Note
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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
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.
path = "$GAMMAPY_DATA/joint-crab/spectra/hess/"
datasets = Datasets()
for id in ["23526", "23559", "23592"]:
dataset = SpectrumDatasetOnOff.read(f"{path}pha_obs{id}.fits")
datasets.append(dataset)
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 dictionnary.
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(-4.16) * Expected Volume: exp(0.00) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|****************** ************************** *| +1.0e-10
Z=-inf(0.00%) | Like=-3089.23..-60.94 [-3089.2332..-335.1691] | it/evals=0/301 eff=0.0000% N=300
Z=-935.9(0.00%) | Like=-917.43..-60.94 [-3089.2332..-335.1691] | it/evals=18/319 eff=94.7368% N=300
Z=-553.9(0.00%) | Like=-548.03..-60.94 [-3089.2332..-335.1691] | it/evals=30/331 eff=96.7742% N=300
Z=-537.6(0.00%) | Like=-531.02..-60.94 [-3089.2332..-335.1691] | it/evals=44/349 eff=89.7959% N=300
Z=-516.6(0.00%) | Like=-508.41..-60.94 [-3089.2332..-335.1691] | it/evals=60/365 eff=92.3077% N=300
Mono-modal Volume: ~exp(-4.30) * Expected Volume: exp(-0.22) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|****************** **************** ******** *| +1.0e-10
Z=-508.7(0.00%) | Like=-502.75..-60.94 [-3089.2332..-335.1691] | it/evals=67/372 eff=93.0556% N=300
Z=-494.7(0.00%) | Like=-489.72..-60.94 [-3089.2332..-335.1691] | it/evals=82/390 eff=91.1111% N=300
Z=-488.9(0.00%) | Like=-482.70..-60.62 [-3089.2332..-335.1691] | it/evals=90/398 eff=91.8367% N=300
Z=-471.4(0.00%) | Like=-464.53..-60.62 [-3089.2332..-335.1691] | it/evals=104/416 eff=89.6552% N=300
Z=-459.3(0.00%) | Like=-453.48..-60.62 [-3089.2332..-335.1691] | it/evals=120/433 eff=90.2256% 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=-440.8(0.00%) | Like=-434.83..-60.62 [-3089.2332..-335.1691] | it/evals=134/450 eff=89.3333% N=300
Z=-411.0(0.00%) | Like=-404.92..-60.62 [-3089.2332..-335.1691] | it/evals=150/467 eff=89.8204% N=300
Z=-393.0(0.00%) | Like=-386.77..-60.62 [-3089.2332..-335.1691] | it/evals=166/487 eff=88.7701% N=300
Z=-375.7(0.00%) | Like=-368.59..-60.62 [-3089.2332..-335.1691] | it/evals=179/505 eff=87.3171% N=300
Z=-374.5(0.00%) | Like=-367.13..-60.62 [-3089.2332..-335.1691] | it/evals=180/506 eff=87.3786% N=300
Z=-361.8(0.00%) | Like=-352.64..-60.62 [-3089.2332..-335.1691] | it/evals=196/525 eff=87.1111% N=300
Mono-modal Volume: ~exp(-4.72) * Expected Volume: exp(-0.67) Quality: ok
index : +1.0| **********************************************| +5.0
amplitude: +1.0e-12| ********************************** ********* *| +1.0e-10
Z=-356.3(0.00%) | Like=-350.52..-60.62 [-3089.2332..-335.1691] | it/evals=201/534 eff=85.8974% N=300
Z=-349.5(0.00%) | Like=-343.27..-60.62 [-3089.2332..-335.1691] | it/evals=210/545 eff=85.7143% N=300
Z=-340.2(0.00%) | Like=-334.93..-60.62 [-334.9746..-193.4020] | it/evals=225/563 eff=85.5513% N=300
Z=-325.7(0.00%) | Like=-320.45..-60.62 [-334.9746..-193.4020] | it/evals=240/580 eff=85.7143% N=300
Z=-315.1(0.00%) | Like=-308.31..-60.62 [-334.9746..-193.4020] | it/evals=256/598 eff=85.9060% N=300
Mono-modal Volume: ~exp(-4.80) * Expected Volume: exp(-0.89) Quality: ok
index : +1.0| ********************************************| +5.0
amplitude: +1.0e-12| **********************************************| +1.0e-10
Z=-306.6(0.00%) | Like=-300.47..-60.62 [-334.9746..-193.4020] | it/evals=268/611 eff=86.1736% N=300
Z=-305.1(0.00%) | Like=-296.88..-60.62 [-334.9746..-193.4020] | it/evals=270/613 eff=86.2620% N=300
Z=-292.8(0.00%) | Like=-285.40..-59.79 [-334.9746..-193.4020] | it/evals=285/631 eff=86.1027% N=300
Z=-278.0(0.00%) | Like=-271.62..-59.79 [-334.9746..-193.4020] | it/evals=300/647 eff=86.4553% N=300
Z=-266.6(0.00%) | Like=-259.91..-59.79 [-334.9746..-193.4020] | it/evals=313/666 eff=85.5191% N=300
Z=-255.9(0.00%) | Like=-250.10..-59.58 [-334.9746..-193.4020] | it/evals=329/684 eff=85.6771% N=300
Z=-255.6(0.00%) | Like=-249.54..-59.58 [-334.9746..-193.4020] | it/evals=330/685 eff=85.7143% N=300
Mono-modal Volume: ~exp(-5.12) * Expected Volume: exp(-1.12) Quality: ok
index : +1.0| *******************************************| +5.0
amplitude: +1.0e-12| *********************************************| +1.0e-10
Z=-253.7(0.00%) | Like=-247.85..-59.58 [-334.9746..-193.4020] | it/evals=335/692 eff=85.4592% N=300
Z=-243.2(0.00%) | Like=-236.79..-59.58 [-334.9746..-193.4020] | it/evals=350/710 eff=85.3659% N=300
Z=-237.4(0.00%) | Like=-230.31..-59.58 [-334.9746..-193.4020] | it/evals=360/720 eff=85.7143% N=300
Z=-228.9(0.00%) | Like=-222.19..-59.58 [-334.9746..-193.4020] | it/evals=373/739 eff=84.9658% N=300
Z=-221.0(0.00%) | Like=-213.13..-59.58 [-334.9746..-193.4020] | it/evals=387/757 eff=84.6827% N=300
Z=-218.5(0.00%) | Like=-212.21..-59.58 [-334.9746..-193.4020] | it/evals=390/763 eff=84.2333% N=300
Z=-213.7(0.00%) | Like=-206.72..-59.58 [-334.9746..-193.4020] | it/evals=400/782 eff=82.9876% N=300
Mono-modal Volume: ~exp(-5.21) * Expected Volume: exp(-1.34) Quality: ok
index : +1.0| ******************************************| +5.0
amplitude: +1.0e-12| ***************************************** *| +1.0e-10
Z=-212.6(0.00%) | Like=-206.33..-59.58 [-334.9746..-193.4020] | it/evals=402/784 eff=83.0579% N=300
Z=-207.0(0.00%) | Like=-200.66..-59.58 [-334.9746..-193.4020] | it/evals=414/803 eff=82.3062% N=300
Z=-205.2(0.00%) | Like=-198.92..-59.58 [-334.9746..-193.4020] | it/evals=420/813 eff=81.8713% N=300
Z=-199.1(0.00%) | Like=-191.96..-59.58 [-193.3711..-131.1112] | it/evals=434/831 eff=81.7326% N=300
Z=-193.9(0.00%) | Like=-187.74..-59.58 [-193.3711..-131.1112] | it/evals=448/849 eff=81.6029% N=300
Z=-193.3(0.00%) | Like=-187.36..-59.58 [-193.3711..-131.1112] | it/evals=450/852 eff=81.5217% N=300
Z=-187.7(0.00%) | Like=-181.15..-59.58 [-193.3711..-131.1112] | it/evals=465/871 eff=81.4361% 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=-186.3(0.00%) | Like=-179.73..-59.58 [-193.3711..-131.1112] | it/evals=469/877 eff=81.2825% N=300
Z=-182.7(0.00%) | Like=-176.57..-59.58 [-193.3711..-131.1112] | it/evals=480/893 eff=80.9444% N=300
Z=-179.5(0.00%) | Like=-173.73..-59.58 [-193.3711..-131.1112] | it/evals=492/911 eff=80.5237% N=300
Z=-176.2(0.00%) | Like=-170.24..-59.58 [-193.3711..-131.1112] | it/evals=506/929 eff=80.4452% N=300
Z=-175.2(0.00%) | Like=-169.08..-59.58 [-193.3711..-131.1112] | it/evals=510/936 eff=80.1887% N=300
Z=-171.3(0.00%) | Like=-164.76..-59.14 [-193.3711..-131.1112] | it/evals=526/955 eff=80.3053% N=300
Mono-modal Volume: ~exp(-5.94) * Expected Volume: exp(-1.79) Quality: ok
index : +1.0| *********************************** | +5.0
amplitude: +1.0e-12| ******************************** **** * *| +1.0e-10
Z=-169.0(0.00%) | Like=-162.79..-59.14 [-193.3711..-131.1112] | it/evals=536/969 eff=80.1196% N=300
Z=-167.8(0.00%) | Like=-161.50..-59.14 [-193.3711..-131.1112] | it/evals=540/975 eff=80.0000% N=300
Z=-163.5(0.00%) | Like=-157.10..-59.06 [-193.3711..-131.1112] | it/evals=555/993 eff=80.0866% N=300
Z=-159.9(0.00%) | Like=-153.98..-59.06 [-193.3711..-131.1112] | it/evals=570/1012 eff=80.0562% N=300
Z=-156.4(0.00%) | Like=-150.22..-59.06 [-193.3711..-131.1112] | it/evals=584/1030 eff=80.0000% N=300
Z=-152.3(0.00%) | Like=-145.78..-59.06 [-193.3711..-131.1112] | it/evals=597/1049 eff=79.7063% N=300
Z=-151.3(0.00%) | Like=-144.95..-59.06 [-193.3711..-131.1112] | it/evals=600/1054 eff=79.5756% N=300
Mono-modal Volume: ~exp(-6.03) * Expected Volume: exp(-2.01) Quality: ok
index : +1.0| ****************************** | +5.0
amplitude: +1.0e-12| ************************************ | +1.0e-10
Z=-150.6(0.00%) | Like=-143.83..-59.06 [-193.3711..-131.1112] | it/evals=603/1059 eff=79.4466% N=300
Z=-147.0(0.00%) | Like=-140.31..-59.06 [-193.3711..-131.1112] | it/evals=616/1078 eff=79.1774% N=300
Z=-143.9(0.00%) | Like=-137.54..-59.06 [-193.3711..-131.1112] | it/evals=628/1096 eff=78.8945% N=300
Z=-143.4(0.00%) | Like=-137.20..-59.06 [-193.3711..-131.1112] | it/evals=630/1100 eff=78.7500% N=300
Z=-141.1(0.00%) | Like=-134.90..-59.06 [-193.3711..-131.1112] | it/evals=646/1118 eff=78.9731% N=300
Z=-138.1(0.00%) | Like=-131.19..-59.06 [-193.3711..-131.1112] | it/evals=660/1136 eff=78.9474% N=300
Mono-modal Volume: ~exp(-6.03) Expected Volume: exp(-2.23) Quality: ok
index : +1.0| +1.9 ************************** +4.0 | +5.0
amplitude: +1.0e-12| ************************************ | +1.0e-10
Z=-135.9(0.00%) | Like=-129.89..-59.06 [-131.0642..-92.0535] | it/evals=671/1153 eff=78.6635% N=300
Z=-133.0(0.00%) | Like=-125.22..-59.06 [-131.0642..-92.0535] | it/evals=682/1171 eff=78.3008% N=300
Z=-129.5(0.00%) | Like=-122.76..-59.04 [-131.0642..-92.0535] | it/evals=690/1184 eff=78.0543% N=300
Z=-127.1(0.00%) | Like=-120.26..-59.04 [-131.0642..-92.0535] | it/evals=700/1202 eff=77.6053% N=300
Z=-124.5(0.00%) | Like=-118.04..-59.04 [-131.0642..-92.0535] | it/evals=712/1221 eff=77.3073% N=300
Z=-123.2(0.00%) | Like=-117.15..-59.04 [-131.0642..-92.0535] | it/evals=720/1233 eff=77.1704% N=300
Z=-121.3(0.00%) | Like=-114.36..-59.03 [-131.0642..-92.0535] | it/evals=734/1255 eff=76.8586% N=300
Mono-modal Volume: ~exp(-6.03) Expected Volume: exp(-2.46) Quality: ok
index : +1.0| +2.0 *********************** +3.8 | +5.0
amplitude: +1.0e-12| ****************************** | +1.0e-10
Z=-118.7(0.00%) | Like=-111.74..-59.03 [-131.0642..-92.0535] | it/evals=744/1271 eff=76.6220% N=300
Z=-117.4(0.00%) | Like=-110.97..-59.03 [-131.0642..-92.0535] | it/evals=750/1282 eff=76.3747% N=300
Z=-115.3(0.00%) | Like=-108.47..-59.03 [-131.0642..-92.0535] | it/evals=761/1300 eff=76.1000% N=300
Z=-113.5(0.00%) | Like=-106.61..-59.03 [-131.0642..-92.0535] | it/evals=771/1319 eff=75.6624% N=300
Z=-111.8(0.00%) | Like=-105.06..-59.03 [-131.0642..-92.0535] | it/evals=780/1330 eff=75.7282% N=300
Z=-109.6(0.00%) | Like=-103.26..-59.03 [-131.0642..-92.0535] | it/evals=793/1349 eff=75.5958% N=300
Mono-modal Volume: ~exp(-6.25) * Expected Volume: exp(-2.68) Quality: ok
index : +1.0| +2.0 ******************* +3.5 | +5.0
amplitude: +1.0e-12| **************************** +7.8e-11| +1.0e-10
Z=-108.2(0.00%) | Like=-101.63..-59.03 [-131.0642..-92.0535] | it/evals=804/1364 eff=75.5639% N=300
Z=-107.4(0.00%) | Like=-100.98..-59.03 [-131.0642..-92.0535] | it/evals=810/1370 eff=75.7009% N=300
Z=-105.2(0.00%) | Like=-98.49..-59.03 [-131.0642..-92.0535] | it/evals=822/1388 eff=75.5515% N=300
Z=-104.0(0.00%) | Like=-97.82..-58.81 [-131.0642..-92.0535] | it/evals=833/1408 eff=75.1805% N=300
Z=-103.4(0.00%) | Like=-97.06..-58.81 [-131.0642..-92.0535] | it/evals=840/1418 eff=75.1342% N=300
Z=-102.0(0.00%) | Like=-95.43..-58.81 [-131.0642..-92.0535] | it/evals=852/1436 eff=75.0000% N=300
Z=-99.8(0.00%) | Like=-93.05..-58.81 [-131.0642..-92.0535] | it/evals=867/1454 eff=75.1300% N=300
Z=-99.3(0.00%) | Like=-92.79..-58.81 [-131.0642..-92.0535] | it/evals=870/1457 eff=75.1945% N=300
Mono-modal Volume: ~exp(-6.48) * Expected Volume: exp(-2.90) Quality: ok
index : +1.0| +2.1 ****************** +3.5 | +5.0
amplitude: +1.0e-12| ************************* +7.1e-11 | +1.0e-10
Z=-99.2(0.00%) | Like=-92.63..-58.81 [-131.0642..-92.0535] | it/evals=871/1459 eff=75.1510% N=300
Z=-97.8(0.00%) | Like=-91.26..-58.81 [-92.0530..-75.1083] | it/evals=884/1478 eff=75.0424% N=300
Z=-96.5(0.00%) | Like=-89.92..-58.81 [-92.0530..-75.1083] | it/evals=897/1498 eff=74.8748% N=300
Z=-96.1(0.00%) | Like=-89.72..-58.81 [-92.0530..-75.1083] | it/evals=900/1503 eff=74.8130% N=300
Z=-95.2(0.00%) | Like=-88.38..-58.81 [-92.0530..-75.1083] | it/evals=909/1521 eff=74.4472% N=300
Z=-94.1(0.00%) | Like=-87.23..-58.81 [-92.0530..-75.1083] | it/evals=917/1539 eff=74.0113% N=300
Z=-92.9(0.00%) | Like=-86.07..-58.81 [-92.0530..-75.1083] | it/evals=926/1559 eff=73.5504% N=300
Z=-92.4(0.00%) | Like=-85.87..-58.81 [-92.0530..-75.1083] | it/evals=930/1563 eff=73.6342% N=300
Mono-modal Volume: ~exp(-6.54) * Expected Volume: exp(-3.13) Quality: ok
index : +1.0| +2.1 **************** +3.4 | +5.0
amplitude: +1.0e-12| +2.6e-11 ********************* +6.8e-11 | +1.0e-10
Z=-91.8(0.00%) | Like=-85.55..-58.81 [-92.0530..-75.1083] | it/evals=938/1578 eff=73.3959% N=300
Z=-90.0(0.00%) | Like=-82.80..-58.79 [-92.0530..-75.1083] | it/evals=952/1596 eff=73.4568% N=300
Z=-88.9(0.00%) | Like=-82.20..-58.79 [-92.0530..-75.1083] | it/evals=960/1610 eff=73.2824% N=300
Z=-87.4(0.00%) | Like=-80.69..-58.79 [-92.0530..-75.1083] | it/evals=972/1629 eff=73.1377% N=300
Z=-86.6(0.00%) | Like=-80.13..-58.79 [-92.0530..-75.1083] | it/evals=982/1646 eff=72.9569% N=300
Z=-85.9(0.00%) | Like=-79.37..-58.79 [-92.0530..-75.1083] | it/evals=990/1660 eff=72.7941% N=300
Z=-85.1(0.00%) | Like=-78.88..-58.79 [-92.0530..-75.1083] | it/evals=1002/1678 eff=72.7141% N=300
Mono-modal Volume: ~exp(-7.56) * Expected Volume: exp(-3.35) Quality: ok
index : +1.0| +2.2 ************** +3.3 | +5.0
amplitude: +1.0e-12| +2.8e-11 ******************* +6.5e-11 | +1.0e-10
Z=-84.9(0.00%) | Like=-78.80..-58.79 [-92.0530..-75.1083] | it/evals=1005/1681 eff=72.7734% N=300
Z=-84.2(0.00%) | Like=-77.86..-58.79 [-92.0530..-75.1083] | it/evals=1019/1698 eff=72.8898% N=300
Z=-84.1(0.00%) | Like=-77.81..-58.79 [-92.0530..-75.1083] | it/evals=1020/1699 eff=72.9092% N=300
Z=-83.0(0.00%) | Like=-76.49..-58.79 [-92.0530..-75.1083] | it/evals=1035/1719 eff=72.9387% N=300
Z=-82.1(0.00%) | Like=-75.82..-58.79 [-92.0530..-75.1083] | it/evals=1049/1738 eff=72.9485% N=300
Z=-82.1(0.00%) | Like=-75.78..-58.79 [-92.0530..-75.1083] | it/evals=1050/1739 eff=72.9673% N=300
Z=-81.5(0.00%) | Like=-75.19..-58.79 [-92.0530..-75.1083] | it/evals=1062/1757 eff=72.8895% N=300
Mono-modal Volume: ~exp(-7.56) Expected Volume: exp(-3.57) Quality: ok
index : +1.0| +2.3 ************ +3.2 | +5.0
amplitude: +1.0e-12| +2.9e-11 ***************** +6.3e-11 | +1.0e-10
Z=-80.9(0.00%) | Like=-74.65..-58.75 [-75.0637..-66.7495] | it/evals=1074/1773 eff=72.9124% N=300
Z=-80.6(0.00%) | Like=-74.47..-58.75 [-75.0637..-66.7495] | it/evals=1080/1780 eff=72.9730% N=300
Z=-80.0(0.00%) | Like=-73.58..-58.75 [-75.0637..-66.7495] | it/evals=1093/1799 eff=72.9153% N=300
Z=-79.3(0.00%) | Like=-72.84..-58.75 [-75.0637..-66.7495] | it/evals=1107/1818 eff=72.9249% N=300
Z=-79.1(0.00%) | Like=-72.80..-58.75 [-75.0637..-66.7495] | it/evals=1110/1822 eff=72.9304% N=300
Z=-78.6(0.00%) | Like=-72.15..-58.75 [-75.0637..-66.7495] | it/evals=1121/1842 eff=72.6978% N=300
Z=-78.1(0.00%) | Like=-71.67..-58.75 [-75.0637..-66.7495] | it/evals=1131/1860 eff=72.5000% N=300
Mono-modal Volume: ~exp(-7.67) * 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=-77.8(0.00%) | Like=-71.40..-58.75 [-75.0637..-66.7495] | it/evals=1139/1868 eff=72.6403% N=300
Z=-77.7(0.00%) | Like=-71.39..-58.75 [-75.0637..-66.7495] | it/evals=1140/1869 eff=72.6577% N=300
Z=-77.2(0.00%) | Like=-70.59..-58.75 [-75.0637..-66.7495] | it/evals=1152/1887 eff=72.5898% N=300
Z=-76.6(0.00%) | Like=-70.17..-58.75 [-75.0637..-66.7495] | it/evals=1164/1905 eff=72.5234% N=300
Z=-76.4(0.00%) | Like=-69.96..-58.75 [-75.0637..-66.7495] | it/evals=1170/1911 eff=72.6257% N=300
Z=-75.9(0.00%) | Like=-69.29..-58.75 [-75.0637..-66.7495] | it/evals=1181/1930 eff=72.4540% N=300
Z=-75.4(0.00%) | Like=-68.80..-58.75 [-75.0637..-66.7495] | it/evals=1191/1948 eff=72.2694% N=300
Z=-75.0(0.00%) | Like=-68.39..-58.75 [-75.0637..-66.7495] | it/evals=1200/1966 eff=72.0288% N=300
Mono-modal Volume: ~exp(-8.33) * Expected Volume: exp(-4.02) Quality: ok
index : +1.0| +2.3 ********* +3.0 | +5.0
amplitude: +1.0e-12| +3.2e-11 ************* +5.8e-11 | +1.0e-10
Z=-74.7(0.01%) | Like=-68.31..-58.75 [-75.0637..-66.7495] | it/evals=1206/1975 eff=72.0000% N=300
Z=-74.3(0.01%) | Like=-68.05..-58.75 [-75.0637..-66.7495] | it/evals=1218/1993 eff=71.9433% N=300
Z=-74.0(0.01%) | Like=-67.82..-58.75 [-75.0637..-66.7495] | it/evals=1230/2008 eff=72.0141% N=300
Z=-73.6(0.02%) | Like=-67.32..-58.75 [-75.0637..-66.7495] | it/evals=1246/2025 eff=72.2319% N=300
Z=-73.2(0.03%) | Like=-66.95..-58.75 [-75.0637..-66.7495] | it/evals=1260/2042 eff=72.3307% N=300
Mono-modal Volume: ~exp(-8.46) * 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=-66.65..-58.75 [-66.7454..-65.2753] | it/evals=1273/2059 eff=72.3707% N=300
Z=-72.5(0.05%) | Like=-66.15..-58.75 [-66.7454..-65.2753] | it/evals=1287/2077 eff=72.4254% N=300
Z=-72.4(0.06%) | Like=-66.05..-58.75 [-66.7454..-65.2753] | it/evals=1290/2080 eff=72.4719% N=300
Z=-72.1(0.09%) | Like=-65.61..-58.75 [-66.7454..-65.2753] | it/evals=1306/2098 eff=72.6363% N=300
Z=-71.8(0.11%) | Like=-65.35..-58.75 [-66.7454..-65.2753] | it/evals=1317/2117 eff=72.4821% N=300
Z=-71.7(0.12%) | Like=-65.22..-58.75 [-65.2366..-65.0721] | it/evals=1320/2123 eff=72.4081% N=300
Z=-71.3(0.18%) | Like=-65.02..-58.75 [-65.0151..-65.0018] | it/evals=1336/2141 eff=72.5693% N=300
Mono-modal Volume: ~exp(-8.46) Expected Volume: exp(-4.47) 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=-71.1(0.23%) | Like=-64.74..-58.75 [-64.7763..-64.7398] | it/evals=1348/2157 eff=72.5902% N=300
Z=-71.0(0.24%) | Like=-64.73..-58.75 [-64.7347..-64.7296]*| it/evals=1350/2161 eff=72.5416% N=300
Z=-70.8(0.29%) | Like=-64.48..-58.75 [-64.5100..-64.4813] | it/evals=1360/2179 eff=72.3789% N=300
Z=-70.6(0.38%) | Like=-64.14..-58.75 [-64.1400..-64.1174] | it/evals=1371/2197 eff=72.2720% N=300
Z=-70.4(0.46%) | Like=-64.00..-58.75 [-64.0405..-64.0039] | it/evals=1380/2208 eff=72.3270% N=300
Z=-70.2(0.54%) | Like=-63.87..-58.75 [-63.8658..-63.8431] | it/evals=1389/2226 eff=72.1184% N=300
Z=-70.0(0.64%) | Like=-63.73..-58.75 [-63.7252..-63.7056] | it/evals=1400/2244 eff=72.0165% N=300
Mono-modal Volume: ~exp(-8.88) * 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(0.73%) | Like=-63.66..-58.75 [-63.6597..-63.6493] | it/evals=1407/2253 eff=72.0430% N=300
Z=-69.9(0.77%) | Like=-63.59..-58.75 [-63.6196..-63.5930] | it/evals=1410/2256 eff=72.0859% N=300
Z=-69.6(0.95%) | Like=-63.39..-58.75 [-63.3920..-63.3821]*| it/evals=1424/2273 eff=72.1744% N=300
Z=-69.4(1.19%) | Like=-63.19..-58.75 [-63.2044..-63.1920] | it/evals=1438/2291 eff=72.2250% N=300
Z=-69.4(1.23%) | Like=-63.15..-58.75 [-63.1461..-63.1393]*| it/evals=1440/2293 eff=72.2529% N=300
Z=-69.2(1.51%) | Like=-62.97..-58.75 [-62.9732..-62.9576] | it/evals=1455/2312 eff=72.3161% N=300
Z=-69.0(1.81%) | Like=-62.82..-58.75 [-62.8236..-62.8007] | it/evals=1468/2330 eff=72.3153% N=300
Z=-69.0(1.85%) | Like=-62.76..-58.75 [-62.7618..-62.7004] | it/evals=1470/2332 eff=72.3425% N=300
Mono-modal Volume: ~exp(-8.92) * Expected Volume: exp(-4.91) 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=-69.0(1.92%) | Like=-62.70..-58.75 [-62.6968..-62.6967]*| it/evals=1474/2336 eff=72.3969% N=300
Z=-68.8(2.31%) | Like=-62.44..-58.75 [-62.4413..-62.4276] | it/evals=1489/2354 eff=72.4927% N=300
Z=-68.6(2.63%) | Like=-62.29..-58.75 [-62.2912..-62.2841]*| it/evals=1500/2369 eff=72.4988% N=300
Z=-68.5(3.09%) | Like=-62.17..-58.75 [-62.1714..-62.1685]*| it/evals=1513/2388 eff=72.4617% N=300
Z=-68.3(3.53%) | Like=-62.03..-58.75 [-62.0337..-62.0297]*| it/evals=1524/2407 eff=72.3303% N=300
Z=-68.3(3.83%) | Like=-61.98..-58.75 [-61.9794..-61.9766]*| it/evals=1530/2417 eff=72.2721% N=300
Mono-modal Volume: ~exp(-9.17) * Expected Volume: exp(-5.14) 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=-68.2(4.37%) | Like=-61.90..-58.75 [-61.9054..-61.8954] | it/evals=1541/2432 eff=72.2795% N=300
Z=-68.0(5.10%) | Like=-61.75..-58.75 [-61.7794..-61.7467] | it/evals=1556/2451 eff=72.3384% N=300
Z=-68.0(5.33%) | Like=-61.70..-58.75 [-61.7175..-61.7005] | it/evals=1560/2455 eff=72.3898% N=300
Z=-67.8(6.11%) | Like=-61.60..-58.75 [-61.6044..-61.6031]*| it/evals=1575/2473 eff=72.4804% N=300
Z=-67.7(6.98%) | Like=-61.47..-58.75 [-61.4835..-61.4668] | it/evals=1589/2491 eff=72.5240% N=300
Z=-67.7(7.02%) | Like=-61.46..-58.75 [-61.4618..-61.4616]*| it/evals=1590/2492 eff=72.5365% N=300
Z=-67.6(7.67%) | Like=-61.36..-58.75 [-61.3555..-61.3408] | it/evals=1603/2511 eff=72.5011% N=300
Mono-modal Volume: ~exp(-9.58) * 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(7.99%) | Like=-61.31..-58.75 [-61.3089..-61.2989] | it/evals=1608/2516 eff=72.5632% N=300
Z=-67.5(8.80%) | Like=-61.18..-58.75 [-61.1758..-61.1728]*| it/evals=1620/2532 eff=72.5806% N=300
Z=-67.3(10.00%) | Like=-61.04..-58.75 [-61.0661..-61.0350] | it/evals=1636/2549 eff=72.7434% N=300
Z=-67.2(11.10%) | Like=-60.94..-58.75 [-60.9534..-60.9387] | it/evals=1650/2569 eff=72.7193% N=300
Z=-67.1(12.46%) | Like=-60.87..-58.75 [-60.8651..-60.8585]*| it/evals=1666/2587 eff=72.8465% N=300
Mono-modal Volume: ~exp(-9.58) Expected Volume: exp(-5.58) 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=-67.0(13.57%) | Like=-60.77..-58.75 [-60.7750..-60.7731]*| it/evals=1678/2604 eff=72.8299% N=300
Z=-67.0(13.79%) | Like=-60.77..-58.75 [-60.7709..-60.7708]*| it/evals=1680/2608 eff=72.7903% N=300
Z=-67.0(14.79%) | Like=-60.71..-58.75 [-60.7134..-60.7014] | it/evals=1691/2628 eff=72.6375% N=300
Z=-66.9(16.09%) | Like=-60.65..-58.75 [-60.6492..-60.6451]*| it/evals=1706/2646 eff=72.7195% N=300
Z=-66.8(16.35%) | Like=-60.62..-58.75 [-60.6178..-60.6160]*| it/evals=1710/2653 eff=72.6732% N=300
Z=-66.8(17.74%) | Like=-60.53..-58.75 [-60.5404..-60.5289] | it/evals=1725/2674 eff=72.6622% N=300
Z=-66.7(19.00%) | Like=-60.42..-58.75 [-60.4238..-60.4027] | it/evals=1739/2692 eff=72.7007% N=300
Z=-66.7(19.13%) | Like=-60.40..-58.75 [-60.4238..-60.4027] | it/evals=1740/2693 eff=72.7121% N=300
Mono-modal Volume: ~exp(-9.91) * 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=-66.7(19.38%) | Like=-60.38..-58.75 [-60.3776..-60.3582] | it/evals=1742/2699 eff=72.6136% N=300
Z=-66.6(20.45%) | Like=-60.32..-58.75 [-60.3242..-60.3231]*| it/evals=1753/2717 eff=72.5279% N=300
Z=-66.5(21.87%) | Like=-60.25..-58.75 [-60.2489..-60.2483]*| it/evals=1766/2737 eff=72.4661% N=300
Z=-66.5(22.28%) | Like=-60.24..-58.75 [-60.2364..-60.2356]*| it/evals=1770/2742 eff=72.4816% N=300
Z=-66.5(23.61%) | Like=-60.17..-58.75 [-60.1710..-60.1701]*| it/evals=1781/2760 eff=72.3984% N=300
Z=-66.4(25.19%) | Like=-60.11..-58.75 [-60.1149..-60.1140]*| it/evals=1794/2778 eff=72.3971% N=300
Z=-66.4(25.84%) | Like=-60.09..-58.75 [-60.0863..-60.0762] | it/evals=1800/2787 eff=72.3764% N=300
Mono-modal Volume: ~exp(-10.05) * Expected Volume: exp(-6.03) 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=-66.3(26.96%) | Like=-60.03..-58.75 [-60.0298..-60.0277]*| it/evals=1809/2805 eff=72.2156% N=300
Z=-66.3(28.77%) | Like=-59.96..-58.75 [-59.9603..-59.9580]*| it/evals=1825/2823 eff=72.3345% N=300
Z=-66.3(29.35%) | Like=-59.95..-58.75 [-59.9479..-59.9476]*| it/evals=1830/2829 eff=72.3606% N=300
Z=-66.2(31.28%) | Like=-59.90..-58.75 [-59.8974..-59.8967]*| it/evals=1845/2847 eff=72.4382% N=300
Z=-66.1(33.04%) | Like=-59.86..-58.75 [-59.8583..-59.8575]*| it/evals=1860/2864 eff=72.5429% N=300
Z=-66.1(34.40%) | Like=-59.81..-58.75 [-59.8087..-59.7914] | it/evals=1871/2882 eff=72.4632% N=300
Mono-modal Volume: ~exp(-10.52) * Expected Volume: exp(-6.25) 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=-66.1(35.01%) | Like=-59.78..-58.75 [-59.7757..-59.7498] | it/evals=1876/2889 eff=72.4604% N=300
Z=-66.0(36.84%) | Like=-59.71..-58.75 [-59.7135..-59.7095]*| it/evals=1890/2905 eff=72.5528% N=300
Z=-66.0(38.58%) | Like=-59.67..-58.75 [-59.6697..-59.6691]*| it/evals=1904/2923 eff=72.5886% N=300
Z=-65.9(40.20%) | Like=-59.64..-58.75 [-59.6405..-59.6394]*| it/evals=1917/2942 eff=72.5587% N=300
Z=-65.9(40.57%) | Like=-59.64..-58.75 [-59.6363..-59.6337]*| it/evals=1920/2945 eff=72.5898% N=300
Z=-65.9(42.30%) | Like=-59.60..-58.75 [-59.6002..-59.5986]*| it/evals=1935/2962 eff=72.6897% N=300
Mono-modal Volume: ~exp(-10.52) 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(43.48%) | Like=-59.58..-58.75 [-59.5774..-59.5768]*| it/evals=1944/2978 eff=72.5915% N=300
Z=-65.9(44.07%) | Like=-59.56..-58.75 [-59.5585..-59.5573]*| it/evals=1950/2986 eff=72.5987% N=300
Z=-65.8(45.74%) | Like=-59.53..-58.75 [-59.5302..-59.5273]*| it/evals=1964/3004 eff=72.6331% N=300
Z=-65.8(47.56%) | Like=-59.49..-58.75 [-59.4927..-59.4772] | it/evals=1979/3022 eff=72.7039% N=300
Z=-65.8(47.69%) | Like=-59.48..-58.75 [-59.4927..-59.4772] | it/evals=1980/3024 eff=72.6872% N=300
Z=-65.7(49.66%) | Like=-59.45..-58.75 [-59.4458..-59.4443]*| it/evals=1997/3043 eff=72.8035% N=300
Z=-65.7(50.95%) | Like=-59.42..-58.75 [-59.4180..-59.4174]*| it/evals=2008/3061 eff=72.7273% N=300
Mono-modal Volume: ~exp(-10.67) * 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=-65.7(51.20%) | Like=-59.41..-58.75 [-59.4123..-59.4080]*| it/evals=2010/3063 eff=72.7470% N=300
Z=-65.7(52.89%) | Like=-59.37..-58.75 [-59.3748..-59.3719]*| it/evals=2024/3081 eff=72.7796% N=300
Z=-65.6(54.66%) | Like=-59.34..-58.75 [-59.3420..-59.3408]*| it/evals=2040/3099 eff=72.8832% N=300
Z=-65.6(56.06%) | Like=-59.33..-58.75 [-59.3253..-59.3222]*| it/evals=2053/3117 eff=72.8789% N=300
Z=-65.6(57.62%) | Like=-59.30..-58.75 [-59.3039..-59.3029]*| it/evals=2068/3135 eff=72.9453% N=300
Z=-65.6(57.83%) | Like=-59.30..-58.75 [-59.2975..-59.2965]*| it/evals=2070/3140 eff=72.8873% N=300
Mono-modal Volume: ~exp(-11.02) * Expected Volume: exp(-6.92) 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=-65.6(58.51%) | Like=-59.28..-58.75 [-59.2802..-59.2801]*| it/evals=2077/3152 eff=72.8261% N=300
Z=-65.5(60.03%) | Like=-59.27..-58.75 [-59.2712..-59.2648]*| it/evals=2092/3170 eff=72.8920% N=300
Z=-65.5(60.82%) | Like=-59.25..-58.75 [-59.2529..-59.2510]*| it/evals=2100/3182 eff=72.8661% N=300
Z=-65.5(61.76%) | Like=-59.23..-58.75 [-59.2295..-59.2227]*| it/evals=2110/3200 eff=72.7586% N=300
Z=-65.5(63.10%) | Like=-59.20..-58.75 [-59.2029..-59.2007]*| it/evals=2124/3218 eff=72.7896% N=300
Z=-65.5(63.64%) | Like=-59.19..-58.75 [-59.1900..-59.1847]*| it/evals=2130/3226 eff=72.7956% N=300
Z=-65.5(64.65%) | Like=-59.16..-58.75 [-59.1639..-59.1541]*| it/evals=2141/3244 eff=72.7242% N=300
Mono-modal Volume: ~exp(-11.02) 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(65.99%) | Like=-59.13..-58.75 [-59.1290..-59.1283]*| it/evals=2155/3260 eff=72.8041% N=300
Z=-65.4(66.45%) | Like=-59.13..-58.75 [-59.1262..-59.1261]*| it/evals=2160/3270 eff=72.7273% N=300
Z=-65.4(67.61%) | Like=-59.11..-58.75 [-59.1135..-59.1121]*| it/evals=2173/3288 eff=72.7242% N=300
Z=-65.4(68.88%) | Like=-59.10..-58.75 [-59.0971..-59.0965]*| it/evals=2187/3308 eff=72.7061% N=300
Z=-65.4(69.13%) | Like=-59.09..-58.75 [-59.0933..-59.0928]*| it/evals=2190/3312 eff=72.7092% N=300
[ultranest] Explored until L=-6e+01
[ultranest] Likelihood function evaluations: 3327
[ultranest] logZ = -65.03 +- 0.1146
[ultranest] Effective samples strategy satisfied (ESS = 1003.1, 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=0.29, need <0.5)
[ultranest] logZ error budget: single: 0.13 bs:0.11 tail:0.26 total:0.29 required:<0.50
[ultranest] done iterating.
logZ = -65.036 +- 0.470
single instance: logZ = -65.036 +- 0.133
bootstrapped : logZ = -65.026 +- 0.389
tail : logZ = +- 0.262
insert order U test : converged: True correlation: inf iterations
index : 2.338 │ ▁▁ ▁▁▁▁▁▁▂▂▃▃▅▅▅▆▆▇▇▇▆▅▅▄▃▃▂▁▂▁▁▁▁▁▁▁ │2.984 2.669 +- 0.085
amplitude : 0.0000000000341│ ▁▁▁▁▁▁▁▁▃▂▄▄▅▆▇▇▆▆▅▅▅▃▃▃▂▂▂▁▁▁▁▁▁▁▁ ▁ │0.0000000000566 0.0000000000443 +- 0.0000000000030
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 : 2.669 +/- 0.09
amplitude : 4.43e-11 +/- 3.0e-12 1 / (TeV s cm2)
reference (frozen): 1.000 TeV
The Sampler class returns a very rich dictionnary.
The most “standard” information about the posterior distributions can
be found in :
print(result_joint.sampler_results["posterior"])
{'mean': [2.6692133354352268, 4.426596025515653e-11], 'stdev': [0.08524752761787628, 3.040091702045256e-12], 'median': [2.6696431277523196, 4.4025444223123914e-11], 'errlo': [2.581895408429762, 4.1265253133575293e-11], 'errup': [2.7546635399483965, 4.744990231334529e-11], 'information_gain_bits': [2.719369819189697, 3.1158196672456064]}
Besides mean, errors, etc, an interesting value is the
information gain which estimates how much the posterior
distribution has shrinked 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 example 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()
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(-3.99) * Expected Volume: exp(0.00) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|********************************* * ***** *** **| +1.0e-10
Z=-inf(0.00%) | Like=-1765.17..-22.32 [-1765.1743..-103.5407] | it/evals=0/301 eff=0.0000% N=300
Z=-176.8(0.00%) | Like=-171.69..-20.84 [-1765.1743..-103.5407] | it/evals=30/333 eff=90.9091% N=300
Z=-163.3(0.00%) | Like=-157.51..-20.84 [-1765.1743..-103.5407] | it/evals=58/365 eff=89.2308% N=300
Z=-162.2(0.00%) | Like=-157.07..-20.84 [-1765.1743..-103.5407] | it/evals=60/367 eff=89.5522% N=300
Mono-modal Volume: ~exp(-4.26) * Expected Volume: exp(-0.22) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|***************************************** ** * | +1.0e-10
Z=-159.1(0.00%) | Like=-153.82..-20.84 [-1765.1743..-103.5407] | it/evals=67/374 eff=90.5405% N=300
Z=-149.9(0.00%) | Like=-144.42..-20.84 [-1765.1743..-103.5407] | it/evals=90/403 eff=87.3786% N=300
Z=-141.5(0.00%) | Like=-136.87..-20.84 [-1765.1743..-103.5407] | it/evals=117/437 eff=85.4015% N=300
Z=-140.9(0.00%) | Like=-135.96..-20.84 [-1765.1743..-103.5407] | it/evals=120/441 eff=85.1064% N=300
Mono-modal Volume: ~exp(-4.33) * Expected Volume: exp(-0.45) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|*********************************** ***** ** * | +1.0e-10
Z=-136.8(0.00%) | Like=-131.65..-20.84 [-1765.1743..-103.5407] | it/evals=134/460 eff=83.7500% N=300
Z=-129.7(0.00%) | Like=-124.56..-20.84 [-1765.1743..-103.5407] | it/evals=150/480 eff=83.3333% N=300
Z=-120.8(0.00%) | Like=-115.09..-20.84 [-1765.1743..-103.5407] | it/evals=175/513 eff=82.1596% N=300
Z=-118.7(0.00%) | Like=-113.45..-20.84 [-1765.1743..-103.5407] | it/evals=180/518 eff=82.5688% N=300
Mono-modal Volume: ~exp(-4.85) * Expected Volume: exp(-0.67) Quality: ok
index : +1.0| ********************************************| +5.0
amplitude: +1.0e-12| ********************************** *** ** * | +1.0e-10
Z=-110.7(0.00%) | Like=-105.24..-20.84 [-1765.1743..-103.5407] | it/evals=201/543 eff=82.7160% N=300
Z=-107.9(0.00%) | Like=-101.84..-20.84 [-102.9686..-64.4702] | it/evals=210/555 eff=82.3529% N=300
Z=-100.3(0.00%) | Like=-94.72..-20.84 [-102.9686..-64.4702] | it/evals=234/589 eff=80.9689% N=300
Z=-98.2(0.00%) | Like=-92.91..-20.84 [-102.9686..-64.4702] | it/evals=240/595 eff=81.3559% N=300
Z=-91.6(0.00%) | Like=-86.07..-20.54 [-102.9686..-64.4702] | it/evals=267/627 eff=81.6514% N=300
Mono-modal Volume: ~exp(-4.85) Expected Volume: exp(-0.89) Quality: ok
index : +1.0| *******************************************| +5.0
amplitude: +1.0e-12| ********************************* ** ** ** | +1.0e-10
Z=-90.7(0.00%) | Like=-85.50..-20.54 [-102.9686..-64.4702] | it/evals=270/631 eff=81.5710% N=300
Z=-85.8(0.00%) | Like=-80.54..-20.54 [-102.9686..-64.4702] | it/evals=290/665 eff=79.4521% N=300
Z=-83.7(0.00%) | Like=-78.61..-20.54 [-102.9686..-64.4702] | it/evals=300/676 eff=79.7872% N=300
Z=-79.6(0.00%) | Like=-75.24..-20.54 [-102.9686..-64.4702] | it/evals=324/710 eff=79.0244% N=300
Z=-79.1(0.00%) | Like=-74.83..-20.54 [-102.9686..-64.4702] | it/evals=330/721 eff=78.3848% N=300
Mono-modal Volume: ~exp(-5.20) * Expected Volume: exp(-1.12) Quality: ok
index : +1.0| ***************************************** | +5.0
amplitude: +1.0e-12| ******************************** ** * | +1.0e-10
Z=-78.7(0.00%) | Like=-74.13..-20.54 [-102.9686..-64.4702] | it/evals=335/727 eff=78.4543% N=300
Z=-75.1(0.00%) | Like=-70.29..-20.54 [-102.9686..-64.4702] | it/evals=360/759 eff=78.4314% N=300
Z=-72.0(0.00%) | Like=-67.24..-20.54 [-102.9686..-64.4702] | it/evals=383/793 eff=77.6876% N=300
Z=-71.3(0.00%) | Like=-66.18..-20.54 [-102.9686..-64.4702] | it/evals=390/802 eff=77.6892% N=300
Mono-modal Volume: ~exp(-5.61) * Expected Volume: exp(-1.34) Quality: ok
index : +1.0| ********************************* * | +5.0
amplitude: +1.0e-12| ******************************* * +7.7e-11 | +1.0e-10
Z=-69.6(0.00%) | Like=-64.53..-20.54 [-102.9686..-64.4702] | it/evals=402/818 eff=77.6062% N=300
Z=-67.7(0.00%) | Like=-62.90..-20.54 [-64.4142..-46.7465] | it/evals=420/839 eff=77.9221% N=300
Z=-65.4(0.00%) | Like=-60.26..-20.54 [-64.4142..-46.7465] | it/evals=441/874 eff=76.8293% N=300
Z=-64.3(0.00%) | Like=-59.36..-20.54 [-64.4142..-46.7465] | it/evals=450/887 eff=76.6610% N=300
Mono-modal Volume: ~exp(-5.90) * Expected Volume: exp(-1.56) Quality: ok
index : +1.0| ******************************** | +5.0
amplitude: +1.0e-12| ******************************* * +7.5e-11 | +1.0e-10
Z=-62.7(0.00%) | Like=-57.86..-20.54 [-64.4142..-46.7465] | it/evals=469/914 eff=76.3844% N=300
Z=-61.8(0.00%) | Like=-57.03..-20.50 [-64.4142..-46.7465] | it/evals=480/934 eff=75.7098% N=300
Z=-59.4(0.00%) | Like=-54.44..-20.50 [-64.4142..-46.7465] | it/evals=504/968 eff=75.4491% N=300
Z=-58.9(0.00%) | Like=-53.74..-20.50 [-64.4142..-46.7465] | it/evals=510/975 eff=75.5556% N=300
Z=-56.5(0.00%) | Like=-51.32..-20.50 [-64.4142..-46.7465] | it/evals=534/1008 eff=75.4237% N=300
Mono-modal Volume: ~exp(-5.90) Expected Volume: exp(-1.79) Quality: ok
index : +1.0| *************************** +3.9 | +5.0
amplitude: +1.0e-12| ***************************** +7.0e-11 | +1.0e-10
Z=-56.0(0.00%) | Like=-50.93..-20.50 [-64.4142..-46.7465] | it/evals=540/1016 eff=75.4190% N=300
Z=-54.4(0.00%) | Like=-49.54..-20.50 [-64.4142..-46.7465] | it/evals=562/1048 eff=75.1337% N=300
Z=-53.9(0.00%) | Like=-48.81..-20.50 [-64.4142..-46.7465] | it/evals=570/1058 eff=75.1979% N=300
Z=-52.3(0.00%) | Like=-47.17..-20.50 [-64.4142..-46.7465] | it/evals=590/1090 eff=74.6835% N=300
Z=-51.6(0.00%) | Like=-46.60..-20.50 [-46.7204..-35.5325] | it/evals=600/1102 eff=74.8130% N=300
Mono-modal Volume: ~exp(-5.95) * Expected Volume: exp(-2.01) Quality: ok
index : +1.0| *********************** +3.7 | +5.0
amplitude: +1.0e-12| **************************** +6.7e-11 | +1.0e-10
Z=-51.4(0.00%) | Like=-46.47..-20.50 [-46.7204..-35.5325] | it/evals=603/1106 eff=74.8139% N=300
Z=-50.0(0.00%) | Like=-44.82..-20.50 [-46.7204..-35.5325] | it/evals=624/1139 eff=74.3743% N=300
Z=-49.6(0.00%) | Like=-44.16..-20.50 [-46.7204..-35.5325] | it/evals=630/1147 eff=74.3802% N=300
Z=-47.9(0.00%) | Like=-42.82..-20.50 [-46.7204..-35.5325] | it/evals=656/1181 eff=74.4608% N=300
Z=-47.7(0.00%) | Like=-42.57..-20.50 [-46.7204..-35.5325] | it/evals=660/1186 eff=74.4921% N=300
Mono-modal Volume: ~exp(-6.51) * 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=-47.1(0.00%) | Like=-42.01..-20.50 [-46.7204..-35.5325] | it/evals=670/1203 eff=74.1971% N=300
Z=-45.9(0.00%) | Like=-40.70..-20.50 [-46.7204..-35.5325] | it/evals=690/1231 eff=74.1139% N=300
Z=-44.5(0.00%) | Like=-39.27..-20.50 [-46.7204..-35.5325] | it/evals=717/1264 eff=74.3776% N=300
Z=-44.3(0.00%) | Like=-39.18..-20.50 [-46.7204..-35.5325] | it/evals=720/1270 eff=74.2268% N=300
Mono-modal Volume: ~exp(-6.74) * 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=-43.5(0.00%) | Like=-38.31..-20.50 [-46.7204..-35.5325] | it/evals=737/1295 eff=74.0704% N=300
Z=-42.8(0.00%) | Like=-37.71..-20.50 [-46.7204..-35.5325] | it/evals=750/1312 eff=74.1107% N=300
Z=-41.6(0.00%) | Like=-36.45..-20.50 [-46.7204..-35.5325] | it/evals=776/1344 eff=74.3295% N=300
Z=-41.4(0.00%) | Like=-36.30..-20.50 [-46.7204..-35.5325] | it/evals=780/1349 eff=74.3565% N=300
Mono-modal Volume: ~exp(-6.83) * Expected Volume: exp(-2.68) Quality: ok
index : +1.0| +2.0 ****************** +3.3 | +5.0
amplitude: +1.0e-12| ********************** +5.9e-11 | +1.0e-10
Z=-40.5(0.00%) | Like=-35.50..-20.50 [-35.5117..-28.3039] | it/evals=804/1381 eff=74.3756% N=300
Z=-40.3(0.00%) | Like=-35.10..-20.50 [-35.5117..-28.3039] | it/evals=810/1387 eff=74.5170% N=300
Z=-39.5(0.00%) | Like=-34.22..-20.50 [-35.5117..-28.3039] | it/evals=832/1420 eff=74.2857% N=300
Z=-39.2(0.00%) | Like=-34.03..-20.50 [-35.5117..-28.3039] | it/evals=840/1429 eff=74.4021% N=300
Z=-38.4(0.00%) | Like=-33.40..-20.50 [-35.5117..-28.3039] | it/evals=866/1461 eff=74.5909% N=300
Z=-38.3(0.00%) | Like=-33.15..-20.50 [-35.5117..-28.3039] | it/evals=870/1471 eff=74.2955% N=300
Mono-modal Volume: ~exp(-6.95) * Expected Volume: exp(-2.90) Quality: ok
index : +1.0| +2.0 **************** +3.3 | +5.0
amplitude: +1.0e-12| ******************* +5.6e-11 | +1.0e-10
Z=-38.3(0.00%) | Like=-33.14..-20.50 [-35.5117..-28.3039] | it/evals=871/1472 eff=74.3174% N=300
Z=-37.5(0.00%) | Like=-32.36..-20.50 [-35.5117..-28.3039] | it/evals=896/1504 eff=74.4186% N=300
Z=-37.4(0.00%) | Like=-32.23..-20.50 [-35.5117..-28.3039] | it/evals=900/1514 eff=74.1351% N=300
Z=-36.6(0.00%) | Like=-31.35..-20.50 [-35.5117..-28.3039] | it/evals=924/1547 eff=74.0978% N=300
Z=-36.5(0.00%) | Like=-31.06..-20.50 [-35.5117..-28.3039] | it/evals=930/1554 eff=74.1627% N=300
Mono-modal Volume: ~exp(-6.96) * 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=-36.2(0.00%) | Like=-30.74..-20.50 [-35.5117..-28.3039] | it/evals=938/1568 eff=73.9748% N=300
Z=-35.4(0.01%) | Like=-29.74..-20.50 [-35.5117..-28.3039] | it/evals=960/1598 eff=73.9599% N=300
Z=-34.6(0.02%) | Like=-29.31..-20.50 [-35.5117..-28.3039] | it/evals=983/1630 eff=73.9098% N=300
Z=-34.5(0.02%) | Like=-29.13..-20.50 [-35.5117..-28.3039] | it/evals=990/1639 eff=73.9358% N=300
Mono-modal Volume: ~exp(-7.12) * 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=-34.0(0.03%) | Like=-28.53..-20.50 [-35.5117..-28.3039] | it/evals=1005/1659 eff=73.9514% N=300
Z=-33.6(0.05%) | Like=-28.29..-20.50 [-28.2955..-27.1187] | it/evals=1020/1678 eff=74.0203% N=300
Z=-33.0(0.09%) | Like=-27.60..-20.50 [-28.2955..-27.1187] | it/evals=1047/1712 eff=74.1501% N=300
Z=-32.9(0.10%) | Like=-27.55..-20.50 [-28.2955..-27.1187] | it/evals=1050/1715 eff=74.2049% N=300
Mono-modal Volume: ~exp(-7.42) * Expected Volume: exp(-3.57) Quality: ok
index : +1.0| +2.2 *********** +3.1 | +5.0
amplitude: +1.0e-12| ************** +4.9e-11 | +1.0e-10
Z=-32.5(0.16%) | Like=-26.97..-20.50 [-27.1142..-26.9650] | it/evals=1072/1743 eff=74.2897% N=300
Z=-32.3(0.19%) | Like=-26.75..-20.50 [-26.7450..-26.7259] | it/evals=1080/1753 eff=74.3290% N=300
Z=-31.7(0.33%) | Like=-26.31..-20.49 [-26.3102..-26.3068]*| it/evals=1107/1786 eff=74.4953% N=300
Z=-31.7(0.34%) | Like=-26.27..-20.49 [-26.2746..-26.2483] | it/evals=1110/1790 eff=74.4966% N=300
Z=-31.3(0.47%) | Like=-25.90..-20.49 [-25.9775..-25.9040] | it/evals=1128/1825 eff=73.9672% N=300
Mono-modal Volume: ~exp(-7.54) * Expected Volume: exp(-3.80) Quality: ok
index : +1.0| +2.2 *********** +3.0 | +5.0
amplitude: +1.0e-12| ************* +4.7e-11 | +1.0e-10
Z=-31.2(0.56%) | Like=-25.76..-20.49 [-25.7728..-25.7562] | it/evals=1139/1842 eff=73.8651% N=300
Z=-31.1(0.57%) | Like=-25.71..-20.49 [-25.7069..-25.6911] | it/evals=1140/1843 eff=73.8820% N=300
Z=-30.7(0.86%) | Like=-25.27..-20.49 [-25.2964..-25.2719] | it/evals=1165/1876 eff=73.9213% N=300
Z=-30.7(0.92%) | Like=-25.20..-20.49 [-25.2138..-25.1953] | it/evals=1170/1881 eff=74.0038% N=300
Z=-30.4(1.23%) | Like=-24.95..-20.49 [-24.9538..-24.9516]*| it/evals=1191/1914 eff=73.7918% N=300
Z=-30.2(1.41%) | Like=-24.82..-20.49 [-24.8247..-24.8202]*| it/evals=1200/1927 eff=73.7554% N=300
Mono-modal Volume: ~exp(-7.91) * Expected Volume: exp(-4.02) Quality: ok
index : +1.0| +2.2 ********** +2.9 | +5.0
amplitude: +1.0e-12| +2.4e-11 *********** +4.6e-11 | +1.0e-10
Z=-30.1(1.52%) | Like=-24.65..-20.49 [-24.6542..-24.6479]*| it/evals=1206/1937 eff=73.6714% N=300
Z=-29.8(2.18%) | Like=-24.27..-20.49 [-24.2735..-24.2462] | it/evals=1230/1964 eff=73.9183% N=300
Z=-29.5(3.08%) | Like=-23.97..-20.49 [-23.9712..-23.9558] | it/evals=1254/1996 eff=73.9387% N=300
Z=-29.4(3.31%) | Like=-23.92..-20.49 [-23.9217..-23.8986] | it/evals=1260/2002 eff=74.0306% N=300
Mono-modal Volume: ~exp(-8.66) * Expected Volume: exp(-4.24) 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(3.81%) | Like=-23.74..-20.49 [-23.7986..-23.7430] | it/evals=1273/2025 eff=73.7971% N=300
Z=-29.1(4.55%) | Like=-23.60..-20.47 [-23.5988..-23.5894]*| it/evals=1290/2044 eff=73.9679% N=300
Z=-28.8(5.80%) | Like=-23.35..-20.47 [-23.3453..-23.3392]*| it/evals=1316/2077 eff=74.0574% N=300
Z=-28.8(6.06%) | Like=-23.31..-20.47 [-23.3140..-23.3088]*| it/evals=1320/2082 eff=74.0741% N=300
Mono-modal Volume: ~exp(-8.66) 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=-28.5(7.67%) | Like=-23.06..-20.47 [-23.0592..-23.0576]*| it/evals=1344/2111 eff=74.2131% N=300
Z=-28.5(8.11%) | Like=-23.02..-20.47 [-23.0174..-23.0109]*| it/evals=1350/2117 eff=74.2983% N=300
Z=-28.3(9.84%) | Like=-22.81..-20.47 [-22.8149..-22.8073]*| it/evals=1372/2150 eff=74.1622% N=300
Z=-28.2(10.40%) | Like=-22.78..-20.47 [-22.7800..-22.7754]*| it/evals=1380/2159 eff=74.2335% N=300
Z=-28.1(11.75%) | Like=-22.67..-20.47 [-22.6666..-22.6619]*| it/evals=1397/2192 eff=73.8372% N=300
Mono-modal Volume: ~exp(-8.66) 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=-28.0(12.88%) | Like=-22.59..-20.47 [-22.6098..-22.5946] | it/evals=1410/2206 eff=73.9769% N=300
Z=-27.9(15.03%) | Like=-22.46..-20.47 [-22.4558..-22.4477]*| it/evals=1435/2238 eff=74.0454% N=300
Z=-27.8(15.41%) | Like=-22.41..-20.47 [-22.4108..-22.4080]*| it/evals=1440/2250 eff=73.8462% N=300
Z=-27.7(17.73%) | Like=-22.28..-20.47 [-22.2819..-22.2806]*| it/evals=1462/2286 eff=73.6153% N=300
Z=-27.7(18.64%) | Like=-22.26..-20.47 [-22.2618..-22.2614]*| it/evals=1470/2300 eff=73.5000% N=300
Mono-modal Volume: ~exp(-9.18) * 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=-27.6(19.02%) | Like=-22.25..-20.47 [-22.2546..-22.2429] | it/evals=1474/2305 eff=73.5162% N=300
Z=-27.5(21.56%) | Like=-22.14..-20.47 [-22.1407..-22.1203] | it/evals=1500/2334 eff=73.7463% N=300
Z=-27.4(24.29%) | Like=-21.99..-20.47 [-22.0064..-21.9894] | it/evals=1526/2366 eff=73.8625% N=300
Z=-27.4(24.72%) | Like=-21.98..-20.47 [-21.9775..-21.9719]*| it/evals=1530/2373 eff=73.8061% N=300
Mono-modal Volume: ~exp(-9.48) * 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(25.96%) | Like=-21.93..-20.47 [-21.9312..-21.9248]*| it/evals=1541/2385 eff=73.9089% N=300
Z=-27.2(28.05%) | Like=-21.85..-20.47 [-21.8505..-21.8410]*| it/evals=1560/2408 eff=74.0038% N=300
Z=-27.1(30.93%) | Like=-21.74..-20.47 [-21.7434..-21.7348]*| it/evals=1586/2440 eff=74.1121% N=300
Z=-27.1(31.40%) | Like=-21.71..-20.47 [-21.7092..-21.6968] | it/evals=1590/2446 eff=74.0913% N=300
Mono-modal Volume: ~exp(-9.48) Expected Volume: exp(-5.36) 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=-27.1(33.60%) | Like=-21.63..-20.47 [-21.6302..-21.6273]*| it/evals=1609/2475 eff=73.9770% N=300
Z=-27.0(34.82%) | Like=-21.59..-20.47 [-21.5949..-21.5947]*| it/evals=1620/2488 eff=74.0402% N=300
Z=-26.9(37.46%) | Like=-21.54..-20.47 [-21.5423..-21.5359]*| it/evals=1643/2521 eff=73.9757% N=300
Z=-26.9(38.26%) | Like=-21.52..-20.47 [-21.5201..-21.5192]*| it/evals=1650/2531 eff=73.9579% N=300
Mono-modal Volume: ~exp(-9.49) * Expected Volume: exp(-5.58) 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=-26.8(41.13%) | Like=-21.42..-20.47 [-21.4162..-21.4113]*| it/evals=1675/2562 eff=74.0495% N=300
Z=-26.8(41.75%) | Like=-21.39..-20.47 [-21.3939..-21.3922]*| it/evals=1680/2567 eff=74.1067% N=300
Z=-26.8(44.38%) | Like=-21.31..-20.47 [-21.3115..-21.3072]*| it/evals=1703/2598 eff=74.1079% N=300
Z=-26.7(45.19%) | Like=-21.29..-20.47 [-21.2871..-21.2859]*| it/evals=1710/2610 eff=74.0260% N=300
Z=-26.7(47.76%) | Like=-21.24..-20.47 [-21.2373..-21.2366]*| it/evals=1733/2643 eff=73.9650% N=300
Z=-26.7(48.56%) | Like=-21.21..-20.47 [-21.2144..-21.2106]*| it/evals=1740/2655 eff=73.8854% N=300
Mono-modal Volume: ~exp(-9.56) * 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=-26.7(48.80%) | Like=-21.21..-20.46 [-21.2078..-21.2055]*| it/evals=1742/2657 eff=73.9075% N=300
Z=-26.6(51.24%) | Like=-21.16..-20.46 [-21.1592..-21.1528]*| it/evals=1765/2690 eff=73.8494% N=300
Z=-26.6(51.73%) | Like=-21.15..-20.46 [-21.1497..-21.1497]*| it/evals=1770/2696 eff=73.8731% N=300
Z=-26.6(54.21%) | Like=-21.07..-20.46 [-21.0725..-21.0716]*| it/evals=1792/2731 eff=73.7145% N=300
Z=-26.5(55.04%) | Like=-21.06..-20.46 [-21.0612..-21.0551]*| it/evals=1800/2751 eff=73.4394% N=300
Mono-modal Volume: ~exp(-9.71) * 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(55.99%) | Like=-21.04..-20.46 [-21.0402..-21.0389]*| it/evals=1809/2764 eff=73.4172% N=300
Z=-26.5(58.19%) | Like=-20.99..-20.46 [-20.9896..-20.9885]*| it/evals=1830/2788 eff=73.5531% N=300
Z=-26.4(60.71%) | Like=-20.96..-20.46 [-20.9610..-20.9461] | it/evals=1854/2822 eff=73.5131% N=300
Z=-26.4(61.33%) | Like=-20.94..-20.46 [-20.9359..-20.9358]*| it/evals=1860/2831 eff=73.4887% N=300
Mono-modal Volume: ~exp(-10.02) * Expected Volume: exp(-6.25) 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=-26.4(62.94%) | Like=-20.91..-20.46 [-20.9067..-20.9059]*| it/evals=1876/2855 eff=73.4247% N=300
Z=-26.4(64.30%) | Like=-20.89..-20.46 [-20.8949..-20.8942]*| it/evals=1890/2874 eff=73.4266% N=300
Z=-26.4(66.34%) | Like=-20.87..-20.46 [-20.8670..-20.8654]*| it/evals=1912/2908 eff=73.3129% N=300
Z=-26.3(67.02%) | Like=-20.86..-20.46 [-20.8571..-20.8563]*| it/evals=1920/2922 eff=73.2265% N=300
Z=-26.3(68.86%) | Like=-20.83..-20.46 [-20.8342..-20.8323]*| it/evals=1941/2955 eff=73.1073% N=300
Mono-modal Volume: ~exp(-10.57) * 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(69.04%) | Like=-20.83..-20.46 [-20.8287..-20.8268]*| it/evals=1943/2957 eff=73.1276% N=300
Z=-26.3(69.62%) | Like=-20.82..-20.46 [-20.8217..-20.8191]*| it/evals=1950/2965 eff=73.1707% N=300
[ultranest] Explored until L=-2e+01
[ultranest] Likelihood function evaluations: 2969
[ultranest] logZ = -25.95 +- 0.09901
[ultranest] Effective samples strategy satisfied (ESS = 1017.3, 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=0.28, need <0.5)
[ultranest] logZ error budget: single: 0.12 bs:0.10 tail:0.26 total:0.28 required:<0.50
[ultranest] done iterating.
logZ = -25.946 +- 0.310
single instance: logZ = -25.946 +- 0.122
bootstrapped : logZ = -25.953 +- 0.165
tail : logZ = +- 0.262
insert order U test : converged: True correlation: inf iterations
index : 2.13 │ ▁▁▁▁▁▁▂▂▂▂▄▄▆▆▇▇▅▆▇▇▆▄▃▄▂▂▂▂▁▁▁▁▁▁ ▁ │3.09 2.57 +- 0.13
amplitude : 0.0000000000209│ ▁▁▁▁▁▁▁▂▂▃▃▄▅▆▆▆▇▆▇▅▄▄▄▃▂▂▂▂▁▁▁▁▁▁▁ ▁ │0.0000000000498 0.0000000000341 +- 0.0000000000039
[ultranest] Sampling 300 live points from prior ...
Mono-modal Volume: ~exp(-4.33) * Expected Volume: exp(0.00) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|****************** *********************** ****| +1.0e-10
Z=-inf(0.00%) | Like=-949.71..-23.06 [-949.7090..-132.7236] | it/evals=0/301 eff=0.0000% N=300
Z=-217.9(0.00%) | Like=-213.00..-19.72 [-949.7090..-132.7236] | it/evals=30/332 eff=93.7500% N=300
Z=-202.2(0.00%) | Like=-197.51..-19.72 [-949.7090..-132.7236] | it/evals=60/364 eff=93.7500% N=300
Mono-modal Volume: ~exp(-4.40) * Expected Volume: exp(-0.22) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|******************************************* ****| +1.0e-10
Z=-198.7(0.00%) | Like=-193.76..-19.72 [-949.7090..-132.7236] | it/evals=67/372 eff=93.0556% N=300
Z=-186.3(0.00%) | Like=-181.01..-19.72 [-949.7090..-132.7236] | it/evals=90/399 eff=90.9091% N=300
Z=-175.7(0.00%) | Like=-170.87..-19.72 [-949.7090..-132.7236] | it/evals=120/431 eff=91.6031% N=300
Mono-modal Volume: ~exp(-4.61) * Expected Volume: exp(-0.45) Quality: ok
index : +1.0| ****************** ****************************| +5.0
amplitude: +1.0e-12|************************************************| +1.0e-10
Z=-166.8(0.00%) | Like=-160.91..-19.72 [-949.7090..-132.7236] | it/evals=134/448 eff=90.5405% N=300
Z=-157.9(0.00%) | Like=-153.15..-19.72 [-949.7090..-132.7236] | it/evals=150/467 eff=89.8204% N=300
Z=-147.0(0.00%) | Like=-142.09..-19.72 [-949.7090..-132.7236] | it/evals=176/499 eff=88.4422% N=300
Z=-146.0(0.00%) | Like=-140.83..-19.72 [-949.7090..-132.7236] | it/evals=180/505 eff=87.8049% N=300
Mono-modal Volume: ~exp(-4.61) Expected Volume: exp(-0.67) Quality: ok
index : +1.0| **********************************************| +5.0
amplitude: +1.0e-12| ***********************************************| +1.0e-10
Z=-139.1(0.00%) | Like=-134.04..-19.72 [-949.7090..-132.7236] | it/evals=203/534 eff=86.7521% N=300
Z=-136.8(0.00%) | Like=-130.70..-19.72 [-132.5953..-64.0299] | it/evals=210/542 eff=86.7769% N=300
Z=-124.6(0.00%) | Like=-119.06..-19.65 [-132.5953..-64.0299] | it/evals=236/574 eff=86.1314% N=300
Z=-122.0(0.00%) | Like=-116.24..-19.65 [-132.5953..-64.0299] | it/evals=240/579 eff=86.0215% N=300
Z=-113.9(0.00%) | Like=-108.25..-19.65 [-132.5953..-64.0299] | it/evals=265/611 eff=85.2090% N=300
Mono-modal Volume: ~exp(-4.99) * Expected Volume: exp(-0.89) Quality: ok
index : +1.0| ********************************************| +5.0
amplitude: +1.0e-12| ******************************************** | +1.0e-10
Z=-112.9(0.00%) | Like=-107.56..-19.37 [-132.5953..-64.0299] | it/evals=268/614 eff=85.3503% N=300
Z=-112.4(0.00%) | Like=-107.26..-19.37 [-132.5953..-64.0299] | it/evals=270/616 eff=85.4430% N=300
Z=-100.6(0.00%) | Like=-94.55..-19.37 [-132.5953..-64.0299] | it/evals=298/648 eff=85.6322% N=300
Z=-99.6(0.00%) | Like=-93.68..-19.37 [-132.5953..-64.0299] | it/evals=300/653 eff=84.9858% N=300
Z=-89.4(0.00%) | Like=-83.38..-19.37 [-132.5953..-64.0299] | it/evals=327/686 eff=84.7150% N=300
Z=-88.4(0.00%) | Like=-82.90..-19.37 [-132.5953..-64.0299] | it/evals=330/689 eff=84.8329% N=300
Mono-modal Volume: ~exp(-5.49) * 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=-82.14..-19.37 [-132.5953..-64.0299] | it/evals=335/696 eff=84.5960% N=300
Z=-80.3(0.00%) | Like=-74.76..-19.37 [-132.5953..-64.0299] | it/evals=360/724 eff=84.9057% N=300
Z=-72.9(0.00%) | Like=-67.02..-19.37 [-132.5953..-64.0299] | it/evals=388/756 eff=85.0877% N=300
Z=-72.4(0.00%) | Like=-66.63..-19.37 [-132.5953..-64.0299] | it/evals=390/758 eff=85.1528% N=300
Mono-modal Volume: ~exp(-5.49) Expected Volume: exp(-1.34) Quality: ok
index : +1.0| ****************************************| +5.0
amplitude: +1.0e-12| *****************************************| +1.0e-10
Z=-68.6(0.00%) | Like=-63.07..-19.37 [-64.0065..-41.5702] | it/evals=411/788 eff=84.2213% N=300
Z=-67.0(0.00%) | Like=-61.73..-19.37 [-64.0065..-41.5702] | it/evals=420/802 eff=83.6653% N=300
Z=-62.8(0.00%) | Like=-57.59..-19.30 [-64.0065..-41.5702] | it/evals=445/836 eff=83.0224% N=300
Z=-62.1(0.00%) | Like=-56.70..-19.30 [-64.0065..-41.5702] | it/evals=450/843 eff=82.8729% N=300
Mono-modal Volume: ~exp(-5.87) * Expected Volume: exp(-1.56) Quality: ok
index : +1.0| ***************************************| +5.0
amplitude: +1.0e-12| ****************************************| +1.0e-10
Z=-59.2(0.00%) | Like=-53.91..-19.30 [-64.0065..-41.5702] | it/evals=469/866 eff=82.8622% N=300
Z=-58.0(0.00%) | Like=-53.23..-19.30 [-64.0065..-41.5702] | it/evals=480/878 eff=83.0450% N=300
Z=-55.7(0.00%) | Like=-50.66..-19.30 [-64.0065..-41.5702] | it/evals=505/912 eff=82.5163% N=300
Z=-55.3(0.00%) | Like=-50.28..-19.23 [-64.0065..-41.5702] | it/evals=510/917 eff=82.6580% N=300
Z=-53.4(0.00%) | Like=-48.00..-19.23 [-64.0065..-41.5702] | it/evals=535/949 eff=82.4345% N=300
Mono-modal Volume: ~exp(-5.87) Expected Volume: exp(-1.79) Quality: ok
index : +1.0| ********************************** | +5.0
amplitude: +1.0e-12| ***************************************| +1.0e-10
Z=-52.7(0.00%) | Like=-47.22..-19.23 [-64.0065..-41.5702] | it/evals=540/954 eff=82.5688% N=300
Z=-50.7(0.00%) | Like=-45.76..-19.23 [-64.0065..-41.5702] | it/evals=561/987 eff=81.6594% N=300
Z=-50.2(0.00%) | Like=-45.42..-19.23 [-64.0065..-41.5702] | it/evals=570/998 eff=81.6619% N=300
Z=-48.4(0.00%) | Like=-43.17..-19.23 [-64.0065..-41.5702] | it/evals=595/1032 eff=81.2842% N=300
Z=-48.0(0.00%) | Like=-42.93..-19.23 [-64.0065..-41.5702] | it/evals=600/1038 eff=81.3008% N=300
Mono-modal Volume: ~exp(-5.87) Expected Volume: exp(-2.01) Quality: ok
index : +1.0| +2.0 ****************************** | +5.0
amplitude: +1.0e-12| **************************************| +1.0e-10
Z=-46.5(0.00%) | Like=-41.31..-19.23 [-41.4910..-31.2719] | it/evals=620/1067 eff=80.8344% N=300
Z=-45.7(0.00%) | Like=-40.47..-19.23 [-41.4910..-31.2719] | it/evals=630/1079 eff=80.8729% N=300
Z=-44.2(0.00%) | Like=-39.24..-19.23 [-41.4910..-31.2719] | it/evals=653/1112 eff=80.4187% N=300
Z=-43.8(0.00%) | Like=-38.66..-19.23 [-41.4910..-31.2719] | it/evals=660/1122 eff=80.2920% N=300
Mono-modal Volume: ~exp(-6.22) * Expected Volume: exp(-2.23) Quality: ok
index : +1.0| +2.0 ************************** +4.1 | +5.0
amplitude: +1.0e-12| +2.6e-11 *********************************** | +1.0e-10
Z=-43.2(0.00%) | Like=-38.20..-19.23 [-41.4910..-31.2719] | it/evals=670/1138 eff=79.9523% N=300
Z=-42.3(0.00%) | Like=-37.57..-19.23 [-41.4910..-31.2719] | it/evals=690/1164 eff=79.8611% N=300
Z=-41.3(0.00%) | Like=-36.46..-19.23 [-41.4910..-31.2719] | it/evals=714/1197 eff=79.5987% N=300
Z=-41.1(0.00%) | Like=-36.08..-19.23 [-41.4910..-31.2719] | it/evals=720/1207 eff=79.3826% N=300
Mono-modal Volume: ~exp(-6.58) * Expected Volume: exp(-2.46) Quality: ok
index : +1.0| +2.1 ************************ +4.0 | +5.0
amplitude: +1.0e-12| +2.9e-11 ********************************* | +1.0e-10
Z=-40.0(0.00%) | Like=-34.70..-19.23 [-41.4910..-31.2719] | it/evals=737/1230 eff=79.2473% N=300
Z=-39.3(0.00%) | Like=-34.14..-19.23 [-41.4910..-31.2719] | it/evals=750/1244 eff=79.4492% N=300
Z=-38.1(0.00%) | Like=-32.88..-19.23 [-41.4910..-31.2719] | it/evals=775/1278 eff=79.2434% N=300
Z=-37.8(0.00%) | Like=-32.49..-19.23 [-41.4910..-31.2719] | it/evals=780/1286 eff=79.1075% N=300
Z=-37.0(0.00%) | Like=-31.82..-19.23 [-41.4910..-31.2719] | it/evals=798/1320 eff=78.2353% N=300
Mono-modal Volume: ~exp(-6.58) * Expected Volume: exp(-2.68) Quality: ok
index : +1.0| +2.2 ********************* +3.8 | +5.0
amplitude: +1.0e-12| +3.1e-11 ***************************** | +1.0e-10
Z=-36.8(0.00%) | Like=-31.61..-19.23 [-41.4910..-31.2719] | it/evals=804/1327 eff=78.2863% N=300
Z=-36.5(0.00%) | Like=-31.34..-19.23 [-41.4910..-31.2719] | it/evals=810/1334 eff=78.3366% N=300
Z=-35.8(0.00%) | Like=-30.64..-19.23 [-31.2523..-26.1586] | it/evals=830/1368 eff=77.7154% N=300
Z=-35.5(0.00%) | Like=-30.25..-19.23 [-31.2523..-26.1586] | it/evals=840/1385 eff=77.4194% N=300
Z=-34.8(0.00%) | Like=-29.59..-19.23 [-31.2523..-26.1586] | it/evals=859/1419 eff=76.7650% N=300
Z=-34.4(0.00%) | Like=-29.09..-19.23 [-31.2523..-26.1586] | it/evals=870/1434 eff=76.7196% N=300
Mono-modal Volume: ~exp(-6.95) * Expected Volume: exp(-2.90) Quality: ok
index : +1.0| +2.2 ******************* +3.7 | +5.0
amplitude: +1.0e-12| +3.5e-11 ************************* | +1.0e-10
Z=-34.4(0.00%) | Like=-29.00..-19.23 [-31.2523..-26.1586] | it/evals=871/1436 eff=76.6725% N=300
Z=-33.6(0.01%) | Like=-28.28..-19.20 [-31.2523..-26.1586] | it/evals=895/1469 eff=76.5612% N=300
Z=-33.4(0.01%) | Like=-28.11..-19.20 [-31.2523..-26.1586] | it/evals=900/1475 eff=76.5957% N=300
Z=-32.6(0.03%) | Like=-27.34..-19.20 [-31.2523..-26.1586] | it/evals=924/1508 eff=76.4901% N=300
Z=-32.4(0.03%) | Like=-27.14..-19.20 [-31.2523..-26.1586] | it/evals=930/1524 eff=75.9804% N=300
Mono-modal Volume: ~exp(-7.16) * Expected Volume: exp(-3.13) Quality: ok
index : +1.0| +2.3 **************** +3.5 | +5.0
amplitude: +1.0e-12| +3.5e-11 *********************** | +1.0e-10
Z=-32.2(0.04%) | Like=-26.99..-19.20 [-31.2523..-26.1586] | it/evals=938/1535 eff=75.9514% N=300
Z=-31.6(0.07%) | Like=-26.33..-19.20 [-31.2523..-26.1586] | it/evals=960/1560 eff=76.1905% N=300
Z=-31.1(0.12%) | Like=-26.00..-19.19 [-26.1453..-25.6013] | it/evals=983/1592 eff=76.0836% N=300
Z=-30.9(0.14%) | Like=-25.80..-19.19 [-26.1453..-25.6013] | it/evals=990/1600 eff=76.1538% N=300
Mono-modal Volume: ~exp(-7.43) * Expected Volume: exp(-3.35) 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=-30.6(0.18%) | Like=-25.46..-19.19 [-25.4617..-25.4574]*| it/evals=1005/1621 eff=76.0787% N=300
Z=-30.3(0.25%) | Like=-25.07..-19.19 [-25.0741..-25.0566] | it/evals=1020/1639 eff=76.1763% N=300
Z=-29.8(0.41%) | Like=-24.56..-19.19 [-24.5629..-24.5340] | it/evals=1045/1673 eff=76.1107% N=300
Z=-29.7(0.45%) | Like=-24.41..-19.19 [-24.4984..-24.4142] | it/evals=1050/1678 eff=76.1974% N=300
Mono-modal Volume: ~exp(-7.86) * 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=-29.3(0.68%) | Like=-24.05..-19.19 [-24.1173..-24.0484] | it/evals=1072/1710 eff=76.0284% N=300
Z=-29.2(0.79%) | Like=-23.92..-19.19 [-23.9220..-23.9177]*| it/evals=1080/1720 eff=76.0563% N=300
Z=-28.8(1.20%) | Like=-23.51..-19.16 [-23.5137..-23.5075]*| it/evals=1105/1752 eff=76.1019% N=300
Z=-28.7(1.29%) | Like=-23.50..-19.16 [-23.4987..-23.4828] | it/evals=1110/1758 eff=76.1317% N=300
Z=-28.3(1.88%) | Like=-23.23..-19.16 [-23.2474..-23.2301] | it/evals=1137/1790 eff=76.3087% N=300
Mono-modal Volume: ~exp(-7.86) Expected Volume: exp(-3.80) Quality: ok
index : +1.0| +2.4 ************ +3.3 | +5.0
amplitude: +1.0e-12| +4.1e-11 ***************** +7.4e-11 | +1.0e-10
Z=-28.3(1.96%) | Like=-23.21..-19.16 [-23.2204..-23.2071] | it/evals=1140/1793 eff=76.3563% N=300
Z=-28.0(2.58%) | Like=-22.92..-19.16 [-22.9463..-22.9213] | it/evals=1165/1825 eff=76.3934% N=300
Z=-27.9(2.73%) | Like=-22.89..-19.16 [-22.9062..-22.8879] | it/evals=1170/1831 eff=76.4206% N=300
Z=-27.7(3.51%) | Like=-22.54..-19.16 [-22.5682..-22.5436] | it/evals=1195/1866 eff=76.3091% N=300
Z=-27.6(3.69%) | Like=-22.51..-19.16 [-22.5127..-22.5085]*| it/evals=1200/1872 eff=76.3359% N=300
Mono-modal Volume: ~exp(-7.88) * 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=-27.6(3.94%) | Like=-22.47..-19.16 [-22.4719..-22.4536] | it/evals=1206/1881 eff=76.2808% N=300
Z=-27.3(5.00%) | Like=-22.19..-19.16 [-22.1902..-22.1867]*| it/evals=1230/1910 eff=76.3975% N=300
Z=-27.1(6.37%) | Like=-21.93..-19.16 [-21.9299..-21.9171] | it/evals=1256/1943 eff=76.4455% N=300
Z=-27.1(6.57%) | Like=-21.91..-19.16 [-21.9089..-21.8754] | it/evals=1260/1947 eff=76.5027% N=300
Mono-modal Volume: ~exp(-8.11) * 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=-27.0(7.36%) | Like=-21.74..-19.16 [-21.7808..-21.7407] | it/evals=1273/1962 eff=76.5945% N=300
Z=-26.8(8.59%) | Like=-21.58..-19.16 [-21.5783..-21.5774]*| it/evals=1290/1980 eff=76.7857% N=300
Z=-26.6(10.41%) | Like=-21.34..-19.16 [-21.3561..-21.3429] | it/evals=1313/2014 eff=76.6044% N=300
Z=-26.6(11.05%) | Like=-21.27..-19.16 [-21.2769..-21.2663] | it/evals=1320/2024 eff=76.5661% N=300
Mono-modal Volume: ~exp(-8.55) * 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=-26.4(12.86%) | Like=-21.14..-19.16 [-21.1387..-21.1363]*| it/evals=1340/2049 eff=76.6152% N=300
Z=-26.3(13.81%) | Like=-21.10..-19.16 [-21.1023..-21.0895] | it/evals=1350/2064 eff=76.5306% N=300
Z=-26.2(15.85%) | Like=-20.96..-19.16 [-20.9577..-20.9575]*| it/evals=1371/2096 eff=76.3363% N=300
Z=-26.1(16.70%) | Like=-20.90..-19.16 [-20.8962..-20.8879]*| it/evals=1380/2110 eff=76.2431% N=300
Z=-26.0(18.93%) | Like=-20.83..-19.16 [-20.8271..-20.8249]*| it/evals=1401/2142 eff=76.0586% N=300
Mono-modal Volume: ~exp(-8.55) Expected Volume: exp(-4.69) Quality: ok
index : +1.0| +2.5 ******** +3.1 | +5.0
amplitude: +1.0e-12| +4.5e-11 *********** +6.6e-11 | +1.0e-10
Z=-26.0(19.89%) | Like=-20.78..-19.16 [-20.7789..-20.7788]*| it/evals=1410/2153 eff=76.0928% N=300
Z=-25.8(22.55%) | Like=-20.70..-19.16 [-20.6952..-20.6933]*| it/evals=1434/2186 eff=76.0339% N=300
Z=-25.8(23.16%) | Like=-20.68..-19.16 [-20.6823..-20.6583] | it/evals=1440/2196 eff=75.9494% N=300
Z=-25.7(25.52%) | Like=-20.54..-19.16 [-20.5423..-20.5391]*| it/evals=1461/2227 eff=75.8173% N=300
Z=-25.7(26.53%) | Like=-20.52..-19.16 [-20.5209..-20.5174]*| it/evals=1470/2237 eff=75.8906% N=300
Mono-modal Volume: ~exp(-8.81) * Expected Volume: exp(-4.91) 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=-25.7(27.05%) | Like=-20.50..-19.16 [-20.5006..-20.4995]*| it/evals=1474/2242 eff=75.9011% N=300
Z=-25.6(29.70%) | Like=-20.43..-19.16 [-20.4307..-20.4226]*| it/evals=1496/2273 eff=75.8236% N=300
Z=-25.6(30.07%) | Like=-20.42..-19.16 [-20.4164..-20.4154]*| it/evals=1500/2277 eff=75.8725% N=300
Z=-25.5(32.95%) | Like=-20.32..-19.16 [-20.3337..-20.3213] | it/evals=1524/2309 eff=75.8586% N=300
Z=-25.4(33.57%) | Like=-20.30..-19.16 [-20.2996..-20.2966]*| it/evals=1530/2319 eff=75.7801% N=300
Mono-modal Volume: ~exp(-8.81) Expected Volume: exp(-5.14) 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=-25.4(35.59%) | Like=-20.25..-19.16 [-20.2511..-20.2484]*| it/evals=1548/2351 eff=75.4754% N=300
Z=-25.3(37.00%) | Like=-20.19..-19.16 [-20.1929..-20.1921]*| it/evals=1560/2374 eff=75.2170% N=300
Z=-25.3(39.85%) | Like=-20.11..-19.16 [-20.1148..-20.1118]*| it/evals=1583/2407 eff=75.1305% N=300
Z=-25.3(40.60%) | Like=-20.09..-19.16 [-20.0884..-20.0866]*| it/evals=1590/2419 eff=75.0354% N=300
Mono-modal Volume: ~exp(-9.17) * 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=-25.2(42.80%) | Like=-20.04..-19.16 [-20.0414..-20.0357]*| it/evals=1608/2446 eff=74.9301% N=300
Z=-25.2(44.31%) | Like=-20.02..-19.16 [-20.0198..-20.0188]*| it/evals=1620/2461 eff=74.9653% N=300
Z=-25.1(47.24%) | Like=-19.94..-19.16 [-19.9380..-19.9346]*| it/evals=1644/2494 eff=74.9316% N=300
Z=-25.1(47.90%) | Like=-19.93..-19.16 [-19.9259..-19.9232]*| it/evals=1650/2503 eff=74.8979% N=300
Mono-modal Volume: ~exp(-9.53) * Expected Volume: exp(-5.58) Quality: ok
index : +1.0| +2.6 ****** +3.0 | +5.0
amplitude: +1.0e-12| +4.8e-11 ******** +6.2e-11 | +1.0e-10
Z=-25.0(50.64%) | Like=-19.88..-19.16 [-19.8773..-19.8740]*| it/evals=1675/2535 eff=74.9441% N=300
Z=-25.0(51.17%) | Like=-19.87..-19.16 [-19.8662..-19.8649]*| it/evals=1680/2541 eff=74.9665% N=300
Z=-25.0(54.01%) | Like=-19.80..-19.16 [-19.7987..-19.7908]*| it/evals=1706/2574 eff=75.0220% N=300
Z=-25.0(54.41%) | Like=-19.79..-19.16 [-19.7881..-19.7881]*| it/evals=1710/2578 eff=75.0658% N=300
Z=-24.9(56.61%) | Like=-19.73..-19.16 [-19.7317..-19.7306]*| it/evals=1731/2611 eff=74.9026% N=300
Z=-24.9(57.52%) | Like=-19.72..-19.16 [-19.7210..-19.7208]*| it/evals=1740/2625 eff=74.8387% N=300
Mono-modal Volume: ~exp(-10.14) * Expected Volume: exp(-5.81) 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(57.74%) | Like=-19.72..-19.16 [-19.7152..-19.7139]*| it/evals=1742/2627 eff=74.8603% N=300
Z=-24.9(60.40%) | Like=-19.65..-19.16 [-19.6495..-19.6468]*| it/evals=1769/2658 eff=75.0212% N=300
Z=-24.9(60.51%) | Like=-19.65..-19.16 [-19.6468..-19.6457]*| it/evals=1770/2660 eff=75.0000% N=300
Z=-24.8(63.05%) | Like=-19.60..-19.16 [-19.6029..-19.6025]*| it/evals=1796/2693 eff=75.0522% N=300
Z=-24.8(63.48%) | Like=-19.60..-19.16 [-19.6009..-19.5998]*| it/evals=1800/2699 eff=75.0313% N=300
Mono-modal Volume: ~exp(-10.14) 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=-24.8(65.49%) | Like=-19.57..-19.16 [-19.5669..-19.5655]*| it/evals=1821/2728 eff=75.0000% N=300
Z=-24.8(66.28%) | Like=-19.55..-19.16 [-19.5490..-19.5486]*| it/evals=1830/2738 eff=75.0615% N=300
Z=-24.7(68.33%) | Like=-19.51..-19.16 [-19.5147..-19.5141]*| it/evals=1853/2770 eff=75.0202% N=300
Z=-24.7(68.94%) | Like=-19.51..-19.16 [-19.5099..-19.5084]*| it/evals=1860/2779 eff=75.0303% N=300
[ultranest] Explored until L=-2e+01
[ultranest] Likelihood function evaluations: 2795
[ultranest] logZ = -24.34 +- 0.07778
[ultranest] Effective samples strategy satisfied (ESS = 1007.9, need >400)
[ultranest] Posterior uncertainty strategy is satisfied (KL: 0.47+-0.06 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 = -24.355 +- 0.343
single instance: logZ = -24.355 +- 0.118
bootstrapped : logZ = -24.337 +- 0.222
tail : logZ = +- 0.262
insert order U test : converged: True correlation: inf iterations
index : 2.29 │ ▁ ▁▁▁▁▁▁▂▃▄▅▆▅▆▇▇▆▆▅▅▂▄▃▃▃▂▁▂▁▁▁▁▁▁ ▁ │3.45 2.82 +- 0.16
amplitude : 0.0000000000344│ ▁▁▁▁▁▁▁▂▃▃▄▄▅▆▆▇▅▆▅▅▅▄▃▃▃▂▁▂▁▁▁▁▁▁ ▁ │0.0000000000795 0.0000000000550 +- 0.0000000000060
[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=-890.54..-13.33 [-890.5404..-100.1738] | it/evals=0/301 eff=0.0000% N=300
Z=-151.2(0.00%) | Like=-146.95..-13.33 [-890.5404..-100.1738] | it/evals=30/331 eff=96.7742% N=300
Z=-143.5(0.00%) | Like=-138.96..-13.33 [-890.5404..-100.1738] | it/evals=59/362 eff=95.1613% N=300
Z=-143.2(0.00%) | Like=-138.88..-13.33 [-890.5404..-100.1738] | it/evals=60/363 eff=95.2381% N=300
Mono-modal Volume: ~exp(-4.39) * Expected Volume: exp(-0.22) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|**************************** ************ *** *| +1.0e-10
Z=-141.4(0.00%) | Like=-136.76..-13.33 [-890.5404..-100.1738] | it/evals=67/370 eff=95.7143% N=300
Z=-133.9(0.00%) | Like=-129.31..-13.33 [-890.5404..-100.1738] | it/evals=90/399 eff=90.9091% N=300
Z=-126.9(0.00%) | Like=-122.23..-13.33 [-890.5404..-100.1738] | it/evals=116/432 eff=87.8788% N=300
Z=-126.1(0.00%) | Like=-121.33..-13.33 [-890.5404..-100.1738] | it/evals=120/436 eff=88.2353% N=300
Mono-modal Volume: ~exp(-4.39) Expected Volume: exp(-0.45) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|***************************************** *****| +1.0e-10
Z=-120.8(0.00%) | Like=-115.97..-13.33 [-890.5404..-100.1738] | it/evals=144/464 eff=87.8049% N=300
Z=-119.5(0.00%) | Like=-114.86..-13.33 [-890.5404..-100.1738] | it/evals=150/470 eff=88.2353% N=300
Z=-112.8(0.00%) | Like=-106.95..-13.33 [-890.5404..-100.1738] | it/evals=178/502 eff=88.1188% N=300
Z=-111.9(0.00%) | Like=-106.71..-13.33 [-890.5404..-100.1738] | it/evals=180/505 eff=87.8049% N=300
Mono-modal Volume: ~exp(-4.63) * Expected Volume: exp(-0.67) Quality: ok
index : +1.0| ***********************************************| +5.0
amplitude: +1.0e-12|***************************************** *** *| +1.0e-10
Z=-106.2(0.00%) | Like=-100.81..-13.33 [-890.5404..-100.1738] | it/evals=201/536 eff=85.1695% N=300
Z=-104.0(0.00%) | Like=-98.65..-13.33 [-100.1365..-48.9157] | it/evals=210/546 eff=85.3659% N=300
Z=-95.4(0.00%) | Like=-90.82..-13.33 [-100.1365..-48.9157] | it/evals=238/577 eff=85.9206% N=300
Z=-95.1(0.00%) | Like=-90.56..-13.33 [-100.1365..-48.9157] | it/evals=240/581 eff=85.4093% N=300
Z=-86.5(0.00%) | Like=-80.91..-13.33 [-100.1365..-48.9157] | it/evals=267/614 eff=85.0318% N=300
Mono-modal Volume: ~exp(-4.77) * Expected Volume: exp(-0.89) Quality: ok
index : +1.0| *********************************************| +5.0
amplitude: +1.0e-12| **************************************** *** *| +1.0e-10
Z=-86.2(0.00%) | Like=-80.75..-13.33 [-100.1365..-48.9157] | it/evals=268/615 eff=85.0794% N=300
Z=-85.7(0.00%) | Like=-80.47..-13.33 [-100.1365..-48.9157] | it/evals=270/618 eff=84.9057% N=300
Z=-78.9(0.00%) | Like=-73.47..-13.33 [-100.1365..-48.9157] | it/evals=293/651 eff=83.4758% N=300
Z=-77.1(0.00%) | Like=-71.94..-13.33 [-100.1365..-48.9157] | it/evals=300/661 eff=83.1025% N=300
Z=-71.9(0.00%) | Like=-66.45..-13.33 [-100.1365..-48.9157] | it/evals=324/694 eff=82.2335% N=300
Z=-70.8(0.00%) | Like=-65.41..-13.33 [-100.1365..-48.9157] | it/evals=330/700 eff=82.5000% N=300
Mono-modal Volume: ~exp(-5.18) * Expected Volume: exp(-1.12) Quality: ok
index : +1.0| *******************************************| +5.0
amplitude: +1.0e-12| *********************************************| +1.0e-10
Z=-69.5(0.00%) | Like=-64.15..-13.33 [-100.1365..-48.9157] | it/evals=335/707 eff=82.3096% N=300
Z=-64.8(0.00%) | Like=-59.71..-13.33 [-100.1365..-48.9157] | it/evals=360/736 eff=82.5688% N=300
Z=-60.5(0.00%) | Like=-55.13..-13.33 [-100.1365..-48.9157] | it/evals=385/769 eff=82.0896% N=300
Z=-59.7(0.00%) | Like=-54.47..-13.33 [-100.1365..-48.9157] | it/evals=390/774 eff=82.2785% N=300
Mono-modal Volume: ~exp(-5.49) * Expected Volume: exp(-1.34) Quality: ok
index : +1.0| *****************************************| +5.0
amplitude: +1.0e-12| ********************************************| +1.0e-10
Z=-58.1(0.00%) | Like=-52.75..-13.33 [-100.1365..-48.9157] | it/evals=402/791 eff=81.8737% N=300
Z=-55.0(0.00%) | Like=-49.58..-13.33 [-100.1365..-48.9157] | it/evals=420/816 eff=81.3953% N=300
Z=-51.7(0.00%) | Like=-46.56..-13.33 [-48.9152..-30.5364] | it/evals=446/848 eff=81.3869% N=300
Z=-51.0(0.00%) | Like=-45.37..-13.33 [-48.9152..-30.5364] | it/evals=450/854 eff=81.2274% N=300
Mono-modal Volume: ~exp(-5.51) * 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=-43.56..-13.33 [-48.9152..-30.5364] | it/evals=469/877 eff=81.2825% N=300
Z=-47.3(0.00%) | Like=-42.01..-13.33 [-48.9152..-30.5364] | it/evals=480/893 eff=80.9444% N=300
Z=-45.0(0.00%) | Like=-40.18..-13.33 [-48.9152..-30.5364] | it/evals=506/927 eff=80.7018% N=300
Z=-44.7(0.00%) | Like=-39.64..-13.33 [-48.9152..-30.5364] | it/evals=510/933 eff=80.5687% N=300
Z=-42.7(0.00%) | Like=-37.67..-13.33 [-48.9152..-30.5364] | it/evals=533/966 eff=80.0300% 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=-42.4(0.00%) | Like=-37.58..-13.33 [-48.9152..-30.5364] | it/evals=536/970 eff=80.0000% N=300
Z=-42.2(0.00%) | Like=-37.28..-13.33 [-48.9152..-30.5364] | it/evals=540/974 eff=80.1187% N=300
Z=-40.7(0.00%) | Like=-35.96..-13.33 [-48.9152..-30.5364] | it/evals=561/1007 eff=79.3494% N=300
Z=-40.2(0.00%) | Like=-35.30..-13.33 [-48.9152..-30.5364] | it/evals=570/1020 eff=79.1667% N=300
Z=-38.0(0.00%) | Like=-32.54..-13.33 [-48.9152..-30.5364] | it/evals=598/1054 eff=79.3103% N=300
Z=-37.8(0.00%) | Like=-32.41..-13.33 [-48.9152..-30.5364] | it/evals=600/1056 eff=79.3651% N=300
Mono-modal Volume: ~exp(-6.26) * Expected Volume: exp(-2.01) Quality: ok
index : +1.0| **************************** +4.1 | +5.0
amplitude: +1.0e-12| ***************************************| +1.0e-10
Z=-37.5(0.00%) | Like=-32.12..-13.33 [-48.9152..-30.5364] | it/evals=603/1060 eff=79.3421% N=300
Z=-35.9(0.00%) | Like=-30.85..-13.33 [-48.9152..-30.5364] | it/evals=626/1093 eff=78.9407% N=300
Z=-35.6(0.00%) | Like=-30.54..-13.33 [-48.9152..-30.5364] | it/evals=630/1101 eff=78.6517% N=300
Z=-34.3(0.00%) | Like=-29.12..-13.33 [-30.5355..-21.9723] | it/evals=654/1135 eff=78.3234% N=300
Z=-33.9(0.00%) | Like=-28.84..-13.33 [-30.5355..-21.9723] | it/evals=660/1142 eff=78.3848% 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=-32.7(0.00%) | Like=-27.70..-13.33 [-30.5355..-21.9723] | it/evals=683/1171 eff=78.4156% N=300
Z=-32.4(0.00%) | Like=-27.38..-13.33 [-30.5355..-21.9723] | it/evals=690/1178 eff=78.5877% N=300
Z=-31.2(0.00%) | Like=-26.30..-13.33 [-30.5355..-21.9723] | it/evals=716/1211 eff=78.5950% N=300
Z=-31.0(0.00%) | Like=-26.23..-13.33 [-30.5355..-21.9723] | it/evals=720/1216 eff=78.6026% N=300
Mono-modal Volume: ~exp(-6.75) * Expected Volume: exp(-2.46) Quality: ok
index : +1.0| +2.0 ********************** +3.8 | +5.0
amplitude: +1.0e-12| **************************************| +1.0e-10
Z=-30.4(0.00%) | Like=-25.39..-13.33 [-30.5355..-21.9723] | it/evals=737/1239 eff=78.4878% N=300
Z=-29.9(0.00%) | Like=-25.01..-13.33 [-30.5355..-21.9723] | it/evals=750/1254 eff=78.6164% N=300
Z=-29.1(0.00%) | Like=-24.06..-13.33 [-30.5355..-21.9723] | it/evals=773/1286 eff=78.3976% N=300
Z=-28.9(0.00%) | Like=-23.95..-13.33 [-30.5355..-21.9723] | it/evals=780/1295 eff=78.3920% N=300
Z=-28.2(0.00%) | Like=-23.22..-13.33 [-30.5355..-21.9723] | it/evals=802/1326 eff=78.1676% N=300
Mono-modal Volume: ~exp(-6.75) Expected Volume: exp(-2.68) Quality: ok
index : +1.0| +2.1 ******************* +3.6 | +5.0
amplitude: +1.0e-12| +2.6e-11 ********************************** | +1.0e-10
Z=-27.9(0.01%) | Like=-23.14..-13.33 [-30.5355..-21.9723] | it/evals=810/1339 eff=77.9596% N=300
Z=-27.3(0.01%) | Like=-22.49..-13.33 [-30.5355..-21.9723] | it/evals=836/1372 eff=77.9851% N=300
Z=-27.2(0.01%) | Like=-22.29..-13.33 [-30.5355..-21.9723] | it/evals=840/1377 eff=77.9944% N=300
Z=-26.7(0.02%) | Like=-21.81..-13.33 [-21.9708..-19.9690] | it/evals=860/1410 eff=77.4775% N=300
Z=-26.4(0.03%) | Like=-21.69..-13.33 [-21.9708..-19.9690] | it/evals=870/1426 eff=77.2647% N=300
Mono-modal Volume: ~exp(-7.04) * Expected Volume: exp(-2.90) Quality: ok
index : +1.0| +2.1 ****************** +3.5 | +5.0
amplitude: +1.0e-12| +2.8e-11 ****************************** | +1.0e-10
Z=-26.4(0.03%) | Like=-21.62..-13.33 [-21.9708..-19.9690] | it/evals=871/1427 eff=77.2848% N=300
Z=-25.9(0.05%) | Like=-21.00..-13.33 [-21.9708..-19.9690] | it/evals=896/1460 eff=77.2414% N=300
Z=-25.8(0.05%) | Like=-20.94..-13.32 [-21.9708..-19.9690] | it/evals=900/1465 eff=77.2532% N=300
Z=-25.4(0.08%) | Like=-20.65..-13.32 [-21.9708..-19.9690] | it/evals=922/1499 eff=76.8974% N=300
Z=-25.2(0.10%) | Like=-20.50..-13.32 [-21.9708..-19.9690] | it/evals=930/1513 eff=76.6694% N=300
Mono-modal Volume: ~exp(-7.04) Expected Volume: exp(-3.13) Quality: ok
index : +1.0| +2.2 **************** +3.4 | +5.0
amplitude: +1.0e-12| +2.9e-11 **************************** | +1.0e-10
Z=-24.9(0.14%) | Like=-19.97..-13.32 [-19.9657..-19.5837] | it/evals=950/1544 eff=76.3666% N=300
Z=-24.7(0.16%) | Like=-19.83..-13.32 [-19.9657..-19.5837] | it/evals=960/1560 eff=76.1905% N=300
Z=-24.3(0.24%) | Like=-19.28..-13.31 [-19.2770..-19.2711]*| it/evals=983/1597 eff=75.7903% N=300
Z=-24.1(0.27%) | Like=-19.18..-13.31 [-19.2147..-19.1778] | it/evals=990/1606 eff=75.8040% N=300
Mono-modal Volume: ~exp(-7.21) * Expected Volume: exp(-3.35) Quality: ok
index : +1.0| +2.2 *************** +3.3 | +5.0
amplitude: +1.0e-12| +3.1e-11 ************************** | +1.0e-10
Z=-23.9(0.35%) | Like=-18.98..-13.31 [-18.9977..-18.9794] | it/evals=1005/1628 eff=75.6777% N=300
Z=-23.7(0.44%) | Like=-18.65..-13.31 [-18.6525..-18.6498]*| it/evals=1020/1645 eff=75.8364% N=300
Z=-23.3(0.66%) | Like=-18.26..-13.31 [-18.2781..-18.2625] | it/evals=1044/1676 eff=75.8721% N=300
Z=-23.2(0.73%) | Like=-18.20..-13.31 [-18.2111..-18.1967] | it/evals=1050/1683 eff=75.9219% N=300
Mono-modal Volume: ~exp(-7.48) * 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=-22.9(1.02%) | Like=-17.88..-13.31 [-17.8790..-17.7622] | it/evals=1072/1709 eff=76.0823% N=300
Z=-22.7(1.15%) | Like=-17.59..-13.31 [-17.5858..-17.5602] | it/evals=1080/1722 eff=75.9494% N=300
Z=-22.4(1.63%) | Like=-17.32..-13.31 [-17.3211..-17.3021] | it/evals=1104/1755 eff=75.8763% N=300
Z=-22.3(1.76%) | Like=-17.25..-13.31 [-17.2478..-17.2409]*| it/evals=1110/1766 eff=75.7162% N=300
Z=-22.1(2.28%) | Like=-17.00..-13.31 [-17.0027..-16.9997]*| it/evals=1130/1799 eff=75.3836% N=300
Mono-modal Volume: ~exp(-7.51) * 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=-22.0(2.53%) | Like=-16.90..-13.31 [-16.9166..-16.9047] | it/evals=1139/1815 eff=75.1815% N=300
Z=-21.9(2.57%) | Like=-16.90..-13.31 [-16.8991..-16.8946]*| it/evals=1140/1816 eff=75.1979% N=300
Z=-21.7(3.38%) | Like=-16.65..-13.31 [-16.6515..-16.6502]*| it/evals=1165/1848 eff=75.2584% N=300
Z=-21.6(3.55%) | Like=-16.63..-13.31 [-16.6301..-16.6282]*| it/evals=1170/1856 eff=75.1928% N=300
Z=-21.4(4.60%) | Like=-16.31..-13.31 [-16.3120..-16.3047]*| it/evals=1194/1888 eff=75.1889% N=300
Z=-21.3(4.90%) | Like=-16.24..-13.31 [-16.2546..-16.2433] | it/evals=1200/1896 eff=75.1880% N=300
Mono-modal Volume: ~exp(-7.75) * Expected Volume: exp(-4.02) Quality: ok
index : +1.0| +2.4 *********** +3.2 | +5.0
amplitude: +1.0e-12| +3.6e-11 ****************** +7.2e-11 | +1.0e-10
Z=-21.3(5.25%) | Like=-16.20..-13.31 [-16.1955..-16.1940]*| it/evals=1206/1904 eff=75.1870% N=300
Z=-21.0(6.46%) | Like=-15.98..-13.31 [-15.9848..-15.9624] | it/evals=1230/1935 eff=75.2294% N=300
Z=-20.8(8.00%) | Like=-15.74..-13.31 [-15.7422..-15.7123] | it/evals=1254/1968 eff=75.1799% N=300
Z=-20.8(8.45%) | Like=-15.67..-13.31 [-15.6719..-15.6607] | it/evals=1260/1975 eff=75.2239% N=300
Mono-modal Volume: ~exp(-8.28) * Expected Volume: exp(-4.24) Quality: ok
index : +1.0| +2.4 ********** +3.1 | +5.0
amplitude: +1.0e-12| +3.8e-11 **************** +7.1e-11 | +1.0e-10
Z=-20.7(9.43%) | Like=-15.52..-13.31 [-15.5227..-15.5129]*| it/evals=1273/1993 eff=75.1920% N=300
Z=-20.5(10.83%) | Like=-15.43..-13.31 [-15.4284..-15.4232]*| it/evals=1290/2014 eff=75.2625% N=300
Z=-20.4(12.97%) | Like=-15.23..-13.31 [-15.2317..-15.2184] | it/evals=1315/2047 eff=75.2719% N=300
Z=-20.3(13.42%) | Like=-15.19..-13.31 [-15.1938..-15.1923]*| it/evals=1320/2054 eff=75.2566% N=300
Mono-modal Volume: ~exp(-8.69) * Expected Volume: exp(-4.47) Quality: ok
index : +1.0| +2.4 ******** +3.1 | +5.0
amplitude: +1.0e-12| +4.0e-11 ************** +6.8e-11 | +1.0e-10
Z=-20.2(15.55%) | Like=-15.02..-13.31 [-15.0160..-15.0154]*| it/evals=1340/2080 eff=75.2809% N=300
Z=-20.1(16.59%) | Like=-14.97..-13.31 [-14.9722..-14.9541] | it/evals=1350/2091 eff=75.3769% N=300
Z=-20.0(19.01%) | Like=-14.85..-13.31 [-14.8458..-14.8403]*| it/evals=1374/2123 eff=75.3703% N=300
Z=-19.9(19.73%) | Like=-14.81..-13.31 [-14.8251..-14.8083] | it/evals=1380/2131 eff=75.3687% N=300
Z=-19.8(22.44%) | Like=-14.72..-13.31 [-14.7171..-14.7127]*| it/evals=1404/2163 eff=75.3623% N=300
Mono-modal Volume: ~exp(-8.69) Expected Volume: exp(-4.69) Quality: ok
index : +1.0| +2.5 ******** +3.0 | +5.0
amplitude: +1.0e-12| +4.1e-11 ************* +6.7e-11 | +1.0e-10
Z=-19.8(23.17%) | Like=-14.70..-13.31 [-14.6963..-14.6844] | it/evals=1410/2176 eff=75.1599% N=300
Z=-19.7(26.08%) | Like=-14.55..-13.31 [-14.5535..-14.5508]*| it/evals=1436/2207 eff=75.3015% N=300
Z=-19.6(26.60%) | Like=-14.52..-13.31 [-14.5183..-14.5169]*| it/evals=1440/2214 eff=75.2351% N=300
Z=-19.5(29.29%) | Like=-14.43..-13.31 [-14.4281..-14.4251]*| it/evals=1462/2246 eff=75.1285% N=300
Z=-19.5(30.34%) | Like=-14.39..-13.31 [-14.3920..-14.3916]*| it/evals=1470/2258 eff=75.0766% N=300
Mono-modal Volume: ~exp(-9.00) * Expected Volume: exp(-4.91) Quality: ok
index : +1.0| +2.5 ****** +3.0 | +5.0
amplitude: +1.0e-12| +4.2e-11 *********** +6.5e-11 | +1.0e-10
Z=-19.5(30.85%) | Like=-14.37..-13.31 [-14.3697..-14.3690]*| it/evals=1474/2263 eff=75.0891% N=300
Z=-19.4(34.27%) | Like=-14.30..-13.31 [-14.3019..-14.2990]*| it/evals=1500/2295 eff=75.1880% N=300
Z=-19.3(37.02%) | Like=-14.19..-13.31 [-14.1938..-14.1937]*| it/evals=1521/2327 eff=75.0370% N=300
Z=-19.3(38.22%) | Like=-14.17..-13.31 [-14.1672..-14.1672]*| it/evals=1530/2339 eff=75.0368% N=300
Mono-modal Volume: ~exp(-9.09) * Expected Volume: exp(-5.14) 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=-19.3(39.51%) | Like=-14.13..-13.31 [-14.1332..-14.1330]*| it/evals=1541/2350 eff=75.1707% N=300
Z=-19.2(42.01%) | Like=-14.08..-13.31 [-14.0786..-14.0755]*| it/evals=1560/2373 eff=75.2533% N=300
Z=-19.1(44.81%) | Like=-14.03..-13.31 [-14.0290..-14.0250]*| it/evals=1582/2406 eff=75.1187% N=300
Z=-19.1(45.78%) | Like=-14.01..-13.31 [-14.0102..-14.0041]*| it/evals=1590/2416 eff=75.1418% N=300
Mono-modal Volume: ~exp(-9.09) Expected Volume: exp(-5.36) Quality: ok
index : +1.0| +2.5 ****** +2.9 | +5.0
amplitude: +1.0e-12| +4.4e-11 ********* +6.2e-11 | +1.0e-10
Z=-19.1(48.37%) | Like=-13.97..-13.31 [-13.9711..-13.9678]*| it/evals=1611/2445 eff=75.1049% N=300
Z=-19.0(49.43%) | Like=-13.95..-13.31 [-13.9504..-13.9473]*| it/evals=1620/2459 eff=75.0347% N=300
Z=-19.0(52.21%) | Like=-13.91..-13.31 [-13.9052..-13.9034]*| it/evals=1644/2491 eff=75.0342% N=300
Z=-19.0(52.91%) | Like=-13.90..-13.31 [-13.8957..-13.8952]*| it/evals=1650/2500 eff=75.0000% N=300
Z=-18.9(55.40%) | Like=-13.85..-13.31 [-13.8470..-13.8438]*| it/evals=1672/2535 eff=74.8098% N=300
Mono-modal Volume: ~exp(-9.39) * Expected Volume: exp(-5.58) Quality: ok
index : +1.0| +2.6 ***** +2.9 | +5.0
amplitude: +1.0e-12| +4.5e-11 ********* +6.1e-11 | +1.0e-10
Z=-18.9(55.78%) | Like=-13.84..-13.31 [-13.8422..-13.8392]*| it/evals=1675/2538 eff=74.8436% N=300
Z=-18.9(56.32%) | Like=-13.83..-13.31 [-13.8287..-13.8278]*| it/evals=1680/2544 eff=74.8663% N=300
Z=-18.8(59.26%) | Like=-13.79..-13.31 [-13.7851..-13.7846]*| it/evals=1707/2575 eff=75.0330% N=300
Z=-18.8(59.59%) | Like=-13.78..-13.31 [-13.7827..-13.7826]*| it/evals=1710/2582 eff=74.9343% N=300
Z=-18.8(61.76%) | Like=-13.76..-13.31 [-13.7570..-13.7553]*| it/evals=1732/2616 eff=74.7841% N=300
Z=-18.8(62.51%) | Like=-13.74..-13.30 [-13.7435..-13.7404]*| it/evals=1740/2631 eff=74.6461% N=300
Mono-modal Volume: ~exp(-10.10) * 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=-18.8(62.67%) | Like=-13.74..-13.30 [-13.7392..-13.7374]*| it/evals=1742/2638 eff=74.5081% N=300
Z=-18.8(64.96%) | Like=-13.70..-13.30 [-13.7024..-13.7023]*| it/evals=1765/2670 eff=74.4726% N=300
Z=-18.7(65.41%) | Like=-13.70..-13.30 [-13.6956..-13.6938]*| it/evals=1770/2676 eff=74.4949% N=300
Z=-18.7(67.68%) | Like=-13.66..-13.30 [-13.6603..-13.6602]*| it/evals=1795/2708 eff=74.5432% N=300
Z=-18.7(68.13%) | Like=-13.65..-13.30 [-13.6498..-13.6498]*| it/evals=1800/2717 eff=74.4725% N=300
Mono-modal Volume: ~exp(-10.14) * Expected Volume: exp(-6.03) Quality: ok
index : +1.0| +2.6 **** +2.9 | +5.0
amplitude: +1.0e-12| +4.7e-11 ******* +5.9e-11 | +1.0e-10
Z=-18.7(68.92%) | Like=-13.63..-13.30 [-13.6341..-13.6335]*| it/evals=1809/2729 eff=74.4751% N=300
[ultranest] Explored until L=-1e+01
[ultranest] Likelihood function evaluations: 2746
[ultranest] logZ = -18.33 +- 0.08357
[ultranest] Effective samples strategy satisfied (ESS = 975.7, 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: 0.12 bs:0.08 tail:0.26 total:0.28 required:<0.50
[ultranest] done iterating.
logZ = -18.325 +- 0.312
single instance: logZ = -18.325 +- 0.117
bootstrapped : logZ = -18.331 +- 0.168
tail : logZ = +- 0.262
insert order U test : converged: True correlation: inf iterations
index : 2.13 │ ▁▁▁▁▁▁▁▁▂▃▃▆▆▇▇▇▇▇▇▆▄▄▃▂▂▂▁▁▁▁▁▁▁▁▁▁▁ │3.51 2.74 +- 0.17
amplitude : 0.0000000000256│ ▁ ▁▁▁▁▂▁▂▃▄▅▅▇▆▆▇▆▇▇▆▃▄▄▃▁▂▁▁▁▁▁▁▁ ▁▁ │0.0000000000847 0.0000000000531 +- 0.0000000000078
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).
Total running time of the script: (0 minutes 49.407 seconds)
