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
import astropy.units as u
from gammapy.datasets import Datasets
from gammapy.datasets import SpectrumDatasetOnOff
from gammapy.modeling.models import (
SkyModel,
UniformPrior,
LogUniformPrior,
)
from gammapy.modeling.sampler import Sampler
Loading the spectral datasets#
Here we will load a few Crab 1D spectral data for which we will do a fit.
Model definition#
Now we want to define the spectral model that will be fitted to the data. The Crab spectra will be fitted here with a simple powerlaw for simplicity.
model = SkyModel.create(spectral_model="pl", name="crab")
Warning
Priors definition: Unlike a traditional fit where priors on the parameters are optional, here it is inherent to the Bayesian approach and are therefore mandatory.
In this case we will set (min,max) prior that will define the
boundaries in which the sampling will be performed.
Note that it is usually recommended to use a LogUniformPrior for
the parameters that have a large amplitude range like the
amplitude parameter.
A UniformPrior means that the samples will be drawn with uniform
probability between the (min,max) values in the linear or log space
in the case of a LogUniformPrior.
model.spectral_model.amplitude.prior = LogUniformPrior(min=1e-12, max=1e-10)
model.spectral_model.index.prior = UniformPrior(min=1, max=5)
datasets.models = [model]
print(datasets.models)
DatasetModels
Component 0: SkyModel
Name : crab
Datasets names : None
Spectral model type : PowerLawSpectralModel
Spatial model type :
Temporal model type :
Parameters:
index : 2.000 +/- 0.00
amplitude : 1.00e-12 +/- 0.0e+00 1 / (TeV s cm2)
reference (frozen): 1.000 TeV
Defining the sampler and options#
As for the Fit object, the Sampler object can receive
different backend (although just one is available for now).
The Sampler comes with “reasonable” default parameters, but you can
change them via the sampler_opts dictionary.
Here is a short description of the most relevant parameters that you
could change :
live_points: minimum number of live points throughout the run. More points allow to discover multiple peaks if existing, but is slower. To test the Prior boundaries and for debugging, a lower number (~100) can be used before a production run with more points (~400 or more).frac_remain: the cut-off condition for the integration, set by the maximum allowed fraction of posterior mass left in the live points vs the dead points. High values (e.g., 0.5) are faster and can be used if the posterior distribution is a relatively simple shape. A low value (1e-1, 1e-2) is optimal for finding peaks, but slower.log_dir: directory where the output files will be stored. If set to None, no files will be written. If set to a string, a directory will be created containing the ongoing status of the run and final results. For time consuming analysis, it is highly recommended to use that option to monitor the run and restart it in case of a crash (withresume=True).
Important note: unlike the MCMC method, you don’t need to define the number of steps for which the sampler will run. The algorithm will automatically stop once a convergence criteria has been reached.
sampler_opts = {
"live_points": 300,
"frac_remain": 0.3,
"log_dir": None,
}
sampler = Sampler(backend="ultranest", sampler_opts=sampler_opts)
Next we can run the sampler on a given dataset. No options are accepted in the run method.
[ultranest] Sampling 300 live points from prior ...
Mono-modal Volume: ~exp(-4.18) * Expected Volume: exp(0.00) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|********************************* *** *** *** *| +1.0e-10
Z=-inf(0.00%) | Like=-3528.85..-61.94 [-3528.8533..-319.1543] | it/evals=0/301 eff=0.0000% N=300
Z=-543.6(0.00%) | Like=-537.93..-61.94 [-3528.8533..-319.1543] | it/evals=21/322 eff=95.4545% N=300
Z=-527.0(0.00%) | Like=-516.98..-61.94 [-3528.8533..-319.1543] | it/evals=30/333 eff=90.9091% N=300
Z=-490.8(0.00%) | Like=-485.39..-61.94 [-3528.8533..-319.1543] | it/evals=52/355 eff=94.5455% N=300
Z=-478.2(0.00%) | Like=-472.45..-61.94 [-3528.8533..-319.1543] | it/evals=60/364 eff=93.7500% N=300
Mono-modal Volume: ~exp(-4.18) Expected Volume: exp(-0.22) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|********************************************* *| +1.0e-10
Z=-463.6(0.00%) | Like=-457.84..-61.94 [-3528.8533..-319.1543] | it/evals=79/384 eff=94.0476% N=300
Z=-457.4(0.00%) | Like=-451.34..-61.94 [-3528.8533..-319.1543] | it/evals=90/401 eff=89.1089% N=300
Z=-424.8(0.00%) | Like=-418.29..-61.94 [-3528.8533..-319.1543] | it/evals=109/425 eff=87.2000% N=300
Z=-412.3(0.00%) | Like=-404.84..-61.94 [-3528.8533..-319.1543] | it/evals=120/436 eff=88.2353% N=300
Mono-modal Volume: ~exp(-4.66) * Expected Volume: exp(-0.45) Quality: ok
index : +1.0| ***********************************************| +5.0
amplitude: +1.0e-12|********************************************** *| +1.0e-10
Z=-394.1(0.00%) | Like=-387.36..-61.94 [-3528.8533..-319.1543] | it/evals=134/453 eff=87.5817% N=300
Z=-375.2(0.00%) | Like=-368.44..-61.94 [-3528.8533..-319.1543] | it/evals=150/474 eff=86.2069% N=300
Z=-356.8(0.00%) | Like=-350.07..-61.83 [-3528.8533..-319.1543] | it/evals=170/497 eff=86.2944% N=300
Z=-346.2(0.00%) | Like=-339.31..-61.83 [-3528.8533..-319.1543] | it/evals=180/508 eff=86.5385% N=300
Z=-334.4(0.00%) | Like=-326.23..-61.83 [-3528.8533..-319.1543] | it/evals=198/530 eff=86.0870% N=300
Mono-modal Volume: ~exp(-4.83) * Expected Volume: exp(-0.67) Quality: ok
index : +1.0| * ********************************************| +5.0
amplitude: +1.0e-12| ********************************************* *| +1.0e-10
Z=-330.3(0.00%) | Like=-324.00..-61.83 [-3528.8533..-319.1543] | it/evals=201/534 eff=85.8974% N=300
Z=-323.8(0.00%) | Like=-315.91..-61.83 [-318.3865..-175.0657] | it/evals=210/546 eff=85.3659% N=300
Z=-302.6(0.00%) | Like=-295.62..-59.29 [-318.3865..-175.0657] | it/evals=229/568 eff=85.4478% N=300
Z=-296.1(0.00%) | Like=-287.17..-59.29 [-318.3865..-175.0657] | it/evals=240/582 eff=85.1064% N=300
Z=-277.3(0.00%) | Like=-269.98..-59.29 [-318.3865..-175.0657] | it/evals=259/604 eff=85.1974% N=300
Mono-modal Volume: ~exp(-4.83) Expected Volume: exp(-0.89) Quality: ok
index : +1.0| ****************************************** *| +5.0
amplitude: +1.0e-12| ********************************************* | +1.0e-10
Z=-268.8(0.00%) | Like=-262.74..-59.29 [-318.3865..-175.0657] | it/evals=270/615 eff=85.7143% N=300
Z=-252.7(0.00%) | Like=-244.62..-59.29 [-318.3865..-175.0657] | it/evals=288/637 eff=85.4599% N=300
Z=-237.9(0.00%) | Like=-231.49..-59.29 [-318.3865..-175.0657] | it/evals=300/652 eff=85.2273% N=300
Z=-226.2(0.00%) | Like=-220.10..-59.29 [-318.3865..-175.0657] | it/evals=316/674 eff=84.4920% N=300
Z=-219.6(0.00%) | Like=-213.49..-59.29 [-318.3865..-175.0657] | it/evals=330/690 eff=84.6154% N=300
Mono-modal Volume: ~exp(-4.98) * Expected Volume: exp(-1.12) Quality: ok
index : +1.0| *****************************************| +5.0
amplitude: +1.0e-12| ********************************************| +1.0e-10
Z=-216.1(0.00%) | Like=-209.97..-59.29 [-318.3865..-175.0657] | it/evals=335/695 eff=84.8101% N=300
Z=-206.6(0.00%) | Like=-199.74..-59.29 [-318.3865..-175.0657] | it/evals=351/717 eff=84.1727% N=300
Z=-202.6(0.00%) | Like=-196.50..-59.29 [-318.3865..-175.0657] | it/evals=360/728 eff=84.1121% N=300
Z=-196.6(0.00%) | Like=-190.71..-59.29 [-318.3865..-175.0657] | it/evals=375/750 eff=83.3333% N=300
Z=-192.4(0.00%) | Like=-186.23..-59.29 [-318.3865..-175.0657] | it/evals=387/774 eff=81.6456% N=300
Z=-191.4(0.00%) | Like=-184.40..-59.29 [-318.3865..-175.0657] | it/evals=390/777 eff=81.7610% N=300
Mono-modal Volume: ~exp(-5.60) * Expected Volume: exp(-1.34) Quality: ok
index : +1.0| ************************************** | +5.0
amplitude: +1.0e-12| ****************************************** | +1.0e-10
Z=-185.7(0.00%) | Like=-179.09..-59.29 [-318.3865..-175.0657] | it/evals=402/796 eff=81.0484% N=300
Z=-181.1(0.00%) | Like=-175.09..-59.29 [-318.3865..-175.0657] | it/evals=419/818 eff=80.8880% N=300
Z=-180.8(0.00%) | Like=-174.79..-59.29 [-174.7888..-125.7187] | it/evals=420/819 eff=80.9249% N=300
Z=-177.4(0.00%) | Like=-171.61..-59.29 [-174.7888..-125.7187] | it/evals=434/842 eff=80.0738% N=300
Z=-173.7(0.00%) | Like=-168.10..-59.29 [-174.7888..-125.7187] | it/evals=450/861 eff=80.2139% N=300
Z=-169.7(0.00%) | Like=-163.54..-59.29 [-174.7888..-125.7187] | it/evals=466/883 eff=79.9314% N=300
Mono-modal Volume: ~exp(-5.92) * Expected Volume: exp(-1.56) Quality: ok
index : +1.0| ********************************* | +5.0
amplitude: +1.0e-12| *************************************** * | +1.0e-10
Z=-168.6(0.00%) | Like=-162.09..-59.29 [-174.7888..-125.7187] | it/evals=469/887 eff=79.8978% N=300
Z=-164.3(0.00%) | Like=-156.87..-59.29 [-174.7888..-125.7187] | it/evals=480/899 eff=80.1336% N=300
Z=-157.6(0.00%) | Like=-151.95..-59.29 [-174.7888..-125.7187] | it/evals=500/921 eff=80.5153% N=300
Z=-155.3(0.00%) | Like=-149.05..-59.00 [-174.7888..-125.7187] | it/evals=510/935 eff=80.3150% N=300
Z=-151.8(0.00%) | Like=-146.09..-59.00 [-174.7888..-125.7187] | it/evals=526/957 eff=80.0609% N=300
Mono-modal Volume: ~exp(-5.93) * Expected Volume: exp(-1.79) Quality: ok
index : +1.0| ****************************** | +5.0
amplitude: +1.0e-12| ************************************* * | +1.0e-10
Z=-150.2(0.00%) | Like=-144.64..-58.82 [-174.7888..-125.7187] | it/evals=536/971 eff=79.8808% N=300
Z=-149.3(0.00%) | Like=-142.97..-58.82 [-174.7888..-125.7187] | it/evals=540/976 eff=79.8817% N=300
Z=-145.1(0.00%) | Like=-138.16..-58.82 [-174.7888..-125.7187] | it/evals=557/999 eff=79.6853% N=300
Z=-142.1(0.00%) | Like=-135.89..-58.82 [-174.7888..-125.7187] | it/evals=570/1016 eff=79.6089% N=300
Z=-138.9(0.00%) | Like=-132.96..-58.82 [-174.7888..-125.7187] | it/evals=588/1038 eff=79.6748% N=300
Z=-136.4(0.00%) | Like=-130.64..-58.82 [-174.7888..-125.7187] | it/evals=600/1057 eff=79.2602% N=300
Mono-modal Volume: ~exp(-5.93) Expected Volume: exp(-2.01) Quality: ok
index : +1.0| ************************** +4.0 | +5.0
amplitude: +1.0e-12| *********************************** | +1.0e-10
Z=-133.1(0.00%) | Like=-127.21..-58.82 [-174.7888..-125.7187] | it/evals=615/1077 eff=79.1506% N=300
Z=-131.1(0.00%) | Like=-125.07..-58.82 [-125.6830..-95.7408] | it/evals=630/1096 eff=79.1457% N=300
Z=-128.0(0.00%) | Like=-121.98..-58.82 [-125.6830..-95.7408] | it/evals=646/1123 eff=78.4933% N=300
Z=-125.1(0.00%) | Like=-119.26..-58.82 [-125.6830..-95.7408] | it/evals=660/1140 eff=78.5714% N=300
Mono-modal Volume: ~exp(-6.25) * Expected Volume: exp(-2.23) Quality: ok
index : +1.0| ************************ +3.8 | +5.0
amplitude: +1.0e-12| ******************************** | +1.0e-10
Z=-124.0(0.00%) | Like=-117.28..-58.82 [-125.6830..-95.7408] | it/evals=670/1157 eff=78.1797% N=300
Z=-120.7(0.00%) | Like=-114.34..-58.82 [-125.6830..-95.7408] | it/evals=687/1179 eff=78.1570% N=300
Z=-120.2(0.00%) | Like=-114.21..-58.82 [-125.6830..-95.7408] | it/evals=690/1185 eff=77.9661% N=300
Z=-118.7(0.00%) | Like=-112.95..-58.82 [-125.6830..-95.7408] | it/evals=703/1207 eff=77.5083% N=300
Z=-117.2(0.00%) | Like=-111.17..-58.82 [-125.6830..-95.7408] | it/evals=719/1230 eff=77.3118% N=300
Z=-117.0(0.00%) | Like=-111.11..-58.82 [-125.6830..-95.7408] | it/evals=720/1232 eff=77.2532% N=300
Z=-115.4(0.00%) | Like=-108.87..-58.82 [-125.6830..-95.7408] | it/evals=733/1254 eff=76.8344% N=300
Mono-modal Volume: ~exp(-6.48) * Expected Volume: exp(-2.46) Quality: ok
index : +1.0| +2.0 ********************** +3.7 | +5.0
amplitude: +1.0e-12| ****************************** | +1.0e-10
Z=-114.7(0.00%) | Like=-108.31..-58.82 [-125.6830..-95.7408] | it/evals=737/1258 eff=76.9311% N=300
Z=-113.2(0.00%) | Like=-107.09..-58.82 [-125.6830..-95.7408] | it/evals=750/1275 eff=76.9231% N=300
Z=-110.6(0.00%) | Like=-104.29..-58.82 [-125.6830..-95.7408] | it/evals=765/1298 eff=76.6533% N=300
Z=-108.8(0.00%) | Like=-102.64..-58.82 [-125.6830..-95.7408] | it/evals=780/1319 eff=76.5456% N=300
Z=-106.3(0.00%) | Like=-100.00..-58.82 [-125.6830..-95.7408] | it/evals=798/1342 eff=76.5835% N=300
Mono-modal Volume: ~exp(-6.48) Expected Volume: exp(-2.68) Quality: ok
index : +1.0| +2.0 ******************* +3.5 | +5.0
amplitude: +1.0e-12| ************************* * +7.7e-11 | +1.0e-10
Z=-105.1(0.00%) | Like=-99.00..-58.82 [-125.6830..-95.7408] | it/evals=810/1358 eff=76.5595% N=300
Z=-102.4(0.00%) | Like=-95.86..-58.82 [-125.6830..-95.7408] | it/evals=831/1381 eff=76.8733% N=300
Z=-101.3(0.00%) | Like=-94.79..-58.82 [-95.6843..-78.4970] | it/evals=840/1395 eff=76.7123% N=300
Z=-99.9(0.00%) | Like=-93.45..-58.82 [-95.6843..-78.4970] | it/evals=852/1417 eff=76.2757% N=300
Z=-98.2(0.00%) | Like=-91.83..-58.82 [-95.6843..-78.4970] | it/evals=868/1440 eff=76.1404% N=300
Z=-98.0(0.00%) | Like=-91.64..-58.82 [-95.6843..-78.4970] | it/evals=870/1442 eff=76.1821% N=300
Mono-modal Volume: ~exp(-6.61) * Expected Volume: exp(-2.90) Quality: ok
index : +1.0| +2.1 ***************** +3.5 | +5.0
amplitude: +1.0e-12| +2.4e-11 ************************ +7.3e-11 | +1.0e-10
Z=-97.9(0.00%) | Like=-91.47..-58.82 [-95.6843..-78.4970] | it/evals=871/1443 eff=76.2030% N=300
Z=-96.4(0.00%) | Like=-90.23..-58.82 [-95.6843..-78.4970] | it/evals=886/1465 eff=76.0515% N=300
Z=-95.6(0.00%) | Like=-89.57..-58.82 [-95.6843..-78.4970] | it/evals=897/1488 eff=75.5051% N=300
Z=-95.4(0.00%) | Like=-89.53..-58.82 [-95.6843..-78.4970] | it/evals=900/1491 eff=75.5668% N=300
Z=-94.0(0.00%) | Like=-87.51..-58.79 [-95.6843..-78.4970] | it/evals=916/1513 eff=75.5153% N=300
Z=-92.8(0.00%) | Like=-86.54..-58.79 [-95.6843..-78.4970] | it/evals=930/1533 eff=75.4258% N=300
Mono-modal Volume: ~exp(-7.32) * Expected Volume: exp(-3.13) Quality: ok
index : +1.0| +2.1 **************** +3.4 | +5.0
amplitude: +1.0e-12| +2.6e-11 ********************* +6.7e-11 | +1.0e-10
Z=-92.1(0.00%) | Like=-85.58..-58.79 [-95.6843..-78.4970] | it/evals=938/1546 eff=75.2809% N=300
Z=-89.9(0.00%) | Like=-83.52..-58.79 [-95.6843..-78.4970] | it/evals=956/1568 eff=75.3943% N=300
Z=-89.6(0.00%) | Like=-83.33..-58.79 [-95.6843..-78.4970] | it/evals=960/1573 eff=75.4124% N=300
Z=-88.5(0.00%) | Like=-82.19..-58.79 [-95.6843..-78.4970] | it/evals=977/1595 eff=75.4440% N=300
Z=-87.6(0.00%) | Like=-81.44..-58.79 [-95.6843..-78.4970] | it/evals=990/1611 eff=75.5149% N=300
Mono-modal Volume: ~exp(-7.32) Expected Volume: exp(-3.35) Quality: ok
index : +1.0| +2.2 ************** +3.3 | +5.0
amplitude: +1.0e-12| +2.8e-11 ******************** +6.7e-11 | +1.0e-10
Z=-86.7(0.00%) | Like=-80.24..-58.79 [-95.6843..-78.4970] | it/evals=1005/1631 eff=75.5071% N=300
Z=-85.6(0.00%) | Like=-79.14..-58.79 [-95.6843..-78.4970] | it/evals=1020/1654 eff=75.3323% N=300
Z=-84.4(0.00%) | Like=-77.85..-58.79 [-78.4578..-68.3823] | it/evals=1039/1676 eff=75.5087% N=300
Z=-83.7(0.00%) | Like=-77.06..-58.79 [-78.4578..-68.3823] | it/evals=1050/1687 eff=75.7030% N=300
Z=-82.7(0.00%) | Like=-76.17..-58.79 [-78.4578..-68.3823] | it/evals=1063/1711 eff=75.3366% N=300
Mono-modal Volume: ~exp(-7.54) * Expected Volume: exp(-3.57) Quality: ok
index : +1.0| +2.3 ************ +3.2 | +5.0
amplitude: +1.0e-12| +2.9e-11 ****************** +6.4e-11 | +1.0e-10
Z=-82.2(0.00%) | Like=-75.53..-58.79 [-78.4578..-68.3823] | it/evals=1072/1720 eff=75.4930% N=300
Z=-81.6(0.00%) | Like=-75.09..-58.79 [-78.4578..-68.3823] | it/evals=1080/1730 eff=75.5245% N=300
Z=-80.7(0.00%) | Like=-74.03..-58.79 [-78.4578..-68.3823] | it/evals=1096/1752 eff=75.4821% N=300
Z=-79.9(0.00%) | Like=-73.37..-58.79 [-78.4578..-68.3823] | it/evals=1109/1774 eff=75.2374% N=300
Z=-79.8(0.00%) | Like=-73.23..-58.79 [-78.4578..-68.3823] | it/evals=1110/1776 eff=75.2033% N=300
Z=-78.9(0.00%) | Like=-72.66..-58.79 [-78.4578..-68.3823] | it/evals=1128/1799 eff=75.2502% N=300
Mono-modal Volume: ~exp(-7.85) * Expected Volume: exp(-3.80) Quality: ok
index : +1.0| +2.3 *********** +3.2 | +5.0
amplitude: +1.0e-12| +3.0e-11 **************** +6.2e-11 | +1.0e-10
Z=-78.5(0.00%) | Like=-72.28..-58.79 [-78.4578..-68.3823] | it/evals=1139/1814 eff=75.2312% N=300
Z=-78.5(0.00%) | Like=-72.28..-58.79 [-78.4578..-68.3823] | it/evals=1140/1815 eff=75.2475% N=300
Z=-77.8(0.00%) | Like=-71.51..-58.79 [-78.4578..-68.3823] | it/evals=1159/1838 eff=75.3576% N=300
Z=-77.4(0.00%) | Like=-71.02..-58.79 [-78.4578..-68.3823] | it/evals=1170/1851 eff=75.4352% N=300
Z=-77.0(0.00%) | Like=-70.65..-58.79 [-78.4578..-68.3823] | it/evals=1182/1873 eff=75.1430% N=300
Z=-76.4(0.00%) | Like=-70.10..-58.79 [-78.4578..-68.3823] | it/evals=1199/1895 eff=75.1724% N=300
Z=-76.4(0.00%) | Like=-70.01..-58.79 [-78.4578..-68.3823] | it/evals=1200/1896 eff=75.1880% N=300
Mono-modal Volume: ~exp(-8.17) * Expected Volume: exp(-4.02) Quality: ok
index : +1.0| +2.3 ********** +3.1 | +5.0
amplitude: +1.0e-12| +3.2e-11 *************** +5.9e-11 | +1.0e-10
Z=-76.2(0.00%) | Like=-69.94..-58.79 [-78.4578..-68.3823] | it/evals=1206/1903 eff=75.2339% N=300
Z=-75.6(0.00%) | Like=-69.13..-58.79 [-78.4578..-68.3823] | it/evals=1223/1925 eff=75.2615% N=300
Z=-75.4(0.00%) | Like=-68.93..-58.79 [-78.4578..-68.3823] | it/evals=1230/1932 eff=75.3676% N=300
Z=-74.8(0.01%) | Like=-68.20..-58.79 [-68.2791..-65.4534] | it/evals=1246/1953 eff=75.3781% N=300
Z=-74.3(0.01%) | Like=-67.67..-58.79 [-68.2791..-65.4534] | it/evals=1260/1974 eff=75.2688% N=300
Mono-modal Volume: ~exp(-8.17) Expected Volume: exp(-4.24) Quality: ok
index : +1.0| +2.4 ********* +3.0 | +5.0
amplitude: +1.0e-12| +3.3e-11 ************* +5.7e-11 | +1.0e-10
Z=-73.8(0.02%) | Like=-67.28..-58.79 [-68.2791..-65.4534] | it/evals=1275/1994 eff=75.2656% N=300
Z=-73.5(0.02%) | Like=-66.94..-58.79 [-68.2791..-65.4534] | it/evals=1286/2016 eff=74.9417% N=300
Z=-73.3(0.03%) | Like=-66.82..-58.79 [-68.2791..-65.4534] | it/evals=1290/2020 eff=75.0000% N=300
Z=-73.0(0.04%) | Like=-66.49..-58.79 [-68.2791..-65.4534] | it/evals=1303/2042 eff=74.7991% N=300
Z=-72.7(0.05%) | Like=-66.33..-58.79 [-68.2791..-65.4534] | it/evals=1314/2065 eff=74.4476% N=300
Z=-72.5(0.06%) | Like=-66.28..-58.79 [-68.2791..-65.4534] | it/evals=1320/2071 eff=74.5342% N=300
Z=-72.3(0.08%) | Like=-65.98..-58.77 [-68.2791..-65.4534] | it/evals=1333/2093 eff=74.3447% N=300
Mono-modal Volume: ~exp(-8.47) * Expected Volume: exp(-4.47) Quality: ok
index : +1.0| +2.4 ******** +3.0 | +5.0
amplitude: +1.0e-12| +3.4e-11 ************ +5.6e-11 | +1.0e-10
Z=-72.1(0.09%) | Like=-65.82..-58.77 [-68.2791..-65.4534] | it/evals=1340/2102 eff=74.3618% N=300
Z=-71.9(0.11%) | Like=-65.42..-58.77 [-65.4186..-65.0964] | it/evals=1350/2113 eff=74.4622% N=300
Z=-71.5(0.16%) | Like=-65.11..-58.77 [-65.4186..-65.0964] | it/evals=1369/2136 eff=74.5643% N=300
Z=-71.3(0.21%) | Like=-64.86..-58.77 [-64.8625..-64.7802] | it/evals=1380/2149 eff=74.6349% N=300
Z=-71.0(0.28%) | Like=-64.44..-58.77 [-64.4496..-64.4350] | it/evals=1395/2171 eff=74.5591% N=300
Mono-modal Volume: ~exp(-8.51) * Expected Volume: exp(-4.69) Quality: ok
index : +1.0| +2.4 ******* +3.0 | +5.0
amplitude: +1.0e-12| +3.5e-11 *********** +5.5e-11 | +1.0e-10
Z=-70.7(0.37%) | Like=-64.19..-58.77 [-64.2150..-64.1873] | it/evals=1407/2193 eff=74.3265% N=300
Z=-70.6(0.39%) | Like=-64.16..-58.77 [-64.1699..-64.1573] | it/evals=1410/2196 eff=74.3671% N=300
Z=-70.3(0.55%) | Like=-63.88..-58.77 [-63.8755..-63.8618] | it/evals=1428/2217 eff=74.4914% N=300
Z=-70.1(0.68%) | Like=-63.67..-58.77 [-63.7052..-63.6708] | it/evals=1440/2229 eff=74.6501% N=300
Z=-69.8(0.94%) | Like=-63.42..-58.77 [-63.4227..-63.4197]*| it/evals=1459/2251 eff=74.7822% N=300
Z=-69.6(1.10%) | Like=-63.28..-58.77 [-63.3280..-63.2837] | it/evals=1470/2264 eff=74.8473% N=300
Mono-modal Volume: ~exp(-8.66) * Expected Volume: exp(-4.91) 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=-69.6(1.17%) | Like=-63.18..-58.77 [-63.2043..-63.1783] | it/evals=1474/2268 eff=74.8984% N=300
Z=-69.3(1.56%) | Like=-62.92..-58.77 [-62.9905..-62.9153] | it/evals=1493/2289 eff=75.0628% N=300
Z=-69.2(1.67%) | Like=-62.81..-58.77 [-62.8140..-62.8122]*| it/evals=1500/2297 eff=75.1127% N=300
Z=-69.0(2.06%) | Like=-62.55..-58.77 [-62.5527..-62.5492]*| it/evals=1517/2319 eff=75.1362% N=300
Z=-68.8(2.45%) | Like=-62.39..-58.77 [-62.3910..-62.3835]*| it/evals=1530/2333 eff=75.2582% N=300
Mono-modal Volume: ~exp(-8.90) * Expected Volume: exp(-5.14) Quality: ok
index : +1.0| +2.5 ****** +2.9 | +5.0
amplitude: +1.0e-12| +3.7e-11 ********* +5.3e-11 | +1.0e-10
Z=-68.7(2.84%) | Like=-62.28..-58.77 [-62.2827..-62.2816]*| it/evals=1541/2350 eff=75.1707% N=300
Z=-68.5(3.39%) | Like=-62.13..-58.77 [-62.1626..-62.1330] | it/evals=1556/2372 eff=75.0965% N=300
Z=-68.5(3.57%) | Like=-62.10..-58.77 [-62.1030..-62.0965]*| it/evals=1560/2380 eff=75.0000% N=300
Z=-68.3(4.08%) | Like=-61.94..-58.75 [-61.9388..-61.9371]*| it/evals=1574/2402 eff=74.8811% N=300
Z=-68.2(4.74%) | Like=-61.77..-58.75 [-61.7936..-61.7741] | it/evals=1588/2424 eff=74.7646% N=300
Z=-68.2(4.80%) | Like=-61.70..-58.75 [-61.7311..-61.7034] | it/evals=1590/2428 eff=74.7180% N=300
Z=-68.0(5.46%) | Like=-61.61..-58.75 [-61.6149..-61.6106]*| it/evals=1602/2450 eff=74.5116% N=300
Mono-modal Volume: ~exp(-9.63) * Expected Volume: exp(-5.36) 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=-68.0(5.77%) | Like=-61.56..-58.75 [-61.5754..-61.5643] | it/evals=1608/2463 eff=74.3412% N=300
Z=-67.9(6.39%) | Like=-61.49..-58.75 [-61.4901..-61.4530] | it/evals=1620/2475 eff=74.4828% N=300
Z=-67.7(7.70%) | Like=-61.31..-58.75 [-61.3134..-61.3077]*| it/evals=1639/2497 eff=74.6017% N=300
Z=-67.6(8.47%) | Like=-61.21..-58.75 [-61.2097..-61.2092]*| it/evals=1650/2512 eff=74.5931% N=300
Z=-67.5(9.95%) | Like=-61.04..-58.75 [-61.0370..-61.0368]*| it/evals=1669/2535 eff=74.6756% N=300
Mono-modal Volume: ~exp(-9.63) Expected Volume: exp(-5.58) Quality: ok
index : +1.0| +2.5 ****** +2.8 | +5.0
amplitude: +1.0e-12| +3.8e-11 ******* +5.1e-11 | +1.0e-10
Z=-67.4(10.90%) | Like=-60.96..-58.75 [-60.9590..-60.9573]*| it/evals=1680/2551 eff=74.6335% N=300
Z=-67.2(12.52%) | Like=-60.84..-58.75 [-60.8441..-60.8356]*| it/evals=1698/2575 eff=74.6374% N=300
Z=-67.2(13.50%) | Like=-60.76..-58.75 [-60.7807..-60.7606] | it/evals=1710/2592 eff=74.6073% N=300
Z=-67.1(15.11%) | Like=-60.65..-58.75 [-60.6473..-60.6445]*| it/evals=1727/2614 eff=74.6327% N=300
Z=-67.0(16.44%) | Like=-60.56..-58.75 [-60.5981..-60.5622] | it/evals=1740/2634 eff=74.5501% N=300
Mono-modal Volume: ~exp(-9.63) 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=-66.9(18.04%) | Like=-60.49..-58.75 [-60.4868..-60.4792]*| it/evals=1755/2654 eff=74.5540% N=300
Z=-66.8(19.61%) | Like=-60.42..-58.75 [-60.4241..-60.4235]*| it/evals=1770/2679 eff=74.4010% N=300
Z=-66.7(21.54%) | Like=-60.34..-58.75 [-60.3405..-60.3403]*| it/evals=1789/2701 eff=74.5106% N=300
Z=-66.6(22.80%) | Like=-60.29..-58.75 [-60.2867..-60.2811]*| it/evals=1800/2725 eff=74.2268% N=300
Mono-modal Volume: ~exp(-10.16) * Expected Volume: exp(-6.03) Quality: ok
index : +1.0| +2.5 **** +2.8 | +5.0
amplitude: +1.0e-12| +3.9e-11 ****** +5.0e-11 | +1.0e-10
Z=-66.6(23.84%) | Like=-60.23..-58.75 [-60.2461..-60.2279] | it/evals=1809/2741 eff=74.1090% N=300
Z=-66.5(25.73%) | Like=-60.17..-58.75 [-60.1696..-60.1678]*| it/evals=1825/2762 eff=74.1267% N=300
Z=-66.5(26.27%) | Like=-60.14..-58.75 [-60.1445..-60.1412]*| it/evals=1830/2767 eff=74.1792% N=300
Z=-66.5(28.04%) | Like=-60.10..-58.75 [-60.1004..-60.0990]*| it/evals=1844/2788 eff=74.1158% N=300
Z=-66.4(29.71%) | Like=-60.03..-58.75 [-60.0349..-60.0262]*| it/evals=1860/2807 eff=74.1923% N=300
Z=-66.3(31.37%) | Like=-59.98..-58.75 [-59.9794..-59.9760]*| it/evals=1874/2830 eff=74.0711% N=300
Mono-modal Volume: ~exp(-10.16) Expected Volume: exp(-6.25) Quality: ok
index : +1.0| +2.5 **** +2.8 | +5.0
amplitude: +1.0e-12| +4.0e-11 ****** +4.9e-11 | +1.0e-10
Z=-66.3(32.93%) | Like=-59.93..-58.75 [-59.9254..-59.9250]*| it/evals=1889/2850 eff=74.0784% N=300
Z=-66.3(33.08%) | Like=-59.92..-58.75 [-59.9250..-59.9245]*| it/evals=1890/2852 eff=74.0596% N=300
Z=-66.2(34.78%) | Like=-59.88..-58.75 [-59.8767..-59.8765]*| it/evals=1905/2874 eff=74.0093% N=300
Z=-66.2(36.71%) | Like=-59.79..-58.75 [-59.7860..-59.7826]*| it/evals=1920/2892 eff=74.0741% N=300
Z=-66.1(38.65%) | Like=-59.74..-58.75 [-59.7388..-59.7381]*| it/evals=1936/2914 eff=74.0627% N=300
Mono-modal Volume: ~exp(-10.57) * Expected Volume: exp(-6.48) 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(39.40%) | Like=-59.72..-58.75 [-59.7197..-59.7187]*| it/evals=1943/2922 eff=74.1037% N=300
Z=-66.1(40.34%) | Like=-59.70..-58.75 [-59.6987..-59.6965]*| it/evals=1950/2930 eff=74.1445% N=300
Z=-66.0(42.71%) | Like=-59.65..-58.75 [-59.6523..-59.6519]*| it/evals=1969/2953 eff=74.2179% N=300
Z=-66.0(44.09%) | Like=-59.62..-58.75 [-59.6212..-59.6212]*| it/evals=1980/2968 eff=74.2129% N=300
Z=-66.0(45.67%) | Like=-59.59..-58.75 [-59.5859..-59.5817]*| it/evals=1993/2990 eff=74.0892% N=300
Z=-65.9(47.44%) | Like=-59.54..-58.75 [-59.5378..-59.5343]*| it/evals=2009/3012 eff=74.0782% N=300
Mono-modal Volume: ~exp(-10.81) * 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=-65.9(47.58%) | Like=-59.53..-58.75 [-59.5343..-59.5328]*| it/evals=2010/3014 eff=74.0604% N=300
Z=-65.9(49.40%) | Like=-59.49..-58.75 [-59.4915..-59.4914]*| it/evals=2026/3037 eff=74.0227% N=300
Z=-65.9(51.08%) | Like=-59.47..-58.75 [-59.4706..-59.4705]*| it/evals=2040/3057 eff=73.9935% N=300
Z=-65.8(52.87%) | Like=-59.43..-58.75 [-59.4299..-59.4266]*| it/evals=2056/3079 eff=73.9834% N=300
Z=-65.8(54.30%) | Like=-59.41..-58.75 [-59.4104..-59.4100]*| it/evals=2069/3101 eff=73.8665% N=300
Z=-65.8(54.43%) | Like=-59.41..-58.75 [-59.4100..-59.4075]*| it/evals=2070/3102 eff=73.8758% N=300
Mono-modal Volume: ~exp(-10.81) Expected Volume: exp(-6.92) Quality: ok
index : +1.0| +2.6 **** +2.8 | +5.0
amplitude: +1.0e-12| +4.1e-11 **** +4.8e-11 | +1.0e-10
Z=-65.8(55.64%) | Like=-59.39..-58.75 [-59.3867..-59.3863]*| it/evals=2081/3123 eff=73.7159% N=300
Z=-65.7(57.16%) | Like=-59.37..-58.75 [-59.3660..-59.3617]*| it/evals=2095/3145 eff=73.6380% N=300
Z=-65.7(57.67%) | Like=-59.35..-58.75 [-59.3548..-59.3535]*| it/evals=2100/3151 eff=73.6584% N=300
Z=-65.7(59.17%) | Like=-59.33..-58.75 [-59.3287..-59.3278]*| it/evals=2115/3173 eff=73.6164% N=300
Z=-65.7(60.70%) | Like=-59.29..-58.75 [-59.2913..-59.2888]*| it/evals=2130/3190 eff=73.7024% N=300
Z=-65.7(61.81%) | Like=-59.28..-58.75 [-59.2787..-59.2747]*| it/evals=2142/3212 eff=73.5577% N=300
Mono-modal Volume: ~exp(-11.16) * Expected Volume: exp(-7.15) Quality: ok
index : +1.0| +2.6 **** +2.8 | +5.0
amplitude: +1.0e-12| +4.1e-11 **** +4.8e-11 | +1.0e-10
Z=-65.7(61.97%) | Like=-59.27..-58.75 [-59.2734..-59.2730]*| it/evals=2144/3214 eff=73.5758% N=300
Z=-65.6(63.52%) | Like=-59.25..-58.75 [-59.2532..-59.2525]*| it/evals=2160/3235 eff=73.5945% N=300
Z=-65.6(64.98%) | Like=-59.22..-58.75 [-59.2184..-59.2176]*| it/evals=2175/3256 eff=73.5792% N=300
Z=-65.6(66.32%) | Like=-59.19..-58.75 [-59.1860..-59.1818]*| it/evals=2190/3277 eff=73.5640% N=300
Z=-65.6(68.01%) | Like=-59.16..-58.75 [-59.1567..-59.1554]*| it/evals=2209/3299 eff=73.6579% N=300
Mono-modal Volume: ~exp(-11.49) * Expected Volume: exp(-7.37) 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=-65.6(68.18%) | Like=-59.15..-58.75 [-59.1550..-59.1537]*| it/evals=2211/3302 eff=73.6509% N=300
Z=-65.6(68.93%) | Like=-59.14..-58.75 [-59.1420..-59.1415]*| it/evals=2220/3311 eff=73.7297% N=300
[ultranest] Explored until L=-6e+01
[ultranest] Likelihood function evaluations: 3324
[ultranest] logZ = -65.17 +- 0.1169
[ultranest] Effective samples strategy satisfied (ESS = 1006.8, 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=0.29, need <0.5)
[ultranest] logZ error budget: single: 0.13 bs:0.12 tail:0.26 total:0.29 required:<0.50
[ultranest] done iterating.
logZ = -65.182 +- 0.370
single instance: logZ = -65.182 +- 0.134
bootstrapped : logZ = -65.174 +- 0.262
tail : logZ = +- 0.262
insert order U test : converged: True correlation: inf iterations
index : 2.365 │ ▁ ▁▁▁▁▁▁▂▂▃▃▅▅▄▆▇▅▅▅▆▅▄▃▃▂▂▂▁▁▁▁▁▁▁▁▁ │3.019 2.676 +- 0.086
amplitude : 0.0000000000339│ ▁▁▁▁▁▁▁▂▃▃▄▆▅▆▆▇▇▇▇▆▇▅▅▃▄▃▂▁▁▁▁▁ ▁ ▁ │0.0000000000570 0.0000000000445 +- 0.0000000000031
Understanding the outputs#
In the Jupyter notebook, you should be able to see an interactive visualisation of how the parameter space shrinks which starts from the (min,max) shrinks down towards the optimal parameters.
The output above is filled with interesting information. Here we provide a short description of the most relevant information provided above. For more detailed information see the UltraNest docs.
During the sampling
Z=-68.8(0.53%) | Like=-63.96..-58.75 [-63.9570..-63.9539]*| it/evals=640/1068 eff=73.7327% N=300
Some important information here is:
Progress (0.53%): the completed fraction of the integral. This is not a time progress bar. Stays at zero for a good fraction of the run.
Efficiency (eff value) of the sampling: this indicates out of the proposed new points, how many were accepted. If your efficiency is too small (<<1%), maybe you should revise your priors (e.g use a LogUniform prior for the normalisation).
Final outputs
The final lines indicate that all three “convergence” strategies are satisfied (samples, posterior uncertainty, and evidence uncertainty).
logZ = -65.104 +- 0.292
The main goal of the Nested sampling algorithm is to estimate Z (the Bayesian evidence) which is given above together with an uncertainty. In a similar way to deltaLogLike and deltaAIC, deltaLogZ values can be used for model comparison. For more information see : on the use of the evidence for model comparison. An interesting comparison of the efficiency and false discovery rate of model selection with deltaLogLike and deltaLogZ is given in Appendix C of Buchner et al., 2014.
Results stored on disk
if log_dir is set to a name where the results will be stored, then
a directory is created containing many useful results and plots.
A description of these outputs is given in the Ultranest
docs.
Results#
Within a Bayesian analysis, the concept of best-fit has to be viewed differently from what is done in a gradient descent fit.
The output of the Bayesian analysis is the posterior distribution and there is no “best-fit” output. One has to define, based on the posteriors, what we want to consider as “best-fit” and several options are possible:
the mean of the distribution
the median
the lowest likelihood value
By default the DatasetModels will be updated with the mean of
the posterior distributions.
print(result_joint.models)
DatasetModels
Component 0: SkyModel
Name : crab
Datasets names : None
Spectral model type : PowerLawSpectralModel
Spatial model type :
Temporal model type :
Parameters:
index : 2.676 +/- 0.09
amplitude : 4.45e-11 +/- 3.1e-12 1 / (TeV s cm2)
reference (frozen): 1.000 TeV
The Sampler class returns a very rich dictionary.
The most “standard” information about the posterior distributions can
be found in :
print(result_joint.sampler_results["posterior"])
{'mean': [2.676353675412606, 4.4477774095761483e-11], 'stdev': [0.08646190568691424, 3.0928581203492323e-12], 'median': [2.67194541361323, 4.44859447204164e-11], 'errlo': [2.5892229640814515, 4.125761420272836e-11], 'errup': [2.7615606702475333, 4.766126816184817e-11], 'information_gain_bits': [2.6964609395963834, 3.103512641828282]}
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()
Spectral model error band from samples#
To compute the spectral error band (“butterfly plots”), we will directly use the samples of the posterior distribution. This is more robust as compared to the traditional method of using the covariance matrix of the parameters which implicitly assumes Gaussian errors while for the posterior distribution there is no shape assumed. This difference can become significant when the parameter errors are non-Gaussian. For this we will need to convert the list of samples back to the spectral model parameters with the relevant units (e.g. normalisation units).
def get_samples_from_posterior(spectral_model, results):
"""
Create a list of spectral parameters with correct units
from the unitless parameters returned by the sampler.
"""
n_samples = results.samples.shape[0]
samples = []
for p in spectral_model.parameters:
try:
idx = spectral_model.parameters.free_unique_parameters.index(p)
samples.append(results.samples[:, idx] * p.unit)
except ValueError:
samples.append(np.ones(n_samples) * p.quantity)
return samples
samples = get_samples_from_posterior(datasets.models[0].spectral_model, result_joint)
Next we can provide these samples to the plot_error
method.
Individual run analysis#
Now we’ll analyse several Crab runs individually so that we can compare them.
result_0 = sampler.run(datasets[0])
result_1 = sampler.run(datasets[1])
result_2 = sampler.run(datasets[2])
[ultranest] Sampling 300 live points from prior ...
Mono-modal Volume: ~exp(-4.27) * Expected Volume: exp(0.00) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|************************* ** **** ****** *** *| +1.0e-10
Z=-inf(0.00%) | Like=-1698.90..-22.30 [-1698.9027..-103.4632] | it/evals=0/301 eff=0.0000% N=300
Z=-178.8(0.00%) | Like=-173.80..-21.50 [-1698.9027..-103.4632] | it/evals=30/331 eff=96.7742% N=300
Z=-167.1(0.00%) | Like=-162.35..-20.69 [-1698.9027..-103.4632] | it/evals=60/365 eff=92.3077% N=300
Mono-modal Volume: ~exp(-4.27) Expected Volume: exp(-0.22) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|********************************* ****** ** *| +1.0e-10
Z=-152.2(0.00%) | Like=-146.74..-20.69 [-1698.9027..-103.4632] | it/evals=90/398 eff=91.8367% N=300
Z=-142.3(0.00%) | Like=-136.72..-20.69 [-1698.9027..-103.4632] | it/evals=120/433 eff=90.2256% N=300
Mono-modal Volume: ~exp(-4.68) * Expected Volume: exp(-0.45) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|********************************* ****** **** *| +1.0e-10
Z=-135.4(0.00%) | Like=-129.73..-20.69 [-1698.9027..-103.4632] | it/evals=134/450 eff=89.3333% N=300
Z=-128.6(0.00%) | Like=-122.94..-20.69 [-1698.9027..-103.4632] | it/evals=150/468 eff=89.2857% N=300
Z=-117.9(0.00%) | Like=-112.68..-20.69 [-1698.9027..-103.4632] | it/evals=180/507 eff=86.9565% N=300
Mono-modal Volume: ~exp(-4.75) * Expected Volume: exp(-0.67) Quality: ok
index : +1.0| **********************************************| +5.0
amplitude: +1.0e-12| ******************************** ****** ***** *| +1.0e-10
Z=-111.3(0.00%) | Like=-106.64..-20.69 [-1698.9027..-103.4632] | it/evals=201/531 eff=87.0130% N=300
Z=-109.4(0.00%) | Like=-104.00..-20.69 [-1698.9027..-103.4632] | it/evals=210/542 eff=86.7769% N=300
Z=-98.3(0.00%) | Like=-93.07..-20.69 [-103.2181..-68.3897] | it/evals=240/576 eff=86.9565% N=300
Mono-modal Volume: ~exp(-4.91) * Expected Volume: exp(-0.89) Quality: ok
index : +1.0| ********************************************| +5.0
amplitude: +1.0e-12| *********************************** ** ***** | +1.0e-10
Z=-91.7(0.00%) | Like=-86.66..-20.69 [-103.2181..-68.3897] | it/evals=268/612 eff=85.8974% N=300
Z=-91.4(0.00%) | Like=-86.45..-20.69 [-103.2181..-68.3897] | it/evals=270/614 eff=85.9873% N=300
Z=-85.3(0.00%) | Like=-80.54..-20.69 [-103.2181..-68.3897] | it/evals=300/653 eff=84.9858% N=300
Z=-81.6(0.00%) | Like=-76.94..-20.69 [-103.2181..-68.3897] | it/evals=329/693 eff=83.7150% N=300
Z=-81.5(0.00%) | Like=-76.76..-20.69 [-103.2181..-68.3897] | it/evals=330/694 eff=83.7563% N=300
Mono-modal Volume: ~exp(-5.29) * Expected Volume: exp(-1.12) Quality: ok
index : +1.0| *******************************************| +5.0
amplitude: +1.0e-12| ******************************* *** ** * | +1.0e-10
Z=-81.0(0.00%) | Like=-76.33..-20.69 [-103.2181..-68.3897] | it/evals=335/701 eff=83.5411% N=300
Z=-78.1(0.00%) | Like=-73.13..-20.69 [-103.2181..-68.3897] | it/evals=360/736 eff=82.5688% N=300
Z=-74.9(0.00%) | Like=-70.16..-20.69 [-103.2181..-68.3897] | it/evals=386/776 eff=81.0924% N=300
Z=-74.5(0.00%) | Like=-69.73..-20.69 [-103.2181..-68.3897] | it/evals=390/782 eff=80.9129% N=300
Mono-modal Volume: ~exp(-5.62) * Expected Volume: exp(-1.34) Quality: ok
index : +1.0| *************************************** | +5.0
amplitude: +1.0e-12| ********************************** * | +1.0e-10
Z=-73.4(0.00%) | Like=-68.71..-20.69 [-103.2181..-68.3897] | it/evals=402/799 eff=80.5611% N=300
Z=-71.6(0.00%) | Like=-66.60..-20.69 [-68.2712..-48.8451] | it/evals=420/822 eff=80.4598% N=300
Z=-68.9(0.00%) | Like=-64.01..-20.69 [-68.2712..-48.8451] | it/evals=444/863 eff=78.8632% N=300
Z=-68.3(0.00%) | Like=-63.41..-20.69 [-68.2712..-48.8451] | it/evals=450/872 eff=78.6713% N=300
Mono-modal Volume: ~exp(-5.62) Expected Volume: exp(-1.56) Quality: ok
index : +1.0| ********************************** | +5.0
amplitude: +1.0e-12| ********************************* * | +1.0e-10
Z=-66.4(0.00%) | Like=-61.28..-20.69 [-68.2712..-48.8451] | it/evals=475/912 eff=77.6144% N=300
Z=-65.9(0.00%) | Like=-60.98..-20.69 [-68.2712..-48.8451] | it/evals=480/924 eff=76.9231% N=300
Z=-64.1(0.00%) | Like=-59.22..-20.69 [-68.2712..-48.8451] | it/evals=501/965 eff=75.3383% N=300
Z=-63.1(0.00%) | Like=-57.82..-20.69 [-68.2712..-48.8451] | it/evals=510/975 eff=75.5556% N=300
Mono-modal Volume: ~exp(-5.62) Expected Volume: exp(-1.79) Quality: ok
index : +1.0| ****************************** +4.1 | +5.0
amplitude: +1.0e-12| ******************************* +7.4e-11 | +1.0e-10
Z=-60.7(0.00%) | Like=-55.82..-20.69 [-68.2712..-48.8451] | it/evals=536/1013 eff=75.1753% N=300
Z=-60.4(0.00%) | Like=-55.57..-20.69 [-68.2712..-48.8451] | it/evals=540/1020 eff=75.0000% N=300
Z=-58.5(0.00%) | Like=-53.36..-20.69 [-68.2712..-48.8451] | it/evals=566/1060 eff=74.4737% N=300
Z=-58.2(0.00%) | Like=-52.81..-20.69 [-68.2712..-48.8451] | it/evals=570/1066 eff=74.4125% N=300
Z=-56.0(0.00%) | Like=-50.82..-20.69 [-68.2712..-48.8451] | it/evals=594/1107 eff=73.6059% N=300
Z=-55.5(0.00%) | Like=-50.38..-20.69 [-68.2712..-48.8451] | it/evals=600/1114 eff=73.7101% N=300
Mono-modal Volume: ~exp(-5.67) * Expected Volume: exp(-2.01) Quality: ok
index : +1.0| ************************** +3.8 | +5.0
amplitude: +1.0e-12| ***************************** +7.0e-11 | +1.0e-10
Z=-55.3(0.00%) | Like=-49.95..-20.69 [-68.2712..-48.8451] | it/evals=603/1122 eff=73.3577% N=300
Z=-53.0(0.00%) | Like=-47.57..-20.69 [-48.8087..-36.8249] | it/evals=630/1160 eff=73.2558% N=300
Z=-51.6(0.00%) | Like=-46.62..-20.54 [-48.8087..-36.8249] | it/evals=651/1201 eff=72.2531% N=300
Z=-51.2(0.00%) | Like=-46.20..-20.54 [-48.8087..-36.8249] | it/evals=660/1213 eff=72.2892% N=300
Mono-modal Volume: ~exp(-6.23) * Expected Volume: exp(-2.23) Quality: ok
index : +1.0| ************************ +3.7 | +5.0
amplitude: +1.0e-12| *************************** +6.8e-11 | +1.0e-10
Z=-50.7(0.00%) | Like=-45.60..-20.54 [-48.8087..-36.8249] | it/evals=670/1228 eff=72.1983% N=300
Z=-49.5(0.00%) | Like=-44.30..-20.54 [-48.8087..-36.8249] | it/evals=690/1253 eff=72.4029% N=300
Z=-47.6(0.00%) | Like=-42.37..-20.54 [-48.8087..-36.8249] | it/evals=719/1293 eff=72.4068% N=300
Z=-47.5(0.00%) | Like=-42.31..-20.54 [-48.8087..-36.8249] | it/evals=720/1294 eff=72.4346% N=300
Mono-modal Volume: ~exp(-6.60) * Expected Volume: exp(-2.46) Quality: ok
index : +1.0| ********************* +3.6 | +5.0
amplitude: +1.0e-12| ************************ +6.4e-11 | +1.0e-10
Z=-46.6(0.00%) | Like=-41.20..-20.54 [-48.8087..-36.8249] | it/evals=737/1314 eff=72.6824% N=300
Z=-45.7(0.00%) | Like=-40.21..-20.54 [-48.8087..-36.8249] | it/evals=750/1332 eff=72.6744% N=300
Z=-44.0(0.00%) | Like=-38.50..-20.54 [-48.8087..-36.8249] | it/evals=780/1371 eff=72.8291% N=300
Mono-modal Volume: ~exp(-6.60) Expected Volume: exp(-2.68) Quality: ok
index : +1.0| +1.9 ******************* +3.5 | +5.0
amplitude: +1.0e-12| ********************** +6.0e-11 | +1.0e-10
Z=-42.8(0.00%) | Like=-37.63..-20.54 [-48.8087..-36.8249] | it/evals=804/1408 eff=72.5632% N=300
Z=-42.6(0.00%) | Like=-37.48..-20.54 [-48.8087..-36.8249] | it/evals=810/1415 eff=72.6457% N=300
Z=-41.4(0.00%) | Like=-36.05..-20.54 [-36.7195..-29.0528] | it/evals=837/1456 eff=72.4048% N=300
Z=-41.3(0.00%) | Like=-35.91..-20.54 [-36.7195..-29.0528] | it/evals=840/1461 eff=72.3514% N=300
Z=-40.3(0.00%) | Like=-34.80..-20.54 [-36.7195..-29.0528] | it/evals=863/1501 eff=71.8568% N=300
Z=-40.0(0.00%) | Like=-34.31..-20.54 [-36.7195..-29.0528] | it/evals=870/1512 eff=71.7822% N=300
Mono-modal Volume: ~exp(-6.89) * Expected Volume: exp(-2.90) Quality: ok
index : +1.0| +2.0 ***************** +3.3 | +5.0
amplitude: +1.0e-12| ******************* +5.7e-11 | +1.0e-10
Z=-39.9(0.00%) | Like=-34.27..-20.54 [-36.7195..-29.0528] | it/evals=871/1513 eff=71.8054% N=300
Z=-38.6(0.00%) | Like=-33.18..-20.54 [-36.7195..-29.0528] | it/evals=900/1545 eff=72.2892% N=300
Z=-37.6(0.00%) | Like=-32.18..-20.54 [-36.7195..-29.0528] | it/evals=927/1584 eff=72.1963% N=300
Z=-37.5(0.00%) | Like=-32.12..-20.54 [-36.7195..-29.0528] | it/evals=930/1589 eff=72.1490% N=300
Mono-modal Volume: ~exp(-7.18) * Expected Volume: exp(-3.13) Quality: ok
index : +1.0| +2.1 **************** +3.3 | +5.0
amplitude: +1.0e-12| ****************** +5.4e-11 | +1.0e-10
Z=-37.3(0.00%) | Like=-31.82..-20.54 [-36.7195..-29.0528] | it/evals=938/1600 eff=72.1538% N=300
Z=-36.5(0.00%) | Like=-30.98..-20.54 [-36.7195..-29.0528] | it/evals=960/1627 eff=72.3436% N=300
Z=-35.7(0.01%) | Like=-30.20..-20.54 [-36.7195..-29.0528] | it/evals=988/1666 eff=72.3280% N=300
Z=-35.6(0.01%) | Like=-30.17..-20.54 [-36.7195..-29.0528] | it/evals=990/1669 eff=72.3156% N=300
Mono-modal Volume: ~exp(-7.18) Expected Volume: exp(-3.35) Quality: ok
index : +1.0| +2.1 ************* +3.2 | +5.0
amplitude: +1.0e-12| **************** +5.1e-11 | +1.0e-10
Z=-34.8(0.02%) | Like=-29.24..-20.54 [-36.7195..-29.0528] | it/evals=1020/1704 eff=72.6496% N=300
Z=-34.1(0.03%) | Like=-28.57..-20.54 [-29.0192..-27.1047] | it/evals=1046/1745 eff=72.3875% N=300
Z=-34.0(0.03%) | Like=-28.44..-20.54 [-29.0192..-27.1047] | it/evals=1050/1749 eff=72.4638% N=300
Mono-modal Volume: ~exp(-7.26) * Expected Volume: exp(-3.57) Quality: ok
index : +1.0| +2.1 ************ +3.1 | +5.0
amplitude: +1.0e-12| *************** +5.0e-11 | +1.0e-10
Z=-33.5(0.06%) | Like=-27.98..-20.54 [-29.0192..-27.1047] | it/evals=1072/1781 eff=72.3835% N=300
Z=-33.3(0.07%) | Like=-27.66..-20.54 [-29.0192..-27.1047] | it/evals=1080/1790 eff=72.4832% N=300
Z=-32.6(0.14%) | Like=-27.04..-20.54 [-27.0902..-26.8390] | it/evals=1108/1829 eff=72.4657% N=300
Z=-32.5(0.15%) | Like=-27.01..-20.54 [-27.0902..-26.8390] | it/evals=1110/1832 eff=72.4543% N=300
Mono-modal Volume: ~exp(-7.64) * Expected Volume: exp(-3.80) Quality: ok
index : +1.0| +2.2 ************ +3.0 | +5.0
amplitude: +1.0e-12| ************* +4.8e-11 | +1.0e-10
Z=-32.0(0.27%) | Like=-26.54..-20.54 [-26.5599..-26.5434] | it/evals=1139/1869 eff=72.5940% N=300
Z=-32.0(0.27%) | Like=-26.54..-20.54 [-26.5399..-26.5267] | it/evals=1140/1870 eff=72.6115% N=300
Z=-31.5(0.45%) | Like=-25.81..-20.54 [-25.8083..-25.8002]*| it/evals=1167/1909 eff=72.5295% N=300
Z=-31.4(0.47%) | Like=-25.73..-20.54 [-25.7333..-25.7151] | it/evals=1170/1913 eff=72.5356% N=300
Z=-30.9(0.80%) | Like=-25.16..-20.54 [-25.2095..-25.1551] | it/evals=1198/1952 eff=72.5182% N=300
Z=-30.9(0.84%) | Like=-25.13..-20.54 [-25.1509..-25.1350] | it/evals=1200/1956 eff=72.4638% N=300
Mono-modal Volume: ~exp(-7.87) * Expected Volume: exp(-4.02) Quality: ok
index : +1.0| +2.2 ********** +3.0 | +5.0
amplitude: +1.0e-12| +2.4e-11 *********** +4.6e-11 | +1.0e-10
Z=-30.7(0.92%) | Like=-25.05..-20.54 [-25.0651..-25.0479] | it/evals=1206/1964 eff=72.4760% N=300
Z=-30.3(1.39%) | Like=-24.61..-20.53 [-24.6418..-24.6052] | it/evals=1230/1993 eff=72.6521% N=300
Z=-29.9(2.06%) | Like=-24.26..-20.51 [-24.2587..-24.2551]*| it/evals=1257/2032 eff=72.5751% N=300
Z=-29.9(2.14%) | Like=-24.23..-20.51 [-24.2316..-24.2072] | it/evals=1260/2035 eff=72.6225% N=300
Mono-modal Volume: ~exp(-7.87) Expected Volume: exp(-4.24) Quality: ok
index : +1.0| +2.2 ********* +2.9 | +5.0
amplitude: +1.0e-12| +2.5e-11 *********** +4.5e-11 | +1.0e-10
Z=-29.6(2.88%) | Like=-23.94..-20.51 [-23.9417..-23.9336]*| it/evals=1282/2071 eff=72.3885% N=300
Z=-29.5(3.19%) | Like=-23.83..-20.51 [-23.8308..-23.8247]*| it/evals=1290/2082 eff=72.3906% N=300
Z=-29.2(4.23%) | Like=-23.57..-20.51 [-23.5654..-23.5581]*| it/evals=1315/2122 eff=72.1734% N=300
Z=-29.1(4.44%) | Like=-23.51..-20.51 [-23.5346..-23.5080] | it/evals=1320/2131 eff=72.0918% N=300
Mono-modal Volume: ~exp(-8.05) * Expected Volume: exp(-4.47) Quality: ok
index : +1.0| +2.3 ******** +2.9 | +5.0
amplitude: +1.0e-12| +2.6e-11 ********** +4.4e-11 | +1.0e-10
Z=-28.9(5.42%) | Like=-23.32..-20.51 [-23.3218..-23.3194]*| it/evals=1340/2167 eff=71.7729% N=300
Z=-28.8(5.97%) | Like=-23.27..-20.51 [-23.2714..-23.2555] | it/evals=1350/2180 eff=71.8085% N=300
Z=-28.5(7.86%) | Like=-22.96..-20.47 [-22.9632..-22.9462] | it/evals=1380/2218 eff=71.9499% N=300
Z=-28.3(9.78%) | Like=-22.68..-20.47 [-22.6832..-22.6785]*| it/evals=1406/2258 eff=71.8080% N=300
Mono-modal Volume: ~exp(-8.55) * Expected Volume: exp(-4.69) Quality: ok
index : +1.0| +2.3 ******** +2.8 | +5.0
amplitude: +1.0e-12| +2.7e-11 ********* +4.3e-11 | +1.0e-10
Z=-28.3(9.83%) | Like=-22.68..-20.47 [-22.6785..-22.6720]*| it/evals=1407/2259 eff=71.8224% N=300
Z=-28.3(10.04%) | Like=-22.65..-20.47 [-22.6652..-22.6531] | it/evals=1410/2264 eff=71.7923% N=300
Z=-28.0(12.58%) | Like=-22.39..-20.47 [-22.4043..-22.3931] | it/evals=1440/2304 eff=71.8563% N=300
Z=-27.8(15.21%) | Like=-22.21..-20.47 [-22.2338..-22.2097] | it/evals=1468/2344 eff=71.8200% N=300
Z=-27.8(15.45%) | Like=-22.20..-20.47 [-22.2045..-22.1813] | it/evals=1470/2347 eff=71.8124% N=300
Mono-modal Volume: ~exp(-9.11) * Expected Volume: exp(-4.91) Quality: ok
index : +1.0| +2.3 ****** +2.8 | +5.0
amplitude: +1.0e-12| +2.8e-11 ******** +4.1e-11 | +1.0e-10
Z=-27.8(15.88%) | Like=-22.16..-20.47 [-22.1612..-22.1505] | it/evals=1474/2352 eff=71.8324% N=300
Z=-27.6(18.64%) | Like=-22.05..-20.47 [-22.0469..-22.0377]*| it/evals=1500/2381 eff=72.0807% N=300
Z=-27.5(21.53%) | Like=-21.92..-20.47 [-21.9233..-21.9220]*| it/evals=1526/2420 eff=71.9811% N=300
Z=-27.5(22.03%) | Like=-21.91..-20.47 [-21.9067..-21.8985]*| it/evals=1530/2424 eff=72.0339% N=300
Mono-modal Volume: ~exp(-9.11) 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(24.63%) | Like=-21.78..-20.47 [-21.7799..-21.7786]*| it/evals=1553/2460 eff=71.8981% N=300
Z=-27.3(25.52%) | Like=-21.76..-20.47 [-21.7624..-21.7612]*| it/evals=1560/2468 eff=71.9557% N=300
Z=-27.2(28.63%) | Like=-21.63..-20.47 [-21.6430..-21.6310] | it/evals=1586/2508 eff=71.8297% N=300
Z=-27.2(29.16%) | Like=-21.62..-20.47 [-21.6228..-21.6062] | it/evals=1590/2513 eff=71.8482% N=300
Mono-modal Volume: ~exp(-9.22) * Expected Volume: exp(-5.36) Quality: ok
index : +1.0| +2.4 ****** +2.8 | +5.0
amplitude: +1.0e-12| +2.9e-11 ****** +4.0e-11 | +1.0e-10
Z=-27.1(31.51%) | Like=-21.57..-20.47 [-21.5675..-21.5673]*| it/evals=1608/2538 eff=71.8499% N=300
Z=-27.1(32.83%) | Like=-21.52..-20.47 [-21.5181..-21.5140]*| it/evals=1620/2554 eff=71.8722% N=300
Z=-27.0(36.50%) | Like=-21.41..-20.47 [-21.4085..-21.4072]*| it/evals=1647/2593 eff=71.8273% N=300
Z=-26.9(36.86%) | Like=-21.40..-20.47 [-21.4019..-21.4017]*| it/evals=1650/2596 eff=71.8641% N=300
Mono-modal Volume: ~exp(-9.81) * 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=-26.9(40.11%) | Like=-21.32..-20.47 [-21.3226..-21.3152]*| it/evals=1675/2636 eff=71.7038% N=300
Z=-26.8(40.79%) | Like=-21.29..-20.47 [-21.2946..-21.2913]*| it/evals=1680/2641 eff=71.7642% N=300
Z=-26.8(44.43%) | Like=-21.23..-20.47 [-21.2312..-21.2298]*| it/evals=1710/2674 eff=72.0303% N=300
Z=-26.7(48.02%) | Like=-21.16..-20.47 [-21.1622..-21.1607]*| it/evals=1740/2713 eff=72.1094% N=300
Mono-modal Volume: ~exp(-9.81) 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=-26.6(51.40%) | Like=-21.10..-20.47 [-21.1037..-21.1020]*| it/evals=1768/2748 eff=72.2222% N=300
Z=-26.6(51.61%) | Like=-21.10..-20.47 [-21.1019..-21.0999]*| it/evals=1770/2752 eff=72.1860% N=300
Z=-26.5(54.50%) | Like=-21.05..-20.47 [-21.0539..-21.0536]*| it/evals=1796/2793 eff=72.0417% N=300
Z=-26.5(54.92%) | Like=-21.05..-20.47 [-21.0519..-21.0518]*| it/evals=1800/2801 eff=71.9712% N=300
Mono-modal Volume: ~exp(-10.19) * 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(55.91%) | Like=-21.03..-20.47 [-21.0336..-21.0321]*| it/evals=1809/2812 eff=72.0143% N=300
Z=-26.5(58.13%) | Like=-20.99..-20.47 [-20.9909..-20.9906]*| it/evals=1830/2835 eff=72.1893% N=300
Z=-26.4(61.28%) | Like=-20.94..-20.46 [-20.9372..-20.9369]*| it/evals=1860/2873 eff=72.2892% N=300
Mono-modal Volume: ~exp(-10.30) * 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(62.88%) | Like=-20.92..-20.46 [-20.9229..-20.9215]*| it/evals=1876/2897 eff=72.2372% N=300
Z=-26.4(64.21%) | Like=-20.90..-20.46 [-20.9015..-20.8996]*| it/evals=1890/2916 eff=72.2477% N=300
Z=-26.3(66.97%) | Like=-20.86..-20.46 [-20.8562..-20.8523]*| it/evals=1920/2954 eff=72.3436% N=300
Mono-modal Volume: ~exp(-10.35) * 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(68.95%) | Like=-20.83..-20.46 [-20.8270..-20.8265]*| it/evals=1943/2989 eff=72.2573% N=300
Z=-26.3(69.54%) | Like=-20.82..-20.46 [-20.8211..-20.8206]*| it/evals=1950/3000 eff=72.2222% N=300
[ultranest] Explored until L=-2e+01
[ultranest] Likelihood function evaluations: 3006
[ultranest] logZ = -25.93 +- 0.08578
[ultranest] Effective samples strategy satisfied (ESS = 974.8, need >400)
[ultranest] Posterior uncertainty strategy is satisfied (KL: 0.46+-0.08 nat, need <0.50 nat)
[ultranest] Evidency uncertainty strategy is satisfied (dlogz=0.28, need <0.5)
[ultranest] logZ error budget: single: 0.12 bs:0.09 tail:0.26 total:0.28 required:<0.50
[ultranest] done iterating.
logZ = -25.941 +- 0.330
single instance: logZ = -25.941 +- 0.123
bootstrapped : logZ = -25.932 +- 0.201
tail : logZ = +- 0.262
insert order U test : converged: True correlation: inf iterations
index : 2.13 │ ▁▁▁▁▁▁▁▂▃▃▃▄▅▆▇▇▆▆▅▅▅▄▄▂▂▁▁▁▁▁▁▁▁▁▁ ▁ │3.10 2.56 +- 0.13
amplitude : 0.0000000000191│ ▁ ▁▁▁▁▁▁▂▁▃▃▄▅▅▇▆▇▇▆▅▅▄▄▃▂▂▁▁▁▁▁▁ ▁ ▁ │0.0000000000499 0.0000000000340 +- 0.0000000000037
[ultranest] Sampling 300 live points from prior ...
Mono-modal Volume: ~exp(-3.79) * Expected Volume: exp(0.00) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|******************************** * *** * ** **| +1.0e-10
Z=-inf(0.00%) | Like=-784.75..-19.63 [-784.7513..-130.7465] | it/evals=0/301 eff=0.0000% N=300
Z=-217.9(0.00%) | Like=-212.52..-19.63 [-784.7513..-130.7465] | it/evals=30/335 eff=85.7143% N=300
Z=-200.1(0.00%) | Like=-194.64..-19.63 [-784.7513..-130.7465] | it/evals=60/367 eff=89.5522% N=300
Mono-modal Volume: ~exp(-3.95) * Expected Volume: exp(-0.22) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|*********************************** *** * ** * | +1.0e-10
Z=-197.5(0.00%) | Like=-192.03..-19.63 [-784.7513..-130.7465] | it/evals=67/375 eff=89.3333% N=300
Z=-185.3(0.00%) | Like=-180.16..-19.63 [-784.7513..-130.7465] | it/evals=90/399 eff=90.9091% N=300
Z=-170.9(0.00%) | Like=-165.65..-19.63 [-784.7513..-130.7465] | it/evals=120/438 eff=86.9565% N=300
Mono-modal Volume: ~exp(-4.07) * Expected Volume: exp(-0.45) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|*********************************** *** * **** | +1.0e-10
Z=-165.5(0.00%) | Like=-160.17..-19.63 [-784.7513..-130.7465] | it/evals=134/455 eff=86.4516% N=300
Z=-159.5(0.00%) | Like=-153.88..-19.63 [-784.7513..-130.7465] | it/evals=150/473 eff=86.7052% N=300
Z=-146.3(0.00%) | Like=-141.20..-19.63 [-784.7513..-130.7465] | it/evals=180/508 eff=86.5385% N=300
Mono-modal Volume: ~exp(-4.52) * Expected Volume: exp(-0.67) Quality: ok
index : +1.0| *********************************************| +5.0
amplitude: +1.0e-12| ********************************** ***** **** | +1.0e-10
Z=-133.3(0.00%) | Like=-126.99..-19.30 [-130.5738..-58.9188] | it/evals=201/530 eff=87.3913% N=300
Z=-128.1(0.00%) | Like=-122.19..-19.30 [-130.5738..-58.9188] | it/evals=210/542 eff=86.7769% N=300
Z=-115.1(0.00%) | Like=-108.91..-19.30 [-130.5738..-58.9188] | it/evals=240/582 eff=85.1064% 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=-104.7(0.00%) | Like=-99.38..-19.30 [-130.5738..-58.9188] | it/evals=268/616 eff=84.8101% N=300
Z=-104.1(0.00%) | Like=-98.33..-19.30 [-130.5738..-58.9188] | it/evals=270/619 eff=84.6395% N=300
Z=-97.0(0.00%) | Like=-92.25..-19.30 [-130.5738..-58.9188] | it/evals=300/658 eff=83.7989% N=300
Z=-84.9(0.00%) | Like=-78.30..-19.30 [-130.5738..-58.9188] | it/evals=330/696 eff=83.3333% N=300
Mono-modal Volume: ~exp(-4.99) * Expected Volume: exp(-1.12) Quality: ok
index : +1.0| ******************************************| +5.0
amplitude: +1.0e-12| ************************************* *****| +1.0e-10
Z=-83.0(0.00%) | Like=-77.52..-19.30 [-130.5738..-58.9188] | it/evals=335/701 eff=83.5411% N=300
Z=-77.9(0.00%) | Like=-71.89..-19.30 [-130.5738..-58.9188] | it/evals=360/732 eff=83.3333% N=300
Z=-69.5(0.00%) | Like=-63.76..-19.30 [-130.5738..-58.9188] | it/evals=387/772 eff=81.9915% N=300
Z=-68.7(0.00%) | Like=-62.78..-19.30 [-130.5738..-58.9188] | it/evals=390/779 eff=81.4196% N=300
Mono-modal Volume: ~exp(-5.36) * Expected Volume: exp(-1.34) Quality: ok
index : +1.0| ****************************************| +5.0
amplitude: +1.0e-12| *************************************** | +1.0e-10
Z=-65.9(0.00%) | Like=-60.46..-19.30 [-130.5738..-58.9188] | it/evals=402/794 eff=81.3765% N=300
Z=-62.3(0.00%) | Like=-56.91..-19.30 [-58.5635..-40.0958] | it/evals=420/820 eff=80.7692% N=300
Z=-58.5(0.00%) | Like=-53.54..-19.30 [-58.5635..-40.0958] | it/evals=450/856 eff=80.9353% N=300
Mono-modal Volume: ~exp(-5.81) * Expected Volume: exp(-1.56) Quality: ok
index : +1.0| ************************************ | +5.0
amplitude: +1.0e-12| ************************************** | +1.0e-10
Z=-56.8(0.00%) | Like=-51.91..-19.30 [-58.5635..-40.0958] | it/evals=469/877 eff=81.2825% N=300
Z=-55.9(0.00%) | Like=-51.04..-19.30 [-58.5635..-40.0958] | it/evals=480/891 eff=81.2183% N=300
Z=-53.6(0.00%) | Like=-48.87..-19.30 [-58.5635..-40.0958] | it/evals=510/929 eff=81.0811% N=300
Mono-modal Volume: ~exp(-5.95) * Expected Volume: exp(-1.79) Quality: ok
index : +1.0| ********************************** | +5.0
amplitude: +1.0e-12| ************************************ *| +1.0e-10
Z=-51.6(0.00%) | Like=-46.88..-19.30 [-58.5635..-40.0958] | it/evals=536/965 eff=80.6015% N=300
Z=-51.4(0.00%) | Like=-46.64..-19.30 [-58.5635..-40.0958] | it/evals=540/970 eff=80.5970% N=300
Z=-48.9(0.00%) | Like=-43.82..-19.30 [-58.5635..-40.0958] | it/evals=570/1011 eff=80.1688% N=300
Z=-46.7(0.00%) | Like=-41.44..-19.26 [-58.5635..-40.0958] | it/evals=600/1048 eff=80.2139% N=300
Mono-modal Volume: ~exp(-5.95) Expected Volume: exp(-2.01) Quality: ok
index : +1.0| +2.0 ***************************** | +5.0
amplitude: +1.0e-12| +2.5e-11 *********************************** *| +1.0e-10
Z=-45.2(0.00%) | Like=-40.16..-19.26 [-58.5635..-40.0958] | it/evals=624/1087 eff=79.2884% N=300
Z=-44.8(0.00%) | Like=-39.99..-19.26 [-40.0847..-30.0092] | it/evals=630/1097 eff=79.0464% N=300
Z=-42.9(0.00%) | Like=-37.73..-19.26 [-40.0847..-30.0092] | it/evals=660/1137 eff=78.8530% N=300
Mono-modal Volume: ~exp(-6.25) * Expected Volume: exp(-2.23) Quality: ok
index : +1.0| +2.1 ************************ +4.1 | +5.0
amplitude: +1.0e-12| +2.7e-11 ********************************* | +1.0e-10
Z=-42.3(0.00%) | Like=-37.10..-19.26 [-40.0847..-30.0092] | it/evals=670/1150 eff=78.8235% N=300
Z=-41.0(0.00%) | Like=-35.78..-19.26 [-40.0847..-30.0092] | it/evals=690/1182 eff=78.2313% N=300
Z=-39.7(0.00%) | Like=-34.97..-19.26 [-40.0847..-30.0092] | it/evals=717/1222 eff=77.7657% N=300
Z=-39.6(0.00%) | Like=-34.89..-19.26 [-40.0847..-30.0092] | it/evals=720/1225 eff=77.8378% N=300
Mono-modal Volume: ~exp(-6.41) * Expected Volume: exp(-2.46) Quality: ok
index : +1.0| +2.1 *********************** +3.9 | +5.0
amplitude: +1.0e-12| +2.9e-11 ******************************* | +1.0e-10
Z=-39.0(0.00%) | Like=-34.00..-19.26 [-40.0847..-30.0092] | it/evals=737/1250 eff=77.5789% N=300
Z=-38.4(0.00%) | Like=-33.11..-19.26 [-40.0847..-30.0092] | it/evals=750/1266 eff=77.6398% N=300
Z=-37.5(0.00%) | Like=-32.60..-19.17 [-40.0847..-30.0092] | it/evals=773/1307 eff=76.7627% N=300
Z=-37.2(0.00%) | Like=-32.12..-19.17 [-40.0847..-30.0092] | it/evals=780/1315 eff=76.8473% N=300
Mono-modal Volume: ~exp(-6.41) 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=-36.2(0.00%) | Like=-31.01..-19.17 [-40.0847..-30.0092] | it/evals=806/1351 eff=76.6889% N=300
Z=-36.0(0.00%) | Like=-30.67..-19.17 [-40.0847..-30.0092] | it/evals=810/1357 eff=76.6320% N=300
Z=-34.9(0.00%) | Like=-29.57..-19.17 [-29.9874..-26.0687] | it/evals=837/1397 eff=76.2990% N=300
Z=-34.8(0.00%) | Like=-29.48..-19.17 [-29.9874..-26.0687] | it/evals=840/1403 eff=76.1559% N=300
Z=-33.7(0.01%) | Like=-28.52..-19.17 [-29.9874..-26.0687] | it/evals=870/1442 eff=76.1821% N=300
Mono-modal Volume: ~exp(-6.99) * Expected Volume: exp(-2.90) Quality: ok
index : +1.0| +2.2 ****************** +3.7 | +5.0
amplitude: +1.0e-12| +3.3e-11 ************************ * | +1.0e-10
Z=-33.7(0.01%) | Like=-28.52..-19.17 [-29.9874..-26.0687] | it/evals=871/1443 eff=76.2030% N=300
Z=-32.9(0.02%) | Like=-27.90..-19.17 [-29.9874..-26.0687] | it/evals=898/1483 eff=75.9087% N=300
Z=-32.9(0.02%) | Like=-27.88..-19.17 [-29.9874..-26.0687] | it/evals=900/1485 eff=75.9494% N=300
Z=-32.1(0.04%) | Like=-26.99..-19.17 [-29.9874..-26.0687] | it/evals=928/1525 eff=75.7551% N=300
Z=-32.1(0.04%) | Like=-26.95..-19.17 [-29.9874..-26.0687] | it/evals=930/1528 eff=75.7329% N=300
Mono-modal Volume: ~exp(-7.25) * Expected Volume: exp(-3.13) Quality: ok
index : +1.0| +2.3 **************** +3.6 | +5.0
amplitude: +1.0e-12| +3.5e-11 *********************** | +1.0e-10
Z=-31.9(0.05%) | Like=-26.65..-19.17 [-29.9874..-26.0687] | it/evals=938/1536 eff=75.8900% N=300
Z=-31.3(0.09%) | Like=-26.13..-19.17 [-29.9874..-26.0687] | it/evals=960/1561 eff=76.1301% N=300
Z=-30.7(0.19%) | Like=-25.44..-19.17 [-25.4438..-25.4290] | it/evals=989/1601 eff=76.0184% N=300
Z=-30.6(0.19%) | Like=-25.43..-19.17 [-25.4438..-25.4290] | it/evals=990/1602 eff=76.0369% N=300
Mono-modal Volume: ~exp(-7.37) * Expected Volume: exp(-3.35) Quality: ok
index : +1.0| +2.3 ************** +3.4 | +5.0
amplitude: +1.0e-12| +3.8e-11 ********************* +7.8e-11| +1.0e-10
Z=-30.3(0.26%) | Like=-25.00..-19.17 [-25.0351..-25.0004] | it/evals=1005/1622 eff=76.0212% N=300
Z=-30.0(0.36%) | Like=-24.77..-19.17 [-24.7750..-24.7708]*| it/evals=1020/1644 eff=75.8929% N=300
Z=-29.5(0.60%) | Like=-24.39..-19.17 [-24.3946..-24.3719] | it/evals=1050/1682 eff=75.9768% N=300
Mono-modal Volume: ~exp(-7.39) * Expected Volume: exp(-3.57) Quality: ok
index : +1.0| +2.4 ************* +3.4 | +5.0
amplitude: +1.0e-12| +3.9e-11 ****************** +7.5e-11 | +1.0e-10
Z=-29.1(0.85%) | Like=-24.07..-19.17 [-24.0668..-24.0615]*| it/evals=1072/1714 eff=75.8133% N=300
Z=-29.0(0.97%) | Like=-23.86..-19.17 [-23.8599..-23.8495] | it/evals=1080/1725 eff=75.7895% N=300
Z=-28.5(1.57%) | Like=-23.40..-19.17 [-23.4020..-23.3994]*| it/evals=1110/1760 eff=76.0274% N=300
Mono-modal Volume: ~exp(-8.28) * Expected Volume: exp(-3.80) Quality: ok
index : +1.0| +2.4 ************ +3.3 | +5.0
amplitude: +1.0e-12| +4.0e-11 **************** +7.2e-11 | +1.0e-10
Z=-28.2(2.37%) | Like=-23.08..-19.17 [-23.0832..-23.0746]*| it/evals=1139/1795 eff=76.1873% N=300
Z=-28.2(2.38%) | Like=-23.07..-19.17 [-23.0746..-23.0650]*| it/evals=1140/1797 eff=76.1523% N=300
Z=-27.8(3.40%) | Like=-22.72..-19.17 [-22.7266..-22.7162] | it/evals=1170/1831 eff=76.4206% N=300
Z=-27.5(4.53%) | Like=-22.51..-19.17 [-22.5236..-22.5135] | it/evals=1197/1870 eff=76.2420% N=300
Z=-27.5(4.63%) | Like=-22.48..-19.17 [-22.4846..-22.4735] | it/evals=1200/1873 eff=76.2873% N=300
Mono-modal Volume: ~exp(-8.28) Expected Volume: exp(-4.02) Quality: ok
index : +1.0| +2.4 *********** +3.3 | +5.0
amplitude: +1.0e-12| +4.1e-11 **************** +7.1e-11 | +1.0e-10
Z=-27.3(5.57%) | Like=-22.29..-19.17 [-22.2852..-22.2814]*| it/evals=1221/1909 eff=75.8856% N=300
Z=-27.2(5.92%) | Like=-22.22..-19.17 [-22.2240..-22.2237]*| it/evals=1230/1919 eff=75.9728% N=300
Z=-27.0(7.37%) | Like=-21.99..-19.17 [-21.9871..-21.9849]*| it/evals=1260/1955 eff=76.1329% N=300
Mono-modal Volume: ~exp(-8.28) Expected Volume: exp(-4.24) Quality: ok
index : +1.0| +2.5 ********** +3.2 | +5.0
amplitude: +1.0e-12| +4.3e-11 ************** +6.9e-11 | +1.0e-10
Z=-26.8(8.84%) | Like=-21.79..-19.17 [-21.7930..-21.7810] | it/evals=1286/1991 eff=76.0497% N=300
Z=-26.8(9.08%) | Like=-21.76..-19.17 [-21.7596..-21.7295] | it/evals=1290/1997 eff=76.0165% N=300
Z=-26.6(11.17%) | Like=-21.44..-19.16 [-21.4494..-21.4376] | it/evals=1320/2036 eff=76.0369% N=300
Mono-modal Volume: ~exp(-8.53) * Expected Volume: exp(-4.47) Quality: ok
index : +1.0| +2.5 ******** +3.1 | +5.0
amplitude: +1.0e-12| +4.4e-11 ************* +6.8e-11 | +1.0e-10
Z=-26.4(12.86%) | Like=-21.27..-19.16 [-21.2727..-21.2590] | it/evals=1340/2071 eff=75.6635% N=300
Z=-26.4(13.64%) | Like=-21.20..-19.16 [-21.2028..-21.1939]*| it/evals=1350/2083 eff=75.7151% N=300
Z=-26.2(15.61%) | Like=-21.02..-19.16 [-21.0173..-21.0126]*| it/evals=1374/2122 eff=75.4116% N=300
Z=-26.2(16.14%) | Like=-20.97..-19.16 [-20.9739..-20.9709]*| it/evals=1380/2133 eff=75.2864% N=300
Mono-modal Volume: ~exp(-8.65) * 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(19.00%) | Like=-20.81..-19.16 [-20.8131..-20.8123]*| it/evals=1407/2169 eff=75.2809% N=300
Z=-26.0(19.24%) | Like=-20.81..-19.16 [-20.8110..-20.8086]*| it/evals=1410/2173 eff=75.2803% N=300
Z=-25.9(22.37%) | Like=-20.69..-19.16 [-20.6930..-20.6736] | it/evals=1438/2212 eff=75.2092% N=300
Z=-25.9(22.56%) | Like=-20.67..-19.16 [-20.6659..-20.6653]*| it/evals=1440/2215 eff=75.1958% N=300
Z=-25.7(25.42%) | Like=-20.52..-19.16 [-20.5234..-20.5179]*| it/evals=1467/2255 eff=75.0384% N=300
Z=-25.7(25.80%) | Like=-20.51..-19.16 [-20.5111..-20.5061]*| it/evals=1470/2258 eff=75.0766% N=300
Mono-modal Volume: ~exp(-8.71) * 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(26.28%) | Like=-20.50..-19.16 [-20.4988..-20.4977]*| it/evals=1474/2264 eff=75.0509% N=300
Z=-25.6(29.63%) | Like=-20.38..-19.16 [-20.3783..-20.3783]*| it/evals=1500/2293 eff=75.2634% N=300
Z=-25.5(33.10%) | Like=-20.31..-19.16 [-20.3135..-20.3099]*| it/evals=1528/2332 eff=75.1969% N=300
Z=-25.5(33.37%) | Like=-20.30..-19.16 [-20.3099..-20.2953] | it/evals=1530/2334 eff=75.2212% N=300
Mono-modal Volume: ~exp(-8.71) 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(36.35%) | Like=-20.19..-19.16 [-20.1949..-20.1894]*| it/evals=1555/2370 eff=75.1208% N=300
Z=-25.4(36.89%) | Like=-20.17..-19.16 [-20.1680..-20.1599]*| it/evals=1560/2379 eff=75.0361% N=300
Z=-25.3(40.29%) | Like=-20.04..-19.16 [-20.0442..-20.0384]*| it/evals=1590/2420 eff=75.0000% N=300
Mono-modal Volume: ~exp(-8.81) * 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(42.54%) | Like=-20.01..-19.16 [-20.0063..-20.0050]*| it/evals=1608/2454 eff=74.6518% N=300
Z=-25.2(43.98%) | Like=-19.99..-19.16 [-19.9906..-19.9899]*| it/evals=1620/2468 eff=74.7232% N=300
Z=-25.1(47.51%) | Like=-19.90..-19.16 [-19.9030..-19.9025]*| it/evals=1650/2507 eff=74.7621% N=300
Mono-modal Volume: ~exp(-9.16) * Expected Volume: exp(-5.58) Quality: ok
index : +1.0| +2.6 ****** +3.0 | +5.0
amplitude: +1.0e-12| +4.9e-11 ******* +6.2e-11 | +1.0e-10
Z=-25.1(50.52%) | Like=-19.86..-19.16 [-19.8581..-19.8561]*| it/evals=1675/2546 eff=74.5770% N=300
Z=-25.0(51.12%) | Like=-19.84..-19.16 [-19.8433..-19.8400]*| it/evals=1680/2555 eff=74.5011% N=300
Z=-25.0(54.07%) | Like=-19.80..-19.16 [-19.8036..-19.7996]*| it/evals=1707/2594 eff=74.4115% N=300
Z=-25.0(54.39%) | Like=-19.79..-19.16 [-19.7911..-19.7881]*| it/evals=1710/2597 eff=74.4449% N=300
Z=-24.9(57.19%) | Like=-19.73..-19.16 [-19.7311..-19.7289]*| it/evals=1737/2637 eff=74.3261% N=300
Z=-24.9(57.51%) | Like=-19.72..-19.16 [-19.7244..-19.7220]*| it/evals=1740/2643 eff=74.2638% N=300
Mono-modal Volume: ~exp(-9.67) * 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(57.71%) | Like=-19.72..-19.16 [-19.7218..-19.7211]*| it/evals=1742/2645 eff=74.2857% N=300
Z=-24.9(60.57%) | Like=-19.67..-19.16 [-19.6683..-19.6678]*| it/evals=1770/2681 eff=74.3385% N=300
Z=-24.8(63.58%) | Like=-19.61..-19.16 [-19.6137..-19.6131]*| it/evals=1800/2720 eff=74.3802% N=300
Mono-modal Volume: ~exp(-10.17) * Expected Volume: exp(-6.03) Quality: ok
index : +1.0| +2.6 ***** +3.0 | +5.0
amplitude: +1.0e-12| +5.0e-11 ****** +6.1e-11 | +1.0e-10
Z=-24.8(64.41%) | Like=-19.61..-19.16 [-19.6071..-19.6057]*| it/evals=1809/2733 eff=74.3527% N=300
Z=-24.8(66.33%) | Like=-19.57..-19.16 [-19.5744..-19.5736]*| it/evals=1830/2771 eff=74.0591% N=300
Z=-24.7(68.56%) | Like=-19.54..-19.16 [-19.5353..-19.5349]*| it/evals=1855/2812 eff=73.8455% N=300
Z=-24.7(68.97%) | Like=-19.53..-19.16 [-19.5288..-19.5271]*| it/evals=1860/2819 eff=73.8388% N=300
[ultranest] Explored until L=-2e+01
[ultranest] Likelihood function evaluations: 2832
[ultranest] logZ = -24.39 +- 0.08903
[ultranest] Effective samples strategy satisfied (ESS = 1007.6, need >400)
[ultranest] Posterior uncertainty strategy is satisfied (KL: 0.46+-0.09 nat, need <0.50 nat)
[ultranest] Evidency uncertainty strategy is satisfied (dlogz=0.28, need <0.5)
[ultranest] logZ error budget: single: 0.12 bs:0.09 tail:0.26 total:0.28 required:<0.50
[ultranest] done iterating.
logZ = -24.367 +- 0.317
single instance: logZ = -24.367 +- 0.118
bootstrapped : logZ = -24.388 +- 0.179
tail : logZ = +- 0.262
insert order U test : converged: True correlation: inf iterations
index : 2.28 │ ▁▁▁▁▁▁▂▂▃▅▅▇▅▇▇▆▇▆▅▅▄▃▃▃▂▁▁▁▁▁▁▁▁ ▁ ▁ │3.59 2.83 +- 0.17
amplitude : 0.0000000000359│ ▁▁▁▁▁▁▂▃▃▅▆▅▆▇▆▇▆▅▅▄▃▃▂▂▁▁▁▁▁▁ ▁▁▁ ▁ │0.0000000000834 0.0000000000549 +- 0.0000000000059
[ultranest] Sampling 300 live points from prior ...
Mono-modal Volume: ~exp(-4.11) * Expected Volume: exp(0.00) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|********************* ***************** ********| +1.0e-10
Z=-inf(0.00%) | Like=-465.62..-13.43 [-465.6219..-86.4647] | it/evals=0/301 eff=0.0000% N=300
Z=-150.9(0.00%) | Like=-146.36..-13.43 [-465.6219..-86.4647] | it/evals=30/334 eff=88.2353% N=300
Z=-142.0(0.00%) | Like=-136.45..-13.43 [-465.6219..-86.4647] | it/evals=60/370 eff=85.7143% N=300
Mono-modal Volume: ~exp(-4.37) * Expected Volume: exp(-0.22) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|************************************* * ********| +1.0e-10
Z=-139.0(0.00%) | Like=-133.92..-13.43 [-465.6219..-86.4647] | it/evals=67/379 eff=84.8101% N=300
Z=-130.0(0.00%) | Like=-125.05..-13.43 [-465.6219..-86.4647] | it/evals=90/403 eff=87.3786% N=300
Z=-117.8(0.00%) | Like=-112.95..-13.43 [-465.6219..-86.4647] | it/evals=120/436 eff=88.2353% N=300
Mono-modal Volume: ~exp(-4.37) Expected Volume: exp(-0.45) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|************************************* **********| +1.0e-10
Z=-111.0(0.00%) | Like=-106.27..-13.43 [-465.6219..-86.4647] | it/evals=146/473 eff=84.3931% N=300
Z=-110.3(0.00%) | Like=-105.95..-13.43 [-465.6219..-86.4647] | it/evals=150/480 eff=83.3333% N=300
Z=-103.6(0.00%) | Like=-98.62..-13.43 [-465.6219..-86.4647] | it/evals=179/520 eff=81.3636% N=300
Z=-103.4(0.00%) | Like=-98.46..-13.43 [-465.6219..-86.4647] | it/evals=180/521 eff=81.4480% N=300
Mono-modal Volume: ~exp(-4.71) * Expected Volume: exp(-0.67) Quality: ok
index : +1.0| **********************************************| +5.0
amplitude: +1.0e-12| ************************************ **********| +1.0e-10
Z=-96.0(0.00%) | Like=-90.01..-13.43 [-465.6219..-86.4647] | it/evals=201/546 eff=81.7073% N=300
Z=-92.0(0.00%) | Like=-86.18..-13.43 [-86.1824..-44.0937] | it/evals=210/556 eff=82.0312% N=300
Z=-84.1(0.00%) | Like=-79.56..-13.43 [-86.1824..-44.0937] | it/evals=240/595 eff=81.3559% 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=-77.1(0.00%) | Like=-71.62..-13.43 [-86.1824..-44.0937] | it/evals=268/631 eff=80.9668% N=300
Z=-76.6(0.00%) | Like=-70.77..-13.43 [-86.1824..-44.0937] | it/evals=270/633 eff=81.0811% N=300
Z=-70.1(0.00%) | Like=-64.94..-13.43 [-86.1824..-44.0937] | it/evals=300/672 eff=80.6452% N=300
Z=-64.3(0.00%) | Like=-58.72..-13.43 [-86.1824..-44.0937] | it/evals=330/707 eff=81.0811% N=300
Mono-modal Volume: ~exp(-5.22) * Expected Volume: exp(-1.12) Quality: ok
index : +1.0| ******************************************| +5.0
amplitude: +1.0e-12| ********************************** **********| +1.0e-10
Z=-63.2(0.00%) | Like=-57.71..-13.43 [-86.1824..-44.0937] | it/evals=335/716 eff=80.5288% N=300
Z=-59.2(0.00%) | Like=-53.66..-13.43 [-86.1824..-44.0937] | it/evals=360/745 eff=80.8989% N=300
Z=-53.2(0.00%) | Like=-48.31..-13.43 [-86.1824..-44.0937] | it/evals=390/781 eff=81.0811% N=300
Mono-modal Volume: ~exp(-5.22) Expected Volume: exp(-1.34) Quality: ok
index : +1.0| ****************************************| +5.0
amplitude: +1.0e-12| ******************************** **********| +1.0e-10
Z=-50.1(0.00%) | Like=-44.98..-13.43 [-86.1824..-44.0937] | it/evals=419/817 eff=81.0445% N=300
Z=-50.0(0.00%) | Like=-44.78..-13.43 [-86.1824..-44.0937] | it/evals=420/819 eff=80.9249% N=300
Z=-47.7(0.00%) | Like=-43.12..-13.43 [-44.0254..-28.4179] | it/evals=449/859 eff=80.3220% N=300
Z=-47.6(0.00%) | Like=-42.90..-13.43 [-44.0254..-28.4179] | it/evals=450/860 eff=80.3571% N=300
Mono-modal Volume: ~exp(-5.70) * Expected Volume: exp(-1.56) Quality: ok
index : +1.0| ************************************* | +5.0
amplitude: +1.0e-12| ******************************* **********| +1.0e-10
Z=-46.1(0.00%) | Like=-41.35..-13.43 [-44.0254..-28.4179] | it/evals=469/890 eff=79.4915% N=300
Z=-45.3(0.00%) | Like=-40.47..-13.43 [-44.0254..-28.4179] | it/evals=480/906 eff=79.2079% N=300
Z=-42.5(0.00%) | Like=-37.55..-13.43 [-44.0254..-28.4179] | it/evals=508/947 eff=78.5162% N=300
Z=-42.3(0.00%) | Like=-37.21..-13.43 [-44.0254..-28.4179] | it/evals=510/950 eff=78.4615% N=300
Mono-modal Volume: ~exp(-6.03) * Expected Volume: exp(-1.79) Quality: ok
index : +1.0| ******************************* | +5.0
amplitude: +1.0e-12| ****************************** **********| +1.0e-10
Z=-40.4(0.00%) | Like=-35.58..-13.43 [-44.0254..-28.4179] | it/evals=536/990 eff=77.6812% N=300
Z=-40.1(0.00%) | Like=-35.10..-13.43 [-44.0254..-28.4179] | it/evals=540/998 eff=77.3639% N=300
Z=-38.3(0.00%) | Like=-33.49..-13.43 [-44.0254..-28.4179] | it/evals=570/1038 eff=77.2358% N=300
Z=-36.7(0.00%) | Like=-31.58..-13.43 [-44.0254..-28.4179] | it/evals=599/1078 eff=76.9923% N=300
Z=-36.6(0.00%) | Like=-31.57..-13.43 [-44.0254..-28.4179] | it/evals=600/1080 eff=76.9231% N=300
Mono-modal Volume: ~exp(-6.11) * Expected Volume: exp(-2.01) Quality: ok
index : +1.0| +1.9 ************************** +4.0 | +5.0
amplitude: +1.0e-12| ****************************************| +1.0e-10
Z=-36.4(0.00%) | Like=-31.27..-13.43 [-44.0254..-28.4179] | it/evals=603/1085 eff=76.8153% N=300
Z=-34.7(0.00%) | Like=-29.66..-13.43 [-44.0254..-28.4179] | it/evals=630/1125 eff=76.3636% N=300
Z=-33.2(0.00%) | Like=-28.32..-13.42 [-28.3644..-20.6025] | it/evals=656/1166 eff=75.7506% N=300
Z=-33.0(0.00%) | Like=-28.13..-13.42 [-28.3644..-20.6025] | it/evals=660/1171 eff=75.7750% N=300
Mono-modal Volume: ~exp(-6.11) Expected Volume: exp(-2.23) Quality: ok
index : +1.0| +2.0 ************************ +3.9 | +5.0
amplitude: +1.0e-12| **************************************| +1.0e-10
Z=-31.8(0.00%) | Like=-26.86..-13.42 [-28.3644..-20.6025] | it/evals=684/1208 eff=75.3304% N=300
Z=-31.5(0.00%) | Like=-26.53..-13.42 [-28.3644..-20.6025] | it/evals=690/1218 eff=75.1634% N=300
Z=-29.9(0.00%) | Like=-24.82..-13.42 [-28.3644..-20.6025] | it/evals=720/1258 eff=75.1566% N=300
Mono-modal Volume: ~exp(-6.62) * Expected Volume: exp(-2.46) Quality: ok
index : +1.0| +2.1 ******************** +3.7 | +5.0
amplitude: +1.0e-12| +2.5e-11 *********************************** | +1.0e-10
Z=-29.1(0.00%) | Like=-24.08..-13.42 [-28.3644..-20.6025] | it/evals=737/1281 eff=75.1274% N=300
Z=-28.6(0.00%) | Like=-23.58..-13.42 [-28.3644..-20.6025] | it/evals=750/1297 eff=75.2257% N=300
Z=-27.7(0.01%) | Like=-23.03..-13.37 [-28.3644..-20.6025] | it/evals=775/1338 eff=74.6628% N=300
Z=-27.6(0.01%) | Like=-22.77..-13.37 [-28.3644..-20.6025] | it/evals=780/1346 eff=74.5698% N=300
Mono-modal Volume: ~exp(-6.65) * Expected Volume: exp(-2.68) Quality: ok
index : +1.0| +2.1 ****************** +3.6 | +5.0
amplitude: +1.0e-12| +2.8e-11 ******************************** | +1.0e-10
Z=-26.9(0.02%) | Like=-22.11..-13.37 [-28.3644..-20.6025] | it/evals=804/1380 eff=74.4444% N=300
Z=-26.7(0.02%) | Like=-21.86..-13.37 [-28.3644..-20.6025] | it/evals=810/1390 eff=74.3119% N=300
Z=-25.9(0.04%) | Like=-21.08..-13.37 [-28.3644..-20.6025] | it/evals=840/1430 eff=74.3363% N=300
Z=-25.3(0.08%) | Like=-20.51..-13.37 [-20.5972..-19.7790] | it/evals=866/1470 eff=74.0171% N=300
Z=-25.2(0.09%) | Like=-20.42..-13.33 [-20.5972..-19.7790] | it/evals=870/1477 eff=73.9167% N=300
Mono-modal Volume: ~exp(-6.71) * Expected Volume: exp(-2.90) Quality: ok
index : +1.0| +2.2 **************** +3.5 | +5.0
amplitude: +1.0e-12| +2.9e-11 ***************************** | +1.0e-10
Z=-25.2(0.09%) | Like=-20.40..-13.33 [-20.5972..-19.7790] | it/evals=871/1481 eff=73.7511% N=300
Z=-24.6(0.16%) | Like=-19.75..-13.33 [-19.7699..-19.6602] | it/evals=899/1521 eff=73.6282% N=300
Z=-24.6(0.16%) | Like=-19.75..-13.33 [-19.7699..-19.6602] | it/evals=900/1524 eff=73.5294% N=300
Z=-24.0(0.29%) | Like=-19.02..-13.33 [-19.1043..-19.0200] | it/evals=929/1563 eff=73.5550% N=300
Z=-24.0(0.30%) | Like=-18.99..-13.33 [-18.9890..-18.9880]*| it/evals=930/1564 eff=73.5759% N=300
Mono-modal Volume: ~exp(-7.41) * Expected Volume: exp(-3.13) Quality: ok
index : +1.0| +2.2 *************** +3.4 | +5.0
amplitude: +1.0e-12| +3.2e-11 ************************* | +1.0e-10
Z=-23.8(0.36%) | Like=-18.81..-13.33 [-18.8117..-18.7755] | it/evals=938/1573 eff=73.6842% N=300
Z=-23.3(0.56%) | Like=-18.32..-13.33 [-18.3173..-18.2680] | it/evals=960/1598 eff=73.9599% N=300
Z=-22.8(0.98%) | Like=-17.83..-13.33 [-17.8425..-17.8303] | it/evals=987/1638 eff=73.7668% N=300
Z=-22.8(1.02%) | Like=-17.77..-13.33 [-17.7709..-17.7638]*| it/evals=990/1644 eff=73.6607% N=300
Mono-modal Volume: ~exp(-7.71) * Expected Volume: exp(-3.35) Quality: ok
index : +1.0| +2.3 ************* +3.3 | +5.0
amplitude: +1.0e-12| +3.4e-11 *********************** +7.9e-11| +1.0e-10
Z=-22.5(1.32%) | Like=-17.58..-13.33 [-17.5808..-17.5746]*| it/evals=1005/1664 eff=73.6804% N=300
Z=-22.3(1.64%) | Like=-17.34..-13.33 [-17.3435..-17.3087] | it/evals=1020/1682 eff=73.8061% N=300
Z=-21.8(2.48%) | Like=-16.95..-13.32 [-16.9486..-16.9350] | it/evals=1050/1721 eff=73.8916% N=300
Mono-modal Volume: ~exp(-8.02) * Expected Volume: exp(-3.57) Quality: ok
index : +1.0| +2.3 *********** +3.2 | +5.0
amplitude: +1.0e-12| +3.6e-11 ********************* +7.6e-11 | +1.0e-10
Z=-21.6(3.25%) | Like=-16.70..-13.32 [-16.6970..-16.6885]*| it/evals=1072/1751 eff=73.8801% N=300
Z=-21.5(3.59%) | Like=-16.62..-13.32 [-16.6160..-16.6033] | it/evals=1080/1761 eff=73.9220% N=300
Z=-21.1(5.00%) | Like=-16.25..-13.32 [-16.2500..-16.2201] | it/evals=1110/1798 eff=74.0988% N=300
Mono-modal Volume: ~exp(-8.21) * Expected Volume: exp(-3.80) Quality: ok
index : +1.0| +2.4 ********** +3.1 | +5.0
amplitude: +1.0e-12| +3.7e-11 ****************** +7.2e-11 | +1.0e-10
Z=-20.8(6.55%) | Like=-15.97..-13.32 [-15.9708..-15.9578] | it/evals=1139/1834 eff=74.2503% N=300
Z=-20.8(6.63%) | Like=-15.96..-13.32 [-15.9708..-15.9578] | it/evals=1140/1835 eff=74.2671% N=300
Z=-20.6(8.65%) | Like=-15.76..-13.32 [-15.7575..-15.7552]*| it/evals=1170/1873 eff=74.3802% N=300
Z=-20.3(10.90%) | Like=-15.53..-13.32 [-15.5254..-15.5216]*| it/evals=1199/1912 eff=74.3797% N=300
Z=-20.3(11.00%) | Like=-15.52..-13.32 [-15.5216..-15.5209]*| it/evals=1200/1913 eff=74.3955% N=300
Mono-modal Volume: ~exp(-8.40) * Expected Volume: exp(-4.02) Quality: ok
index : +1.0| +2.4 ********** +3.1 | +5.0
amplitude: +1.0e-12| +3.9e-11 **************** +7.0e-11 | +1.0e-10
Z=-20.3(11.48%) | Like=-15.46..-13.32 [-15.4606..-15.4420] | it/evals=1206/1922 eff=74.3527% N=300
Z=-20.1(13.70%) | Like=-15.24..-13.32 [-15.2417..-15.2411]*| it/evals=1230/1952 eff=74.4552% N=300
Z=-19.9(16.35%) | Like=-15.11..-13.32 [-15.1125..-15.1107]*| it/evals=1257/1992 eff=74.2908% N=300
Z=-19.9(16.66%) | Like=-15.07..-13.32 [-15.0688..-15.0627]*| it/evals=1260/1995 eff=74.3363% N=300
Mono-modal Volume: ~exp(-8.47) * Expected Volume: exp(-4.24) 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=-19.8(18.08%) | Like=-14.98..-13.32 [-14.9795..-14.9747]*| it/evals=1273/2013 eff=74.3141% N=300
Z=-19.7(19.82%) | Like=-14.90..-13.32 [-14.8978..-14.8925]*| it/evals=1290/2033 eff=74.4374% N=300
Z=-19.6(23.16%) | Like=-14.75..-13.32 [-14.7503..-14.7439]*| it/evals=1320/2072 eff=74.4921% N=300
Mono-modal Volume: ~exp(-8.80) * Expected Volume: exp(-4.47) Quality: ok
index : +1.0| +2.5 ******** +3.0 | +5.0
amplitude: +1.0e-12| +4.1e-11 ************* +6.6e-11 | +1.0e-10
Z=-19.5(25.51%) | Like=-14.64..-13.32 [-14.6439..-14.6364]*| it/evals=1340/2099 eff=74.4858% N=300
Z=-19.4(26.68%) | Like=-14.61..-13.32 [-14.6180..-14.6057] | it/evals=1350/2112 eff=74.5033% N=300
Z=-19.3(30.39%) | Like=-14.51..-13.32 [-14.5073..-14.4971] | it/evals=1380/2147 eff=74.7158% N=300
Mono-modal Volume: ~exp(-8.86) * Expected Volume: exp(-4.69) Quality: ok
index : +1.0| +2.5 ****** +3.0 | +5.0
amplitude: +1.0e-12| +4.2e-11 *********** +6.5e-11 | +1.0e-10
Z=-19.2(33.79%) | Like=-14.38..-13.32 [-14.3798..-14.3766]*| it/evals=1407/2184 eff=74.6815% N=300
Z=-19.2(34.17%) | Like=-14.37..-13.32 [-14.3730..-14.3660]*| it/evals=1410/2188 eff=74.6822% N=300
Z=-19.1(37.87%) | Like=-14.26..-13.32 [-14.2581..-14.2570]*| it/evals=1439/2228 eff=74.6369% N=300
Z=-19.1(38.00%) | Like=-14.26..-13.32 [-14.2570..-14.2552]*| it/evals=1440/2229 eff=74.6501% N=300
Z=-19.0(41.09%) | Like=-14.18..-13.31 [-14.1812..-14.1810]*| it/evals=1466/2269 eff=74.4540% N=300
Z=-19.0(41.59%) | Like=-14.18..-13.31 [-14.1778..-14.1762]*| it/evals=1470/2273 eff=74.5058% N=300
Mono-modal Volume: ~exp(-9.24) * Expected Volume: exp(-4.91) Quality: ok
index : +1.0| +2.5 ****** +3.0 | +5.0
amplitude: +1.0e-12| +4.4e-11 *********** +6.4e-11 | +1.0e-10
Z=-19.0(42.11%) | Like=-14.16..-13.31 [-14.1636..-14.1631]*| it/evals=1474/2277 eff=74.5574% N=300
Z=-18.9(45.39%) | Like=-14.11..-13.31 [-14.1098..-14.1046]*| it/evals=1500/2311 eff=74.5898% N=300
Z=-18.8(48.93%) | Like=-14.04..-13.31 [-14.0394..-14.0384]*| it/evals=1530/2348 eff=74.7070% N=300
Mono-modal Volume: ~exp(-9.44) * Expected Volume: exp(-5.14) 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=-18.8(50.20%) | Like=-14.02..-13.31 [-14.0206..-14.0159]*| it/evals=1541/2361 eff=74.7695% N=300
Z=-18.7(52.34%) | Like=-13.97..-13.31 [-13.9677..-13.9665]*| it/evals=1560/2389 eff=74.6769% N=300
Z=-18.7(55.32%) | Like=-13.92..-13.31 [-13.9158..-13.9114]*| it/evals=1587/2427 eff=74.6121% N=300
Z=-18.7(55.64%) | Like=-13.90..-13.31 [-13.9040..-13.9033]*| it/evals=1590/2430 eff=74.6479% N=300
Mono-modal Volume: ~exp(-9.78) * Expected Volume: exp(-5.36) 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=-18.6(57.60%) | Like=-13.87..-13.31 [-13.8710..-13.8617]*| it/evals=1608/2459 eff=74.4789% N=300
Z=-18.6(58.90%) | Like=-13.84..-13.31 [-13.8426..-13.8420]*| it/evals=1620/2474 eff=74.5170% N=300
Z=-18.6(61.62%) | Like=-13.81..-13.31 [-13.8102..-13.8094]*| it/evals=1647/2513 eff=74.4239% N=300
Z=-18.6(61.90%) | Like=-13.81..-13.31 [-13.8078..-13.8076]*| it/evals=1650/2517 eff=74.4249% N=300
Mono-modal Volume: ~exp(-9.83) * Expected Volume: exp(-5.58) Quality: ok
index : +1.0| +2.6 ***** +2.9 | +5.0
amplitude: +1.0e-12| +4.6e-11 ******** +6.1e-11 | +1.0e-10
Z=-18.5(64.26%) | Like=-13.78..-13.31 [-13.7764..-13.7716]*| it/evals=1675/2550 eff=74.4444% N=300
Z=-18.5(64.74%) | Like=-13.77..-13.31 [-13.7651..-13.7647]*| it/evals=1680/2555 eff=74.5011% N=300
Z=-18.5(67.48%) | Like=-13.73..-13.31 [-13.7330..-13.7320]*| it/evals=1710/2590 eff=74.6725% N=300
Z=-18.5(69.59%) | Like=-13.71..-13.31 [-13.7073..-13.7063]*| it/evals=1736/2630 eff=74.5064% N=300
Z=-18.5(69.91%) | Like=-13.70..-13.31 [-13.7043..-13.7038]*| it/evals=1740/2635 eff=74.5182% N=300
[ultranest] Explored until L=-1e+01
[ultranest] Likelihood function evaluations: 2636
[ultranest] logZ = -18.11 +- 0.08038
[ultranest] Effective samples strategy satisfied (ESS = 985.4, need >400)
[ultranest] Posterior uncertainty strategy is satisfied (KL: 0.46+-0.08 nat, need <0.50 nat)
[ultranest] Evidency uncertainty strategy is satisfied (dlogz=0.27, need <0.5)
[ultranest] logZ error budget: single: 0.11 bs:0.08 tail:0.26 total:0.27 required:<0.50
[ultranest] done iterating.
logZ = -18.093 +- 0.315
single instance: logZ = -18.093 +- 0.113
bootstrapped : logZ = -18.110 +- 0.176
tail : logZ = +- 0.262
insert order U test : converged: True correlation: inf iterations
index : 2.16 │ ▁▁ ▁▁▁▂▃▄▄▆▇▆▇▇▇▇▆▅▄▄▃▂▂▁▁▁▁▁▁▁ ▁ ▁ │3.59 2.74 +- 0.17
amplitude : 0.0000000000268│ ▁ ▁▁▁▁▁▂▃▃▄▇▅▇▇▇▇▆▆▅▄▃▃▂▂▁▂▁▁▁▁ ▁ ▁▁▁ │0.0000000000871 0.0000000000529 +- 0.0000000000077
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 40.331 seconds)
