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(-4.18) * Expected Volume: exp(0.00) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|**************************** *************** ** | +1.0e-10
Z=-inf(0.00%) | Like=-4179.26..-60.45 [-4179.2604..-322.6077] | it/evals=0/301 eff=0.0000% N=300
Z=-540.8(0.00%) | Like=-533.24..-60.45 [-4179.2604..-322.6077] | it/evals=25/328 eff=89.2857% N=300
Z=-534.0(0.00%) | Like=-526.75..-60.45 [-4179.2604..-322.6077] | it/evals=30/333 eff=90.9091% N=300
Z=-504.9(0.00%) | Like=-497.72..-60.45 [-4179.2604..-322.6077] | it/evals=54/360 eff=90.0000% N=300
Z=-500.2(0.00%) | Like=-495.43..-60.45 [-4179.2604..-322.6077] | it/evals=60/366 eff=90.9091% N=300
Mono-modal Volume: ~exp(-4.46) * Expected Volume: exp(-0.22) Quality: ok
index : +1.0|************************************** *********| +5.0
amplitude: +1.0e-12|**************************** ******* *** *** ** | +1.0e-10
Z=-488.8(0.00%) | Like=-480.92..-60.45 [-4179.2604..-322.6077] | it/evals=67/374 eff=90.5405% N=300
Z=-468.3(0.00%) | Like=-462.23..-60.45 [-4179.2604..-322.6077] | it/evals=89/401 eff=88.1188% N=300
Z=-467.6(0.00%) | Like=-461.41..-60.45 [-4179.2604..-322.6077] | it/evals=90/402 eff=88.2353% N=300
Z=-440.3(0.00%) | Like=-433.48..-60.45 [-4179.2604..-322.6077] | it/evals=115/429 eff=89.1473% N=300
Z=-435.0(0.00%) | Like=-429.70..-60.45 [-4179.2604..-322.6077] | it/evals=120/435 eff=88.8889% N=300
Mono-modal Volume: ~exp(-4.46) 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=-401.02..-60.45 [-4179.2604..-322.6077] | it/evals=142/460 eff=88.7500% N=300
Z=-400.0(0.00%) | Like=-393.84..-60.45 [-4179.2604..-322.6077] | it/evals=150/468 eff=89.2857% N=300
Z=-378.3(0.00%) | Like=-372.41..-60.45 [-4179.2604..-322.6077] | it/evals=172/495 eff=88.2051% N=300
Z=-369.3(0.00%) | Like=-362.44..-60.45 [-4179.2604..-322.6077] | it/evals=180/504 eff=88.2353% N=300
Mono-modal Volume: ~exp(-4.68) * Expected Volume: exp(-0.67) Quality: ok
index : +1.0| **********************************************| +5.0
amplitude: +1.0e-12| *************************** ******* ******* ** | +1.0e-10
Z=-341.0(0.00%) | Like=-334.75..-60.45 [-4179.2604..-322.6077] | it/evals=201/531 eff=87.0130% N=300
Z=-330.8(0.00%) | Like=-323.75..-60.45 [-4179.2604..-322.6077] | it/evals=210/543 eff=86.4198% N=300
Z=-310.5(0.00%) | Like=-304.05..-60.45 [-322.4455..-178.2019] | it/evals=231/570 eff=85.5556% N=300
Z=-304.4(0.00%) | Like=-298.51..-60.45 [-322.4455..-178.2019] | it/evals=240/582 eff=85.1064% N=300
Z=-286.2(0.00%) | Like=-279.40..-59.13 [-322.4455..-178.2019] | it/evals=262/609 eff=84.7896% N=300
Mono-modal Volume: ~exp(-4.72) * Expected Volume: exp(-0.89) Quality: ok
index : +1.0| ********************************************| +5.0
amplitude: +1.0e-12| ****************************************** ** | +1.0e-10
Z=-279.9(0.00%) | Like=-273.83..-59.13 [-322.4455..-178.2019] | it/evals=268/618 eff=84.2767% N=300
Z=-278.8(0.00%) | Like=-272.92..-59.13 [-322.4455..-178.2019] | it/evals=270/621 eff=84.1121% N=300
Z=-258.0(0.00%) | Like=-251.87..-59.13 [-322.4455..-178.2019] | it/evals=290/650 eff=82.8571% N=300
Z=-252.4(0.00%) | Like=-244.25..-59.13 [-322.4455..-178.2019] | it/evals=300/663 eff=82.6446% N=300
Z=-227.8(0.00%) | Like=-220.67..-59.13 [-322.4455..-178.2019] | it/evals=323/690 eff=82.8205% N=300
Z=-222.6(0.00%) | Like=-215.42..-59.13 [-322.4455..-178.2019] | it/evals=330/700 eff=82.5000% N=300
Mono-modal Volume: ~exp(-4.91) * Expected Volume: exp(-1.12) Quality: ok
index : +1.0| ******************************************| +5.0
amplitude: +1.0e-12| ******************************************* | +1.0e-10
Z=-218.0(0.00%) | Like=-211.16..-59.13 [-322.4455..-178.2019] | it/evals=335/705 eff=82.7160% N=300
Z=-204.1(0.00%) | Like=-198.04..-59.13 [-322.4455..-178.2019] | it/evals=355/732 eff=82.1759% N=300
Z=-202.0(0.00%) | Like=-196.46..-59.13 [-322.4455..-178.2019] | it/evals=360/737 eff=82.3799% N=300
Z=-194.6(0.00%) | Like=-188.81..-59.13 [-322.4455..-178.2019] | it/evals=377/765 eff=81.0753% N=300
Z=-191.0(0.00%) | Like=-185.46..-59.13 [-322.4455..-178.2019] | it/evals=390/789 eff=79.7546% N=300
Mono-modal Volume: ~exp(-5.06) * Expected Volume: exp(-1.34) Quality: ok
index : +1.0| **************************************** | +5.0
amplitude: +1.0e-12| ****************************************** | +1.0e-10
Z=-187.7(0.00%) | Like=-182.02..-59.13 [-322.4455..-178.2019] | it/evals=402/809 eff=78.9784% N=300
Z=-180.3(0.00%) | Like=-173.86..-59.13 [-178.1619..-123.9505] | it/evals=420/834 eff=78.6517% N=300
Z=-173.8(0.00%) | Like=-166.13..-59.13 [-178.1619..-123.9505] | it/evals=440/861 eff=78.4314% N=300
Z=-169.2(0.00%) | Like=-162.98..-59.13 [-178.1619..-123.9505] | it/evals=450/879 eff=77.7202% N=300
Z=-166.3(0.00%) | Like=-160.10..-59.13 [-178.1619..-123.9505] | it/evals=464/908 eff=76.3158% N=300
Mono-modal Volume: ~exp(-5.64) * Expected Volume: exp(-1.56) Quality: ok
index : +1.0| ******************************** | +5.0
amplitude: +1.0e-12| ***************************************** | +1.0e-10
Z=-164.0(0.00%) | Like=-157.94..-59.04 [-178.1619..-123.9505] | it/evals=469/914 eff=76.3844% N=300
Z=-161.0(0.00%) | Like=-155.06..-59.04 [-178.1619..-123.9505] | it/evals=480/929 eff=76.3116% N=300
Z=-155.1(0.00%) | Like=-149.33..-59.04 [-178.1619..-123.9505] | it/evals=501/956 eff=76.3720% N=300
Z=-153.0(0.00%) | Like=-146.91..-59.04 [-178.1619..-123.9505] | it/evals=510/969 eff=76.2332% N=300
Z=-148.4(0.00%) | Like=-142.29..-59.04 [-178.1619..-123.9505] | it/evals=530/997 eff=76.0402% N=300
Mono-modal Volume: ~exp(-5.74) * Expected Volume: exp(-1.79) Quality: ok
index : +1.0| ****************************** | +5.0
amplitude: +1.0e-12| *************************************** | +1.0e-10
Z=-147.1(0.00%) | Like=-141.04..-59.04 [-178.1619..-123.9505] | it/evals=536/1005 eff=76.0284% N=300
Z=-146.2(0.00%) | Like=-140.20..-59.04 [-178.1619..-123.9505] | it/evals=540/1009 eff=76.1636% N=300
Z=-142.0(0.00%) | Like=-135.90..-59.04 [-178.1619..-123.9505] | it/evals=560/1037 eff=75.9837% N=300
Z=-140.0(0.00%) | Like=-134.17..-59.04 [-178.1619..-123.9505] | it/evals=570/1050 eff=76.0000% N=300
Z=-135.7(0.00%) | Like=-129.51..-59.04 [-178.1619..-123.9505] | it/evals=591/1077 eff=76.0618% N=300
Z=-134.2(0.00%) | Like=-128.72..-59.04 [-178.1619..-123.9505] | it/evals=600/1090 eff=75.9494% N=300
Mono-modal Volume: ~exp(-6.02) * Expected Volume: exp(-2.01) Quality: ok
index : +1.0| ************************** +4.0 | +5.0
amplitude: +1.0e-12| *********************************** | +1.0e-10
Z=-133.9(0.00%) | Like=-128.07..-58.75 [-178.1619..-123.9505] | it/evals=603/1093 eff=76.0404% N=300
Z=-129.8(0.00%) | Like=-123.80..-58.75 [-123.9347..-94.1425] | it/evals=624/1120 eff=76.0976% N=300
Z=-128.8(0.00%) | Like=-122.80..-58.75 [-123.9347..-94.1425] | it/evals=630/1129 eff=75.9952% N=300
Z=-125.5(0.00%) | Like=-119.32..-58.75 [-123.9347..-94.1425] | it/evals=648/1157 eff=75.6126% N=300
Z=-122.9(0.00%) | Like=-116.55..-58.75 [-123.9347..-94.1425] | it/evals=660/1171 eff=75.7750% N=300
Mono-modal Volume: ~exp(-6.30) * Expected Volume: exp(-2.23) Quality: ok
index : +1.0| +1.9 *********************** +3.8 | +5.0
amplitude: +1.0e-12| ******************************** | +1.0e-10
Z=-121.5(0.00%) | Like=-115.68..-58.75 [-123.9347..-94.1425] | it/evals=670/1184 eff=75.7919% N=300
Z=-118.2(0.00%) | Like=-112.15..-58.75 [-123.9347..-94.1425] | it/evals=690/1213 eff=75.5750% N=300
Z=-115.4(0.00%) | Like=-109.04..-58.75 [-123.9347..-94.1425] | it/evals=711/1242 eff=75.4777% N=300
Z=-114.0(0.00%) | Like=-107.92..-58.75 [-123.9347..-94.1425] | it/evals=720/1253 eff=75.5509% N=300
Mono-modal Volume: ~exp(-6.42) * Expected Volume: exp(-2.46) Quality: ok
index : +1.0| +2.0 ********************* +3.6 | +5.0
amplitude: +1.0e-12| **************************** +7.9e-11| +1.0e-10
Z=-112.2(0.00%) | Like=-106.00..-58.75 [-123.9347..-94.1425] | it/evals=737/1280 eff=75.2041% N=300
Z=-110.5(0.00%) | Like=-104.24..-58.75 [-123.9347..-94.1425] | it/evals=750/1295 eff=75.3769% N=300
Z=-108.3(0.00%) | Like=-102.28..-58.75 [-123.9347..-94.1425] | it/evals=768/1322 eff=75.1468% N=300
Z=-107.0(0.00%) | Like=-100.89..-58.75 [-123.9347..-94.1425] | it/evals=780/1339 eff=75.0722% N=300
Z=-104.4(0.00%) | Like=-97.92..-58.75 [-123.9347..-94.1425] | it/evals=800/1369 eff=74.8363% N=300
Mono-modal Volume: ~exp(-6.67) * Expected Volume: exp(-2.68) Quality: ok
index : +1.0| +2.0 ******************* +3.5 | +5.0
amplitude: +1.0e-12| ************************** +7.5e-11 | +1.0e-10
Z=-103.9(0.00%) | Like=-97.44..-58.75 [-123.9347..-94.1425] | it/evals=804/1373 eff=74.9301% N=300
Z=-103.2(0.00%) | Like=-96.78..-58.75 [-123.9347..-94.1425] | it/evals=810/1382 eff=74.8614% N=300
Z=-101.4(0.00%) | Like=-95.40..-58.75 [-123.9347..-94.1425] | it/evals=827/1409 eff=74.5717% N=300
Z=-100.2(0.00%) | Like=-93.73..-58.75 [-94.1273..-77.0818] | it/evals=840/1429 eff=74.4021% N=300
Z=-98.4(0.00%) | Like=-92.18..-58.75 [-94.1273..-77.0818] | it/evals=856/1456 eff=74.0484% N=300
Z=-97.3(0.00%) | Like=-91.11..-58.75 [-94.1273..-77.0818] | it/evals=870/1474 eff=74.1056% N=300
Mono-modal Volume: ~exp(-6.79) * Expected Volume: exp(-2.90) Quality: ok
index : +1.0| +2.1 ***************** +3.5 | +5.0
amplitude: +1.0e-12| +2.4e-11 ************************ +7.2e-11 | +1.0e-10
Z=-97.2(0.00%) | Like=-90.96..-58.75 [-94.1273..-77.0818] | it/evals=871/1475 eff=74.1277% N=300
Z=-95.0(0.00%) | Like=-88.78..-58.75 [-94.1273..-77.0818] | it/evals=892/1502 eff=74.2097% N=300
Z=-94.4(0.00%) | Like=-88.36..-58.75 [-94.1273..-77.0818] | it/evals=900/1512 eff=74.2574% N=300
Z=-92.7(0.00%) | Like=-86.05..-58.75 [-94.1273..-77.0818] | it/evals=921/1539 eff=74.3341% N=300
Z=-91.9(0.00%) | Like=-85.60..-58.75 [-94.1273..-77.0818] | it/evals=930/1552 eff=74.2812% N=300
Mono-modal Volume: ~exp(-7.00) * Expected Volume: exp(-3.13) Quality: ok
index : +1.0| +2.1 **************** +3.3 | +5.0
amplitude: +1.0e-12| +2.6e-11 ********************* +6.9e-11 | +1.0e-10
Z=-91.2(0.00%) | Like=-84.48..-58.75 [-94.1273..-77.0818] | it/evals=938/1564 eff=74.2089% N=300
Z=-89.0(0.00%) | Like=-82.26..-58.75 [-94.1273..-77.0818] | it/evals=958/1592 eff=74.1486% N=300
Z=-88.8(0.00%) | Like=-82.11..-58.75 [-94.1273..-77.0818] | it/evals=960/1594 eff=74.1886% N=300
Z=-87.2(0.00%) | Like=-80.98..-58.75 [-94.1273..-77.0818] | it/evals=980/1622 eff=74.1301% N=300
Z=-86.6(0.00%) | Like=-80.44..-58.75 [-94.1273..-77.0818] | it/evals=990/1638 eff=73.9910% N=300
Mono-modal Volume: ~exp(-7.63) * Expected Volume: exp(-3.35) Quality: ok
index : +1.0| +2.2 ************** +3.3 | +5.0
amplitude: +1.0e-12| +2.7e-11 ******************** +6.6e-11 | +1.0e-10
Z=-85.7(0.00%) | Like=-79.42..-58.75 [-94.1273..-77.0818] | it/evals=1005/1663 eff=73.7344% N=300
Z=-84.9(0.00%) | Like=-78.81..-58.75 [-94.1273..-77.0818] | it/evals=1020/1681 eff=73.8595% N=300
Z=-83.8(0.00%) | Like=-77.56..-58.75 [-94.1273..-77.0818] | it/evals=1040/1709 eff=73.8112% N=300
Z=-83.3(0.00%) | Like=-77.04..-58.75 [-77.0542..-67.0564] | it/evals=1050/1724 eff=73.7360% N=300
Mono-modal Volume: ~exp(-7.63) Expected Volume: exp(-3.57) Quality: ok
index : +1.0| +2.2 ************* +3.2 | +5.0
amplitude: +1.0e-12| +2.9e-11 ****************** +6.4e-11 | +1.0e-10
Z=-82.0(0.00%) | Like=-75.74..-58.75 [-77.0542..-67.0564] | it/evals=1073/1750 eff=74.0000% N=300
Z=-81.7(0.00%) | Like=-75.42..-58.75 [-77.0542..-67.0564] | it/evals=1080/1760 eff=73.9726% N=300
Z=-80.8(0.00%) | Like=-74.44..-58.75 [-77.0542..-67.0564] | it/evals=1096/1787 eff=73.7054% N=300
Z=-80.0(0.00%) | Like=-73.15..-58.75 [-77.0542..-67.0564] | it/evals=1110/1806 eff=73.7052% N=300
Z=-78.8(0.00%) | Like=-72.41..-58.75 [-77.0542..-67.0564] | it/evals=1128/1834 eff=73.5332% N=300
Mono-modal Volume: ~exp(-7.63) Expected Volume: exp(-3.80) Quality: ok
index : +1.0| +2.3 *********** +3.1 | +5.0
amplitude: +1.0e-12| +3.1e-11 **************** +6.1e-11 | +1.0e-10
Z=-78.3(0.00%) | Like=-71.90..-58.75 [-77.0542..-67.0564] | it/evals=1140/1850 eff=73.5484% N=300
Z=-77.4(0.00%) | Like=-70.96..-58.75 [-77.0542..-67.0564] | it/evals=1160/1877 eff=73.5574% N=300
Z=-77.0(0.00%) | Like=-70.37..-58.75 [-77.0542..-67.0564] | it/evals=1170/1890 eff=73.5849% N=300
Z=-76.1(0.00%) | Like=-69.57..-58.75 [-77.0542..-67.0564] | it/evals=1189/1918 eff=73.4858% N=300
Z=-75.6(0.00%) | Like=-69.18..-58.75 [-77.0542..-67.0564] | it/evals=1200/1932 eff=73.5294% N=300
Mono-modal Volume: ~exp(-7.95) * Expected Volume: exp(-4.02) Quality: ok
index : +1.0| +2.3 ********** +3.1 | +5.0
amplitude: +1.0e-12| +3.2e-11 ************** +5.9e-11 | +1.0e-10
Z=-75.4(0.00%) | Like=-68.90..-58.75 [-77.0542..-67.0564] | it/evals=1206/1941 eff=73.4918% N=300
Z=-74.5(0.01%) | Like=-68.01..-58.75 [-77.0542..-67.0564] | it/evals=1228/1969 eff=73.5770% N=300
Z=-74.5(0.01%) | Like=-67.94..-58.75 [-77.0542..-67.0564] | it/evals=1230/1973 eff=73.5206% N=300
Z=-73.7(0.02%) | Like=-67.06..-58.75 [-77.0542..-67.0564] | it/evals=1248/2000 eff=73.4118% N=300
Z=-73.3(0.03%) | Like=-66.74..-58.75 [-67.0554..-65.4955] | it/evals=1260/2015 eff=73.4694% N=300
Mono-modal Volume: ~exp(-7.98) * Expected Volume: exp(-4.24) 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=-72.9(0.04%) | Like=-66.45..-58.75 [-67.0554..-65.4955] | it/evals=1273/2032 eff=73.4988% N=300
Z=-72.4(0.07%) | Like=-66.00..-58.75 [-67.0554..-65.4955] | it/evals=1289/2059 eff=73.2803% N=300
Z=-72.4(0.07%) | Like=-65.99..-58.75 [-67.0554..-65.4955] | it/evals=1290/2060 eff=73.2955% N=300
Z=-71.9(0.11%) | Like=-65.46..-58.75 [-65.4646..-65.2194] | it/evals=1309/2087 eff=73.2513% N=300
Z=-71.6(0.15%) | Like=-65.16..-58.75 [-65.1603..-65.1516]*| it/evals=1320/2100 eff=73.3333% N=300
Z=-71.2(0.23%) | Like=-64.80..-58.75 [-64.8244..-64.8041] | it/evals=1339/2127 eff=73.2895% N=300
Mono-modal Volume: ~exp(-8.57) * Expected Volume: exp(-4.47) Quality: ok
index : +1.0| +2.4 ******** +3.0 | +5.0
amplitude: +1.0e-12| +3.5e-11 *********** +5.5e-11 | +1.0e-10
Z=-71.1(0.23%) | Like=-64.80..-58.75 [-64.7950..-64.7881]*| it/evals=1340/2129 eff=73.2641% N=300
Z=-70.9(0.28%) | Like=-64.65..-58.75 [-64.6538..-64.6264] | it/evals=1350/2144 eff=73.2104% N=300
Z=-70.5(0.41%) | Like=-64.28..-58.75 [-64.2758..-64.2724]*| it/evals=1371/2171 eff=73.2763% N=300
Z=-70.4(0.48%) | Like=-64.09..-58.75 [-64.0914..-64.0579] | it/evals=1380/2183 eff=73.2873% N=300
Z=-70.0(0.71%) | Like=-63.55..-58.75 [-63.5526..-63.5472]*| it/evals=1403/2210 eff=73.4555% N=300
Mono-modal Volume: ~exp(-8.75) * Expected Volume: exp(-4.69) Quality: ok
index : +1.0| +2.4 ******* +2.9 | +5.0
amplitude: +1.0e-12| +3.6e-11 ********** +5.4e-11 | +1.0e-10
Z=-69.9(0.77%) | Like=-63.52..-58.75 [-63.5435..-63.5218] | it/evals=1407/2219 eff=73.3194% N=300
Z=-69.8(0.81%) | Like=-63.47..-58.75 [-63.4687..-63.4649]*| it/evals=1410/2223 eff=73.3229% N=300
Z=-69.5(1.18%) | Like=-63.15..-58.75 [-63.1526..-63.1519]*| it/evals=1432/2250 eff=73.4359% N=300
Z=-69.4(1.33%) | Like=-63.04..-58.75 [-63.0405..-63.0364]*| it/evals=1440/2261 eff=73.4319% N=300
Z=-69.1(1.70%) | Like=-62.88..-58.75 [-62.8823..-62.8659] | it/evals=1458/2289 eff=73.3032% N=300
Z=-68.9(2.02%) | Like=-62.68..-58.75 [-62.6912..-62.6775] | it/evals=1470/2303 eff=73.3899% N=300
Mono-modal Volume: ~exp(-9.26) * Expected Volume: exp(-4.91) Quality: ok
index : +1.0| +2.5 ****** +2.9 | +5.0
amplitude: +1.0e-12| +3.6e-11 ********* +5.3e-11 | +1.0e-10
Z=-68.9(2.14%) | Like=-62.65..-58.75 [-62.6498..-62.6413]*| it/evals=1474/2308 eff=73.4064% N=300
Z=-68.7(2.76%) | Like=-62.44..-58.75 [-62.4958..-62.4418] | it/evals=1495/2335 eff=73.4644% N=300
Z=-68.6(2.94%) | Like=-62.37..-58.75 [-62.3672..-62.3479] | it/evals=1500/2340 eff=73.5294% N=300
Z=-68.4(3.79%) | Like=-62.06..-58.75 [-62.0811..-62.0561] | it/evals=1522/2367 eff=73.6333% N=300
Z=-68.3(4.18%) | Like=-61.97..-58.75 [-61.9708..-61.9445] | it/evals=1530/2379 eff=73.5931% N=300
Mono-modal Volume: ~exp(-9.29) * Expected Volume: exp(-5.14) 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=-68.1(4.70%) | Like=-61.83..-58.75 [-61.8343..-61.8316]*| it/evals=1541/2398 eff=73.4509% N=300
Z=-68.0(5.68%) | Like=-61.73..-58.75 [-61.7253..-61.7123] | it/evals=1560/2419 eff=73.6196% N=300
Z=-67.7(6.99%) | Like=-61.55..-58.75 [-61.5488..-61.5327] | it/evals=1584/2445 eff=73.8462% N=300
Z=-67.7(7.30%) | Like=-61.51..-58.75 [-61.5066..-61.5053]*| it/evals=1590/2452 eff=73.8848% N=300
Mono-modal Volume: ~exp(-9.46) * Expected Volume: exp(-5.36) Quality: ok
index : +1.0| +2.5 ****** +2.9 | +5.0
amplitude: +1.0e-12| +3.8e-11 ******** +5.1e-11 | +1.0e-10
Z=-67.6(8.26%) | Like=-61.35..-58.75 [-61.3451..-61.3399]*| it/evals=1608/2475 eff=73.9310% N=300
Z=-67.5(9.13%) | Like=-61.23..-58.75 [-61.2335..-61.2239]*| it/evals=1620/2490 eff=73.9726% N=300
Z=-67.3(10.46%) | Like=-61.09..-58.75 [-61.1008..-61.0898] | it/evals=1638/2517 eff=73.8836% N=300
Z=-67.2(11.51%) | Like=-61.01..-58.75 [-61.0157..-61.0054] | it/evals=1650/2535 eff=73.8255% N=300
Z=-67.1(12.90%) | Like=-60.92..-58.75 [-60.9177..-60.9170]*| it/evals=1668/2562 eff=73.7401% N=300
Mono-modal Volume: ~exp(-9.76) * Expected Volume: exp(-5.58) Quality: ok
index : +1.0| +2.5 ***** +2.8 | +5.0
amplitude: +1.0e-12| +3.8e-11 ******* +5.1e-11 | +1.0e-10
Z=-67.1(13.56%) | Like=-60.84..-58.75 [-60.8433..-60.8427]*| it/evals=1675/2572 eff=73.7236% N=300
Z=-67.0(13.93%) | Like=-60.83..-58.75 [-60.8331..-60.8170] | it/evals=1680/2577 eff=73.7813% N=300
Z=-66.9(15.71%) | Like=-60.71..-58.75 [-60.7070..-60.7060]*| it/evals=1701/2604 eff=73.8281% N=300
Z=-66.9(16.42%) | Like=-60.63..-58.75 [-60.6330..-60.6275]*| it/evals=1710/2617 eff=73.8023% N=300
Z=-66.8(18.03%) | Like=-60.53..-58.75 [-60.5319..-60.5285]*| it/evals=1726/2644 eff=73.6348% N=300
Z=-66.7(19.40%) | Like=-60.45..-58.75 [-60.4460..-60.4416]*| it/evals=1740/2666 eff=73.5418% N=300
Mono-modal Volume: ~exp(-9.76) Expected Volume: exp(-5.81) 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(21.50%) | Like=-60.36..-58.75 [-60.3621..-60.3562]*| it/evals=1760/2692 eff=73.5786% N=300
Z=-66.6(22.57%) | Like=-60.28..-58.75 [-60.2839..-60.2837]*| it/evals=1770/2709 eff=73.4745% N=300
Z=-66.5(24.64%) | Like=-60.19..-58.75 [-60.1869..-60.1850]*| it/evals=1788/2736 eff=73.3990% N=300
Z=-66.4(26.03%) | Like=-60.15..-58.75 [-60.1542..-60.1518]*| it/evals=1800/2755 eff=73.3198% N=300
Mono-modal Volume: ~exp(-9.94) * 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=-66.4(27.14%) | Like=-60.13..-58.75 [-60.1280..-60.1275]*| it/evals=1809/2767 eff=73.3279% N=300
Z=-66.3(29.58%) | Like=-60.01..-58.75 [-60.0119..-60.0088]*| it/evals=1830/2794 eff=73.3761% N=300
Z=-66.2(31.76%) | Like=-59.95..-58.75 [-59.9481..-59.9477]*| it/evals=1849/2821 eff=73.3439% N=300
Z=-66.2(33.05%) | Like=-59.91..-58.75 [-59.9132..-59.9126]*| it/evals=1860/2842 eff=73.1707% N=300
Mono-modal Volume: ~exp(-9.94) Expected Volume: exp(-6.25) 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=-66.1(34.97%) | Like=-59.84..-58.75 [-59.8421..-59.8419]*| it/evals=1877/2867 eff=73.1204% N=300
Z=-66.1(36.45%) | Like=-59.79..-58.75 [-59.7861..-59.7849]*| it/evals=1890/2887 eff=73.0576% N=300
Z=-66.0(39.21%) | Like=-59.74..-58.75 [-59.7355..-59.7344]*| it/evals=1912/2914 eff=73.1446% N=300
Z=-66.0(40.14%) | Like=-59.72..-58.75 [-59.7176..-59.7098]*| it/evals=1920/2930 eff=73.0038% N=300
Z=-65.9(41.88%) | Like=-59.67..-58.75 [-59.6667..-59.6656]*| it/evals=1934/2957 eff=72.7889% N=300
Mono-modal Volume: ~exp(-10.28) * Expected Volume: exp(-6.48) 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(42.84%) | Like=-59.64..-58.75 [-59.6352..-59.6333]*| it/evals=1943/2969 eff=72.7988% N=300
Z=-65.9(43.75%) | Like=-59.62..-58.75 [-59.6162..-59.6140]*| it/evals=1950/2977 eff=72.8427% N=300
Z=-65.8(46.32%) | Like=-59.54..-58.75 [-59.5425..-59.5398]*| it/evals=1972/3004 eff=72.9290% N=300
Z=-65.8(47.32%) | Like=-59.53..-58.75 [-59.5282..-59.5259]*| it/evals=1980/3015 eff=72.9282% N=300
Z=-65.8(49.26%) | Like=-59.50..-58.75 [-59.4954..-59.4934]*| it/evals=1997/3042 eff=72.8301% N=300
Mono-modal Volume: ~exp(-10.91) * Expected Volume: exp(-6.70) 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=-65.8(50.84%) | Like=-59.45..-58.75 [-59.4537..-59.4503]*| it/evals=2010/3062 eff=72.7734% N=300
Z=-65.7(53.36%) | Like=-59.39..-58.75 [-59.3942..-59.3918]*| it/evals=2032/3088 eff=72.8838% N=300
Z=-65.7(54.20%) | Like=-59.37..-58.75 [-59.3678..-59.3672]*| it/evals=2040/3101 eff=72.8311% N=300
Z=-65.6(56.57%) | Like=-59.33..-58.75 [-59.3274..-59.3251]*| it/evals=2061/3128 eff=72.8784% N=300
Z=-65.6(57.53%) | Like=-59.31..-58.75 [-59.3141..-59.3138]*| it/evals=2070/3145 eff=72.7592% N=300
Mono-modal Volume: ~exp(-10.98) * 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=-65.6(58.29%) | Like=-59.31..-58.75 [-59.3078..-59.3046]*| it/evals=2077/3155 eff=72.7496% N=300
Z=-65.6(60.12%) | Like=-59.28..-58.75 [-59.2795..-59.2793]*| it/evals=2095/3181 eff=72.7178% N=300
Z=-65.6(60.62%) | Like=-59.27..-58.75 [-59.2729..-59.2723]*| it/evals=2100/3187 eff=72.7399% N=300
Z=-65.5(62.77%) | Like=-59.25..-58.75 [-59.2471..-59.2460]*| it/evals=2122/3215 eff=72.7959% N=300
Z=-65.5(63.49%) | Like=-59.23..-58.75 [-59.2308..-59.2306]*| it/evals=2130/3227 eff=72.7708% N=300
Mono-modal Volume: ~exp(-11.28) * Expected Volume: exp(-7.15) 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=-65.5(64.82%) | Like=-59.22..-58.75 [-59.2173..-59.2142]*| it/evals=2144/3246 eff=72.7766% N=300
Z=-65.5(66.28%) | Like=-59.19..-58.75 [-59.1904..-59.1901]*| it/evals=2160/3266 eff=72.8254% N=300
Z=-65.5(68.00%) | Like=-59.17..-58.75 [-59.1700..-59.1690]*| it/evals=2180/3293 eff=72.8366% N=300
Z=-65.5(68.81%) | Like=-59.16..-58.75 [-59.1577..-59.1566]*| it/evals=2190/3305 eff=72.8785% N=300
[ultranest] Explored until L=-6e+01
[ultranest] Likelihood function evaluations: 3324
[ultranest] logZ = -65.04 +- 0.1073
[ultranest] Effective samples strategy satisfied (ESS = 1013.8, 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=0.28, need <0.5)
[ultranest] logZ error budget: single: 0.13 bs:0.11 tail:0.26 total:0.28 required:<0.50
[ultranest] done iterating.
logZ = -65.077 +- 0.368
single instance: logZ = -65.077 +- 0.133
bootstrapped : logZ = -65.036 +- 0.259
tail : logZ = +- 0.262
insert order U test : converged: True correlation: inf iterations
index : 2.365 │ ▁ ▁▁▁▁▁▂▂▂▃▃▅▅▅▇▆▇▇▇▆▆▅▄▄▂▂▂▂▁▁▁▁▁▁▁▁ │2.985 2.670 +- 0.086
amplitude : 0.0000000000339│ ▁▁▁▁▁▁▂▂▃▃▄▅▆▇▇▆▇▅▅▆▅▄▄▃▂▂▁▁▁▁▁▁▁▁ ▁▁ │0.0000000000573 0.0000000000443 +- 0.0000000000032
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 : 2.670 +/- 0.09
amplitude : 4.43e-11 +/- 3.2e-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': [2.6696870373952, 4.432392638820674e-11], 'stdev': [0.08601231612577827, 3.151614370404866e-12], 'median': [2.670167025558571, 4.419710996114369e-11], 'errlo': [2.580937030429043, 4.121851028354166e-11], 'errup': [2.7554564833041804, 4.747965578021952e-11], 'information_gain_bits': [2.696002962237506, 3.1275579179085202]}
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(-3.96) * Expected Volume: exp(0.00) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|************************************ ** *******| +1.0e-10
Z=-inf(0.00%) | Like=-1948.19..-20.96 [-1948.1922..-116.0908] | it/evals=0/301 eff=0.0000% N=300
Z=-177.7(0.00%) | Like=-173.04..-20.96 [-1948.1922..-116.0908] | it/evals=30/332 eff=93.7500% N=300
Z=-166.1(0.00%) | Like=-161.05..-20.96 [-1948.1922..-116.0908] | it/evals=60/366 eff=90.9091% N=300
Mono-modal Volume: ~exp(-4.24) * Expected Volume: exp(-0.22) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|************************************ ** ** * **| +1.0e-10
Z=-163.7(0.00%) | Like=-159.06..-20.96 [-1948.1922..-116.0908] | it/evals=67/374 eff=90.5405% N=300
Z=-156.4(0.00%) | Like=-151.52..-20.96 [-1948.1922..-116.0908] | it/evals=90/399 eff=90.9091% N=300
Z=-145.8(0.00%) | Like=-140.66..-20.96 [-1948.1922..-116.0908] | it/evals=120/435 eff=88.8889% N=300
Mono-modal Volume: ~exp(-4.24) Expected Volume: exp(-0.45) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|*************************************** ** * **| +1.0e-10
Z=-135.5(0.00%) | Like=-130.67..-20.57 [-1948.1922..-116.0908] | it/evals=150/475 eff=85.7143% N=300
Z=-124.8(0.00%) | Like=-119.65..-20.57 [-1948.1922..-116.0908] | it/evals=180/511 eff=85.3081% N=300
Mono-modal Volume: ~exp(-4.55) * Expected Volume: exp(-0.67) Quality: ok
index : +1.0| *********************************************| +5.0
amplitude: +1.0e-12| ********************************* **** ** ***| +1.0e-10
Z=-119.7(0.00%) | Like=-115.03..-20.57 [-116.0484..-68.3785] | it/evals=201/538 eff=84.4538% N=300
Z=-117.5(0.00%) | Like=-112.20..-20.57 [-116.0484..-68.3785] | it/evals=210/551 eff=83.6653% N=300
Z=-108.9(0.00%) | Like=-103.84..-20.57 [-116.0484..-68.3785] | it/evals=240/591 eff=82.4742% N=300
Mono-modal Volume: ~exp(-4.55) Expected Volume: exp(-0.89) Quality: ok
index : +1.0| ********************************************| +5.0
amplitude: +1.0e-12| *********************************** ** ** * | +1.0e-10
Z=-97.8(0.00%) | Like=-92.53..-20.57 [-116.0484..-68.3785] | it/evals=270/629 eff=82.0669% N=300
Z=-90.6(0.00%) | Like=-85.47..-20.57 [-116.0484..-68.3785] | it/evals=300/668 eff=81.5217% N=300
Z=-84.0(0.00%) | Like=-78.95..-20.57 [-116.0484..-68.3785] | it/evals=330/717 eff=79.1367% N=300
Mono-modal Volume: ~exp(-5.11) * Expected Volume: exp(-1.12) Quality: ok
index : +1.0| ******************************************| +5.0
amplitude: +1.0e-12| **************************** ****** * | +1.0e-10
Z=-83.2(0.00%) | Like=-78.03..-20.57 [-116.0484..-68.3785] | it/evals=335/726 eff=78.6385% N=300
Z=-79.3(0.00%) | Like=-74.64..-20.57 [-116.0484..-68.3785] | it/evals=360/765 eff=77.4194% N=300
Z=-76.3(0.00%) | Like=-71.58..-20.57 [-116.0484..-68.3785] | it/evals=389/817 eff=75.2418% N=300
Z=-76.2(0.00%) | Like=-71.51..-20.57 [-116.0484..-68.3785] | it/evals=390/818 eff=75.2896% N=300
Mono-modal Volume: ~exp(-5.51) * Expected Volume: exp(-1.34) Quality: ok
index : +1.0| ************************************** | +5.0
amplitude: +1.0e-12| **************************** **** * +7.9e-11| +1.0e-10
Z=-74.4(0.00%) | Like=-68.78..-20.57 [-116.0484..-68.3785] | it/evals=402/838 eff=74.7212% N=300
Z=-71.9(0.00%) | Like=-66.59..-20.57 [-68.3419..-47.0023] | it/evals=420/863 eff=74.6004% N=300
Z=-67.9(0.00%) | Like=-63.02..-20.57 [-68.3419..-47.0023] | it/evals=450/903 eff=74.6269% N=300
Mono-modal Volume: ~exp(-5.66) * Expected Volume: exp(-1.56) Quality: ok
index : +1.0| ********************************** | +5.0
amplitude: +1.0e-12| ******************************** +7.5e-11 | +1.0e-10
Z=-66.1(0.00%) | Like=-61.19..-20.57 [-68.3419..-47.0023] | it/evals=469/928 eff=74.6815% N=300
Z=-65.1(0.00%) | Like=-60.00..-20.57 [-68.3419..-47.0023] | it/evals=480/947 eff=74.1886% N=300
Z=-62.2(0.00%) | Like=-56.79..-20.57 [-68.3419..-47.0023] | it/evals=510/986 eff=74.3440% N=300
Mono-modal Volume: ~exp(-5.66) Expected Volume: exp(-1.79) Quality: ok
index : +1.0| ***************************** +4.0 | +5.0
amplitude: +1.0e-12| ******************************* +7.3e-11 | +1.0e-10
Z=-59.4(0.00%) | Like=-54.08..-20.57 [-68.3419..-47.0023] | it/evals=537/1028 eff=73.7637% N=300
Z=-59.1(0.00%) | Like=-53.68..-20.57 [-68.3419..-47.0023] | it/evals=540/1031 eff=73.8714% N=300
Z=-56.2(0.00%) | Like=-50.95..-20.57 [-68.3419..-47.0023] | it/evals=568/1082 eff=72.6343% N=300
Z=-56.0(0.00%) | Like=-50.61..-20.57 [-68.3419..-47.0023] | it/evals=570/1086 eff=72.5191% N=300
Z=-53.0(0.00%) | Like=-47.89..-20.57 [-68.3419..-47.0023] | it/evals=600/1131 eff=72.2022% N=300
Mono-modal Volume: ~exp(-5.94) * Expected Volume: exp(-2.01) Quality: ok
index : +1.0| ************************* +3.8 | +5.0
amplitude: +1.0e-12| **************************** +6.9e-11 | +1.0e-10
Z=-52.8(0.00%) | Like=-47.85..-20.57 [-68.3419..-47.0023] | it/evals=603/1136 eff=72.1292% N=300
Z=-51.0(0.00%) | Like=-45.78..-20.57 [-46.9916..-35.1384] | it/evals=630/1172 eff=72.2477% N=300
Z=-48.6(0.00%) | Like=-43.19..-20.57 [-46.9916..-35.1384] | it/evals=660/1211 eff=72.4479% N=300
Mono-modal Volume: ~exp(-6.22) * Expected Volume: exp(-2.23) Quality: ok
index : +1.0| ********************** +3.6 | +5.0
amplitude: +1.0e-12| ************************* +6.3e-11 | +1.0e-10
Z=-47.8(0.00%) | Like=-42.40..-20.57 [-46.9916..-35.1384] | it/evals=670/1228 eff=72.1983% N=300
Z=-46.6(0.00%) | Like=-41.53..-20.57 [-46.9916..-35.1384] | it/evals=690/1257 eff=72.1003% N=300
Z=-44.7(0.00%) | Like=-39.60..-20.57 [-46.9916..-35.1384] | it/evals=720/1296 eff=72.2892% N=300
Mono-modal Volume: ~exp(-6.53) * Expected Volume: exp(-2.46) Quality: ok
index : +1.0| ******************** +3.5 | +5.0
amplitude: +1.0e-12| ********************** +6.1e-11 | +1.0e-10
Z=-43.8(0.00%) | Like=-38.49..-20.57 [-46.9916..-35.1384] | it/evals=737/1318 eff=72.3969% N=300
Z=-43.0(0.00%) | Like=-37.74..-20.57 [-46.9916..-35.1384] | it/evals=750/1333 eff=72.6041% N=300
Z=-41.4(0.00%) | Like=-36.22..-20.57 [-46.9916..-35.1384] | it/evals=780/1379 eff=72.2892% N=300
Mono-modal Volume: ~exp(-6.53) Expected Volume: exp(-2.68) Quality: ok
index : +1.0| +2.0 ****************** +3.4 | +5.0
amplitude: +1.0e-12| ********************* +5.9e-11 | +1.0e-10
Z=-40.3(0.00%) | Like=-35.19..-20.57 [-46.9916..-35.1384] | it/evals=810/1417 eff=72.5157% N=300
Z=-38.9(0.00%) | Like=-33.49..-20.57 [-35.1127..-28.3093] | it/evals=840/1462 eff=72.2892% N=300
Z=-37.8(0.00%) | Like=-32.52..-20.57 [-35.1127..-28.3093] | it/evals=870/1501 eff=72.4396% N=300
Mono-modal Volume: ~exp(-6.91) * Expected Volume: exp(-2.90) Quality: ok
index : +1.0| +2.0 **************** +3.3 | +5.0
amplitude: +1.0e-12| ****************** +5.4e-11 | +1.0e-10
Z=-37.8(0.00%) | Like=-32.49..-20.57 [-35.1127..-28.3093] | it/evals=871/1502 eff=72.4626% N=300
Z=-36.8(0.00%) | Like=-31.51..-20.57 [-35.1127..-28.3093] | it/evals=900/1537 eff=72.7567% N=300
Z=-35.8(0.01%) | Like=-30.46..-20.57 [-35.1127..-28.3093] | it/evals=929/1585 eff=72.2957% N=300
Z=-35.8(0.01%) | Like=-30.44..-20.57 [-35.1127..-28.3093] | it/evals=930/1586 eff=72.3173% N=300
Mono-modal Volume: ~exp(-7.14) * Expected Volume: exp(-3.13) Quality: ok
index : +1.0| +2.1 *************** +3.2 | +5.0
amplitude: +1.0e-12| ***************** +5.3e-11 | +1.0e-10
Z=-35.5(0.01%) | Like=-30.21..-20.57 [-35.1127..-28.3093] | it/evals=938/1597 eff=72.3207% N=300
Z=-35.0(0.01%) | Like=-29.76..-20.57 [-35.1127..-28.3093] | it/evals=960/1628 eff=72.2892% N=300
Z=-34.2(0.03%) | Like=-28.93..-20.57 [-35.1127..-28.3093] | it/evals=990/1669 eff=72.3156% N=300
Mono-modal Volume: ~exp(-7.60) * Expected Volume: exp(-3.35) Quality: ok
index : +1.0| +2.1 ************* +3.1 | +5.0
amplitude: +1.0e-12| **************** +5.1e-11 | +1.0e-10
Z=-33.9(0.04%) | Like=-28.61..-20.57 [-35.1127..-28.3093] | it/evals=1005/1687 eff=72.4585% N=300
Z=-33.5(0.05%) | Like=-28.23..-20.57 [-28.2991..-27.1217] | it/evals=1020/1705 eff=72.5979% N=300
Z=-32.9(0.09%) | Like=-27.66..-20.49 [-28.2991..-27.1217] | it/evals=1050/1749 eff=72.4638% N=300
Mono-modal Volume: ~exp(-7.60) 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=-32.4(0.16%) | Like=-27.04..-20.49 [-27.0683..-26.9028] | it/evals=1077/1793 eff=72.1366% N=300
Z=-32.3(0.17%) | Like=-26.98..-20.49 [-27.0683..-26.9028] | it/evals=1080/1796 eff=72.1925% N=300
Z=-31.8(0.28%) | Like=-26.32..-20.49 [-26.3226..-26.3077] | it/evals=1110/1836 eff=72.2656% N=300
Mono-modal Volume: ~exp(-7.83) * Expected Volume: exp(-3.80) Quality: ok
index : +1.0| +2.2 ********** +3.0 | +5.0
amplitude: +1.0e-12| ************* +4.8e-11 | +1.0e-10
Z=-31.2(0.49%) | Like=-25.71..-20.49 [-25.7085..-25.6956] | it/evals=1139/1882 eff=71.9975% N=300
Z=-31.2(0.49%) | Like=-25.70..-20.49 [-25.7085..-25.6956] | it/evals=1140/1883 eff=72.0152% N=300
Z=-30.7(0.82%) | Like=-25.24..-20.49 [-25.2733..-25.2448] | it/evals=1170/1930 eff=71.7791% N=300
Z=-30.3(1.25%) | Like=-24.90..-20.49 [-24.9112..-24.8972] | it/evals=1200/1976 eff=71.5990% N=300
Mono-modal Volume: ~exp(-8.40) * Expected Volume: exp(-4.02) Quality: ok
index : +1.0| +2.2 ********** +3.0 | +5.0
amplitude: +1.0e-12| +2.4e-11 *********** +4.6e-11 | +1.0e-10
Z=-30.2(1.37%) | Like=-24.84..-20.49 [-24.8412..-24.8379]*| it/evals=1206/1984 eff=71.6152% N=300
Z=-29.9(1.83%) | Like=-24.39..-20.48 [-24.4164..-24.3939] | it/evals=1230/2019 eff=71.5532% N=300
Z=-29.5(2.71%) | Like=-24.10..-20.48 [-24.1033..-24.1027]*| it/evals=1260/2056 eff=71.7540% N=300
Mono-modal Volume: ~exp(-8.40) Expected Volume: exp(-4.24) Quality: ok
index : +1.0| +2.3 ******** +2.9 | +5.0
amplitude: +1.0e-12| +2.5e-11 *********** +4.5e-11 | +1.0e-10
Z=-29.2(3.53%) | Like=-23.69..-20.48 [-23.6865..-23.6808]*| it/evals=1286/2101 eff=71.4048% N=300
Z=-29.2(3.72%) | Like=-23.65..-20.48 [-23.6778..-23.6524] | it/evals=1290/2107 eff=71.3890% N=300
Z=-28.9(5.20%) | Like=-23.36..-20.48 [-23.3753..-23.3617] | it/evals=1320/2145 eff=71.5447% N=300
Mono-modal Volume: ~exp(-8.44) * Expected Volume: exp(-4.47) 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(6.31%) | Like=-23.15..-20.48 [-23.1504..-23.1404]*| it/evals=1340/2177 eff=71.3905% N=300
Z=-28.6(6.94%) | Like=-23.10..-20.48 [-23.1199..-23.1038] | it/evals=1350/2188 eff=71.5042% N=300
Z=-28.3(8.93%) | Like=-22.84..-20.48 [-22.8427..-22.8375]*| it/evals=1380/2231 eff=71.4656% N=300
Mono-modal Volume: ~exp(-8.80) * Expected Volume: exp(-4.69) 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=-28.1(11.02%) | Like=-22.67..-20.48 [-22.6759..-22.6656] | it/evals=1407/2265 eff=71.6031% N=300
Z=-28.1(11.26%) | Like=-22.63..-20.48 [-22.6490..-22.6261] | it/evals=1410/2268 eff=71.6463% N=300
Z=-27.9(13.94%) | Like=-22.40..-20.48 [-22.3964..-22.3920]*| it/evals=1440/2302 eff=71.9281% N=300
Z=-27.7(16.82%) | Like=-22.22..-20.48 [-22.2161..-22.2097]*| it/evals=1470/2345 eff=71.8826% N=300
Mono-modal Volume: ~exp(-9.03) * Expected Volume: exp(-4.91) 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=-27.7(17.27%) | Like=-22.20..-20.48 [-22.1976..-22.1885]*| it/evals=1474/2349 eff=71.9375% N=300
Z=-27.5(19.67%) | Like=-22.08..-20.48 [-22.0802..-22.0772]*| it/evals=1500/2380 eff=72.1154% N=300
Z=-27.4(22.95%) | Like=-21.90..-20.48 [-21.8954..-21.8881]*| it/evals=1530/2420 eff=72.1698% N=300
Mono-modal Volume: ~exp(-9.03) Expected Volume: exp(-5.14) 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=-27.3(25.91%) | Like=-21.79..-20.48 [-21.7912..-21.7807] | it/evals=1558/2464 eff=71.9963% N=300
Z=-27.2(26.15%) | Like=-21.77..-20.48 [-21.7679..-21.7645]*| it/evals=1560/2469 eff=71.9225% N=300
Z=-27.1(29.49%) | Like=-21.62..-20.48 [-21.6189..-21.6102]*| it/evals=1588/2516 eff=71.6606% N=300
Z=-27.1(29.76%) | Like=-21.61..-20.48 [-21.6057..-21.6049]*| it/evals=1590/2520 eff=71.6216% N=300
Mono-modal Volume: ~exp(-9.43) * Expected Volume: exp(-5.36) Quality: ok
index : +1.0| +2.4 ***** +2.7 | +5.0
amplitude: +1.0e-12| +2.9e-11 ****** +4.0e-11 | +1.0e-10
Z=-27.0(31.94%) | Like=-21.51..-20.48 [-21.5150..-21.5085]*| it/evals=1608/2547 eff=71.5621% N=300
Z=-27.0(33.65%) | Like=-21.44..-20.48 [-21.4416..-21.4315] | it/evals=1620/2561 eff=71.6497% N=300
Z=-26.9(37.68%) | Like=-21.35..-20.48 [-21.3539..-21.3495]*| it/evals=1650/2593 eff=71.9581% N=300
Mono-modal Volume: ~exp(-9.87) * Expected Volume: exp(-5.58) 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(40.76%) | Like=-21.26..-20.46 [-21.2578..-21.2577]*| it/evals=1675/2631 eff=71.8576% N=300
Z=-26.8(41.41%) | Like=-21.24..-20.46 [-21.2356..-21.2331]*| it/evals=1680/2638 eff=71.8563% N=300
Z=-26.7(45.16%) | Like=-21.17..-20.46 [-21.1737..-21.1677]*| it/evals=1710/2683 eff=71.7583% N=300
Z=-26.6(48.75%) | Like=-21.11..-20.46 [-21.1086..-21.1085]*| it/evals=1740/2719 eff=71.9305% N=300
Mono-modal Volume: ~exp(-9.87) Expected Volume: exp(-5.81) 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=-26.5(52.26%) | Like=-21.03..-20.46 [-21.0349..-21.0324]*| it/evals=1770/2758 eff=72.0098% N=300
Z=-26.5(55.67%) | Like=-20.98..-20.46 [-20.9788..-20.9777]*| it/evals=1800/2798 eff=72.0576% N=300
Mono-modal Volume: ~exp(-10.29) * Expected Volume: exp(-6.03) 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=-26.5(56.64%) | Like=-20.97..-20.46 [-20.9714..-20.9692]*| it/evals=1809/2811 eff=72.0430% N=300
Z=-26.4(58.86%) | Like=-20.93..-20.46 [-20.9328..-20.9321]*| it/evals=1830/2839 eff=72.0756% N=300
Z=-26.4(61.95%) | Like=-20.89..-20.46 [-20.8882..-20.8878]*| it/evals=1860/2878 eff=72.1490% N=300
Mono-modal Volume: ~exp(-10.29) Expected Volume: exp(-6.25) Quality: ok
index : +1.0| +2.5 **** +2.7 | +5.0
amplitude: +1.0e-12| +3.1e-11 **** +3.8e-11 | +1.0e-10
Z=-26.3(64.91%) | Like=-20.84..-20.46 [-20.8376..-20.8376]*| it/evals=1890/2919 eff=72.1649% N=300
Z=-26.3(67.78%) | Like=-20.80..-20.46 [-20.8034..-20.8013]*| it/evals=1920/2962 eff=72.1262% N=300
Mono-modal Volume: ~exp(-10.76) * Expected Volume: exp(-6.48) 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=-26.3(69.79%) | Like=-20.78..-20.46 [-20.7784..-20.7770]*| it/evals=1943/2999 eff=71.9896% N=300
[ultranest] Explored until L=-2e+01
[ultranest] Likelihood function evaluations: 3001
[ultranest] logZ = -25.9 +- 0.1062
[ultranest] Effective samples strategy satisfied (ESS = 984.8, need >400)
[ultranest] Posterior uncertainty strategy is satisfied (KL: 0.47+-0.08 nat, need <0.50 nat)
[ultranest] Evidency uncertainty strategy is satisfied (dlogz=0.28, need <0.5)
[ultranest] logZ error budget: single: 0.12 bs:0.11 tail:0.26 total:0.28 required:<0.50
[ultranest] done iterating.
logZ = -25.898 +- 0.385
single instance: logZ = -25.898 +- 0.122
bootstrapped : logZ = -25.900 +- 0.282
tail : logZ = +- 0.262
insert order U test : converged: True correlation: inf iterations
index : 2.15 │ ▁▁▁▁▁▁▁▂▃▃▃▅▅▇▇▇▇▆▆▆▄▄▃▂▂▁▁▁▁▁▁▁▁ ▁▁▁ │3.13 2.57 +- 0.12
amplitude : 0.0000000000198│ ▁ ▁▁▁▁▂▂▂▃▄▄▆▆▆▇▆▇▇▇▅▅▃▂▂▂▁▁▁▁▁▁▁ ▁ │0.0000000000504 0.0000000000342 +- 0.0000000000037
[ultranest] Sampling 300 live points from prior ...
Mono-modal Volume: ~exp(-3.91) * Expected Volume: exp(0.00) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|********************************* ********* *** | +1.0e-10
Z=-inf(0.00%) | Like=-806.02..-19.38 [-806.0201..-126.0053] | it/evals=0/302 eff=0.0000% N=300
Z=-220.6(0.00%) | Like=-215.82..-19.38 [-806.0201..-126.0053] | it/evals=30/333 eff=90.9091% N=300
Z=-202.6(0.00%) | Like=-198.07..-19.38 [-806.0201..-126.0053] | it/evals=60/372 eff=83.3333% N=300
Mono-modal Volume: ~exp(-3.91) Expected Volume: exp(-0.22) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|************************************* ***** *** | +1.0e-10
Z=-189.4(0.00%) | Like=-183.94..-19.38 [-806.0201..-126.0053] | it/evals=90/406 eff=84.9057% N=300
Z=-173.2(0.00%) | Like=-166.33..-19.38 [-806.0201..-126.0053] | it/evals=120/444 eff=83.3333% N=300
Mono-modal Volume: ~exp(-4.02) * Expected Volume: exp(-0.45) Quality: ok
index : +1.0| ***********************************************| +5.0
amplitude: +1.0e-12| ********************************************** | +1.0e-10
Z=-164.5(0.00%) | Like=-158.81..-19.38 [-806.0201..-126.0053] | it/evals=134/461 eff=83.2298% N=300
Z=-155.7(0.00%) | Like=-150.25..-19.38 [-806.0201..-126.0053] | it/evals=150/480 eff=83.3333% N=300
Z=-143.3(0.00%) | Like=-136.90..-19.38 [-806.0201..-126.0053] | it/evals=180/519 eff=82.1918% N=300
Mono-modal Volume: ~exp(-4.73) * Expected Volume: exp(-0.67) Quality: ok
index : +1.0| **********************************************| +5.0
amplitude: +1.0e-12| ********************************************* | +1.0e-10
Z=-133.4(0.00%) | Like=-128.11..-19.38 [-806.0201..-126.0053] | it/evals=201/550 eff=80.4000% N=300
Z=-130.3(0.00%) | Like=-125.01..-19.38 [-126.0033..-60.2491] | it/evals=210/559 eff=81.0811% N=300
Z=-116.7(0.00%) | Like=-111.08..-19.38 [-126.0033..-60.2491] | it/evals=240/597 eff=80.8081% 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=-103.8(0.00%) | Like=-97.92..-19.26 [-126.0033..-60.2491] | it/evals=268/630 eff=81.2121% N=300
Z=-102.5(0.00%) | Like=-96.20..-19.26 [-126.0033..-60.2491] | it/evals=270/633 eff=81.0811% N=300
Z=-93.3(0.00%) | Like=-88.12..-19.26 [-126.0033..-60.2491] | it/evals=300/668 eff=81.5217% N=300
Z=-83.7(0.00%) | Like=-77.38..-19.26 [-126.0033..-60.2491] | it/evals=330/705 eff=81.4815% N=300
Mono-modal Volume: ~exp(-4.84) Expected Volume: exp(-1.12) Quality: ok
index : +1.0| *****************************************| +5.0
amplitude: +1.0e-12| ****************************************** | +1.0e-10
Z=-75.1(0.00%) | Like=-69.55..-19.26 [-126.0033..-60.2491] | it/evals=360/746 eff=80.7175% N=300
Z=-67.2(0.00%) | Like=-61.79..-19.26 [-126.0033..-60.2491] | it/evals=390/785 eff=80.4124% 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=-65.1(0.00%) | Like=-59.78..-19.26 [-60.1556..-38.5609] | it/evals=402/801 eff=80.2395% N=300
Z=-61.3(0.00%) | Like=-56.10..-19.26 [-60.1556..-38.5609] | it/evals=420/829 eff=79.3951% N=300
Z=-57.3(0.00%) | Like=-52.21..-19.26 [-60.1556..-38.5609] | it/evals=450/868 eff=79.2254% N=300
Mono-modal Volume: ~exp(-5.76) * Expected Volume: exp(-1.56) Quality: ok
index : +1.0| +1.9 ********************************** | +5.0
amplitude: +1.0e-12| ************************************** | +1.0e-10
Z=-55.3(0.00%) | Like=-50.16..-19.26 [-60.1556..-38.5609] | it/evals=469/895 eff=78.8235% N=300
Z=-53.9(0.00%) | Like=-48.64..-19.26 [-60.1556..-38.5609] | it/evals=480/908 eff=78.9474% N=300
Z=-50.7(0.00%) | Like=-45.40..-19.26 [-60.1556..-38.5609] | it/evals=510/950 eff=78.4615% N=300
Mono-modal Volume: ~exp(-6.13) * Expected Volume: exp(-1.79) Quality: ok
index : +1.0| +2.0 **************************** | +5.0
amplitude: +1.0e-12| +2.4e-11 ************************************ | +1.0e-10
Z=-47.5(0.00%) | Like=-42.56..-19.26 [-60.1556..-38.5609] | it/evals=536/987 eff=78.0204% N=300
Z=-47.2(0.00%) | Like=-42.00..-19.26 [-60.1556..-38.5609] | it/evals=540/995 eff=77.6978% N=300
Z=-45.1(0.00%) | Like=-40.39..-19.26 [-60.1556..-38.5609] | it/evals=570/1036 eff=77.4457% N=300
Z=-43.5(0.00%) | Like=-38.55..-19.26 [-38.5498..-29.3252] | it/evals=600/1071 eff=77.8210% N=300
Mono-modal Volume: ~exp(-6.13) Expected Volume: exp(-2.01) Quality: ok
index : +1.0| +2.0 ************************** +4.1 | +5.0
amplitude: +1.0e-12| +2.6e-11 *********************************** | +1.0e-10
Z=-41.9(0.00%) | Like=-37.28..-19.26 [-38.5498..-29.3252] | it/evals=630/1110 eff=77.7778% N=300
Z=-40.8(0.00%) | Like=-35.93..-19.26 [-38.5498..-29.3252] | it/evals=656/1158 eff=76.4569% N=300
Z=-40.6(0.00%) | Like=-35.75..-19.26 [-38.5498..-29.3252] | it/evals=660/1164 eff=76.3889% N=300
Mono-modal Volume: ~exp(-6.53) * Expected Volume: exp(-2.23) Quality: ok
index : +1.0| +2.1 *********************** +4.0 | +5.0
amplitude: +1.0e-12| +2.8e-11 ********************************* | +1.0e-10
Z=-40.2(0.00%) | Like=-35.36..-19.22 [-38.5498..-29.3252] | it/evals=670/1177 eff=76.3968% N=300
Z=-39.2(0.00%) | Like=-34.33..-19.22 [-38.5498..-29.3252] | it/evals=690/1204 eff=76.3274% N=300
Z=-38.0(0.00%) | Like=-33.21..-19.22 [-38.5498..-29.3252] | it/evals=720/1245 eff=76.1905% N=300
Mono-modal Volume: ~exp(-6.61) * 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=-37.3(0.00%) | Like=-32.11..-19.22 [-38.5498..-29.3252] | it/evals=737/1272 eff=75.8230% N=300
Z=-36.7(0.00%) | Like=-31.69..-19.22 [-38.5498..-29.3252] | it/evals=750/1286 eff=76.0649% N=300
Z=-35.5(0.00%) | Like=-30.53..-19.22 [-38.5498..-29.3252] | it/evals=780/1332 eff=75.5814% N=300
Mono-modal Volume: ~exp(-6.90) * 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=-34.8(0.00%) | Like=-29.87..-19.22 [-38.5498..-29.3252] | it/evals=804/1367 eff=75.3515% N=300
Z=-34.6(0.00%) | Like=-29.63..-19.22 [-38.5498..-29.3252] | it/evals=810/1376 eff=75.2788% N=300
Z=-33.5(0.01%) | Like=-28.21..-19.22 [-29.2956..-26.1234] | it/evals=840/1413 eff=75.4717% N=300
Z=-32.6(0.03%) | Like=-27.47..-19.22 [-29.2956..-26.1234] | it/evals=870/1457 eff=75.1945% N=300
Mono-modal Volume: ~exp(-7.02) * 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=-32.5(0.03%) | Like=-27.46..-19.22 [-29.2956..-26.1234] | it/evals=871/1458 eff=75.2159% N=300
Z=-31.8(0.06%) | Like=-26.86..-19.22 [-29.2956..-26.1234] | it/evals=900/1500 eff=75.0000% N=300
Z=-31.1(0.10%) | Like=-26.20..-19.22 [-29.2956..-26.1234] | it/evals=930/1544 eff=74.7588% N=300
Mono-modal Volume: ~exp(-7.02) Expected Volume: exp(-3.13) Quality: ok
index : +1.0| +2.3 *************** +3.5 | +5.0
amplitude: +1.0e-12| +3.6e-11 ********************* +7.8e-11| +1.0e-10
Z=-30.5(0.19%) | Like=-25.58..-19.22 [-25.5837..-25.5533] | it/evals=959/1588 eff=74.4565% N=300
Z=-30.5(0.20%) | Like=-25.55..-19.22 [-25.5837..-25.5533] | it/evals=960/1589 eff=74.4763% N=300
Z=-30.0(0.33%) | Like=-24.94..-19.22 [-24.9749..-24.9381] | it/evals=990/1630 eff=74.4361% N=300
Mono-modal Volume: ~exp(-7.40) * Expected Volume: exp(-3.35) Quality: ok
index : +1.0| +2.3 ************* +3.4 | +5.0
amplitude: +1.0e-12| +3.8e-11 ******************** +7.7e-11 | +1.0e-10
Z=-29.7(0.44%) | Like=-24.65..-19.22 [-24.6474..-24.6467]*| it/evals=1005/1658 eff=74.0059% N=300
Z=-29.4(0.57%) | Like=-24.30..-19.22 [-24.3756..-24.2964] | it/evals=1020/1683 eff=73.7527% N=300
Z=-28.9(0.90%) | Like=-23.89..-19.22 [-23.8932..-23.8654] | it/evals=1050/1720 eff=73.9437% N=300
Mono-modal Volume: ~exp(-7.95) * Expected Volume: exp(-3.57) Quality: ok
index : +1.0| +2.4 ************* +3.4 | +5.0
amplitude: +1.0e-12| +4.0e-11 ****************** +7.4e-11 | +1.0e-10
Z=-28.6(1.29%) | Like=-23.46..-19.22 [-23.4590..-23.4554]*| it/evals=1072/1757 eff=73.5758% N=300
Z=-28.4(1.46%) | Like=-23.37..-19.22 [-23.3720..-23.3624]*| it/evals=1080/1766 eff=73.6698% N=300
Z=-28.0(2.21%) | Like=-22.98..-19.16 [-22.9824..-22.9338] | it/evals=1110/1810 eff=73.5099% N=300
Mono-modal Volume: ~exp(-7.95) Expected Volume: exp(-3.80) Quality: ok
index : +1.0| +2.4 ********** +3.2 | +5.0
amplitude: +1.0e-12| +4.1e-11 **************** +7.2e-11 | +1.0e-10
Z=-27.6(3.17%) | Like=-22.55..-19.16 [-22.5534..-22.5522]*| it/evals=1140/1851 eff=73.5010% N=300
Z=-27.3(4.45%) | Like=-22.17..-19.16 [-22.1801..-22.1696] | it/evals=1170/1891 eff=73.5387% N=300
Z=-27.0(6.00%) | Like=-21.97..-19.16 [-21.9706..-21.9671]*| it/evals=1200/1933 eff=73.4844% N=300
Mono-modal Volume: ~exp(-8.21) * Expected Volume: exp(-4.02) Quality: ok
index : +1.0| +2.5 ********** +3.2 | +5.0
amplitude: +1.0e-12| +4.3e-11 ************** +7.0e-11 | +1.0e-10
Z=-26.9(6.34%) | Like=-21.91..-19.16 [-21.9112..-21.8996] | it/evals=1206/1941 eff=73.4918% N=300
Z=-26.7(7.83%) | Like=-21.67..-19.16 [-21.6812..-21.6674] | it/evals=1230/1976 eff=73.3890% N=300
Z=-26.5(9.96%) | Like=-21.47..-19.16 [-21.4684..-21.4654]*| it/evals=1259/2023 eff=73.0702% N=300
Z=-26.5(10.05%) | Like=-21.47..-19.16 [-21.4654..-21.4608]*| it/evals=1260/2024 eff=73.0858% N=300
Mono-modal Volume: ~exp(-8.21) Expected Volume: exp(-4.24) Quality: ok
index : +1.0| +2.5 ********** +3.2 | +5.0
amplitude: +1.0e-12| +4.4e-11 ************* +6.8e-11 | +1.0e-10
Z=-26.3(12.56%) | Like=-21.18..-19.16 [-21.1801..-21.1788]*| it/evals=1289/2070 eff=72.8249% N=300
Z=-26.3(12.68%) | Like=-21.18..-19.16 [-21.1788..-21.1776]*| it/evals=1290/2071 eff=72.8402% N=300
Z=-26.1(15.57%) | Like=-21.03..-19.16 [-21.0314..-21.0185] | it/evals=1320/2110 eff=72.9282% N=300
Mono-modal Volume: ~exp(-8.62) * Expected Volume: exp(-4.47) Quality: ok
index : +1.0| +2.5 ******** +3.1 | +5.0
amplitude: +1.0e-12| +4.5e-11 *********** +6.7e-11 | +1.0e-10
Z=-25.9(17.56%) | Like=-20.90..-19.16 [-20.8964..-20.8901]*| it/evals=1340/2138 eff=72.9053% N=300
Z=-25.9(18.46%) | Like=-20.85..-19.16 [-20.8506..-20.8357] | it/evals=1350/2152 eff=72.8942% N=300
Z=-25.7(21.71%) | Like=-20.69..-19.16 [-20.6928..-20.6853]*| it/evals=1380/2193 eff=72.9002% N=300
Mono-modal Volume: ~exp(-9.03) * Expected Volume: exp(-4.69) Quality: ok
index : +1.0| +2.6 ******* +3.1 | +5.0
amplitude: +1.0e-12| +4.6e-11 ********** +6.5e-11 | +1.0e-10
Z=-25.6(24.79%) | Like=-20.53..-19.16 [-20.5254..-20.5245]*| it/evals=1407/2224 eff=73.1289% N=300
Z=-25.6(25.16%) | Like=-20.52..-19.16 [-20.5170..-20.5125]*| it/evals=1410/2228 eff=73.1328% N=300
Z=-25.4(28.66%) | Like=-20.35..-19.16 [-20.3492..-20.3463]*| it/evals=1440/2269 eff=73.1336% N=300
Z=-25.3(32.56%) | Like=-20.26..-19.16 [-20.2593..-20.2572]*| it/evals=1470/2312 eff=73.0616% N=300
Mono-modal Volume: ~exp(-9.11) * Expected Volume: exp(-4.91) Quality: ok
index : +1.0| +2.6 ****** +3.1 | +5.0
amplitude: +1.0e-12| +4.7e-11 ********* +6.4e-11 | +1.0e-10
Z=-25.3(33.03%) | Like=-20.24..-19.16 [-20.2538..-20.2402] | it/evals=1474/2319 eff=73.0064% N=300
Z=-25.2(36.21%) | Like=-20.17..-19.16 [-20.1705..-20.1620]*| it/evals=1500/2350 eff=73.1707% N=300
Z=-25.1(39.86%) | Like=-20.09..-19.16 [-20.0865..-20.0832]*| it/evals=1530/2393 eff=73.1008% N=300
Mono-modal Volume: ~exp(-9.33) * Expected Volume: exp(-5.14) 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(41.24%) | Like=-20.03..-19.16 [-20.0327..-20.0304]*| it/evals=1541/2405 eff=73.2067% N=300
Z=-25.0(43.59%) | Like=-19.98..-19.16 [-19.9796..-19.9775]*| it/evals=1560/2428 eff=73.3083% N=300
Z=-24.9(47.14%) | Like=-19.92..-19.16 [-19.9239..-19.9233]*| it/evals=1590/2473 eff=73.1707% N=300
Mono-modal Volume: ~exp(-9.37) * Expected Volume: exp(-5.36) Quality: ok
index : +1.0| +2.6 ****** +3.0 | +5.0
amplitude: +1.0e-12| +4.9e-11 ******* +6.2e-11 | +1.0e-10
Z=-24.9(49.24%) | Like=-19.85..-19.16 [-19.8463..-19.8463]*| it/evals=1608/2501 eff=73.0577% N=300
Z=-24.9(50.58%) | Like=-19.83..-19.16 [-19.8285..-19.8274]*| it/evals=1620/2520 eff=72.9730% N=300
Z=-24.8(53.97%) | Like=-19.77..-19.16 [-19.7688..-19.7657]*| it/evals=1650/2564 eff=72.8799% N=300
Mono-modal Volume: ~exp(-10.09) * Expected Volume: exp(-5.58) Quality: ok
index : +1.0| +2.6 ***** +3.0 | +5.0
amplitude: +1.0e-12| +4.9e-11 ******* +6.2e-11 | +1.0e-10
Z=-24.7(56.80%) | Like=-19.72..-19.16 [-19.7201..-19.7156]*| it/evals=1675/2600 eff=72.8261% N=300
Z=-24.7(57.39%) | Like=-19.71..-19.16 [-19.7112..-19.7103]*| it/evals=1680/2606 eff=72.8534% N=300
Z=-24.7(60.38%) | Like=-19.66..-19.16 [-19.6573..-19.6569]*| it/evals=1710/2650 eff=72.7660% N=300
Z=-24.6(63.19%) | Like=-19.63..-19.16 [-19.6291..-19.6252]*| it/evals=1739/2700 eff=72.4583% N=300
Z=-24.6(63.30%) | Like=-19.63..-19.16 [-19.6252..-19.6235]*| it/evals=1740/2701 eff=72.4698% N=300
Mono-modal Volume: ~exp(-10.09) 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=-24.6(66.19%) | Like=-19.57..-19.16 [-19.5713..-19.5694]*| it/evals=1770/2736 eff=72.6601% N=300
Z=-24.6(68.83%) | Like=-19.54..-19.16 [-19.5407..-19.5401]*| it/evals=1800/2775 eff=72.7273% N=300
Mono-modal Volume: ~exp(-10.12) * Expected Volume: exp(-6.03) Quality: ok
index : +1.0| +2.7 **** +3.0 | +5.0
amplitude: +1.0e-12| +5.0e-11 ****** +6.0e-11 | +1.0e-10
Z=-24.5(69.60%) | Like=-19.52..-19.16 [-19.5215..-19.5210]*| it/evals=1809/2790 eff=72.6506% N=300
[ultranest] Explored until L=-2e+01
[ultranest] Likelihood function evaluations: 2796
[ultranest] logZ = -24.17 +- 0.08925
[ultranest] Effective samples strategy satisfied (ESS = 989.0, need >400)
[ultranest] Posterior uncertainty strategy is satisfied (KL: 0.46+-0.06 nat, need <0.50 nat)
[ultranest] Evidency uncertainty strategy is satisfied (dlogz=0.28, need <0.5)
[ultranest] logZ error budget: single: 0.12 bs:0.09 tail:0.26 total:0.28 required:<0.50
[ultranest] done iterating.
logZ = -24.177 +- 0.310
single instance: logZ = -24.177 +- 0.116
bootstrapped : logZ = -24.172 +- 0.165
tail : logZ = +- 0.262
insert order U test : converged: True correlation: inf iterations
index : 2.27 │ ▁▁▁▁▁▁▂▃▃▅▆▅▆▆▆▆▇▇▆▆▄▄▄▂▁▂▂▁▁▁▁▁▁▁ ▁▁ │3.51 2.81 +- 0.17
amplitude : 0.0000000000332│ ▁ ▁ ▁▁▁▁▂▂▃▅▆▅▅▆▆▆▇▇▆▅▄▄▃▂▂▂▁▁▁▁▁▁▁ ▁ │0.0000000000787 0.0000000000549 +- 0.0000000000059
[ultranest] Sampling 300 live points from prior ...
Mono-modal Volume: ~exp(-3.84) * Expected Volume: exp(0.00) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|************************ ***** ***** ********** | +1.0e-10
Z=-inf(0.00%) | Like=-1230.19..-16.15 [-1230.1930..-90.6507] | it/evals=0/301 eff=0.0000% N=300
Z=-153.0(0.00%) | Like=-148.60..-14.42 [-1230.1930..-90.6507] | it/evals=30/333 eff=90.9091% N=300
Z=-142.5(0.00%) | Like=-137.67..-14.42 [-1230.1930..-90.6507] | it/evals=60/368 eff=88.2353% N=300
Mono-modal Volume: ~exp(-4.10) * Expected Volume: exp(-0.22) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|******************* **** *********** ********** | +1.0e-10
Z=-140.9(0.00%) | Like=-136.37..-14.42 [-1230.1930..-90.6507] | it/evals=67/378 eff=85.8974% N=300
Z=-133.2(0.00%) | Like=-128.41..-14.42 [-1230.1930..-90.6507] | it/evals=90/403 eff=87.3786% N=300
Z=-124.5(0.00%) | Like=-117.75..-14.42 [-1230.1930..-90.6507] | it/evals=120/443 eff=83.9161% N=300
Mono-modal Volume: ~exp(-4.51) * Expected Volume: exp(-0.45) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|******************* **** *********** ***********| +1.0e-10
Z=-118.3(0.00%) | Like=-112.79..-14.42 [-1230.1930..-90.6507] | it/evals=134/459 eff=84.2767% N=300
Z=-112.7(0.00%) | Like=-108.08..-14.42 [-1230.1930..-90.6507] | it/evals=150/480 eff=83.3333% N=300
Z=-106.3(0.00%) | Like=-101.76..-14.42 [-1230.1930..-90.6507] | it/evals=180/517 eff=82.9493% N=300
Mono-modal Volume: ~exp(-4.51) Expected Volume: exp(-0.67) Quality: ok
index : +1.0| ***********************************************| +5.0
amplitude: +1.0e-12| ****************** ****************************| +1.0e-10
Z=-99.0(0.00%) | Like=-93.95..-13.52 [-1230.1930..-90.6507] | it/evals=210/555 eff=82.3529% N=300
Z=-92.5(0.00%) | Like=-87.36..-13.52 [-90.5937..-48.6677] | it/evals=240/597 eff=80.8081% N=300
Mono-modal Volume: ~exp(-4.82) * Expected Volume: exp(-0.89) Quality: ok
index : +1.0| ********************************************| +5.0
amplitude: +1.0e-12| ***********************************************| +1.0e-10
Z=-86.1(0.00%) | Like=-80.35..-13.52 [-90.5937..-48.6677] | it/evals=268/634 eff=80.2395% N=300
Z=-85.4(0.00%) | Like=-79.93..-13.52 [-90.5937..-48.6677] | it/evals=270/639 eff=79.6460% N=300
Z=-78.3(0.00%) | Like=-73.17..-13.52 [-90.5937..-48.6677] | it/evals=300/672 eff=80.6452% N=300
Z=-72.9(0.00%) | Like=-67.70..-13.52 [-90.5937..-48.6677] | it/evals=330/704 eff=81.6832% N=300
Mono-modal Volume: ~exp(-5.00) * Expected Volume: exp(-1.12) Quality: ok
index : +1.0| *******************************************| +5.0
amplitude: +1.0e-12| **********************************************| +1.0e-10
Z=-71.9(0.00%) | Like=-66.72..-13.52 [-90.5937..-48.6677] | it/evals=335/711 eff=81.5085% N=300
Z=-65.9(0.00%) | Like=-59.77..-13.45 [-90.5937..-48.6677] | it/evals=360/740 eff=81.8182% N=300
Z=-59.7(0.00%) | Like=-54.06..-13.45 [-90.5937..-48.6677] | it/evals=390/774 eff=82.2785% N=300
Mono-modal Volume: ~exp(-5.00) Expected Volume: exp(-1.34) Quality: ok
index : +1.0| *****************************************| +5.0
amplitude: +1.0e-12| ********************************************| +1.0e-10
Z=-55.2(0.00%) | Like=-49.75..-13.45 [-90.5937..-48.6677] | it/evals=420/814 eff=81.7121% N=300
Z=-50.6(0.00%) | Like=-45.54..-13.45 [-48.5246..-32.7665] | it/evals=450/856 eff=80.9353% N=300
Mono-modal Volume: ~exp(-5.11) * Expected Volume: exp(-1.56) Quality: ok
index : +1.0| ************************************* * | +5.0
amplitude: +1.0e-12| *******************************************| +1.0e-10
Z=-48.8(0.00%) | Like=-44.13..-13.45 [-48.5246..-32.7665] | it/evals=469/885 eff=80.1709% N=300
Z=-47.7(0.00%) | Like=-42.74..-13.45 [-48.5246..-32.7665] | it/evals=480/896 eff=80.5369% N=300
Z=-45.2(0.00%) | Like=-40.53..-13.39 [-48.5246..-32.7665] | it/evals=510/943 eff=79.3157% N=300
Mono-modal Volume: ~exp(-5.84) * Expected Volume: exp(-1.79) Quality: ok
index : +1.0| ********************************** | +5.0
amplitude: +1.0e-12| ******************************************| +1.0e-10
Z=-43.6(0.00%) | Like=-38.78..-13.36 [-48.5246..-32.7665] | it/evals=536/983 eff=78.4773% N=300
Z=-43.4(0.00%) | Like=-38.54..-13.36 [-48.5246..-32.7665] | it/evals=540/988 eff=78.4884% N=300
Z=-41.3(0.00%) | Like=-36.39..-13.32 [-48.5246..-32.7665] | it/evals=570/1026 eff=78.5124% N=300
Z=-39.2(0.00%) | Like=-34.17..-13.32 [-48.5246..-32.7665] | it/evals=600/1062 eff=78.7402% N=300
Mono-modal Volume: ~exp(-6.00) * Expected Volume: exp(-2.01) Quality: ok
index : +1.0| ****************************** | +5.0
amplitude: +1.0e-12| ****************************************| +1.0e-10
Z=-39.0(0.00%) | Like=-33.98..-13.32 [-48.5246..-32.7665] | it/evals=603/1065 eff=78.8235% N=300
Z=-37.4(0.00%) | Like=-32.36..-13.32 [-32.7591..-23.1381] | it/evals=630/1094 eff=79.3451% N=300
Z=-35.6(0.00%) | Like=-30.44..-13.32 [-32.7591..-23.1381] | it/evals=660/1131 eff=79.4224% N=300
Mono-modal Volume: ~exp(-6.00) Expected Volume: exp(-2.23) Quality: ok
index : +1.0| +1.9 ************************** +4.0 | +5.0
amplitude: +1.0e-12| ***************************************| +1.0e-10
Z=-33.9(0.00%) | Like=-28.86..-13.32 [-32.7591..-23.1381] | it/evals=690/1170 eff=79.3103% N=300
Z=-32.6(0.00%) | Like=-27.56..-13.32 [-32.7591..-23.1381] | it/evals=720/1211 eff=79.0340% N=300
Mono-modal Volume: ~exp(-6.28) * Expected Volume: exp(-2.46) Quality: ok
index : +1.0| +2.0 ************************ +3.8 | +5.0
amplitude: +1.0e-12| **************************************| +1.0e-10
Z=-31.8(0.00%) | Like=-26.61..-13.32 [-32.7591..-23.1381] | it/evals=737/1240 eff=78.4043% N=300
Z=-31.2(0.00%) | Like=-26.32..-13.32 [-32.7591..-23.1381] | it/evals=750/1253 eff=78.6988% N=300
Z=-30.2(0.00%) | Like=-25.19..-13.32 [-32.7591..-23.1381] | it/evals=780/1287 eff=79.0274% N=300
Mono-modal Volume: ~exp(-6.43) * Expected Volume: exp(-2.68) Quality: ok
index : +1.0| +2.1 ******************** +3.6 | +5.0
amplitude: +1.0e-12| +2.4e-11 ************************************ | +1.0e-10
Z=-29.4(0.00%) | Like=-24.51..-13.32 [-32.7591..-23.1381] | it/evals=804/1322 eff=78.6693% N=300
Z=-29.2(0.00%) | Like=-24.29..-13.32 [-32.7591..-23.1381] | it/evals=810/1328 eff=78.7938% N=300
Z=-28.3(0.01%) | Like=-23.21..-13.32 [-32.7591..-23.1381] | it/evals=840/1369 eff=78.5781% N=300
Z=-27.4(0.01%) | Like=-22.29..-13.32 [-23.1287..-20.1595] | it/evals=870/1412 eff=78.2374% N=300
Mono-modal Volume: ~exp(-7.03) * Expected Volume: exp(-2.90) Quality: ok
index : +1.0| +2.1 ******************* +3.6 | +5.0
amplitude: +1.0e-12| +2.6e-11 ********************************* | +1.0e-10
Z=-27.3(0.01%) | Like=-22.27..-13.32 [-23.1287..-20.1595] | it/evals=871/1413 eff=78.2570% N=300
Z=-26.6(0.03%) | Like=-21.57..-13.32 [-23.1287..-20.1595] | it/evals=900/1450 eff=78.2609% N=300
Z=-25.8(0.07%) | Like=-20.58..-13.32 [-23.1287..-20.1595] | it/evals=930/1489 eff=78.2170% N=300
Mono-modal Volume: ~exp(-7.03) Expected Volume: exp(-3.13) Quality: ok
index : +1.0| +2.2 **************** +3.5 | +5.0
amplitude: +1.0e-12| +2.9e-11 ****************************** | +1.0e-10
Z=-25.1(0.14%) | Like=-19.96..-13.32 [-20.1461..-19.8184] | it/evals=960/1529 eff=78.1123% N=300
Z=-24.5(0.26%) | Like=-19.55..-13.31 [-19.5492..-19.5153] | it/evals=990/1567 eff=78.1373% N=300
Mono-modal Volume: ~exp(-7.03) Expected Volume: exp(-3.35) Quality: ok
index : +1.0| +2.2 *************** +3.4 | +5.0
amplitude: +1.0e-12| +3.2e-11 ************************** | +1.0e-10
Z=-24.1(0.38%) | Like=-19.01..-13.31 [-19.0342..-19.0082] | it/evals=1015/1613 eff=77.3039% N=300
Z=-24.0(0.42%) | Like=-18.84..-13.31 [-18.8754..-18.8426] | it/evals=1020/1622 eff=77.1558% N=300
Z=-23.5(0.69%) | Like=-18.37..-13.31 [-18.3969..-18.3655] | it/evals=1049/1670 eff=76.5693% N=300
Z=-23.4(0.70%) | Like=-18.36..-13.31 [-18.3608..-18.3543]*| it/evals=1050/1673 eff=76.4749% N=300
Mono-modal Volume: ~exp(-7.09) * Expected Volume: exp(-3.57) Quality: ok
index : +1.0| +2.3 ************* +3.3 | +5.0
amplitude: +1.0e-12| +3.3e-11 ************************ | +1.0e-10
Z=-23.1(0.99%) | Like=-18.12..-13.31 [-18.1173..-18.1014] | it/evals=1072/1710 eff=76.0284% N=300
Z=-23.0(1.11%) | Like=-18.03..-13.31 [-18.0348..-18.0042] | it/evals=1080/1718 eff=76.1636% N=300
Z=-22.6(1.65%) | Like=-17.57..-13.31 [-17.5975..-17.5734] | it/evals=1110/1762 eff=75.9234% N=300
Mono-modal Volume: ~exp(-7.41) * Expected Volume: exp(-3.80) 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=-22.3(2.32%) | Like=-17.23..-13.31 [-17.2271..-17.2209]*| it/evals=1139/1810 eff=75.4305% N=300
Z=-22.3(2.35%) | Like=-17.22..-13.31 [-17.2209..-17.2006] | it/evals=1140/1812 eff=75.3968% N=300
Z=-21.9(3.34%) | Like=-16.87..-13.31 [-16.8664..-16.8341] | it/evals=1170/1856 eff=75.1928% N=300
Z=-21.6(4.52%) | Like=-16.53..-13.31 [-16.5784..-16.5347] | it/evals=1200/1903 eff=74.8596% N=300
Mono-modal Volume: ~exp(-8.11) * Expected Volume: exp(-4.02) Quality: ok
index : +1.0| +2.3 *********** +3.2 | +5.0
amplitude: +1.0e-12| +3.6e-11 ******************** +7.5e-11 | +1.0e-10
Z=-21.6(4.74%) | Like=-16.49..-13.31 [-16.4938..-16.4749] | it/evals=1206/1911 eff=74.8603% N=300
Z=-21.3(5.93%) | Like=-16.27..-13.31 [-16.2726..-16.2635]*| it/evals=1230/1941 eff=74.9543% N=300
Z=-21.1(7.43%) | Like=-15.93..-13.31 [-15.9306..-15.9305]*| it/evals=1260/1982 eff=74.9108% 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| +3.7e-11 ****************** +7.2e-11 | +1.0e-10
Z=-21.0(8.16%) | Like=-15.84..-13.31 [-15.8379..-15.8357]*| it/evals=1273/2003 eff=74.7504% N=300
Z=-20.8(9.38%) | Like=-15.70..-13.31 [-15.6968..-15.6855] | it/evals=1290/2025 eff=74.7826% N=300
Z=-20.6(11.88%) | Like=-15.48..-13.31 [-15.4800..-15.4738]*| it/evals=1320/2063 eff=74.8724% N=300
Mono-modal Volume: ~exp(-8.61) * Expected Volume: exp(-4.47) 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=-20.5(13.60%) | Like=-15.32..-13.31 [-15.3174..-15.3139]*| it/evals=1340/2093 eff=74.7351% N=300
Z=-20.4(14.48%) | Like=-15.25..-13.31 [-15.2524..-15.2425]*| it/evals=1350/2106 eff=74.7508% N=300
Z=-20.2(17.28%) | Like=-15.05..-13.31 [-15.0462..-15.0404]*| it/evals=1380/2151 eff=74.5543% N=300
Mono-modal Volume: ~exp(-8.76) * Expected Volume: exp(-4.69) 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=-20.1(20.21%) | Like=-14.87..-13.31 [-14.8708..-14.8696]*| it/evals=1407/2185 eff=74.6419% N=300
Z=-20.1(20.54%) | Like=-14.86..-13.31 [-14.8649..-14.8577]*| it/evals=1410/2188 eff=74.6822% N=300
Z=-19.9(23.84%) | Like=-14.68..-13.31 [-14.6806..-14.6768]*| it/evals=1440/2230 eff=74.6114% N=300
Z=-19.8(27.69%) | Like=-14.56..-13.31 [-14.5556..-14.5503]*| it/evals=1470/2266 eff=74.7711% N=300
Mono-modal Volume: ~exp(-8.89) * Expected Volume: exp(-4.91) 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=-19.7(28.17%) | Like=-14.54..-13.31 [-14.5445..-14.5445]*| it/evals=1474/2270 eff=74.8223% N=300
Z=-19.6(31.57%) | Like=-14.47..-13.31 [-14.4676..-14.4672]*| it/evals=1500/2300 eff=75.0000% N=300
Z=-19.5(35.11%) | Like=-14.37..-13.31 [-14.3672..-14.3632]*| it/evals=1530/2342 eff=74.9265% N=300
Mono-modal Volume: ~exp(-9.35) * Expected Volume: exp(-5.14) Quality: ok
index : +1.0| +2.5 ****** +3.0 | +5.0
amplitude: +1.0e-12| +4.3e-11 *********** +6.5e-11 | +1.0e-10
Z=-19.5(36.40%) | Like=-14.32..-13.31 [-14.3228..-14.3183]*| it/evals=1541/2360 eff=74.8058% N=300
Z=-19.4(38.95%) | Like=-14.26..-13.31 [-14.2597..-14.2559]*| it/evals=1560/2382 eff=74.9280% N=300
Z=-19.3(42.54%) | Like=-14.17..-13.31 [-14.1742..-14.1720]*| it/evals=1590/2421 eff=74.9646% N=300
Mono-modal Volume: ~exp(-9.35) Expected Volume: exp(-5.36) Quality: ok
index : +1.0| +2.5 ****** +3.0 | +5.0
amplitude: +1.0e-12| +4.4e-11 *********** +6.3e-11 | +1.0e-10
Z=-19.2(46.19%) | Like=-14.12..-13.31 [-14.1157..-14.1147]*| it/evals=1620/2465 eff=74.8268% N=300
Z=-19.2(49.64%) | Like=-14.04..-13.31 [-14.0401..-14.0376]*| it/evals=1650/2502 eff=74.9319% N=300
Mono-modal Volume: ~exp(-9.44) * Expected Volume: exp(-5.58) 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=-19.1(52.55%) | Like=-13.99..-13.31 [-13.9926..-13.9907]*| it/evals=1675/2532 eff=75.0448% N=300
Z=-19.1(53.15%) | Like=-13.98..-13.31 [-13.9840..-13.9831]*| it/evals=1680/2537 eff=75.1006% N=300
Z=-19.1(56.29%) | Like=-13.91..-13.31 [-13.9144..-13.9119]*| it/evals=1710/2576 eff=75.1318% N=300
Z=-19.0(59.39%) | Like=-13.86..-13.31 [-13.8576..-13.8557]*| it/evals=1740/2625 eff=74.8387% N=300
Mono-modal Volume: ~exp(-9.80) * Expected Volume: exp(-5.81) 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=-19.0(59.61%) | Like=-13.86..-13.31 [-13.8553..-13.8515]*| it/evals=1742/2627 eff=74.8603% N=300
Z=-18.9(62.36%) | Like=-13.82..-13.31 [-13.8206..-13.8204]*| it/evals=1770/2664 eff=74.8731% N=300
Z=-18.9(65.20%) | Like=-13.77..-13.30 [-13.7691..-13.7690]*| it/evals=1800/2713 eff=74.5959% N=300
Mono-modal Volume: ~exp(-9.80) Expected Volume: exp(-6.03) 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=-18.9(67.86%) | Like=-13.74..-13.30 [-13.7377..-13.7370]*| it/evals=1830/2752 eff=74.6330% N=300
[ultranest] Explored until L=-1e+01
[ultranest] Likelihood function evaluations: 2791
[ultranest] logZ = -18.46 +- 0.09035
[ultranest] Effective samples strategy satisfied (ESS = 994.3, 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: 0.12 bs:0.09 tail:0.26 total:0.28 required:<0.50
[ultranest] done iterating.
logZ = -18.476 +- 0.339
single instance: logZ = -18.476 +- 0.118
bootstrapped : logZ = -18.458 +- 0.215
tail : logZ = +- 0.262
insert order U test : converged: True correlation: inf iterations
index : 2.13 │ ▁▁▁▁▁▁▂▂▃▄▅▅▅▅▇▇▆▆▅▄▄▄▂▂▁▁▁▁▁ ▁▁ ▁ │3.63 2.76 +- 0.18
amplitude : 0.0000000000265│ ▁▁▁▁▁▁▁▂▂▄▄▅▆▆▇▇▆▇▅▅▅▄▃▄▂▂▁▁▁▁▁▁▁▁ ▁ │0.0000000000877 0.0000000000536 +- 0.0000000000079
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 33.572 seconds)
