum number of live points throughout the run. More points allow to discover multiple peaks if existing, but is slower. To test the Prior boundaries and for debugging, a lower number (~100) can be used before a production run with more points (~400 or more).frac_remain: the cut-off condition for the integration, set by the maximum allowed fraction of posterior mass left in the live points vs the dead points. High values (e.g., 0.5) are faster and can be used if the posterior distribution is a relatively simple shape. A low value (1e-1, 1e-2) is optimal for finding peaks, but slower.log_dir: directory where the output files will be stored. If set to None, no files will be written. If set to a string, a directory will be created containing the ongoing status of the run and final results. For time consuming analysis, it is highly recommended to use that option to monitor the run and restart it in case of a crash (withresume=True).
Important note: unlike the MCMC method, you don’t need to define the number of steps for which the sampler will run. The algorithm will automatically stop once a convergence criteria has been reached.
sampler_opts = {
"live_points": 300,
"frac_remain": 0.3,
"log_dir": None,
}
sampler = Sampler(backend="ultranest", sampler_opts=sampler_opts)
Next we can run the sampler on a given dataset. No options are accepted in the run method.
[ultranest] Sampling 300 live points from prior ...
Mono-modal Volume: ~exp(-4.15) * Expected Volume: exp(0.00) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|**************************** *** **** ** ** ***| +1.0e-10
Z=-inf(0.00%) | Like=-3585.15..-66.01 [-3585.1453..-317.9211] | it/evals=0/301 eff=0.0000% N=300
Z=-550.8(0.00%) | Like=-544.83..-66.01 [-3585.1453..-317.9211] | it/evals=18/320 eff=90.0000% N=300
Z=-531.5(0.00%) | Like=-525.73..-59.59 [-3585.1453..-317.9211] | it/evals=30/333 eff=90.9091% N=300
Z=-510.0(0.00%) | Like=-504.86..-59.59 [-3585.1453..-317.9211] | it/evals=48/352 eff=92.3077% N=300
Z=-496.9(0.00%) | Like=-491.88..-59.59 [-3585.1453..-317.9211] | it/evals=60/364 eff=93.7500% N=300
Mono-modal Volume: ~exp(-4.15) Expected Volume: exp(-0.22) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|***************************** ** ******* ** ****| +1.0e-10
Z=-478.6(0.00%) | Like=-472.17..-59.59 [-3585.1453..-317.9211] | it/evals=76/381 eff=93.8272% N=300
Z=-463.0(0.00%) | Like=-455.65..-59.59 [-3585.1453..-317.9211] | it/evals=90/398 eff=91.8367% N=300
Z=-451.6(0.00%) | Like=-446.35..-59.59 [-3585.1453..-317.9211] | it/evals=103/416 eff=88.7931% N=300
Z=-430.0(0.00%) | Like=-424.56..-59.59 [-3585.1453..-317.9211] | it/evals=118/436 eff=86.7647% N=300
Z=-428.7(0.00%) | Like=-422.11..-59.59 [-3585.1453..-317.9211] | it/evals=120/438 eff=86.9565% N=300
Mono-modal Volume: ~exp(-4.43) * Expected Volume: exp(-0.45) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|***************************** ********** ** ***| +1.0e-10
Z=-408.5(0.00%) | Like=-402.78..-59.59 [-3585.1453..-317.9211] | it/evals=134/454 eff=87.0130% N=300
Z=-393.5(0.00%) | Like=-387.12..-59.59 [-3585.1453..-317.9211] | it/evals=150/472 eff=87.2093% N=300
Z=-377.1(0.00%) | Like=-371.39..-59.59 [-3585.1453..-317.9211] | it/evals=165/492 eff=85.9375% N=300
Z=-369.3(0.00%) | Like=-362.77..-59.59 [-3585.1453..-317.9211] | it/evals=178/511 eff=84.3602% N=300
Z=-367.9(0.00%) | Like=-360.12..-59.59 [-3585.1453..-317.9211] | it/evals=180/514 eff=84.1121% N=300
Z=-343.3(0.00%) | Like=-337.06..-59.59 [-3585.1453..-317.9211] | it/evals=196/533 eff=84.1202% N=300
Mono-modal Volume: ~exp(-4.84) * Expected Volume: exp(-0.67) Quality: ok
index : +1.0| *********************************************| +5.0
amplitude: +1.0e-12| **************************** ** ********** ***| +1.0e-10
Z=-340.7(0.00%) | Like=-333.99..-59.59 [-3585.1453..-317.9211] | it/evals=201/539 eff=84.1004% N=300
Z=-327.0(0.00%) | Like=-319.70..-59.59 [-3585.1453..-317.9211] | it/evals=210/553 eff=83.0040% N=300
Z=-315.8(0.00%) | Like=-308.24..-59.59 [-316.9827..-180.7588] | it/evals=225/571 eff=83.0258% N=300
Z=-300.9(0.00%) | Like=-294.91..-59.59 [-316.9827..-180.7588] | it/evals=240/587 eff=83.6237% N=300
Z=-290.6(0.00%) | Like=-281.38..-59.59 [-316.9827..-180.7588] | it/evals=255/606 eff=83.3333% N=300
Mono-modal Volume: ~exp(-4.84) Expected Volume: exp(-0.89) Quality: ok
index : +1.0| ********************************************| +5.0
amplitude: +1.0e-12| *************************** ** *********** ***| +1.0e-10
Z=-278.8(0.00%) | Like=-272.15..-59.59 [-316.9827..-180.7588] | it/evals=270/623 eff=83.5913% N=300
Z=-258.8(0.00%) | Like=-252.03..-59.59 [-316.9827..-180.7588] | it/evals=287/642 eff=83.9181% N=300
Z=-249.3(0.00%) | Like=-242.99..-59.59 [-316.9827..-180.7588] | it/evals=300/658 eff=83.7989% N=300
Z=-242.0(0.00%) | Like=-233.94..-59.59 [-316.9827..-180.7588] | it/evals=312/677 eff=82.7586% N=300
Z=-228.8(0.00%) | Like=-222.37..-59.59 [-316.9827..-180.7588] | it/evals=326/696 eff=82.3232% N=300
Z=-226.6(0.00%) | Like=-220.18..-59.59 [-316.9827..-180.7588] | it/evals=330/702 eff=82.0896% N=300
Mono-modal Volume: ~exp(-5.24) * Expected Volume: exp(-1.12) Quality: ok
index : +1.0| ******************************************| +5.0
amplitude: +1.0e-12| **************************************** ***| +1.0e-10
Z=-223.1(0.00%) | Like=-217.02..-59.59 [-316.9827..-180.7588] | it/evals=335/714 eff=80.9179% N=300
Z=-214.6(0.00%) | Like=-206.12..-59.59 [-316.9827..-180.7588] | it/evals=351/733 eff=81.0624% N=300
Z=-209.4(0.00%) | Like=-202.59..-59.59 [-316.9827..-180.7588] | it/evals=360/745 eff=80.8989% N=300
Z=-201.7(0.00%) | Like=-195.10..-59.59 [-316.9827..-180.7588] | it/evals=375/763 eff=80.9935% N=300
Z=-195.6(0.00%) | Like=-188.81..-59.59 [-316.9827..-180.7588] | it/evals=389/783 eff=80.5383% N=300
Z=-195.0(0.00%) | Like=-188.64..-59.59 [-316.9827..-180.7588] | it/evals=390/784 eff=80.5785% N=300
Mono-modal Volume: ~exp(-5.28) * Expected Volume: exp(-1.34) Quality: ok
index : +1.0| **************************************** | +5.0
amplitude: +1.0e-12| *************************************** ***| +1.0e-10
Z=-192.3(0.00%) | Like=-185.79..-59.59 [-316.9827..-180.7588] | it/evals=402/800 eff=80.4000% N=300
Z=-187.3(0.00%) | Like=-181.34..-59.59 [-316.9827..-180.7588] | it/evals=418/819 eff=80.5395% N=300
Z=-186.4(0.00%) | Like=-180.12..-59.59 [-180.1784..-133.8131] | it/evals=420/821 eff=80.6142% N=300
Z=-183.4(0.00%) | Like=-177.32..-59.59 [-180.1784..-133.8131] | it/evals=432/840 eff=80.0000% N=300
Z=-180.2(0.00%) | Like=-174.82..-59.59 [-180.1784..-133.8131] | it/evals=444/859 eff=79.4275% N=300
Z=-179.0(0.00%) | Like=-173.20..-59.59 [-180.1784..-133.8131] | it/evals=450/866 eff=79.5053% N=300
Z=-175.4(0.00%) | Like=-169.21..-59.59 [-180.1784..-133.8131] | it/evals=465/886 eff=79.3515% N=300
Mono-modal Volume: ~exp(-5.71) * Expected Volume: exp(-1.56) Quality: ok
index : +1.0| *********************************** | +5.0
amplitude: +1.0e-12| *************************************** * *| +1.0e-10
Z=-174.1(0.00%) | Like=-167.87..-59.59 [-180.1784..-133.8131] | it/evals=469/892 eff=79.2230% N=300
Z=-171.1(0.00%) | Like=-165.14..-59.59 [-180.1784..-133.8131] | it/evals=480/905 eff=79.3388% N=300
Z=-166.2(0.00%) | Like=-159.83..-59.59 [-180.1784..-133.8131] | it/evals=495/925 eff=79.2000% N=300
Z=-162.6(0.00%) | Like=-156.79..-59.59 [-180.1784..-133.8131] | it/evals=508/944 eff=78.8820% N=300
Z=-162.3(0.00%) | Like=-156.66..-59.59 [-180.1784..-133.8131] | it/evals=510/947 eff=78.8253% N=300
Z=-159.8(0.00%) | Like=-154.03..-59.59 [-180.1784..-133.8131] | it/evals=522/967 eff=78.2609% N=300
Z=-157.4(0.00%) | Like=-151.04..-59.59 [-180.1784..-133.8131] | it/evals=533/986 eff=77.6968% N=300
Mono-modal Volume: ~exp(-5.79) * Expected Volume: exp(-1.79) Quality: ok
index : +1.0| ******************************* | +5.0
amplitude: +1.0e-12| *************************************** | +1.0e-10
Z=-156.6(0.00%) | Like=-150.73..-59.59 [-180.1784..-133.8131] | it/evals=536/992 eff=77.4566% N=300
Z=-155.9(0.00%) | Like=-149.94..-59.59 [-180.1784..-133.8131] | it/evals=540/996 eff=77.5862% N=300
Z=-154.0(0.00%) | Like=-147.84..-59.59 [-180.1784..-133.8131] | it/evals=550/1015 eff=76.9231% N=300
Z=-150.9(0.00%) | Like=-144.87..-59.59 [-180.1784..-133.8131] | it/evals=565/1034 eff=76.9755% N=300
Z=-149.8(0.00%) | Like=-143.63..-59.59 [-180.1784..-133.8131] | it/evals=570/1040 eff=77.0270% N=300
Z=-146.2(0.00%) | Like=-139.45..-59.59 [-180.1784..-133.8131] | it/evals=585/1058 eff=77.1768% N=300
Z=-143.0(0.00%) | Like=-136.72..-59.48 [-180.1784..-133.8131] | it/evals=599/1077 eff=77.0914% N=300
Z=-142.8(0.00%) | Like=-136.46..-59.48 [-180.1784..-133.8131] | it/evals=600/1078 eff=77.1208% N=300
Mono-modal Volume: ~exp(-6.06) * Expected Volume: exp(-2.01) Quality: ok
index : +1.0| **************************** +4.1 | +5.0
amplitude: +1.0e-12| ************************************** | +1.0e-10
Z=-142.2(0.00%) | Like=-136.27..-59.48 [-180.1784..-133.8131] | it/evals=603/1081 eff=77.2087% N=300
Z=-139.6(0.00%) | Like=-133.79..-59.48 [-133.7911..-99.1427] | it/evals=617/1099 eff=77.2215% N=300
Z=-137.3(0.00%) | Like=-130.88..-59.48 [-133.7911..-99.1427] | it/evals=630/1116 eff=77.2059% N=300
Z=-135.0(0.00%) | Like=-128.68..-59.48 [-133.7911..-99.1427] | it/evals=643/1136 eff=76.9139% N=300
Z=-132.7(0.00%) | Like=-126.21..-59.48 [-133.7911..-99.1427] | it/evals=656/1156 eff=76.6355% N=300
Z=-131.9(0.00%) | Like=-125.75..-59.48 [-133.7911..-99.1427] | it/evals=660/1160 eff=76.7442% N=300
Mono-modal Volume: ~exp(-6.33) * Expected Volume: exp(-2.23) Quality: ok
index : +1.0| ************************* +3.9 | +5.0
amplitude: +1.0e-12| ********************************** | +1.0e-10
Z=-129.8(0.00%) | Like=-123.23..-59.48 [-133.7911..-99.1427] | it/evals=670/1174 eff=76.6590% N=300
Z=-125.8(0.00%) | Like=-119.33..-59.48 [-133.7911..-99.1427] | it/evals=685/1193 eff=76.7077% N=300
Z=-124.9(0.00%) | Like=-118.23..-59.48 [-133.7911..-99.1427] | it/evals=690/1198 eff=76.8374% N=300
Z=-123.0(0.00%) | Like=-116.70..-59.48 [-133.7911..-99.1427] | it/evals=702/1217 eff=76.5540% N=300
Z=-121.3(0.00%) | Like=-114.94..-59.48 [-133.7911..-99.1427] | it/evals=713/1236 eff=76.1752% N=300
Z=-120.0(0.00%) | Like=-113.60..-59.48 [-133.7911..-99.1427] | it/evals=720/1245 eff=76.1905% N=300
Z=-118.3(0.00%) | Like=-112.08..-59.48 [-133.7911..-99.1427] | it/evals=732/1264 eff=75.9336% N=300
Mono-modal Volume: ~exp(-6.33) Expected Volume: exp(-2.46) Quality: ok
index : +1.0| +2.0 ********************* +3.7 | +5.0
amplitude: +1.0e-12| ******************************* | +1.0e-10
Z=-117.3(0.00%) | Like=-110.99..-59.48 [-133.7911..-99.1427] | it/evals=740/1281 eff=75.4332% N=300
Z=-115.8(0.00%) | Like=-109.64..-59.48 [-133.7911..-99.1427] | it/evals=750/1296 eff=75.3012% N=300
Z=-114.1(0.00%) | Like=-107.78..-59.48 [-133.7911..-99.1427] | it/evals=763/1315 eff=75.1724% N=300
Z=-112.0(0.00%) | Like=-105.49..-59.48 [-133.7911..-99.1427] | it/evals=777/1334 eff=75.1451% N=300
Z=-111.5(0.00%) | Like=-105.35..-59.48 [-133.7911..-99.1427] | it/evals=780/1337 eff=75.2170% N=300
Z=-110.3(0.00%) | Like=-103.94..-59.48 [-133.7911..-99.1427] | it/evals=790/1356 eff=74.8106% N=300
Mono-modal Volume: ~exp(-6.76) * Expected Volume: exp(-2.68) Quality: ok
index : +1.0| +2.0 ******************** +3.6 | +5.0
amplitude: +1.0e-12| *************************** +7.6e-11 | +1.0e-10
Z=-108.7(0.00%) | Like=-102.43..-59.35 [-133.7911..-99.1427] | it/evals=804/1375 eff=74.7907% N=300
Z=-108.0(0.00%) | Like=-101.58..-59.35 [-133.7911..-99.1427] | it/evals=810/1383 eff=74.7922% N=300
Z=-106.2(0.00%) | Like=-100.00..-59.35 [-133.7911..-99.1427] | it/evals=827/1402 eff=75.0454% N=300
Z=-104.7(0.00%) | Like=-98.12..-59.35 [-98.9346..-78.8090] | it/evals=840/1420 eff=75.0000% N=300
Z=-102.9(0.00%) | Like=-96.43..-59.35 [-98.9346..-78.8090] | it/evals=856/1440 eff=75.0877% N=300
Z=-101.5(0.00%) | Like=-95.00..-59.35 [-98.9346..-78.8090] | it/evals=867/1459 eff=74.8059% N=300
Z=-101.2(0.00%) | Like=-94.90..-59.35 [-98.9346..-78.8090] | it/evals=870/1463 eff=74.8065% N=300
Mono-modal Volume: ~exp(-6.85) * Expected Volume: exp(-2.90) Quality: ok
index : +1.0| +2.1 ******************* +3.5 | +5.0
amplitude: +1.0e-12| ************************** +7.4e-11 | +1.0e-10
Z=-101.1(0.00%) | Like=-94.83..-59.35 [-98.9346..-78.8090] | it/evals=871/1464 eff=74.8282% N=300
Z=-99.0(0.00%) | Like=-92.01..-59.35 [-98.9346..-78.8090] | it/evals=888/1482 eff=75.1269% N=300
Z=-96.8(0.00%) | Like=-89.84..-59.35 [-98.9346..-78.8090] | it/evals=900/1499 eff=75.0626% N=300
Z=-95.1(0.00%) | Like=-88.68..-59.31 [-98.9346..-78.8090] | it/evals=915/1518 eff=75.1232% N=300
Z=-93.6(0.00%) | Like=-86.94..-59.31 [-98.9346..-78.8090] | it/evals=930/1535 eff=75.3036% N=300
Mono-modal Volume: ~exp(-7.02) * Expected Volume: exp(-3.13) Quality: ok
index : +1.0| +2.1 **************** +3.4 | +5.0
amplitude: +1.0e-12| +2.5e-11 *********************** +6.9e-11 | +1.0e-10
Z=-92.8(0.00%) | Like=-86.37..-59.31 [-98.9346..-78.8090] | it/evals=938/1548 eff=75.1603% N=300
Z=-91.3(0.00%) | Like=-84.99..-59.31 [-98.9346..-78.8090] | it/evals=955/1567 eff=75.3749% N=300
Z=-90.9(0.00%) | Like=-84.43..-59.31 [-98.9346..-78.8090] | it/evals=960/1573 eff=75.4124% N=300
Z=-89.6(0.00%) | Like=-83.31..-59.31 [-98.9346..-78.8090] | it/evals=977/1592 eff=75.6192% N=300
Z=-88.7(0.00%) | Like=-82.26..-59.31 [-98.9346..-78.8090] | it/evals=990/1611 eff=75.5149% N=300
Z=-87.7(0.00%) | Like=-81.22..-59.31 [-98.9346..-78.8090] | it/evals=1002/1631 eff=75.2817% N=300
Mono-modal Volume: ~exp(-7.15) * Expected Volume: exp(-3.35) Quality: ok
index : +1.0| +2.2 ************** +3.3 | +5.0
amplitude: +1.0e-12| +2.8e-11 ******************** +6.7e-11 | +1.0e-10
Z=-87.5(0.00%) | Like=-81.07..-59.31 [-98.9346..-78.8090] | it/evals=1005/1636 eff=75.2246% N=300
Z=-86.6(0.00%) | Like=-79.98..-59.31 [-98.9346..-78.8090] | it/evals=1017/1655 eff=75.0554% N=300
Z=-86.4(0.00%) | Like=-79.86..-59.31 [-98.9346..-78.8090] | it/evals=1020/1659 eff=75.0552% N=300
Z=-85.4(0.00%) | Like=-78.83..-59.31 [-98.9346..-78.8090] | it/evals=1034/1678 eff=75.0363% N=300
Z=-84.5(0.00%) | Like=-78.24..-59.31 [-78.7909..-70.1490] | it/evals=1048/1698 eff=74.9642% N=300
Z=-84.4(0.00%) | Like=-78.15..-59.31 [-78.7909..-70.1490] | it/evals=1050/1702 eff=74.8930% N=300
Z=-83.7(0.00%) | Like=-77.43..-59.31 [-78.7909..-70.1490] | it/evals=1064/1721 eff=74.8768% N=300
Mono-modal Volume: ~exp(-7.25) * Expected Volume: exp(-3.57) Quality: ok
index : +1.0| +2.2 ************* +3.2 | +5.0
amplitude: +1.0e-12| +2.9e-11 ****************** +6.5e-11 | +1.0e-10
Z=-83.3(0.00%) | Like=-76.94..-59.31 [-78.7909..-70.1490] | it/evals=1072/1735 eff=74.7038% N=300
Z=-82.9(0.00%) | Like=-76.56..-59.31 [-78.7909..-70.1490] | it/evals=1080/1748 eff=74.5856% N=300
Z=-82.1(0.00%) | Like=-75.82..-58.88 [-78.7909..-70.1490] | it/evals=1096/1766 eff=74.7613% N=300
Z=-81.5(0.00%) | Like=-75.03..-58.88 [-78.7909..-70.1490] | it/evals=1108/1785 eff=74.6128% N=300
Z=-81.4(0.00%) | Like=-74.94..-58.88 [-78.7909..-70.1490] | it/evals=1110/1789 eff=74.5467% N=300
Z=-80.7(0.00%) | Like=-74.38..-58.88 [-78.7909..-70.1490] | it/evals=1125/1808 eff=74.6021% N=300
Mono-modal Volume: ~exp(-7.71) * Expected Volume: exp(-3.80) Quality: ok
index : +1.0| +2.3 ************ +3.2 | +5.0
amplitude: +1.0e-12| +3.0e-11 ***************** +6.2e-11 | +1.0e-10
Z=-80.1(0.00%) | Like=-73.64..-58.88 [-78.7909..-70.1490] | it/evals=1139/1826 eff=74.6396% N=300
Z=-80.0(0.00%) | Like=-73.60..-58.88 [-78.7909..-70.1490] | it/evals=1140/1827 eff=74.6562% N=300
Z=-79.5(0.00%) | Like=-73.10..-58.88 [-78.7909..-70.1490] | it/evals=1152/1846 eff=74.5149% N=300
Z=-79.0(0.00%) | Like=-72.81..-58.88 [-78.7909..-70.1490] | it/evals=1166/1866 eff=74.4572% N=300
Z=-78.8(0.00%) | Like=-72.29..-58.88 [-78.7909..-70.1490] | it/evals=1170/1872 eff=74.4275% N=300
Z=-78.2(0.00%) | Like=-71.70..-58.88 [-78.7909..-70.1490] | it/evals=1186/1891 eff=74.5443% N=300
Z=-77.7(0.00%) | Like=-71.28..-58.88 [-78.7909..-70.1490] | it/evals=1200/1907 eff=74.6733% N=300
Mono-modal Volume: ~exp(-7.71) Expected Volume: exp(-4.02) Quality: ok
index : +1.0| +2.3 *********** +3.1 | +5.0
amplitude: +1.0e-12| +3.1e-11 *************** +6.0e-11 | +1.0e-10
Z=-77.3(0.00%) | Like=-70.91..-58.88 [-78.7909..-70.1490] | it/evals=1210/1924 eff=74.5074% N=300
Z=-76.8(0.00%) | Like=-70.28..-58.88 [-78.7909..-70.1490] | it/evals=1226/1945 eff=74.5289% N=300
Z=-76.6(0.00%) | Like=-70.11..-58.88 [-70.1376..-65.9592] | it/evals=1230/1949 eff=74.5907% N=300
Z=-76.2(0.00%) | Like=-69.57..-58.88 [-70.1376..-65.9592] | it/evals=1241/1970 eff=74.3114% N=300
Z=-75.7(0.00%) | Like=-69.29..-58.88 [-70.1376..-65.9592] | it/evals=1257/1989 eff=74.4227% N=300
Z=-75.6(0.00%) | Like=-69.11..-58.88 [-70.1376..-65.9592] | it/evals=1260/1993 eff=74.4241% N=300
Mono-modal Volume: ~exp(-8.00) * Expected Volume: exp(-4.24) Quality: ok
index : +1.0| +2.3 ********** +3.1 | +5.0
amplitude: +1.0e-12| +3.3e-11 ************** +5.9e-11 | +1.0e-10
Z=-75.2(0.01%) | Like=-68.76..-58.88 [-70.1376..-65.9592] | it/evals=1273/2007 eff=74.5753% N=300
Z=-74.8(0.01%) | Like=-68.37..-58.88 [-70.1376..-65.9592] | it/evals=1287/2026 eff=74.5655% N=300
Z=-74.7(0.01%) | Like=-68.26..-58.88 [-70.1376..-65.9592] | it/evals=1290/2031 eff=74.5234% N=300
Z=-74.3(0.01%) | Like=-67.88..-58.83 [-70.1376..-65.9592] | it/evals=1304/2050 eff=74.5143% N=300
Z=-73.9(0.02%) | Like=-67.33..-58.83 [-70.1376..-65.9592] | it/evals=1320/2069 eff=74.6184% N=300
Z=-73.5(0.03%) | Like=-66.75..-58.83 [-70.1376..-65.9592] | it/evals=1334/2089 eff=74.5668% N=300
Mono-modal Volume: ~exp(-8.13) * Expected Volume: exp(-4.47) Quality: ok
index : +1.0| +2.4 ********* +3.0 | +5.0
amplitude: +1.0e-12| +3.4e-11 ************* +5.7e-11 | +1.0e-10
Z=-73.3(0.04%) | Like=-66.61..-58.83 [-70.1376..-65.9592] | it/evals=1340/2100 eff=74.4444% N=300
Z=-73.0(0.05%) | Like=-66.41..-58.83 [-70.1376..-65.9592] | it/evals=1349/2119 eff=74.1616% N=300
Z=-73.0(0.05%) | Like=-66.41..-58.83 [-70.1376..-65.9592] | it/evals=1350/2120 eff=74.1758% N=300
Z=-72.6(0.07%) | Like=-66.06..-58.83 [-70.1376..-65.9592] | it/evals=1364/2140 eff=74.1304% N=300
Z=-72.4(0.10%) | Like=-65.70..-58.83 [-65.9125..-65.3491] | it/evals=1375/2161 eff=73.8850% N=300
Z=-72.2(0.11%) | Like=-65.60..-58.83 [-65.9125..-65.3491] | it/evals=1380/2169 eff=73.8363% N=300
Z=-71.9(0.16%) | Like=-65.20..-58.83 [-65.2011..-65.2006]*| it/evals=1395/2189 eff=73.8486% N=300
Mono-modal Volume: ~exp(-8.44) * Expected Volume: exp(-4.69) Quality: ok
index : +1.0| +2.4 ******** +3.0 | +5.0
amplitude: +1.0e-12| +3.5e-11 *********** +5.6e-11 | +1.0e-10
Z=-71.6(0.21%) | Like=-64.90..-58.83 [-64.9003..-64.8989]*| it/evals=1407/2208 eff=73.7421% N=300
Z=-71.5(0.22%) | Like=-64.85..-58.83 [-64.8468..-64.8319] | it/evals=1410/2212 eff=73.7448% N=300
Z=-71.2(0.31%) | Like=-64.60..-58.83 [-64.6200..-64.6046] | it/evals=1425/2231 eff=73.7960% N=300
Z=-70.9(0.41%) | Like=-64.34..-58.83 [-64.3435..-64.3290] | it/evals=1438/2251 eff=73.7058% N=300
Z=-70.9(0.43%) | Like=-64.31..-58.83 [-64.3112..-64.3091]*| it/evals=1440/2253 eff=73.7327% N=300
Z=-70.7(0.51%) | Like=-64.15..-58.77 [-64.1450..-64.1394]*| it/evals=1451/2272 eff=73.5801% N=300
Z=-70.5(0.63%) | Like=-64.04..-58.77 [-64.0579..-64.0432] | it/evals=1463/2291 eff=73.4807% N=300
Z=-70.4(0.70%) | Like=-64.00..-58.77 [-64.0006..-63.9699] | it/evals=1470/2302 eff=73.4266% N=300
Mono-modal Volume: ~exp(-9.01) * Expected Volume: exp(-4.91) Quality: ok
index : +1.0| +2.4 ******* +3.0 | +5.0
amplitude: +1.0e-12| +3.5e-11 ********** +5.4e-11 | +1.0e-10
Z=-70.3(0.74%) | Like=-63.96..-58.77 [-63.9582..-63.9496]*| it/evals=1474/2308 eff=73.4064% N=300
Z=-70.1(0.91%) | Like=-63.69..-58.77 [-63.6886..-63.6853]*| it/evals=1487/2327 eff=73.3596% N=300
Z=-69.9(1.09%) | Like=-63.50..-58.77 [-63.5162..-63.4969] | it/evals=1500/2346 eff=73.3138% N=300
Z=-69.7(1.35%) | Like=-63.26..-58.77 [-63.2630..-63.2602]*| it/evals=1515/2365 eff=73.3656% N=300
Z=-69.5(1.61%) | Like=-63.07..-58.77 [-63.0667..-63.0443] | it/evals=1529/2383 eff=73.4037% N=300
Z=-69.5(1.64%) | Like=-63.04..-58.77 [-63.0667..-63.0443] | it/evals=1530/2384 eff=73.4165% N=300
Mono-modal Volume: ~exp(-9.01) Expected Volume: exp(-5.14) Quality: ok
index : +1.0| +2.4 ****** +2.9 | +5.0
amplitude: +1.0e-12| +3.6e-11 ********* +5.3e-11 | +1.0e-10
Z=-69.4(1.90%) | Like=-62.87..-58.77 [-62.8680..-62.8559] | it/evals=1542/2402 eff=73.3587% N=300
Z=-69.2(2.35%) | Like=-62.68..-58.77 [-62.6780..-62.6762]*| it/evals=1558/2421 eff=73.4559% N=300
Z=-69.1(2.39%) | Like=-62.67..-58.77 [-62.6735..-62.6735]*| it/evals=1560/2424 eff=73.4463% N=300
Z=-69.0(2.73%) | Like=-62.48..-58.77 [-62.4822..-62.4630] | it/evals=1571/2443 eff=73.3084% N=300
Z=-68.9(3.21%) | Like=-62.30..-58.77 [-62.3194..-62.3016] | it/evals=1584/2462 eff=73.2655% N=300
Z=-68.8(3.40%) | Like=-62.27..-58.75 [-62.2656..-62.2304] | it/evals=1590/2471 eff=73.2381% N=300
Z=-68.6(3.90%) | Like=-62.08..-58.75 [-62.0805..-62.0748]*| it/evals=1603/2490 eff=73.1963% N=300
Mono-modal Volume: ~exp(-9.19) * Expected Volume: exp(-5.36) Quality: ok
index : +1.0| +2.5 ****** +2.9 | +5.0
amplitude: +1.0e-12| +3.7e-11 ******** +5.2e-11 | +1.0e-10
Z=-68.6(4.10%) | Like=-61.99..-58.75 [-62.0125..-61.9856] | it/evals=1608/2499 eff=73.1241% N=300
Z=-68.5(4.70%) | Like=-61.93..-58.75 [-61.9270..-61.9265]*| it/evals=1620/2514 eff=73.1707% N=300
Z=-68.3(5.23%) | Like=-61.82..-58.75 [-61.8202..-61.8060] | it/evals=1633/2532 eff=73.1631% N=300
Z=-68.2(5.91%) | Like=-61.72..-58.75 [-61.7174..-61.6970] | it/evals=1646/2550 eff=73.1556% N=300
Z=-68.2(6.15%) | Like=-61.67..-58.75 [-61.6900..-61.6733] | it/evals=1650/2556 eff=73.1383% N=300
Z=-68.0(7.13%) | Like=-61.50..-58.75 [-61.5044..-61.5006]*| it/evals=1666/2575 eff=73.2308% N=300
Mono-modal Volume: ~exp(-9.39) * Expected Volume: exp(-5.58) Quality: ok
index : +1.0| +2.5 ****** +2.9 | +5.0
amplitude: +1.0e-12| +3.8e-11 ******** +5.2e-11 | +1.0e-10
Z=-67.9(7.77%) | Like=-61.46..-58.75 [-61.4559..-61.4542]*| it/evals=1675/2586 eff=73.2721% N=300
Z=-67.9(8.13%) | Like=-61.42..-58.75 [-61.4190..-61.4059] | it/evals=1680/2592 eff=73.2984% N=300
Z=-67.8(8.94%) | Like=-61.28..-58.75 [-61.2779..-61.2600] | it/evals=1693/2612 eff=73.2266% N=300
Z=-67.7(10.12%) | Like=-61.16..-58.75 [-61.1594..-61.1582]*| it/evals=1707/2631 eff=73.2304% N=300
Z=-67.7(10.38%) | Like=-61.14..-58.75 [-61.1381..-61.1249] | it/evals=1710/2637 eff=73.1707% N=300
Z=-67.6(11.44%) | Like=-61.02..-58.75 [-61.0175..-61.0110]*| it/evals=1724/2656 eff=73.1749% N=300
Z=-67.5(12.64%) | Like=-60.90..-58.75 [-60.8960..-60.8860] | it/evals=1738/2675 eff=73.1789% N=300
Z=-67.4(12.81%) | Like=-60.87..-58.75 [-60.8685..-60.8670]*| it/evals=1740/2678 eff=73.1707% N=300
Mono-modal Volume: ~exp(-9.74) * Expected Volume: exp(-5.81) Quality: ok
index : +1.0| +2.5 ***** +2.8 | +5.0
amplitude: +1.0e-12| +3.9e-11 ******* +5.1e-11 | +1.0e-10
Z=-67.4(12.97%) | Like=-60.85..-58.75 [-60.8670..-60.8528] | it/evals=1742/2683 eff=73.1011% N=300
Z=-67.3(14.34%) | Like=-60.76..-58.75 [-60.7589..-60.7452] | it/evals=1756/2702 eff=73.1057% N=300
Z=-67.2(15.79%) | Like=-60.65..-58.75 [-60.6535..-60.6494]*| it/evals=1770/2720 eff=73.1405% N=300
Z=-67.1(17.14%) | Like=-60.58..-58.75 [-60.5952..-60.5784] | it/evals=1784/2739 eff=73.1447% N=300
Z=-67.1(18.55%) | Like=-60.54..-58.75 [-60.5503..-60.5384] | it/evals=1798/2759 eff=73.1192% N=300
Z=-67.1(18.75%) | Like=-60.53..-58.75 [-60.5302..-60.5201] | it/evals=1800/2762 eff=73.1113% N=300
Mono-modal Volume: ~exp(-10.40) * Expected Volume: exp(-6.03) Quality: ok
index : +1.0| +2.5 **** +2.8 | +5.0
amplitude: +1.0e-12| +3.9e-11 ****** +5.0e-11 | +1.0e-10
Z=-67.0(19.56%) | Like=-60.48..-58.75 [-60.4816..-60.4749]*| it/evals=1809/2773 eff=73.1500% N=300
Z=-66.9(21.29%) | Like=-60.40..-58.75 [-60.4010..-60.4007]*| it/evals=1826/2792 eff=73.2745% N=300
Z=-66.9(21.80%) | Like=-60.39..-58.75 [-60.3924..-60.3875]*| it/evals=1830/2797 eff=73.2879% N=300
Z=-66.8(23.32%) | Like=-60.33..-58.75 [-60.3255..-60.3183]*| it/evals=1844/2817 eff=73.2618% N=300
Z=-66.8(25.09%) | Like=-60.24..-58.75 [-60.2412..-60.2310] | it/evals=1860/2836 eff=73.3438% N=300
Z=-66.7(26.46%) | Like=-60.19..-58.75 [-60.1919..-60.1906]*| it/evals=1873/2854 eff=73.3359% N=300
Mono-modal Volume: ~exp(-10.40) Expected Volume: exp(-6.25) Quality: ok
index : +1.0| +2.5 **** +2.8 | +5.0
amplitude: +1.0e-12| +3.9e-11 ****** +5.0e-11 | +1.0e-10
Z=-66.6(27.68%) | Like=-60.15..-58.75 [-60.1507..-60.1389] | it/evals=1884/2872 eff=73.2504% N=300
Z=-66.6(28.30%) | Like=-60.13..-58.75 [-60.1264..-60.1256]*| it/evals=1890/2879 eff=73.2842% N=300
Z=-66.6(29.47%) | Like=-60.10..-58.75 [-60.1003..-60.0922]*| it/evals=1901/2897 eff=73.1998% N=300
Z=-66.5(31.10%) | Like=-60.04..-58.75 [-60.0367..-60.0319]*| it/evals=1915/2915 eff=73.2314% N=300
Z=-66.5(31.66%) | Like=-60.02..-58.75 [-60.0296..-60.0178] | it/evals=1920/2921 eff=73.2545% N=300
Z=-66.4(33.81%) | Like=-59.95..-58.75 [-59.9515..-59.9470]*| it/evals=1938/2939 eff=73.4369% N=300
Mono-modal Volume: ~exp(-10.47) * Expected Volume: exp(-6.48) Quality: ok
index : +1.0| +2.5 **** +2.8 | +5.0
amplitude: +1.0e-12| +4.0e-11 ****** +4.9e-11 | +1.0e-10
Z=-66.4(34.40%) | Like=-59.93..-58.75 [-59.9308..-59.9177] | it/evals=1943/2946 eff=73.4316% N=300
Z=-66.4(35.13%) | Like=-59.90..-58.75 [-59.9034..-59.9024]*| it/evals=1950/2955 eff=73.4463% N=300
Z=-66.4(36.91%) | Like=-59.85..-58.75 [-59.8533..-59.8504]*| it/evals=1966/2976 eff=73.4679% N=300
Z=-66.3(38.39%) | Like=-59.81..-58.75 [-59.8111..-59.8072]*| it/evals=1980/2994 eff=73.4967% N=300
Z=-66.3(40.05%) | Like=-59.76..-58.75 [-59.7608..-59.7570]*| it/evals=1995/3012 eff=73.5619% N=300
Mono-modal Volume: ~exp(-10.47) Expected Volume: exp(-6.70) Quality: ok
index : +1.0| +2.6 **** +2.8 | +5.0
amplitude: +1.0e-12| +4.0e-11 ***** +4.9e-11 | +1.0e-10
Z=-66.2(41.93%) | Like=-59.72..-58.75 [-59.7169..-59.7126]*| it/evals=2010/3029 eff=73.6534% N=300
Z=-66.2(43.52%) | Like=-59.66..-58.75 [-59.6584..-59.6570]*| it/evals=2024/3049 eff=73.6268% N=300
Z=-66.2(44.84%) | Like=-59.63..-58.75 [-59.6287..-59.6270]*| it/evals=2035/3068 eff=73.5188% N=300
Z=-66.1(45.38%) | Like=-59.61..-58.75 [-59.6203..-59.6095] | it/evals=2040/3073 eff=73.5665% N=300
Z=-66.1(46.99%) | Like=-59.57..-58.75 [-59.5731..-59.5666]*| it/evals=2053/3091 eff=73.5579% N=300
Z=-66.1(48.21%) | Like=-59.56..-58.75 [-59.5565..-59.5547]*| it/evals=2063/3111 eff=73.3903% N=300
Z=-66.1(48.99%) | Like=-59.54..-58.75 [-59.5352..-59.5345]*| it/evals=2070/3119 eff=73.4303% N=300
Mono-modal Volume: ~exp(-10.86) * Expected Volume: exp(-6.92) Quality: ok
index : +1.0| +2.6 **** +2.8 | +5.0
amplitude: +1.0e-12| +4.1e-11 **** +4.8e-11 | +1.0e-10
Z=-66.1(49.78%) | Like=-59.53..-58.75 [-59.5265..-59.5260]*| it/evals=2077/3127 eff=73.4701% N=300
Z=-66.0(51.74%) | Like=-59.50..-58.75 [-59.4981..-59.4976]*| it/evals=2093/3146 eff=73.5418% N=300
Z=-66.0(52.41%) | Like=-59.48..-58.75 [-59.4792..-59.4790]*| it/evals=2100/3154 eff=73.5809% N=300
Z=-66.0(54.02%) | Like=-59.45..-58.75 [-59.4547..-59.4523]*| it/evals=2115/3173 eff=73.6164% N=300
Z=-65.9(55.55%) | Like=-59.43..-58.75 [-59.4335..-59.4316]*| it/evals=2130/3192 eff=73.6515% N=300
Mono-modal Volume: ~exp(-10.86) Expected Volume: exp(-7.15) Quality: ok
index : +1.0| +2.6 **** +2.8 | +5.0
amplitude: +1.0e-12| +4.1e-11 **** +4.8e-11 | +1.0e-10
Z=-65.9(56.93%) | Like=-59.40..-58.75 [-59.3973..-59.3966]*| it/evals=2144/3209 eff=73.7023% N=300
Z=-65.9(58.17%) | Like=-59.37..-58.75 [-59.3678..-59.3637]*| it/evals=2156/3228 eff=73.6339% N=300
Z=-65.9(58.55%) | Like=-59.36..-58.75 [-59.3588..-59.3525]*| it/evals=2160/3233 eff=73.6447% N=300
Z=-65.9(59.96%) | Like=-59.34..-58.75 [-59.3385..-59.3340]*| it/evals=2174/3251 eff=73.6699% N=300
Z=-65.8(61.30%) | Like=-59.32..-58.75 [-59.3160..-59.3075]*| it/evals=2188/3269 eff=73.6948% N=300
Z=-65.8(61.50%) | Like=-59.31..-58.75 [-59.3059..-59.3038]*| it/evals=2190/3272 eff=73.6878% N=300
Z=-65.8(62.52%) | Like=-59.29..-58.75 [-59.2894..-59.2854]*| it/evals=2200/3291 eff=73.5540% N=300
Mono-modal Volume: ~exp(-11.05) * Expected Volume: exp(-7.37) Quality: ok
index : +1.0| +2.6 **** +2.8 | +5.0
amplitude: +1.0e-12| +4.1e-11 **** +4.7e-11 | +1.0e-10
Z=-65.8(63.56%) | Like=-59.27..-58.75 [-59.2651..-59.2637]*| it/evals=2211/3308 eff=73.5040% N=300
Z=-65.8(64.36%) | Like=-59.25..-58.75 [-59.2536..-59.2531]*| it/evals=2220/3320 eff=73.5099% N=300
Z=-65.8(65.58%) | Like=-59.24..-58.75 [-59.2359..-59.2313]*| it/evals=2234/3339 eff=73.5110% N=300
Z=-65.8(66.83%) | Like=-59.20..-58.75 [-59.2024..-59.1995]*| it/evals=2249/3358 eff=73.5448% N=300
Z=-65.8(66.93%) | Like=-59.20..-58.75 [-59.1995..-59.1988]*| it/evals=2250/3359 eff=73.5534% N=300
Z=-65.7(67.84%) | Like=-59.19..-58.75 [-59.1879..-59.1867]*| it/evals=2261/3378 eff=73.4568% N=300
Z=-65.7(69.19%) | Like=-59.16..-58.75 [-59.1641..-59.1633]*| it/evals=2277/3397 eff=73.5228% N=300
Mono-modal Volume: ~exp(-11.28) * Expected Volume: exp(-7.59) Quality: ok
index : +1.0| +2.6 ** +2.7 | +5.0
amplitude: +1.0e-12| +4.2e-11 **** +4.7e-11 | +1.0e-10
Z=-65.7(69.29%) | Like=-59.16..-58.75 [-59.1633..-59.1630]*| it/evals=2278/3398 eff=73.5313% N=300
Z=-65.7(69.46%) | Like=-59.16..-58.75 [-59.1624..-59.1622]*| it/evals=2280/3401 eff=73.5247% N=300
[ultranest] Explored until L=-6e+01
[ultranest] Likelihood function evaluations: 3409
[ultranest] logZ = -65.34 +- 0.09777
[ultranest] Effective samples strategy satisfied (ESS = 1018.0, need >400)
[ultranest] Posterior uncertainty strategy is satisfied (KL: 0.46+-0.10 nat, need <0.50 nat)
[ultranest] Evidency uncertainty strategy is satisfied (dlogz=0.28, need <0.5)
[ultranest] logZ error budget: single: 0.14 bs:0.10 tail:0.26 total:0.28 required:<0.50
[ultranest] done iterating.
logZ = -65.358 +- 0.333
single instance: logZ = -65.358 +- 0.136
bootstrapped : logZ = -65.337 +- 0.205
tail : logZ = +- 0.262
insert order U test : converged: True correlation: inf iterations
index : 2.314 │ ▁ ▁▁▁▁▁▁▂▂▃▃▄▅▆▇▇▇▆▆▆▃▄▃▃▂▂▁▁▁▁▁▁ ▁▁ │3.043 2.675 +- 0.088
amplitude : 0.0000000000340│ ▁▁▁▁▁▁▁▁▂▃▄▃▃▅▄▅▅▅▇▅▆▆▃▄▃▃▃▂▂▁▁▁▁▁▁▁▁ │0.0000000000552 0.0000000000444 +- 0.0000000000031
Understanding the outputs#
In the Jupyter notebook, you should be able to see an interactive visualisation of how the parameter space shrinks which starts from the (min,max) shrinks down towards the optimal parameters.
The output above is filled with interesting information. Here we provide a short description of the most relevant information provided above. For more detailed information see the UltraNest docs.
During the sampling
Z=-68.8(0.53%) | Like=-63.96..-58.75 [-63.9570..-63.9539]*| it/evals=640/1068 eff=73.7327% N=300
Some important information here is:
Progress (0.53%): the completed fraction of the integral. This is not a time progress bar. Stays at zero for a good fraction of the run.
Efficiency (eff value) of the sampling: this indicates out of the proposed new points, how many were accepted. If your efficiency is too small (<<1%), maybe you should revise your priors (e.g use a LogUniform prior for the normalisation).
Final outputs
The final lines indicate that all three “convergence” strategies are satisfied (samples, posterior uncertainty, and evidence uncertainty).
logZ = -65.104 +- 0.292
The main goal of the Nested sampling algorithm is to estimate Z (the Bayesian evidence) which is given above together with an uncertainty. In a similar way to deltaLogLike and deltaAIC, deltaLogZ values can be used for model comparison. For more information see : on the use of the evidence for model comparison. An interesting comparison of the efficiency and false discovery rate of model selection with deltaLogLike and deltaLogZ is given in Appendix C of Buchner et al., 2014.
Results stored on disk
if log_dir is set to a name where the results will be stored, then
a directory is created containing many useful results and plots.
A description of these outputs is given in the Ultranest
docs.
Results#
Within a Bayesian analysis, the concept of best-fit has to be viewed differently from what is done in a gradient descent fit.
The output of the Bayesian analysis is the posterior distribution and there is no “best-fit” output. One has to define, based on the posteriors, what we want to consider as “best-fit” and several options are possible:
the mean of the distribution
the median
the lowest likelihood value
By default the DatasetModels will be updated with the mean of
the posterior distributions.
print(result_joint.models)
DatasetModels
Component 0: SkyModel
Name : crab
Datasets names : None
Spectral model type : PowerLawSpectralModel
Spatial model type :
Temporal model type :
Parameters:
index : 2.675 +/- 0.09
amplitude : 4.44e-11 +/- 3.1e-12 1 / (TeV s cm2)
reference (frozen): 1.000 TeV
The Sampler class returns a very rich dictionary.
The most “standard” information about the posterior distributions can
be found in :
print(result_joint.sampler_results["posterior"])
{'mean': [2.675436460142479, 4.439813533813754e-11], 'stdev': [0.0878231893882909, 3.1429937448568174e-12], 'median': [2.672216968377601, 4.447323170408541e-11], 'errlo': [2.590640475275041, 4.119075134718645e-11], 'errup': [2.7663773918188084, 4.7619672518543923e-11], 'information_gain_bits': [2.67556924517269, 3.1159381407964073]}
Besides mean, errors, etc, an interesting value is the
information gain which estimates how much the posterior
distribution has shrunk with respect to the prior (i.e. how much
we’ve learned). A value < 1 means that the parameter is poorly
constrained within the prior range (we haven’t learned much with respect to our prior assumption).
For a physical interpretation of the information gain see this
example.
The SamplerResult dictionary contains also other interesting
information :
print(result_joint.sampler_results.keys())
dict_keys(['niter', 'logz', 'logzerr', 'logz_bs', 'logz_single', 'logzerr_tail', 'logzerr_bs', 'ess', 'H', 'Herr', 'posterior', 'weighted_samples', 'samples', 'maximum_likelihood', 'ncall', 'paramnames', 'logzerr_single', 'insertion_order_MWW_test'])
Of particular interest, the samples used in the process to approximate the posterior distribution can be accessed via :
for i, n in enumerate(model.parameters.free_parameters.names):
s = result_joint.samples[:, i]
fig, ax = plt.subplots()
ax.hist(s, bins=30)
ax.axvline(np.mean(s), ls="--", color="red")
ax.set_xlabel(n)
plt.show()
While the above plots are interesting, the real strength of the Bayesian analysis is to visualise all parameters correlations which is usually done using “corner plots”. Ultranest corner plot function is a wrapper around the corner package. See the above link for optional keywords. Other packages exist for corner plots, like chainconsumer which is discussed later in this tutorial.
from ultranest.plot import cornerplot
cornerplot(
result_joint.sampler_results,
plot_datapoints=True,
plot_density=True,
bins=20,
title_fmt=".2e",
smooth=False,
)
plt.show()
Spectral model error band from samples#
To compute the spectral error band (“butterfly plots”), we will directly use the samples of the posterior distribution. This is more robust as compared to the traditional method of using the covariance matrix of the parameters which implicitly assumes Gaussian errors while for the posterior distribution there is no shape assumed. This difference can become significant when the parameter errors are non-Gaussian. For this we will need to convert the list of samples back to the spectral model parameters with the relevant units (e.g. normalisation units).
def get_samples_from_posterior(spectral_model, results):
"""
Create a list of spectral parameters with correct units
from the unitless parameters returned by the sampler.
"""
n_samples = results.samples.shape[0]
samples = []
for p in spectral_model.parameters:
try:
idx = spectral_model.parameters.free_unique_parameters.index(p)
samples.append(results.samples[:, idx] * p.unit)
except ValueError:
samples.append(np.ones(n_samples) * p.quantity)
return samples
samples = get_samples_from_posterior(datasets.models[0].spectral_model, result_joint)
Next we can provide these samples to the plot_error
method.
Individual run analysis#
Now we’ll analyse several Crab runs individually so that we can compare them.
result_0 = sampler.run(datasets[0])
result_1 = sampler.run(datasets[1])
result_2 = sampler.run(datasets[2])
[ultranest] Sampling 300 live points from prior ...
Mono-modal Volume: ~exp(-3.82) * Expected Volume: exp(0.00) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|****************** ******************* *** **** | +1.0e-10
Z=-inf(0.00%) | Like=-1262.51..-21.91 [-1262.5135..-102.8689] | it/evals=0/301 eff=0.0000% N=300
Z=-177.7(0.00%) | Like=-172.26..-21.91 [-1262.5135..-102.8689] | it/evals=30/333 eff=90.9091% N=300
Z=-163.3(0.00%) | Like=-157.97..-21.91 [-1262.5135..-102.8689] | it/evals=59/367 eff=88.0597% N=300
Z=-162.9(0.00%) | Like=-157.92..-21.91 [-1262.5135..-102.8689] | it/evals=60/372 eff=83.3333% N=300
Mono-modal Volume: ~exp(-4.29) * Expected Volume: exp(-0.22) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|****************** ********* ************* **** | +1.0e-10
Z=-161.0(0.00%) | Like=-156.64..-21.91 [-1262.5135..-102.8689] | it/evals=67/380 eff=83.7500% N=300
Z=-153.5(0.00%) | Like=-148.18..-21.91 [-1262.5135..-102.8689] | it/evals=90/404 eff=86.5385% N=300
Z=-145.8(0.00%) | Like=-141.17..-20.94 [-1262.5135..-102.8689] | it/evals=118/439 eff=84.8921% N=300
Z=-145.2(0.00%) | Like=-140.26..-20.94 [-1262.5135..-102.8689] | it/evals=120/441 eff=85.1064% N=300
Mono-modal Volume: ~exp(-4.40) * Expected Volume: exp(-0.45) Quality: ok
index : +1.0| ***********************************************| +5.0
amplitude: +1.0e-12|**************************** ****************** | +1.0e-10
Z=-139.8(0.00%) | Like=-134.94..-20.94 [-1262.5135..-102.8689] | it/evals=134/460 eff=83.7500% N=300
Z=-132.4(0.00%) | Like=-127.00..-20.94 [-1262.5135..-102.8689] | it/evals=150/481 eff=82.8729% N=300
Z=-121.5(0.00%) | Like=-116.23..-20.94 [-1262.5135..-102.8689] | it/evals=175/515 eff=81.3953% N=300
Z=-119.9(0.00%) | Like=-114.49..-20.94 [-1262.5135..-102.8689] | it/evals=180/521 eff=81.4480% N=300
Mono-modal Volume: ~exp(-4.59) * Expected Volume: exp(-0.67) Quality: ok
index : +1.0| *********************************************| +5.0
amplitude: +1.0e-12| *************************** ********* * *******| +1.0e-10
Z=-113.4(0.00%) | Like=-107.78..-20.94 [-1262.5135..-102.8689] | it/evals=201/552 eff=79.7619% N=300
Z=-110.1(0.00%) | Like=-104.37..-20.94 [-1262.5135..-102.8689] | it/evals=210/564 eff=79.5455% N=300
Z=-103.3(0.00%) | Like=-98.48..-20.94 [-102.5915..-69.6834] | it/evals=237/600 eff=79.0000% N=300
Z=-102.8(0.00%) | Like=-98.24..-20.94 [-102.5915..-69.6834] | it/evals=240/605 eff=78.6885% N=300
Z=-99.5(0.00%) | Like=-94.15..-20.94 [-102.5915..-69.6834] | it/evals=264/640 eff=77.6471% N=300
Mono-modal Volume: ~exp(-4.90) * Expected Volume: exp(-0.89) Quality: ok
index : +1.0| ********************************************| +5.0
amplitude: +1.0e-12| ************************************* * **** * | +1.0e-10
Z=-98.5(0.00%) | Like=-93.63..-20.94 [-102.5915..-69.6834] | it/evals=268/645 eff=77.6812% N=300
Z=-98.2(0.00%) | Like=-93.20..-20.94 [-102.5915..-69.6834] | it/evals=270/648 eff=77.5862% N=300
Z=-92.8(0.00%) | Like=-87.60..-20.94 [-102.5915..-69.6834] | it/evals=300/682 eff=78.5340% N=300
Z=-89.2(0.00%) | Like=-84.42..-20.94 [-102.5915..-69.6834] | it/evals=322/717 eff=77.2182% N=300
Z=-88.0(0.00%) | Like=-83.16..-20.94 [-102.5915..-69.6834] | it/evals=330/728 eff=77.1028% N=300
Mono-modal Volume: ~exp(-4.90) Expected Volume: exp(-1.12) Quality: ok
index : +1.0| *******************************************| +5.0
amplitude: +1.0e-12| ************************************** * ** | +1.0e-10
Z=-85.4(0.00%) | Like=-80.59..-20.94 [-102.5915..-69.6834] | it/evals=350/760 eff=76.0870% N=300
Z=-84.0(0.00%) | Like=-79.00..-20.94 [-102.5915..-69.6834] | it/evals=360/773 eff=76.1099% N=300
Z=-79.7(0.00%) | Like=-74.70..-20.94 [-102.5915..-69.6834] | it/evals=386/807 eff=76.1341% N=300
Z=-79.2(0.00%) | Like=-74.13..-20.94 [-102.5915..-69.6834] | it/evals=390/814 eff=75.8755% N=300
Mono-modal Volume: ~exp(-5.25) * Expected Volume: exp(-1.34) Quality: ok
index : +1.0| ************************************** ** | +5.0
amplitude: +1.0e-12| *************************************** | +1.0e-10
Z=-77.8(0.00%) | Like=-72.33..-20.66 [-102.5915..-69.6834] | it/evals=402/830 eff=75.8491% N=300
Z=-75.2(0.00%) | Like=-69.91..-20.66 [-102.5915..-69.6834] | it/evals=420/854 eff=75.8123% N=300
Z=-70.9(0.00%) | Like=-65.57..-20.66 [-69.6607..-49.7593] | it/evals=446/889 eff=75.7216% N=300
Z=-70.2(0.00%) | Like=-64.69..-20.66 [-69.6607..-49.7593] | it/evals=450/900 eff=75.0000% N=300
Mono-modal Volume: ~exp(-5.48) * Expected Volume: exp(-1.56) Quality: ok
index : +1.0| ********************************* | +5.0
amplitude: +1.0e-12| ********************************** +7.8e-11| +1.0e-10
Z=-67.7(0.00%) | Like=-62.66..-20.66 [-69.6607..-49.7593] | it/evals=469/927 eff=74.8006% N=300
Z=-66.5(0.00%) | Like=-61.53..-20.66 [-69.6607..-49.7593] | it/evals=480/944 eff=74.5342% N=300
Z=-64.5(0.00%) | Like=-59.51..-20.66 [-69.6607..-49.7593] | it/evals=501/980 eff=73.6765% N=300
Z=-63.8(0.00%) | Like=-59.12..-20.66 [-69.6607..-49.7593] | it/evals=510/989 eff=74.0203% N=300
Z=-62.2(0.00%) | Like=-57.10..-20.66 [-69.6607..-49.7593] | it/evals=531/1030 eff=72.7397% N=300
Mono-modal Volume: ~exp(-5.83) * Expected Volume: exp(-1.79) Quality: ok
index : +1.0| ******************************* +4.1 | +5.0
amplitude: +1.0e-12| ********************************* +7.6e-11 | +1.0e-10
Z=-61.8(0.00%) | Like=-56.76..-20.66 [-69.6607..-49.7593] | it/evals=536/1036 eff=72.8261% N=300
Z=-61.4(0.00%) | Like=-56.23..-20.66 [-69.6607..-49.7593] | it/evals=540/1040 eff=72.9730% N=300
Z=-59.3(0.00%) | Like=-54.29..-20.66 [-69.6607..-49.7593] | it/evals=566/1075 eff=73.0323% N=300
Z=-59.0(0.00%) | Like=-53.89..-20.66 [-69.6607..-49.7593] | it/evals=570/1079 eff=73.1707% N=300
Z=-57.1(0.00%) | Like=-52.10..-20.66 [-69.6607..-49.7593] | it/evals=598/1116 eff=73.2843% N=300
Z=-56.9(0.00%) | Like=-51.98..-20.66 [-69.6607..-49.7593] | it/evals=600/1118 eff=73.3496% N=300
Mono-modal Volume: ~exp(-6.05) * Expected Volume: exp(-2.01) Quality: ok
index : +1.0| **************************** +3.9 | +5.0
amplitude: +1.0e-12| ***************************** +7.1e-11 | +1.0e-10
Z=-56.8(0.00%) | Like=-51.80..-20.66 [-69.6607..-49.7593] | it/evals=603/1122 eff=73.3577% N=300
Z=-54.9(0.00%) | Like=-49.72..-20.51 [-49.7187..-36.2200] | it/evals=630/1152 eff=73.9437% N=300
Z=-53.2(0.00%) | Like=-47.96..-20.51 [-49.7187..-36.2200] | it/evals=656/1186 eff=74.0406% N=300
Z=-52.9(0.00%) | Like=-47.46..-20.51 [-49.7187..-36.2200] | it/evals=660/1190 eff=74.1573% N=300
Mono-modal Volume: ~exp(-6.05) Expected Volume: exp(-2.23) Quality: ok
index : +1.0| ************************* +3.8 | +5.0
amplitude: +1.0e-12| *************************** +6.5e-11 | +1.0e-10
Z=-51.3(0.00%) | Like=-45.74..-20.51 [-49.7187..-36.2200] | it/evals=680/1223 eff=73.6728% N=300
Z=-50.5(0.00%) | Like=-44.56..-20.51 [-49.7187..-36.2200] | it/evals=690/1236 eff=73.7179% N=300
Z=-48.9(0.00%) | Like=-43.68..-20.51 [-49.7187..-36.2200] | it/evals=712/1268 eff=73.5537% N=300
Z=-48.5(0.00%) | Like=-43.15..-20.51 [-49.7187..-36.2200] | it/evals=720/1278 eff=73.6196% N=300
Mono-modal Volume: ~exp(-6.05) Expected Volume: exp(-2.46) Quality: ok
index : +1.0| ********************** +3.6 | +5.0
amplitude: +1.0e-12| ************************* +6.4e-11 | +1.0e-10
Z=-47.0(0.00%) | Like=-41.44..-20.51 [-49.7187..-36.2200] | it/evals=741/1310 eff=73.3663% N=300
Z=-46.4(0.00%) | Like=-41.01..-20.51 [-49.7187..-36.2200] | it/evals=750/1329 eff=72.8863% N=300
Z=-45.0(0.00%) | Like=-39.55..-20.51 [-49.7187..-36.2200] | it/evals=773/1365 eff=72.5822% N=300
Z=-44.6(0.00%) | Like=-39.01..-20.51 [-49.7187..-36.2200] | it/evals=780/1376 eff=72.4907% N=300
Mono-modal Volume: ~exp(-6.65) * Expected Volume: exp(-2.68) Quality: ok
index : +1.0| +1.9 ******************* +3.4 | +5.0
amplitude: +1.0e-12| ********************* +5.8e-11 | +1.0e-10
Z=-43.2(0.00%) | Like=-37.60..-20.51 [-49.7187..-36.2200] | it/evals=804/1413 eff=72.2372% N=300
Z=-42.8(0.00%) | Like=-37.34..-20.51 [-49.7187..-36.2200] | it/evals=810/1425 eff=72.0000% N=300
Z=-41.6(0.00%) | Like=-36.04..-20.51 [-36.1023..-29.1058] | it/evals=835/1459 eff=72.0449% N=300
Z=-41.4(0.00%) | Like=-35.69..-20.51 [-36.1023..-29.1058] | it/evals=840/1465 eff=72.1030% N=300
Z=-40.2(0.00%) | Like=-34.86..-20.51 [-36.1023..-29.1058] | it/evals=867/1498 eff=72.3706% N=300
Z=-40.1(0.00%) | Like=-34.74..-20.51 [-36.1023..-29.1058] | it/evals=870/1503 eff=72.3192% N=300
Mono-modal Volume: ~exp(-6.65) Expected Volume: exp(-2.90) Quality: ok
index : +1.0| +2.0 ****************** +3.4 | +5.0
amplitude: +1.0e-12| ********************* +5.8e-11 | +1.0e-10
Z=-39.3(0.00%) | Like=-34.05..-20.51 [-36.1023..-29.1058] | it/evals=893/1534 eff=72.3663% N=300
Z=-39.0(0.00%) | Like=-33.71..-20.51 [-36.1023..-29.1058] | it/evals=900/1546 eff=72.2311% N=300
Z=-38.3(0.00%) | Like=-32.83..-20.51 [-36.1023..-29.1058] | it/evals=922/1584 eff=71.8069% N=300
Z=-38.1(0.00%) | Like=-32.66..-20.51 [-36.1023..-29.1058] | it/evals=930/1596 eff=71.7593% N=300
Mono-modal Volume: ~exp(-6.65) Expected Volume: exp(-3.13) Quality: ok
index : +1.0| +2.0 **************** +3.3 | +5.0
amplitude: +1.0e-12| ******************* +5.6e-11 | +1.0e-10
Z=-37.4(0.00%) | Like=-32.13..-20.51 [-36.1023..-29.1058] | it/evals=952/1628 eff=71.6867% N=300
Z=-37.2(0.00%) | Like=-31.65..-20.51 [-36.1023..-29.1058] | it/evals=960/1641 eff=71.5884% N=300
Z=-36.2(0.01%) | Like=-30.55..-20.51 [-36.1023..-29.1058] | it/evals=988/1676 eff=71.8023% N=300
Z=-36.1(0.01%) | Like=-30.49..-20.51 [-36.1023..-29.1058] | it/evals=990/1678 eff=71.8433% N=300
Mono-modal Volume: ~exp(-7.44) * Expected Volume: exp(-3.35) Quality: ok
index : +1.0| +2.1 *************** +3.2 | +5.0
amplitude: +1.0e-12| ***************** +5.3e-11 | +1.0e-10
Z=-35.6(0.01%) | Like=-30.10..-20.51 [-36.1023..-29.1058] | it/evals=1005/1703 eff=71.6322% N=300
Z=-35.2(0.01%) | Like=-29.77..-20.51 [-36.1023..-29.1058] | it/evals=1020/1720 eff=71.8310% N=300
Z=-34.6(0.02%) | Like=-29.05..-20.50 [-29.0959..-27.2746] | it/evals=1045/1755 eff=71.8213% N=300
Z=-34.4(0.03%) | Like=-28.89..-20.50 [-29.0959..-27.2746] | it/evals=1050/1763 eff=71.7703% N=300
Mono-modal Volume: ~exp(-7.44) Expected Volume: exp(-3.57) Quality: ok
index : +1.0| +2.1 ************* +3.1 | +5.0
amplitude: +1.0e-12| ************** +5.0e-11 | +1.0e-10
Z=-33.8(0.05%) | Like=-28.26..-20.50 [-29.0959..-27.2746] | it/evals=1075/1793 eff=72.0027% N=300
Z=-33.7(0.05%) | Like=-28.11..-20.50 [-29.0959..-27.2746] | it/evals=1080/1799 eff=72.0480% N=300
Z=-33.2(0.09%) | Like=-27.52..-20.50 [-29.0959..-27.2746] | it/evals=1101/1834 eff=71.7731% N=300
Z=-32.9(0.11%) | Like=-27.33..-20.50 [-29.0959..-27.2746] | it/evals=1110/1848 eff=71.7054% N=300
Z=-32.4(0.19%) | Like=-26.75..-20.50 [-26.7486..-26.7458]*| it/evals=1133/1884 eff=71.5278% N=300
Mono-modal Volume: ~exp(-7.82) * Expected Volume: exp(-3.80) Quality: ok
index : +1.0| +2.2 ************ +3.0 | +5.0
amplitude: +1.0e-12| ************** +4.9e-11 | +1.0e-10
Z=-32.3(0.22%) | Like=-26.60..-20.50 [-26.6210..-26.6020] | it/evals=1139/1895 eff=71.4107% N=300
Z=-32.3(0.23%) | Like=-26.59..-20.50 [-26.5876..-26.5839]*| it/evals=1140/1896 eff=71.4286% N=300
Z=-31.8(0.37%) | Like=-26.14..-20.50 [-26.1404..-26.1398]*| it/evals=1165/1932 eff=71.3848% N=300
Z=-31.7(0.41%) | Like=-26.07..-20.50 [-26.0921..-26.0675] | it/evals=1170/1939 eff=71.3850% N=300
Z=-31.2(0.64%) | Like=-25.65..-20.50 [-25.6549..-25.6497]*| it/evals=1195/1974 eff=71.3859% N=300
Z=-31.2(0.69%) | Like=-25.57..-20.50 [-25.5705..-25.5570] | it/evals=1200/1980 eff=71.4286% N=300
Mono-modal Volume: ~exp(-7.82) Expected Volume: exp(-4.02) Quality: ok
index : +1.0| +2.2 ********** +3.0 | +5.0
amplitude: +1.0e-12| ************* +4.7e-11 | +1.0e-10
Z=-30.9(0.87%) | Like=-25.31..-20.50 [-25.3131..-25.3087]*| it/evals=1217/2012 eff=71.0864% N=300
Z=-30.7(1.07%) | Like=-25.05..-20.50 [-25.0542..-25.0392] | it/evals=1230/2028 eff=71.1806% N=300
Z=-30.3(1.51%) | Like=-24.69..-20.50 [-24.6922..-24.6700] | it/evals=1254/2064 eff=71.0884% N=300
Z=-30.3(1.62%) | Like=-24.59..-20.50 [-24.5853..-24.5828]*| it/evals=1260/2074 eff=71.0259% N=300
Mono-modal Volume: ~exp(-7.88) * Expected Volume: exp(-4.24) Quality: ok
index : +1.0| +2.2 ********** +2.9 | +5.0
amplitude: +1.0e-12| +2.5e-11 *********** +4.5e-11 | +1.0e-10
Z=-30.1(1.95%) | Like=-24.40..-20.50 [-24.4032..-24.4030]*| it/evals=1273/2098 eff=70.8009% N=300
Z=-29.8(2.43%) | Like=-24.18..-20.50 [-24.1957..-24.1846] | it/evals=1290/2119 eff=70.9181% N=300
Z=-29.5(3.22%) | Like=-23.92..-20.50 [-23.9227..-23.9181]*| it/evals=1315/2153 eff=70.9660% N=300
Z=-29.5(3.42%) | Like=-23.86..-20.50 [-23.8625..-23.8618]*| it/evals=1320/2159 eff=71.0059% N=300
Mono-modal Volume: ~exp(-8.37) * Expected Volume: exp(-4.47) Quality: ok
index : +1.0| +2.3 ******** +2.9 | +5.0
amplitude: +1.0e-12| +2.5e-11 ********** +4.4e-11 | +1.0e-10
Z=-29.3(4.31%) | Like=-23.69..-20.49 [-23.7089..-23.6940] | it/evals=1340/2185 eff=71.0875% N=300
Z=-29.2(4.81%) | Like=-23.64..-20.49 [-23.6410..-23.6309] | it/evals=1350/2198 eff=71.1275% N=300
Z=-28.9(6.22%) | Like=-23.35..-20.49 [-23.3690..-23.3499] | it/evals=1378/2233 eff=71.2882% N=300
Z=-28.9(6.36%) | Like=-23.34..-20.49 [-23.3354..-23.3322]*| it/evals=1380/2238 eff=71.2074% N=300
Z=-28.7(7.97%) | Like=-23.05..-20.49 [-23.0518..-23.0494]*| it/evals=1406/2272 eff=71.2982% N=300
Mono-modal Volume: ~exp(-8.46) * Expected Volume: exp(-4.69) Quality: ok
index : +1.0| +2.3 ******** +2.9 | +5.0
amplitude: +1.0e-12| +2.6e-11 ********* +4.3e-11 | +1.0e-10
Z=-28.7(8.06%) | Like=-23.05..-20.49 [-23.0494..-23.0470]*| it/evals=1407/2273 eff=71.3127% N=300
Z=-28.7(8.30%) | Like=-23.04..-20.49 [-23.0393..-23.0368]*| it/evals=1410/2277 eff=71.3202% N=300
Z=-28.5(10.15%) | Like=-22.87..-20.49 [-22.8745..-22.8682]*| it/evals=1436/2311 eff=71.4073% N=300
Z=-28.4(10.44%) | Like=-22.85..-20.49 [-22.8509..-22.8506]*| it/evals=1440/2317 eff=71.3932% N=300
Z=-28.2(12.69%) | Like=-22.64..-20.49 [-22.6404..-22.6378]*| it/evals=1469/2351 eff=71.6236% N=300
Z=-28.2(12.78%) | Like=-22.64..-20.49 [-22.6378..-22.6271] | it/evals=1470/2353 eff=71.6025% N=300
Mono-modal Volume: ~exp(-8.89) * Expected Volume: exp(-4.91) Quality: ok
index : +1.0| +2.3 ******** +2.8 | +5.0
amplitude: +1.0e-12| +2.7e-11 ********* +4.2e-11 | +1.0e-10
Z=-28.2(13.10%) | Like=-22.59..-20.49 [-22.5870..-22.5860]*| it/evals=1474/2357 eff=71.6578% N=300
Z=-28.0(15.40%) | Like=-22.42..-20.46 [-22.4244..-22.4214]*| it/evals=1500/2392 eff=71.7017% N=300
Z=-27.9(17.98%) | Like=-22.27..-20.46 [-22.2710..-22.2580] | it/evals=1526/2428 eff=71.7105% N=300
Z=-27.8(18.37%) | Like=-22.24..-20.46 [-22.2402..-22.2392]*| it/evals=1530/2432 eff=71.7636% N=300
Mono-modal Volume: ~exp(-9.22) * Expected Volume: exp(-5.14) Quality: ok
index : +1.0| +2.3 ****** +2.8 | +5.0
amplitude: +1.0e-12| +2.8e-11 ******** +4.1e-11 | +1.0e-10
Z=-27.8(19.53%) | Like=-22.19..-20.46 [-22.1874..-22.1868]*| it/evals=1541/2445 eff=71.8415% N=300
Z=-27.7(21.47%) | Like=-22.10..-20.46 [-22.0977..-22.0913]*| it/evals=1560/2469 eff=71.9225% N=300
Z=-27.6(24.05%) | Like=-21.97..-20.46 [-21.9685..-21.9432] | it/evals=1585/2503 eff=71.9473% N=300
Z=-27.5(24.56%) | Like=-21.91..-20.46 [-21.9110..-21.9096]*| it/evals=1590/2510 eff=71.9457% N=300
Mono-modal Volume: ~exp(-9.22) Expected Volume: exp(-5.36) Quality: ok
index : +1.0| +2.4 ****** +2.8 | +5.0
amplitude: +1.0e-12| +2.8e-11 ******* +4.1e-11 | +1.0e-10
Z=-27.4(27.04%) | Like=-21.85..-20.46 [-21.8514..-21.8493]*| it/evals=1613/2541 eff=71.9768% N=300
Z=-27.4(27.79%) | Like=-21.83..-20.46 [-21.8275..-21.8139] | it/evals=1620/2554 eff=71.8722% N=300
Z=-27.3(31.06%) | Like=-21.70..-20.46 [-21.7045..-21.6904] | it/evals=1649/2588 eff=72.0717% N=300
Z=-27.3(31.14%) | Like=-21.69..-20.46 [-21.7045..-21.6904] | it/evals=1650/2589 eff=72.0839% N=300
Mono-modal Volume: ~exp(-9.22) Expected Volume: exp(-5.58) Quality: ok
index : +1.0| +2.4 ****** +2.8 | +5.0
amplitude: +1.0e-12| +2.9e-11 ****** +4.0e-11 | +1.0e-10
Z=-27.2(34.12%) | Like=-21.56..-20.46 [-21.5645..-21.5627]*| it/evals=1675/2621 eff=72.1672% N=300
Z=-27.2(34.75%) | Like=-21.56..-20.46 [-21.5580..-21.5559]*| it/evals=1680/2631 eff=72.0721% N=300
Z=-27.1(37.78%) | Like=-21.48..-20.46 [-21.4785..-21.4766]*| it/evals=1704/2667 eff=71.9899% N=300
Z=-27.1(38.37%) | Like=-21.47..-20.46 [-21.4743..-21.4732]*| it/evals=1710/2675 eff=72.0000% N=300
Z=-27.0(41.22%) | Like=-21.42..-20.46 [-21.4151..-21.4097]*| it/evals=1734/2709 eff=71.9801% N=300
Z=-27.0(41.98%) | Like=-21.40..-20.46 [-21.4018..-21.4013]*| it/evals=1740/2723 eff=71.8118% N=300
Mono-modal Volume: ~exp(-9.67) * Expected Volume: exp(-5.81) Quality: ok
index : +1.0| +2.4 ***** +2.7 | +5.0
amplitude: +1.0e-12| +2.9e-11 ****** +3.9e-11 | +1.0e-10
Z=-27.0(42.19%) | Like=-21.40..-20.46 [-21.3973..-21.3951]*| it/evals=1742/2725 eff=71.8351% N=300
Z=-26.9(45.06%) | Like=-21.31..-20.46 [-21.3083..-21.3064]*| it/evals=1766/2759 eff=71.8178% N=300
Z=-26.9(45.58%) | Like=-21.29..-20.46 [-21.2934..-21.2932]*| it/evals=1770/2764 eff=71.8344% N=300
Z=-26.9(48.45%) | Like=-21.23..-20.46 [-21.2349..-21.2309]*| it/evals=1796/2798 eff=71.8975% N=300
Z=-26.9(48.90%) | Like=-21.23..-20.46 [-21.2289..-21.2283]*| it/evals=1800/2803 eff=71.9137% N=300
Mono-modal Volume: ~exp(-9.83) * Expected Volume: exp(-6.03) Quality: ok
index : +1.0| +2.4 **** +2.7 | +5.0
amplitude: +1.0e-12| +3.0e-11 ****** +3.9e-11 | +1.0e-10
Z=-26.8(49.96%) | Like=-21.21..-20.46 [-21.2124..-21.2032]*| it/evals=1809/2815 eff=71.9284% N=300
Z=-26.8(52.28%) | Like=-21.16..-20.46 [-21.1620..-21.1556]*| it/evals=1830/2843 eff=71.9622% N=300
Z=-26.7(55.12%) | Like=-21.09..-20.46 [-21.0875..-21.0862]*| it/evals=1858/2877 eff=72.0993% N=300
Z=-26.7(55.30%) | Like=-21.08..-20.46 [-21.0840..-21.0807]*| it/evals=1860/2880 eff=72.0930% N=300
Mono-modal Volume: ~exp(-10.08) * Expected Volume: exp(-6.25) Quality: ok
index : +1.0| +2.4 **** +2.7 | +5.0
amplitude: +1.0e-12| +3.0e-11 ***** +3.8e-11 | +1.0e-10
Z=-26.7(57.00%) | Like=-21.06..-20.46 [-21.0606..-21.0603]*| it/evals=1876/2904 eff=72.0430% N=300
Z=-26.7(58.43%) | Like=-21.03..-20.46 [-21.0335..-21.0331]*| it/evals=1890/2921 eff=72.1099% N=300
Z=-26.6(61.10%) | Like=-20.96..-20.46 [-20.9642..-20.9544]*| it/evals=1917/2955 eff=72.2034% N=300
Z=-26.6(61.41%) | Like=-20.94..-20.46 [-20.9388..-20.9381]*| it/evals=1920/2962 eff=72.1262% N=300
Mono-modal Volume: ~exp(-10.18) * Expected Volume: exp(-6.48) Quality: ok
index : +1.0| +2.4 **** +2.7 | +5.0
amplitude: +1.0e-12| +3.1e-11 **** +3.8e-11 | +1.0e-10
Z=-26.6(63.74%) | Like=-20.90..-20.46 [-20.8996..-20.8986]*| it/evals=1943/2990 eff=72.2305% N=300
Z=-26.6(64.41%) | Like=-20.89..-20.46 [-20.8872..-20.8842]*| it/evals=1950/2999 eff=72.2490% N=300
Z=-26.5(66.96%) | Like=-20.85..-20.46 [-20.8527..-20.8526]*| it/evals=1978/3033 eff=72.3747% N=300
Z=-26.5(67.15%) | Like=-20.85..-20.46 [-20.8518..-20.8489]*| it/evals=1980/3037 eff=72.3420% N=300
Z=-26.5(69.03%) | Like=-20.83..-20.46 [-20.8270..-20.8266]*| it/evals=2001/3075 eff=72.1081% N=300
Mono-modal Volume: ~exp(-10.20) * Expected Volume: exp(-6.70) Quality: ok
index : +1.0| +2.5 **** +2.7 | +5.0
amplitude: +1.0e-12| +3.1e-11 **** +3.7e-11 | +1.0e-10
Z=-26.5(69.83%) | Like=-20.82..-20.46 [-20.8157..-20.8155]*| it/evals=2010/3087 eff=72.1206% N=300
[ultranest] Explored until L=-2e+01
[ultranest] Likelihood function evaluations: 3090
[ultranest] logZ = -26.12 +- 0.05455
[ultranest] Effective samples strategy satisfied (ESS = 1018.1, need >400)
[ultranest] Posterior uncertainty strategy is satisfied (KL: 0.46+-0.07 nat, need <0.50 nat)
[ultranest] Evidency uncertainty strategy is satisfied (dlogz=0.27, need <0.5)
[ultranest] logZ error budget: single: 0.12 bs:0.05 tail:0.26 total:0.27 required:<0.50
[ultranest] done iterating.
logZ = -26.140 +- 0.303
single instance: logZ = -26.140 +- 0.124
bootstrapped : logZ = -26.116 +- 0.152
tail : logZ = +- 0.262
insert order U test : converged: True correlation: inf iterations
index : 2.15 │ ▁▁▁▁▁▁▁▂▃▃▄▅▇▇▇▆▇▇▇▆▆▃▄▃▃▃▂▁▁▁▁▁▁▁▁▁▁ │3.08 2.57 +- 0.13
amplitude : 0.0000000000206│ ▁▁▁▁▁▁▂▃▄▄▅▆▇▇▆▅▅▄▄▃▂▂▁▁▁▁▁▁▁▁ ▁▁ ▁ │0.0000000000551 0.0000000000338 +- 0.0000000000040
[ultranest] Sampling 300 live points from prior ...
Mono-modal Volume: ~exp(-3.75) * Expected Volume: exp(0.00) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|****************************** *** ********** **| +1.0e-10
Z=-inf(0.00%) | Like=-975.79..-19.96 [-975.7858..-128.6487] | it/evals=0/301 eff=0.0000% N=300
Z=-218.3(0.00%) | Like=-213.23..-19.96 [-975.7858..-128.6487] | it/evals=30/334 eff=88.2353% N=300
Z=-197.3(0.00%) | Like=-191.14..-19.96 [-975.7858..-128.6487] | it/evals=60/367 eff=89.5522% N=300
Mono-modal Volume: ~exp(-4.43) * Expected Volume: exp(-0.22) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|****************************** *** ********** **| +1.0e-10
Z=-194.4(0.00%) | Like=-188.21..-19.96 [-975.7858..-128.6487] | it/evals=67/376 eff=88.1579% N=300
Z=-183.2(0.00%) | Like=-178.11..-19.96 [-975.7858..-128.6487] | it/evals=90/402 eff=88.2353% N=300
Z=-168.5(0.00%) | Like=-163.59..-19.19 [-975.7858..-128.6487] | it/evals=120/434 eff=89.5522% N=300
Mono-modal Volume: ~exp(-4.50) * Expected Volume: exp(-0.45) Quality: ok
index : +1.0| ***********************************************| +5.0
amplitude: +1.0e-12| ********************************* ********** **| +1.0e-10
Z=-160.4(0.00%) | Like=-154.79..-19.19 [-975.7858..-128.6487] | it/evals=134/453 eff=87.5817% N=300
Z=-155.5(0.00%) | Like=-150.44..-19.19 [-975.7858..-128.6487] | it/evals=150/473 eff=86.7052% N=300
Z=-144.2(0.00%) | Like=-139.25..-19.19 [-975.7858..-128.6487] | it/evals=179/508 eff=86.0577% N=300
Z=-144.0(0.00%) | Like=-139.15..-19.19 [-975.7858..-128.6487] | it/evals=180/509 eff=86.1244% N=300
Mono-modal Volume: ~exp(-4.50) Expected Volume: exp(-0.67) Quality: ok
index : +1.0| **********************************************| +5.0
amplitude: +1.0e-12| ******************************************** **| +1.0e-10
Z=-134.3(0.00%) | Like=-128.50..-19.19 [-128.4992..-65.3669] | it/evals=202/539 eff=84.5188% N=300
Z=-130.3(0.00%) | Like=-124.33..-19.19 [-128.4992..-65.3669] | it/evals=210/551 eff=83.6653% N=300
Z=-119.1(0.00%) | Like=-114.11..-19.18 [-128.4992..-65.3669] | it/evals=234/585 eff=82.1053% N=300
Z=-117.4(0.00%) | Like=-111.74..-19.18 [-128.4992..-65.3669] | it/evals=240/592 eff=82.1918% N=300
Z=-106.7(0.00%) | Like=-100.32..-19.18 [-128.4992..-65.3669] | it/evals=266/627 eff=81.3456% N=300
Mono-modal Volume: ~exp(-5.00) * Expected Volume: exp(-0.89) Quality: ok
index : +1.0| *******************************************| +5.0
amplitude: +1.0e-12| ****************************************** **| +1.0e-10
Z=-105.6(0.00%) | Like=-99.66..-19.18 [-128.4992..-65.3669] | it/evals=268/630 eff=81.2121% N=300
Z=-104.8(0.00%) | Like=-98.99..-19.18 [-128.4992..-65.3669] | it/evals=270/632 eff=81.3253% N=300
Z=-96.2(0.00%) | Like=-90.77..-19.18 [-128.4992..-65.3669] | it/evals=296/667 eff=80.6540% N=300
Z=-95.4(0.00%) | Like=-90.08..-19.18 [-128.4992..-65.3669] | it/evals=300/671 eff=80.8625% N=300
Z=-89.3(0.00%) | Like=-84.38..-19.18 [-128.4992..-65.3669] | it/evals=326/706 eff=80.2956% N=300
Z=-88.6(0.00%) | Like=-83.02..-19.18 [-128.4992..-65.3669] | it/evals=330/710 eff=80.4878% N=300
Mono-modal Volume: ~exp(-5.02) * Expected Volume: exp(-1.12) Quality: ok
index : +1.0| ******************************************| +5.0
amplitude: +1.0e-12| *******************************************| +1.0e-10
Z=-87.4(0.00%) | Like=-81.12..-19.18 [-128.4992..-65.3669] | it/evals=335/716 eff=80.5288% N=300
Z=-78.1(0.00%) | Like=-72.29..-19.18 [-128.4992..-65.3669] | it/evals=360/748 eff=80.3571% N=300
Z=-72.1(0.00%) | Like=-66.85..-19.18 [-128.4992..-65.3669] | it/evals=389/783 eff=80.5383% N=300
Z=-71.9(0.00%) | Like=-66.51..-19.18 [-128.4992..-65.3669] | it/evals=390/784 eff=80.5785% N=300
Mono-modal Volume: ~exp(-5.33) * Expected Volume: exp(-1.34) Quality: ok
index : +1.0| ****************************************| +5.0
amplitude: +1.0e-12| **************************************** | +1.0e-10
Z=-69.9(0.00%) | Like=-64.94..-19.18 [-65.3042..-40.6830] | it/evals=402/798 eff=80.7229% N=300
Z=-66.5(0.00%) | Like=-61.06..-19.18 [-65.3042..-40.6830] | it/evals=420/823 eff=80.3059% N=300
Z=-62.1(0.00%) | Like=-56.49..-19.18 [-65.3042..-40.6830] | it/evals=449/857 eff=80.6104% N=300
Z=-61.9(0.00%) | Like=-56.33..-19.18 [-65.3042..-40.6830] | it/evals=450/858 eff=80.6452% N=300
Mono-modal Volume: ~exp(-5.55) * Expected Volume: exp(-1.56) Quality: ok
index : +1.0| **************************************| +5.0
amplitude: +1.0e-12| ************************************ * | +1.0e-10
Z=-59.6(0.00%) | Like=-54.66..-19.18 [-65.3042..-40.6830] | it/evals=469/883 eff=80.4460% N=300
Z=-58.4(0.00%) | Like=-53.18..-19.18 [-65.3042..-40.6830] | it/evals=480/897 eff=80.4020% N=300
Z=-55.8(0.00%) | Like=-50.56..-19.18 [-65.3042..-40.6830] | it/evals=507/932 eff=80.2215% N=300
Z=-55.5(0.00%) | Like=-50.15..-19.18 [-65.3042..-40.6830] | it/evals=510/936 eff=80.1887% N=300
Z=-53.2(0.00%) | Like=-48.02..-19.18 [-65.3042..-40.6830] | it/evals=532/972 eff=79.1667% N=300
Mono-modal Volume: ~exp(-5.71) * Expected Volume: exp(-1.79) Quality: ok
index : +1.0| ********************************** | +5.0
amplitude: +1.0e-12| *********************************** * | +1.0e-10
Z=-52.8(0.00%) | Like=-47.75..-19.18 [-65.3042..-40.6830] | it/evals=536/978 eff=79.0560% N=300
Z=-52.5(0.00%) | Like=-47.33..-19.18 [-65.3042..-40.6830] | it/evals=540/983 eff=79.0630% N=300
Z=-50.2(0.00%) | Like=-44.87..-19.18 [-65.3042..-40.6830] | it/evals=562/1018 eff=78.2730% N=300
Z=-49.3(0.00%) | Like=-43.94..-19.18 [-65.3042..-40.6830] | it/evals=570/1028 eff=78.2967% N=300
Z=-47.0(0.00%) | Like=-42.01..-19.18 [-65.3042..-40.6830] | it/evals=597/1064 eff=78.1414% N=300
Z=-46.8(0.00%) | Like=-41.72..-19.18 [-65.3042..-40.6830] | it/evals=600/1067 eff=78.2269% N=300
Mono-modal Volume: ~exp(-5.74) * Expected Volume: exp(-2.01) Quality: ok
index : +1.0| +2.0 **************************** | +5.0
amplitude: +1.0e-12| +2.4e-11 ********************************** *| +1.0e-10
Z=-46.6(0.00%) | Like=-41.25..-19.18 [-65.3042..-40.6830] | it/evals=603/1071 eff=78.2101% N=300
Z=-44.8(0.00%) | Like=-39.63..-19.18 [-40.6347..-29.7786] | it/evals=626/1106 eff=77.6675% N=300
Z=-44.5(0.00%) | Like=-39.43..-19.18 [-40.6347..-29.7786] | it/evals=630/1111 eff=77.6819% N=300
Z=-43.0(0.00%) | Like=-37.86..-19.18 [-40.6347..-29.7786] | it/evals=653/1150 eff=76.8235% N=300
Z=-42.6(0.00%) | Like=-37.35..-19.18 [-40.6347..-29.7786] | it/evals=660/1159 eff=76.8335% N=300
Mono-modal Volume: ~exp(-6.08) * Expected Volume: exp(-2.23) Quality: ok
index : +1.0| +2.1 ************************ +4.1 | +5.0
amplitude: +1.0e-12| +2.7e-11 ********************************* | +1.0e-10
Z=-41.9(0.00%) | Like=-36.87..-19.18 [-40.6347..-29.7786] | it/evals=670/1172 eff=76.8349% N=300
Z=-40.8(0.00%) | Like=-35.71..-19.18 [-40.6347..-29.7786] | it/evals=690/1197 eff=76.9231% N=300
Z=-39.5(0.00%) | Like=-34.21..-19.17 [-40.6347..-29.7786] | it/evals=712/1232 eff=76.3948% N=300
Z=-39.0(0.00%) | Like=-33.73..-19.17 [-40.6347..-29.7786] | it/evals=720/1249 eff=75.8693% N=300
Z=-38.2(0.00%) | Like=-32.87..-19.17 [-40.6347..-29.7786] | it/evals=735/1284 eff=74.6951% N=300
Mono-modal Volume: ~exp(-6.71) * Expected Volume: exp(-2.46) Quality: ok
index : +1.0| +2.1 ********************* +3.8 | +5.0
amplitude: +1.0e-12| +3.0e-11 ******************************* | +1.0e-10
Z=-38.1(0.00%) | Like=-32.85..-19.17 [-40.6347..-29.7786] | it/evals=737/1288 eff=74.5951% N=300
Z=-37.4(0.00%) | Like=-32.28..-19.17 [-40.6347..-29.7786] | it/evals=750/1311 eff=74.1840% N=300
Z=-36.4(0.00%) | Like=-31.16..-19.17 [-40.6347..-29.7786] | it/evals=776/1348 eff=74.0458% N=300
Z=-36.2(0.00%) | Like=-31.02..-19.17 [-40.6347..-29.7786] | it/evals=780/1357 eff=73.7938% N=300
Mono-modal Volume: ~exp(-6.71) Expected Volume: exp(-2.68) Quality: ok
index : +1.0| +2.2 ******************* +3.7 | +5.0
amplitude: +1.0e-12| +3.2e-11 **************************** | +1.0e-10
Z=-35.2(0.00%) | Like=-30.26..-19.17 [-40.6347..-29.7786] | it/evals=804/1391 eff=73.6939% N=300
Z=-35.0(0.00%) | Like=-29.93..-19.17 [-40.6347..-29.7786] | it/evals=810/1398 eff=73.7705% N=300
Z=-34.2(0.00%) | Like=-29.26..-19.17 [-29.7600..-25.8921] | it/evals=835/1433 eff=73.6981% N=300
Z=-34.1(0.00%) | Like=-29.18..-19.17 [-29.7600..-25.8921] | it/evals=840/1439 eff=73.7489% N=300
Z=-33.5(0.01%) | Like=-28.48..-19.17 [-29.7600..-25.8921] | it/evals=863/1474 eff=73.5094% N=300
Z=-33.3(0.01%) | Like=-28.25..-19.17 [-29.7600..-25.8921] | it/evals=870/1487 eff=73.2940% N=300
Mono-modal Volume: ~exp(-6.73) * Expected Volume: exp(-2.90) Quality: ok
index : +1.0| +2.3 ***************** +3.6 | +5.0
amplitude: +1.0e-12| +3.4e-11 ************************* | +1.0e-10
Z=-33.3(0.01%) | Like=-28.20..-19.17 [-29.7600..-25.8921] | it/evals=871/1488 eff=73.3165% N=300
Z=-32.6(0.02%) | Like=-27.65..-19.17 [-29.7600..-25.8921] | it/evals=896/1523 eff=73.2625% N=300
Z=-32.5(0.02%) | Like=-27.59..-19.17 [-29.7600..-25.8921] | it/evals=900/1529 eff=73.2303% N=300
Z=-32.1(0.04%) | Like=-27.22..-19.17 [-29.7600..-25.8921] | it/evals=919/1564 eff=72.7057% N=300
Z=-31.9(0.04%) | Like=-26.89..-19.17 [-29.7600..-25.8921] | it/evals=930/1586 eff=72.3173% N=300
Mono-modal Volume: ~exp(-7.16) * Expected Volume: exp(-3.13) Quality: ok
index : +1.0| +2.3 **************** +3.6 | +5.0
amplitude: +1.0e-12| +3.5e-11 *********************** | +1.0e-10
Z=-31.7(0.05%) | Like=-26.81..-19.17 [-29.7600..-25.8921] | it/evals=938/1597 eff=72.3207% N=300
Z=-31.3(0.08%) | Like=-26.49..-19.17 [-29.7600..-25.8921] | it/evals=960/1628 eff=72.2892% N=300
Z=-30.8(0.13%) | Like=-25.91..-19.17 [-29.7600..-25.8921] | it/evals=989/1662 eff=72.6138% N=300
Z=-30.8(0.13%) | Like=-25.88..-19.17 [-25.8847..-25.4152] | it/evals=990/1663 eff=72.6339% N=300
Mono-modal Volume: ~exp(-7.55) * Expected Volume: exp(-3.35) Quality: ok
index : +1.0| +2.3 *************** +3.5 | +5.0
amplitude: +1.0e-12| +3.7e-11 ******************** +7.6e-11 | +1.0e-10
Z=-30.6(0.18%) | Like=-25.56..-19.17 [-25.8847..-25.4152] | it/evals=1005/1679 eff=72.8789% N=300
Z=-30.3(0.23%) | Like=-25.21..-19.17 [-25.2900..-25.2122] | it/evals=1020/1700 eff=72.8571% N=300
Z=-29.8(0.38%) | Like=-24.54..-19.17 [-24.5360..-24.4982] | it/evals=1045/1735 eff=72.8223% N=300
Z=-29.7(0.42%) | Like=-24.41..-19.17 [-24.4095..-24.3442] | it/evals=1050/1743 eff=72.7651% N=300
Mono-modal Volume: ~exp(-7.55) Expected Volume: exp(-3.57) Quality: ok
index : +1.0| +2.4 ************* +3.4 | +5.0
amplitude: +1.0e-12| +3.9e-11 ******************* +7.5e-11 | +1.0e-10
Z=-29.2(0.67%) | Like=-23.99..-19.16 [-24.0115..-23.9909] | it/evals=1075/1774 eff=72.9308% N=300
Z=-29.2(0.72%) | Like=-23.93..-19.16 [-23.9279..-23.9061] | it/evals=1080/1780 eff=72.9730% N=300
Z=-28.7(1.14%) | Like=-23.43..-19.16 [-23.4637..-23.4266] | it/evals=1108/1814 eff=73.1836% N=300
Z=-28.7(1.17%) | Like=-23.40..-19.16 [-23.4173..-23.3976] | it/evals=1110/1817 eff=73.1707% N=300
Z=-28.4(1.56%) | Like=-23.17..-19.16 [-23.1710..-23.1597] | it/evals=1127/1852 eff=72.6160% N=300
Mono-modal Volume: ~exp(-7.76) * Expected Volume: exp(-3.80) Quality: ok
index : +1.0| +2.4 ************ +3.3 | +5.0
amplitude: +1.0e-12| +4.0e-11 **************** +7.3e-11 | +1.0e-10
Z=-28.2(1.85%) | Like=-22.99..-19.16 [-22.9888..-22.9865]*| it/evals=1139/1869 eff=72.5940% N=300
Z=-28.2(1.88%) | Like=-22.99..-19.16 [-22.9865..-22.9712] | it/evals=1140/1870 eff=72.6115% N=300
Z=-27.9(2.71%) | Like=-22.80..-19.16 [-22.7968..-22.7911]*| it/evals=1167/1905 eff=72.7103% N=300
Z=-27.9(2.79%) | Like=-22.78..-19.16 [-22.7764..-22.7722]*| it/evals=1170/1908 eff=72.7612% N=300
Z=-27.6(3.57%) | Like=-22.44..-19.16 [-22.4566..-22.4414] | it/evals=1194/1943 eff=72.6719% N=300
Z=-27.5(3.78%) | Like=-22.37..-19.16 [-22.3663..-22.3565]*| it/evals=1200/1951 eff=72.6832% N=300
Mono-modal Volume: ~exp(-8.12) * Expected Volume: exp(-4.02) Quality: ok
index : +1.0| +2.4 *********** +3.3 | +5.0
amplitude: +1.0e-12| +4.2e-11 *************** +7.1e-11 | +1.0e-10
Z=-27.5(4.06%) | Like=-22.32..-19.16 [-22.3163..-22.3129]*| it/evals=1206/1961 eff=72.6069% N=300
Z=-27.2(5.10%) | Like=-22.08..-19.16 [-22.0833..-22.0664] | it/evals=1230/1991 eff=72.7380% N=300
Z=-27.0(6.24%) | Like=-21.82..-19.16 [-21.8218..-21.8210]*| it/evals=1250/2027 eff=72.3798% N=300
Z=-27.0(6.90%) | Like=-21.75..-19.16 [-21.7646..-21.7501] | it/evals=1260/2039 eff=72.4554% N=300
Mono-modal Volume: ~exp(-8.12) Expected Volume: exp(-4.24) Quality: ok
index : +1.0| +2.5 ********** +3.2 | +5.0
amplitude: +1.0e-12| +4.3e-11 ************* +6.9e-11 | +1.0e-10
Z=-26.8(8.29%) | Like=-21.57..-19.16 [-21.5913..-21.5694] | it/evals=1281/2070 eff=72.3729% N=300
Z=-26.7(8.88%) | Like=-21.51..-19.16 [-21.5243..-21.5060] | it/evals=1290/2083 eff=72.3500% N=300
Z=-26.5(10.41%) | Like=-21.32..-19.16 [-21.3232..-21.3170]*| it/evals=1310/2119 eff=72.0176% N=300
Z=-26.5(11.15%) | Like=-21.26..-19.16 [-21.2618..-21.2542]*| it/evals=1320/2132 eff=72.0524% N=300
Z=-26.3(12.68%) | Like=-21.12..-19.16 [-21.1244..-21.1104] | it/evals=1339/2167 eff=71.7193% N=300
Mono-modal Volume: ~exp(-8.19) * Expected Volume: exp(-4.47) Quality: ok
index : +1.0| +2.5 ******** +3.2 | +5.0
amplitude: +1.0e-12| +4.4e-11 ************ +6.7e-11 | +1.0e-10
Z=-26.3(12.73%) | Like=-21.11..-19.16 [-21.1244..-21.1104] | it/evals=1340/2169 eff=71.6961% N=300
Z=-26.3(13.70%) | Like=-21.06..-19.16 [-21.0588..-21.0555]*| it/evals=1350/2183 eff=71.6941% N=300
Z=-26.1(16.02%) | Like=-20.91..-19.16 [-20.9071..-20.9051]*| it/evals=1374/2218 eff=71.6371% N=300
Z=-26.1(16.56%) | Like=-20.84..-19.16 [-20.8429..-20.8387]*| it/evals=1380/2227 eff=71.6139% N=300
Mono-modal Volume: ~exp(-8.67) * Expected Volume: exp(-4.69) Quality: ok
index : +1.0| +2.5 ******** +3.1 | +5.0
amplitude: +1.0e-12| +4.6e-11 *********** +6.6e-11 | +1.0e-10
Z=-25.9(19.36%) | Like=-20.70..-19.16 [-20.6962..-20.6884]*| it/evals=1407/2262 eff=71.7125% N=300
Z=-25.9(19.73%) | Like=-20.67..-19.16 [-20.6795..-20.6672] | it/evals=1410/2266 eff=71.7192% N=300
Z=-25.7(23.28%) | Like=-20.48..-19.16 [-20.4874..-20.4761] | it/evals=1440/2300 eff=72.0000% N=300
Z=-25.6(26.23%) | Like=-20.38..-19.16 [-20.3750..-20.3735]*| it/evals=1463/2335 eff=71.8919% N=300
Z=-25.6(27.13%) | Like=-20.35..-19.16 [-20.3513..-20.3488]*| it/evals=1470/2344 eff=71.9178% N=300
Mono-modal Volume: ~exp(-8.91) * Expected Volume: exp(-4.91) Quality: ok
index : +1.0| +2.6 ******* +3.1 | +5.0
amplitude: +1.0e-12| +4.7e-11 ********* +6.5e-11 | +1.0e-10
Z=-25.6(27.65%) | Like=-20.34..-19.16 [-20.3410..-20.3288] | it/evals=1474/2349 eff=71.9375% N=300
Z=-25.4(31.06%) | Like=-20.22..-19.16 [-20.2181..-20.2171]*| it/evals=1500/2383 eff=72.0115% N=300
Z=-25.4(33.94%) | Like=-20.14..-19.16 [-20.1391..-20.1380]*| it/evals=1521/2419 eff=71.7791% N=300
Z=-25.3(35.20%) | Like=-20.10..-19.16 [-20.1012..-20.0989]*| it/evals=1530/2432 eff=71.7636% N=300
Mono-modal Volume: ~exp(-9.15) * Expected Volume: exp(-5.14) Quality: ok
index : +1.0| +2.6 ****** +3.0 | +5.0
amplitude: +1.0e-12| +4.8e-11 ********* +6.4e-11 | +1.0e-10
Z=-25.3(36.62%) | Like=-20.09..-19.16 [-20.0852..-20.0808]*| it/evals=1541/2451 eff=71.6411% N=300
Z=-25.2(39.22%) | Like=-20.03..-19.16 [-20.0323..-20.0314]*| it/evals=1560/2481 eff=71.5268% N=300
Z=-25.1(42.81%) | Like=-19.95..-19.16 [-19.9495..-19.9494]*| it/evals=1588/2514 eff=71.7254% N=300
Z=-25.1(43.07%) | Like=-19.95..-19.16 [-19.9466..-19.9456]*| it/evals=1590/2518 eff=71.6862% N=300
Mono-modal Volume: ~exp(-9.23) * Expected Volume: exp(-5.36) Quality: ok
index : +1.0| +2.6 ****** +3.0 | +5.0
amplitude: +1.0e-12| +4.8e-11 ******** +6.3e-11 | +1.0e-10
Z=-25.1(45.45%) | Like=-19.90..-19.16 [-19.8991..-19.8982]*| it/evals=1608/2543 eff=71.6897% N=300
Z=-25.0(46.89%) | Like=-19.88..-19.16 [-19.8765..-19.8721]*| it/evals=1620/2562 eff=71.6180% N=300
Z=-25.0(49.98%) | Like=-19.83..-19.16 [-19.8316..-19.8278]*| it/evals=1647/2597 eff=71.7022% N=300
Z=-25.0(50.34%) | Like=-19.82..-19.16 [-19.8244..-19.8218]*| it/evals=1650/2603 eff=71.6457% N=300
Mono-modal Volume: ~exp(-9.84) * Expected Volume: exp(-5.58) Quality: ok
index : +1.0| +2.6 ***** +3.0 | +5.0
amplitude: +1.0e-12| +4.9e-11 ******* +6.1e-11 | +1.0e-10
Z=-24.9(53.33%) | Like=-19.78..-19.16 [-19.7764..-19.7763]*| it/evals=1675/2634 eff=71.7652% N=300
Z=-24.9(53.90%) | Like=-19.77..-19.16 [-19.7659..-19.7632]*| it/evals=1680/2639 eff=71.8256% N=300
Z=-24.8(57.09%) | Like=-19.71..-19.16 [-19.7124..-19.7117]*| it/evals=1709/2674 eff=71.9882% N=300
Z=-24.8(57.19%) | Like=-19.71..-19.16 [-19.7117..-19.7114]*| it/evals=1710/2675 eff=72.0000% N=300
Z=-24.8(60.27%) | Like=-19.67..-19.16 [-19.6659..-19.6658]*| it/evals=1739/2709 eff=72.1876% N=300
Z=-24.8(60.37%) | Like=-19.67..-19.16 [-19.6658..-19.6584]*| it/evals=1740/2711 eff=72.1692% N=300
Mono-modal Volume: ~exp(-9.84) Expected Volume: exp(-5.81) Quality: ok
index : +1.0| +2.6 ***** +3.0 | +5.0
amplitude: +1.0e-12| +5.0e-11 ****** +6.1e-11 | +1.0e-10
Z=-24.8(62.40%) | Like=-19.63..-19.16 [-19.6273..-19.6272]*| it/evals=1760/2742 eff=72.0721% N=300
Z=-24.7(63.36%) | Like=-19.61..-19.16 [-19.6142..-19.6135]*| it/evals=1770/2756 eff=72.0684% N=300
Z=-24.7(65.72%) | Like=-19.58..-19.16 [-19.5826..-19.5824]*| it/evals=1795/2790 eff=72.0884% N=300
Z=-24.7(66.17%) | Like=-19.58..-19.16 [-19.5761..-19.5731]*| it/evals=1800/2795 eff=72.1443% N=300
Mono-modal Volume: ~exp(-10.16) * Expected Volume: exp(-6.03) Quality: ok
index : +1.0| +2.7 ***** +3.0 | +5.0
amplitude: +1.0e-12| +5.0e-11 ****** +6.1e-11 | +1.0e-10
Z=-24.7(66.96%) | Like=-19.55..-19.16 [-19.5543..-19.5541]*| it/evals=1809/2807 eff=72.1580% N=300
Z=-24.7(68.81%) | Like=-19.53..-19.16 [-19.5338..-19.5338]*| it/evals=1830/2836 eff=72.1609% N=300
[ultranest] Explored until L=-2e+01
[ultranest] Likelihood function evaluations: 2852
[ultranest] logZ = -24.3 +- 0.08125
[ultranest] Effective samples strategy satisfied (ESS = 971.3, need >400)
[ultranest] Posterior uncertainty strategy is satisfied (KL: 0.46+-0.08 nat, need <0.50 nat)
[ultranest] Evidency uncertainty strategy is satisfied (dlogz=0.27, need <0.5)
[ultranest] logZ error budget: single: 0.12 bs:0.08 tail:0.26 total:0.27 required:<0.50
[ultranest] done iterating.
logZ = -24.282 +- 0.304
single instance: logZ = -24.282 +- 0.118
bootstrapped : logZ = -24.300 +- 0.154
tail : logZ = +- 0.262
insert order U test : converged: True correlation: inf iterations
index : 2.29 │ ▁▁▁▁▁▁▂▂▃▅▄▅▇▇▇▇▅▆▅▅▃▃▂▂▂▁▁▁▁▁▁▁ ▁ ▁ │3.62 2.83 +- 0.17
amplitude : 0.0000000000363│ ▁▁▁▁▁▂▂▂▃▆▆▇▆▇▇▆▇▆▆▄▄▃▂▂▁▁▁▁▁▁▁ ▁ │0.0000000000842 0.0000000000555 +- 0.0000000000058
[ultranest] Sampling 300 live points from prior ...
Mono-modal Volume: ~exp(-3.80) * Expected Volume: exp(0.00) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|******************************** * **** ***** **| +1.0e-10
Z=-inf(0.00%) | Like=-1071.51..-13.32 [-1071.5071..-91.6614] | it/evals=0/301 eff=0.0000% N=300
Z=-151.2(0.00%) | Like=-146.71..-13.32 [-1071.5071..-91.6614] | it/evals=30/331 eff=96.7742% N=300
Z=-141.5(0.00%) | Like=-136.18..-13.32 [-1071.5071..-91.6614] | it/evals=60/366 eff=90.9091% N=300
Mono-modal Volume: ~exp(-3.80) Expected Volume: exp(-0.22) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|******************************** * **** ***** **| +1.0e-10
Z=-131.8(0.00%) | Like=-127.19..-13.32 [-1071.5071..-91.6614] | it/evals=85/397 eff=87.6289% N=300
Z=-129.8(0.00%) | Like=-125.00..-13.32 [-1071.5071..-91.6614] | it/evals=90/402 eff=88.2353% N=300
Z=-121.4(0.00%) | Like=-116.53..-13.32 [-1071.5071..-91.6614] | it/evals=117/437 eff=85.4015% N=300
Z=-120.0(0.00%) | Like=-114.83..-13.32 [-1071.5071..-91.6614] | it/evals=120/441 eff=85.1064% N=300
Mono-modal Volume: ~exp(-4.16) * Expected Volume: exp(-0.45) Quality: ok
index : +1.0| ***********************************************| +5.0
amplitude: +1.0e-12|*************************************** ***** **| +1.0e-10
Z=-115.3(0.00%) | Like=-110.08..-13.32 [-1071.5071..-91.6614] | it/evals=134/461 eff=83.2298% N=300
Z=-110.7(0.00%) | Like=-104.85..-13.32 [-1071.5071..-91.6614] | it/evals=150/479 eff=83.7989% N=300
Z=-103.5(0.00%) | Like=-98.57..-13.32 [-1071.5071..-91.6614] | it/evals=178/514 eff=83.1776% N=300
Z=-103.0(0.00%) | Like=-98.13..-13.32 [-1071.5071..-91.6614] | it/evals=180/516 eff=83.3333% N=300
Mono-modal Volume: ~exp(-4.70) * Expected Volume: exp(-0.67) Quality: ok
index : +1.0| **********************************************| +5.0
amplitude: +1.0e-12| ************************************** ***** **| +1.0e-10
Z=-98.1(0.00%) | Like=-92.86..-13.32 [-1071.5071..-91.6614] | it/evals=201/550 eff=80.4000% N=300
Z=-95.6(0.00%) | Like=-90.04..-13.32 [-91.5040..-45.2776] | it/evals=210/560 eff=80.7692% N=300
Z=-87.8(0.00%) | Like=-82.83..-13.32 [-91.5040..-45.2776] | it/evals=240/596 eff=81.0811% N=300
Mono-modal Volume: ~exp(-4.73) * Expected Volume: exp(-0.89) Quality: ok
index : +1.0| ********************************************| +5.0
amplitude: +1.0e-12| ************************************* ********| +1.0e-10
Z=-80.9(0.00%) | Like=-75.38..-13.32 [-91.5040..-45.2776] | it/evals=268/627 eff=81.9572% N=300
Z=-80.4(0.00%) | Like=-75.35..-13.32 [-91.5040..-45.2776] | it/evals=270/630 eff=81.8182% N=300
Z=-74.4(0.00%) | Like=-69.43..-13.32 [-91.5040..-45.2776] | it/evals=296/665 eff=81.0959% N=300
Z=-73.7(0.00%) | Like=-68.94..-13.32 [-91.5040..-45.2776] | it/evals=300/670 eff=81.0811% N=300
Z=-67.4(0.00%) | Like=-62.26..-13.32 [-91.5040..-45.2776] | it/evals=327/705 eff=80.7407% N=300
Z=-66.8(0.00%) | Like=-61.66..-13.32 [-91.5040..-45.2776] | it/evals=330/711 eff=80.2920% N=300
Mono-modal Volume: ~exp(-4.73) Expected Volume: exp(-1.12) Quality: ok
index : +1.0| ******************************************| +5.0
amplitude: +1.0e-12| *********************************************| +1.0e-10
Z=-62.3(0.00%) | Like=-57.04..-13.32 [-91.5040..-45.2776] | it/evals=351/743 eff=79.2325% N=300
Z=-60.9(0.00%) | Like=-56.20..-13.32 [-91.5040..-45.2776] | it/evals=360/755 eff=79.1209% N=300
Z=-58.0(0.00%) | Like=-52.99..-13.32 [-91.5040..-45.2776] | it/evals=384/790 eff=78.3673% N=300
Z=-57.1(0.00%) | Like=-51.43..-13.32 [-91.5040..-45.2776] | it/evals=390/799 eff=78.1563% N=300
Mono-modal Volume: ~exp(-5.45) * Expected Volume: exp(-1.34) Quality: ok
index : +1.0| *****************************************| +5.0
amplitude: +1.0e-12| ********************************************| +1.0e-10
Z=-54.9(0.00%) | Like=-49.74..-13.32 [-91.5040..-45.2776] | it/evals=402/816 eff=77.9070% N=300
Z=-52.5(0.00%) | Like=-46.75..-13.32 [-91.5040..-45.2776] | it/evals=420/837 eff=78.2123% N=300
Z=-48.9(0.00%) | Like=-44.18..-13.32 [-45.2073..-29.4629] | it/evals=446/871 eff=78.1086% N=300
Z=-48.6(0.00%) | Like=-43.88..-13.32 [-45.2073..-29.4629] | it/evals=450/875 eff=78.2609% N=300
Mono-modal Volume: ~exp(-5.58) * Expected Volume: exp(-1.56) Quality: ok
index : +1.0| ************************************ | +5.0
amplitude: +1.0e-12| ******************************************| +1.0e-10
Z=-47.0(0.00%) | Like=-41.70..-13.32 [-45.2073..-29.4629] | it/evals=469/899 eff=78.2972% N=300
Z=-45.8(0.00%) | Like=-40.50..-13.32 [-45.2073..-29.4629] | it/evals=480/913 eff=78.3034% N=300
Z=-43.3(0.00%) | Like=-38.43..-13.32 [-45.2073..-29.4629] | it/evals=506/948 eff=78.0864% N=300
Z=-43.0(0.00%) | Like=-38.08..-13.32 [-45.2073..-29.4629] | it/evals=510/955 eff=77.8626% N=300
Z=-40.9(0.00%) | Like=-35.97..-13.32 [-45.2073..-29.4629] | it/evals=533/990 eff=77.2464% N=300
Mono-modal Volume: ~exp(-6.22) * Expected Volume: exp(-1.79) Quality: ok
index : +1.0| ******************************** | +5.0
amplitude: +1.0e-12| *****************************************| +1.0e-10
Z=-40.6(0.00%) | Like=-35.70..-13.32 [-45.2073..-29.4629] | it/evals=536/995 eff=77.1223% N=300
Z=-40.3(0.00%) | Like=-35.43..-13.32 [-45.2073..-29.4629] | it/evals=540/1001 eff=77.0328% N=300
Z=-38.7(0.00%) | Like=-34.16..-13.32 [-45.2073..-29.4629] | it/evals=569/1036 eff=77.3098% N=300
Z=-38.7(0.00%) | Like=-34.04..-13.32 [-45.2073..-29.4629] | it/evals=570/1037 eff=77.3406% N=300
Z=-37.2(0.00%) | Like=-32.34..-13.32 [-45.2073..-29.4629] | it/evals=600/1072 eff=77.7202% N=300
Mono-modal Volume: ~exp(-6.22) Expected Volume: exp(-2.01) Quality: ok
index : +1.0| +1.9 *************************** +4.2 | +5.0
amplitude: +1.0e-12| ****************************************| +1.0e-10
Z=-35.8(0.00%) | Like=-30.37..-13.32 [-45.2073..-29.4629] | it/evals=623/1105 eff=77.3913% N=300
Z=-35.2(0.00%) | Like=-29.87..-13.32 [-45.2073..-29.4629] | it/evals=630/1112 eff=77.5862% N=300
Z=-33.6(0.00%) | Like=-28.59..-13.32 [-29.4543..-21.5855] | it/evals=654/1148 eff=77.1226% N=300
Z=-33.3(0.00%) | Like=-28.30..-13.32 [-29.4543..-21.5855] | it/evals=660/1156 eff=77.1028% N=300
Mono-modal Volume: ~exp(-6.26) * Expected Volume: exp(-2.23) Quality: ok
index : +1.0| +2.0 ************************ +3.9 | +5.0
amplitude: +1.0e-12| **************************************| +1.0e-10
Z=-32.8(0.00%) | Like=-27.77..-13.32 [-29.4543..-21.5855] | it/evals=670/1169 eff=77.1001% N=300
Z=-31.6(0.00%) | Like=-26.62..-13.32 [-29.4543..-21.5855] | it/evals=690/1195 eff=77.0950% N=300
Z=-30.4(0.00%) | Like=-25.37..-13.32 [-29.4543..-21.5855] | it/evals=714/1230 eff=76.7742% N=300
Z=-30.2(0.00%) | Like=-25.23..-13.32 [-29.4543..-21.5855] | it/evals=720/1238 eff=76.7591% N=300
Mono-modal Volume: ~exp(-6.90) * Expected Volume: exp(-2.46) Quality: ok
index : +1.0| +2.1 ********************* +3.7 | +5.0
amplitude: +1.0e-12| +2.5e-11 ************************************ | +1.0e-10
Z=-29.4(0.00%) | Like=-24.45..-13.32 [-29.4543..-21.5855] | it/evals=737/1264 eff=76.4523% N=300
Z=-28.9(0.00%) | Like=-24.06..-13.32 [-29.4543..-21.5855] | it/evals=750/1282 eff=76.3747% N=300
Z=-27.9(0.01%) | Like=-23.18..-13.32 [-29.4543..-21.5855] | it/evals=780/1318 eff=76.6208% N=300
Mono-modal Volume: ~exp(-6.90) Expected Volume: exp(-2.68) Quality: ok
index : +1.0| +2.1 ****************** +3.5 | +5.0
amplitude: +1.0e-12| +2.6e-11 ******************************** | +1.0e-10
Z=-27.3(0.01%) | Like=-22.65..-13.32 [-29.4543..-21.5855] | it/evals=804/1353 eff=76.3533% N=300
Z=-27.1(0.01%) | Like=-22.39..-13.32 [-29.4543..-21.5855] | it/evals=810/1363 eff=76.1994% N=300
Z=-26.3(0.03%) | Like=-21.48..-13.32 [-21.5854..-19.9323] | it/evals=839/1398 eff=76.4117% N=300
Z=-26.3(0.03%) | Like=-21.47..-13.32 [-21.5854..-19.9323] | it/evals=840/1399 eff=76.4331% N=300
Z=-25.7(0.06%) | Like=-20.88..-13.32 [-21.5854..-19.9323] | it/evals=865/1434 eff=76.2787% N=300
Z=-25.6(0.07%) | Like=-20.76..-13.32 [-21.5854..-19.9323] | it/evals=870/1440 eff=76.3158% N=300
Mono-modal Volume: ~exp(-7.17) * Expected Volume: exp(-2.90) Quality: ok
index : +1.0| +2.2 **************** +3.5 | +5.0
amplitude: +1.0e-12| +2.9e-11 ****************************** | +1.0e-10
Z=-25.6(0.07%) | Like=-20.71..-13.32 [-21.5854..-19.9323] | it/evals=871/1441 eff=76.3365% N=300
Z=-25.0(0.12%) | Like=-20.04..-13.32 [-21.5854..-19.9323] | it/evals=896/1475 eff=76.2553% N=300
Z=-24.9(0.13%) | Like=-19.87..-13.32 [-19.9153..-19.7262] | it/evals=900/1479 eff=76.3359% N=300
Z=-24.3(0.24%) | Like=-19.31..-13.32 [-19.3148..-19.2865] | it/evals=925/1513 eff=76.2572% N=300
Z=-24.2(0.26%) | Like=-19.17..-13.32 [-19.1679..-19.1613]*| it/evals=930/1523 eff=76.0425% N=300
Mono-modal Volume: ~exp(-7.49) * Expected Volume: exp(-3.13) Quality: ok
index : +1.0| +2.2 *************** +3.4 | +5.0
amplitude: +1.0e-12| +3.1e-11 ************************** | +1.0e-10
Z=-24.0(0.31%) | Like=-19.00..-13.32 [-19.0155..-19.0029] | it/evals=938/1532 eff=76.1364% N=300
Z=-23.6(0.50%) | Like=-18.57..-13.32 [-18.5674..-18.5397] | it/evals=960/1561 eff=76.1301% N=300
Z=-23.1(0.77%) | Like=-18.22..-13.32 [-18.2233..-18.2099] | it/evals=985/1595 eff=76.0618% N=300
Z=-23.0(0.82%) | Like=-18.11..-13.32 [-18.1356..-18.1133] | it/evals=990/1600 eff=76.1538% N=300
Mono-modal Volume: ~exp(-7.66) * Expected Volume: exp(-3.35) Quality: ok
index : +1.0| +2.3 ************* +3.3 | +5.0
amplitude: +1.0e-12| +3.3e-11 ************************ | +1.0e-10
Z=-22.8(1.03%) | Like=-17.88..-13.32 [-17.8840..-17.8647] | it/evals=1005/1620 eff=76.1364% N=300
Z=-22.5(1.32%) | Like=-17.66..-13.32 [-17.6618..-17.6489] | it/evals=1020/1645 eff=75.8364% N=300
Z=-22.2(1.83%) | Like=-17.31..-13.32 [-17.3087..-17.2890] | it/evals=1045/1680 eff=75.7246% N=300
Z=-22.1(1.96%) | Like=-17.26..-13.32 [-17.2636..-17.2563]*| it/evals=1050/1688 eff=75.6484% N=300
Mono-modal Volume: ~exp(-7.83) * Expected Volume: exp(-3.57) Quality: ok
index : +1.0| +2.3 ************ +3.2 | +5.0
amplitude: +1.0e-12| +3.5e-11 ********************** +7.7e-11| +1.0e-10
Z=-21.9(2.50%) | Like=-17.09..-13.32 [-17.0921..-17.0814] | it/evals=1072/1718 eff=75.5994% N=300
Z=-21.8(2.76%) | Like=-16.96..-13.32 [-16.9632..-16.9510] | it/evals=1080/1729 eff=75.5773% N=300
Z=-21.5(3.68%) | Like=-16.74..-13.32 [-16.7425..-16.7297] | it/evals=1108/1763 eff=75.7348% N=300
Z=-21.5(3.75%) | Like=-16.72..-13.32 [-16.7161..-16.7012] | it/evals=1110/1765 eff=75.7679% N=300
Z=-21.2(4.94%) | Like=-16.39..-13.32 [-16.3996..-16.3865] | it/evals=1137/1799 eff=75.8506% N=300
Mono-modal Volume: ~exp(-7.83) Expected Volume: exp(-3.80) Quality: ok
index : +1.0| +2.4 *********** +3.2 | +5.0
amplitude: +1.0e-12| +3.6e-11 ******************* +7.4e-11 | +1.0e-10
Z=-21.2(5.11%) | Like=-16.37..-13.32 [-16.3741..-16.3484] | it/evals=1140/1804 eff=75.7979% N=300
Z=-21.0(6.29%) | Like=-16.16..-13.32 [-16.1606..-16.1594]*| it/evals=1164/1837 eff=75.7319% N=300
Z=-20.9(6.54%) | Like=-16.14..-13.32 [-16.1410..-16.1234] | it/evals=1170/1846 eff=75.6792% N=300
Z=-20.7(8.11%) | Like=-15.89..-13.31 [-15.8935..-15.8772] | it/evals=1197/1880 eff=75.7595% N=300
Z=-20.7(8.29%) | Like=-15.87..-13.31 [-15.8712..-15.8688]*| it/evals=1200/1884 eff=75.7576% N=300
Mono-modal Volume: ~exp(-7.83) Expected Volume: exp(-4.02) Quality: ok
index : +1.0| +2.4 ********** +3.1 | +5.0
amplitude: +1.0e-12| +3.7e-11 ****************** +7.2e-11 | +1.0e-10
Z=-20.5(9.82%) | Like=-15.65..-13.31 [-15.6528..-15.6463]*| it/evals=1221/1916 eff=75.5569% N=300
Z=-20.5(10.51%) | Like=-15.61..-13.31 [-15.6092..-15.6052]*| it/evals=1230/1935 eff=75.2294% N=300
Z=-20.3(12.43%) | Like=-15.41..-13.31 [-15.4060..-15.3995]*| it/evals=1255/1969 eff=75.1947% N=300
Z=-20.3(12.89%) | Like=-15.38..-13.31 [-15.3815..-15.3759]*| it/evals=1260/1979 eff=75.0447% N=300
Mono-modal Volume: ~exp(-8.13) * Expected Volume: exp(-4.24) Quality: ok
index : +1.0| +2.4 ******** +3.1 | +5.0
amplitude: +1.0e-12| +3.9e-11 **************** +7.0e-11 | +1.0e-10
Z=-20.2(14.09%) | Like=-15.31..-13.31 [-15.3107..-15.3052]*| it/evals=1273/1995 eff=75.1032% N=300
Z=-20.1(15.71%) | Like=-15.18..-13.31 [-15.1819..-15.1703] | it/evals=1290/2015 eff=75.2187% N=300
Z=-19.9(18.03%) | Like=-15.04..-13.31 [-15.0352..-15.0336]*| it/evals=1314/2052 eff=75.0000% N=300
Z=-19.9(18.73%) | Like=-14.99..-13.31 [-14.9866..-14.9785]*| it/evals=1320/2062 eff=74.9149% N=300
Mono-modal Volume: ~exp(-8.71) * Expected Volume: exp(-4.47) Quality: ok
index : +1.0| +2.5 ******** +3.0 | +5.0
amplitude: +1.0e-12| +4.1e-11 ************** +6.8e-11 | +1.0e-10
Z=-19.8(20.68%) | Like=-14.86..-13.31 [-14.8754..-14.8646] | it/evals=1340/2093 eff=74.7351% N=300
Z=-19.7(21.75%) | Like=-14.83..-13.31 [-14.8273..-14.8224]*| it/evals=1350/2106 eff=74.7508% N=300
Z=-19.6(24.90%) | Like=-14.72..-13.31 [-14.7185..-14.7115]*| it/evals=1376/2140 eff=74.7826% N=300
Z=-19.6(25.33%) | Like=-14.69..-13.31 [-14.6899..-14.6873]*| it/evals=1380/2146 eff=74.7562% N=300
Z=-19.5(27.63%) | Like=-14.61..-13.31 [-14.6090..-14.5993]*| it/evals=1400/2182 eff=74.3889% N=300
Mono-modal Volume: ~exp(-8.86) * Expected Volume: exp(-4.69) Quality: ok
index : +1.0| +2.5 ******** +3.0 | +5.0
amplitude: +1.0e-12| +4.2e-11 ************* +6.6e-11 | +1.0e-10
Z=-19.5(28.43%) | Like=-14.57..-13.31 [-14.5702..-14.5665]*| it/evals=1407/2190 eff=74.4444% N=300
Z=-19.5(28.73%) | Like=-14.56..-13.31 [-14.5629..-14.5622]*| it/evals=1410/2195 eff=74.4063% N=300
Z=-19.4(31.78%) | Like=-14.47..-13.31 [-14.4715..-14.4648]*| it/evals=1437/2229 eff=74.4946% N=300
Z=-19.4(32.10%) | Like=-14.46..-13.31 [-14.4578..-14.4565]*| it/evals=1440/2233 eff=74.4956% N=300
Z=-19.3(35.16%) | Like=-14.35..-13.31 [-14.3451..-14.3393]*| it/evals=1466/2268 eff=74.4919% N=300
Z=-19.2(35.69%) | Like=-14.33..-13.31 [-14.3265..-14.3256]*| it/evals=1470/2272 eff=74.5436% N=300
Mono-modal Volume: ~exp(-8.92) * Expected Volume: exp(-4.91) Quality: ok
index : +1.0| +2.5 ****** +3.0 | +5.0
amplitude: +1.0e-12| +4.3e-11 *********** +6.4e-11 | +1.0e-10
Z=-19.2(36.20%) | Like=-14.31..-13.31 [-14.3139..-14.3100]*| it/evals=1474/2277 eff=74.5574% N=300
Z=-19.2(39.43%) | Like=-14.22..-13.31 [-14.2225..-14.2213]*| it/evals=1500/2310 eff=74.6269% N=300
Z=-19.1(41.90%) | Like=-14.16..-13.31 [-14.1596..-14.1473] | it/evals=1521/2347 eff=74.3039% N=300
Z=-19.1(43.04%) | Like=-14.12..-13.31 [-14.1250..-14.1249]*| it/evals=1530/2358 eff=74.3440% N=300
Mono-modal Volume: ~exp(-9.25) * Expected Volume: exp(-5.14) Quality: ok
index : +1.0| +2.5 ****** +2.9 | +5.0
amplitude: +1.0e-12| +4.4e-11 ********** +6.3e-11 | +1.0e-10
Z=-19.0(44.34%) | Like=-14.09..-13.31 [-14.0863..-14.0855]*| it/evals=1541/2373 eff=74.3367% N=300
Z=-19.0(46.68%) | Like=-14.05..-13.31 [-14.0468..-14.0411]*| it/evals=1560/2398 eff=74.3565% N=300
Z=-18.9(49.40%) | Like=-13.99..-13.31 [-13.9938..-13.9922]*| it/evals=1582/2432 eff=74.2026% N=300
Z=-18.9(50.29%) | Like=-13.98..-13.31 [-13.9840..-13.9834]*| it/evals=1590/2443 eff=74.1951% N=300
Mono-modal Volume: ~exp(-9.59) * Expected Volume: exp(-5.36) Quality: ok
index : +1.0| +2.6 ***** +2.9 | +5.0
amplitude: +1.0e-12| +4.5e-11 ********* +6.2e-11 | +1.0e-10
Z=-18.9(52.40%) | Like=-13.94..-13.31 [-13.9418..-13.9378]*| it/evals=1608/2473 eff=73.9991% N=300
Z=-18.8(53.71%) | Like=-13.90..-13.30 [-13.9035..-13.9021]*| it/evals=1620/2490 eff=73.9726% N=300
Z=-18.8(57.03%) | Like=-13.85..-13.30 [-13.8505..-13.8495]*| it/evals=1649/2527 eff=74.0458% N=300
Z=-18.8(57.12%) | Like=-13.85..-13.30 [-13.8495..-13.8483]*| it/evals=1650/2529 eff=74.0242% N=300
Mono-modal Volume: ~exp(-9.96) * Expected Volume: exp(-5.58) Quality: ok
index : +1.0| +2.6 ***** +2.9 | +5.0
amplitude: +1.0e-12| +4.6e-11 ******** +6.1e-11 | +1.0e-10
Z=-18.7(59.69%) | Like=-13.80..-13.30 [-13.8040..-13.8033]*| it/evals=1675/2559 eff=74.1479% N=300
Z=-18.7(60.23%) | Like=-13.80..-13.30 [-13.7995..-13.7987]*| it/evals=1680/2565 eff=74.1722% N=300
Z=-18.7(62.37%) | Like=-13.77..-13.30 [-13.7677..-13.7671]*| it/evals=1701/2599 eff=73.9887% N=300
Z=-18.7(63.29%) | Like=-13.76..-13.30 [-13.7586..-13.7583]*| it/evals=1710/2612 eff=73.9619% N=300
Z=-18.6(65.56%) | Like=-13.73..-13.30 [-13.7276..-13.7269]*| it/evals=1734/2647 eff=73.8816% N=300
Z=-18.6(66.10%) | Like=-13.72..-13.30 [-13.7193..-13.7193]*| it/evals=1740/2653 eff=73.9482% N=300
Mono-modal Volume: ~exp(-9.96) Expected Volume: exp(-5.81) Quality: ok
index : +1.0| +2.6 **** +2.9 | +5.0
amplitude: +1.0e-12| +4.6e-11 ******** +6.0e-11 | +1.0e-10
Z=-18.6(67.62%) | Like=-13.70..-13.30 [-13.7038..-13.7029]*| it/evals=1757/2685 eff=73.6688% N=300
Z=-18.6(68.67%) | Like=-13.69..-13.30 [-13.6923..-13.6917]*| it/evals=1770/2708 eff=73.5050% N=300
[ultranest] Explored until L=-1e+01
[ultranest] Likelihood function evaluations: 2735
[ultranest] logZ = -18.23 +- 0.07033
[ultranest] Effective samples strategy satisfied (ESS = 1004.1, need >400)
[ultranest] Posterior uncertainty strategy is satisfied (KL: 0.45+-0.07 nat, need <0.50 nat)
[ultranest] Evidency uncertainty strategy is satisfied (dlogz=0.27, need <0.5)
[ultranest] logZ error budget: single: 0.11 bs:0.07 tail:0.26 total:0.27 required:<0.50
[ultranest] done iterating.
logZ = -18.223 +- 0.287
single instance: logZ = -18.223 +- 0.114
bootstrapped : logZ = -18.225 +- 0.118
tail : logZ = +- 0.262
insert order U test : converged: True correlation: inf iterations
index : 2.21 │ ▁▁▁▁▁▂▂▃▃▄▅▅▆▇▆▇▇▇▆▆▅▄▃▃▃▂▁▁▁▁▁▁▁▁▁▁▁ │3.44 2.75 +- 0.17
amplitude : 0.0000000000287│ ▁▁▁▁▁▂▃▃▃▄▆▅▆▆▇▇▅▅▄▄▃▂▂▂▁▁▁▁▁▁▁▁▁▁ ▁ │0.0000000000911 0.0000000000533 +- 0.0000000000080
Comparing the posterior distribution of all runs#
For a comparison of different posterior distributions, we can use the package chainconsumer. As this is not a Gammapy dependency, you’ll need to install it. More info here : https://samreay.github.io/ChainConsumer/
# Uncomment this if you have installed `chainconsumer`.
# from chainconsumer import Chain, ChainConfig, ChainConsumer, PlotConfig, Truth, make_sample
# from pandas import DataFrame
# c = ChainConsumer()
# def create_chain(result, name, color="k"):
# return Chain(
# samples=DataFrame(result, columns=["index", "amplitude"]),
# name=name,
# color=color,
# smooth=7,
# shade=False,
# linewidth=1.0,
# cmap="magma",
# show_contour_labels=True,
# kde= True
# )
# c.add_chain(create_chain(result_joint.samples, "joint"))
# c.add_chain(create_chain(result_0.samples, "run0", "g"))
# c.add_chain(create_chain(result_1.samples, "run1", "b"))
# c.add_chain(create_chain(result_2.samples, "run2", "y"))
# fig = c.plotter.plot()
# plt.show()
Corner plot comparison#
Corner plot comparing the three Crab runs.#
We can see the joint analysis allows to better constrain the parameters than the individual runs (more observation time is of course better). One can note as well that one of the run has a notably different amplitude (possibly due to calibrations or/and atmospheric issues).
Highest density intervals#
Given the samples, one can also compute the highest density interval (HDI) which is also known as the smallest credible interval (SCI). See more details here. This is the smallest interval in which a given probability (e.g. 68%) is contained.
For unimodal distributions, the HDI is a single continuous interval containing the mode whereas for multimodal distributions, the HDI can be a set of disconnected intervals. The HDI can be particularly helpful with multimodal distributions as opposed to the mean and quantiles approaches which will not report the important information. Here, we showcase the HDI using the Arviz package. Check out the many possibilities offered by Arviz, a package to analyze the samples posterior distributions.
from arviz import hdi
import scipy.stats as stats
# Multi-modal samples example
weight = 0.3
n_samples = 10000
mu1 = 5.5e-11
sigma1 = 0.7e-11
mu2 = 3.5e-11
sigma2 = 0.3e-11
weight = 0.7
rng = np.random.default_rng(42)
component_mask = rng.uniform(size=n_samples) < weight
samples = np.empty(n_samples)
samples[component_mask] = rng.normal(mu1, sigma1, component_mask.sum())
samples[~component_mask] = rng.normal(mu2, sigma2, (~component_mask).sum())
fig, (ax1, ax2) = plt.subplots(
2, 1, sharex=True, figsize=(9, 7), gridspec_kw={"height_ratios": [5, 2]}
)
# Highest density intervals
hdis = hdi(samples, prob=0.68, method="multimodal")
ax1.hist(
samples,
bins=50,
histtype="step",
color="k",
alpha=0.5,
)
yl = ax1.get_ylim()
for k in range(hdis.shape[0]):
label = "68% HDI" if k == 0 else None
ax2.hlines(
1 + 3 * 0.015, hdis[k, 0], hdis[k, 1], lw=15, color="k", alpha=0.5, label=label
)
# Percentile
percentile = np.percentile(samples, q=[16, 84])
ax2.hlines(
1 + 2 * 0.015,
percentile[0],
percentile[1],
lw=15,
color="y",
alpha=0.5,
label="16-84% percentile",
)
# Mean and standard deviation
mean = np.mean(samples)
std = np.std(samples)
ax1.plot([mean, mean], yl, label="mean", color="r", ls="--")
ax2.hlines(
1 + 1 * 0.015,
mean - std,
mean + std,
lw=15,
color="r",
alpha=0.5,
label=r"mean $\pm$ std",
)
# Median and median absolute deviation
median = np.median(samples)
mad = stats.median_abs_deviation(samples)
ax1.plot([median, median], yl, label="median", color="b", ls="--")
ax2.hlines(
1,
median - mad,
median + mad,
lw=15,
color="b",
alpha=0.5,
label=r"median $\pm$ mad",
)
ax2.legend(loc=6)
ax1.legend(loc="upper left")
ax1.set_xlim(1e-11, 8e-11)
ax2.set_ylim(0.98, 1.06)
ax2.set_xlabel("Amplitude")
ax2.tick_params(left=False, labelleft=False)
plt.show()
Total running time of the script: (0 minutes 48.354 seconds)
