Note
Go to the end to download the full example code or to run this example in your browser via Binder.
Bayesian analysis with nested sampling#
A demonstration of a Bayesian analysis using the nested sampling technique.
Context#
1. Bayesian analysis#
Bayesian inference uses prior knowledge, in the form of a prior distribution, in order to estimate posterior probabilities which we traditionally visualise in the form of corner plots. These distributions contain more information than a maximum likelihood fit as they reveal not only the “best model” but provide a more accurate representation of errors and correlation between parameters. In particular, non-Gaussian degeneracies are complex to estimate with a maximum likelihood approach.
2. Limitations of the Markov Chain Monte Carlo approach#
A well-known approach to estimate this posterior distribution is the Markov Chain Monte Carlo (MCMC). This uses an ensemble of walkers to produce a chain of samples that after a convergence period will reach a stationary state. Once convergence is reached, the successive elements of the chain are samples of the target posterior distribution. However, the weakness of the MCMC approach lies in the “Once convergence” part. If the walkers are started far from the best likelihood region, the convergence time can be long or never reached if the walkers fall in a local minima. The choice of the initialisation point can become critical for complex models with a high number of dimensions and the ability of these walkers to escape a local minimum or to accurately describe a complex likelihood space is not guaranteed.
3. Nested sampling approach#
To overcome these issues, the nested sampling (NS) algorithm has gained traction in physics and astronomy. It is a Monte Carlo algorithm for computing an integral of the likelihood function over the prior model parameter space introduced in Skilling, 2004. The method performs this integral by evolving a collection of points through the parameter space (see recent reviews from Ashton et al., 2022, and Buchner, 2023). Without going into too many details, one important specificity of the NS method is that it starts from the entire parameter space and evolves a collection of live points to map all minima (including multiple modes if any), whereas Markov Chain Monte Carlo methods require an initialisation point and the walkers will explore the local likelihood. The ability of these walkers to escape a local minimum or to accurately describe a complex likelihood space is not guaranteed. This is a fundamental difference with MCMC or Minuit which will only ever probe the vicinity along their minimisation paths and do not have an overview of the global likelihood landscape. The analysis using the NS framework is more CPU time consuming than a standard classical fit, but it provides the full posterior distribution for all parameters, which is out of reach with traditional fitting techniques (N*(N-1)/2 contour plots to generate). In addition, it is more robust to the choice of initialisation, requires less human intervention and is therefore readily integrated in pipeline analysis. In Gammapy, we used the NS implementation of the UltraNest package (see here for more information), one of the leading package in Astronomy (already used in Cosmology and in X-rays). For a nice visualisation of the NS method see here : sampling visualisation. And for a tutorial of UltraNest applied to X-ray fitting with concrete examples and questions see : BXA Tutorial.
Note: please cite UltraNest if used for a paper
If you are using the “UltraNest” library for a paper, please follow its citation scheme: Cite UltraNest.
Proposed approach#
In this example, we will perform a Bayesian analysis with multiple 1D spectra of the Crab nebula data and investigate their posterior distributions.
Setup#
As usual, we’ll start with some setup …
import matplotlib.pyplot as plt
import numpy as np
import astropy.units as u
from gammapy.datasets import Datasets
from gammapy.datasets import SpectrumDatasetOnOff
from gammapy.modeling.models import (
SkyModel,
UniformPrior,
LogUniformPrior,
)
from gammapy.modeling.sampler import Sampler
Loading the spectral datasets#
Here we will load a few Crab 1D spectral data for which we will do a fit.
Model definition#
Now we want to define the spectral model that will be fitted to the data. The Crab spectra will be fitted here with a simple powerlaw for simplicity.
model = SkyModel.create(spectral_model="pl", name="crab")
Warning
Priors definition: Unlike a traditional fit where priors on the parameters are optional, here it is inherent to the Bayesian approach and are therefore mandatory.
In this case we will set (min,max) prior that will define the
boundaries in which the sampling will be performed.
Note that it is usually recommended to use a LogUniformPrior for
the parameters that have a large amplitude range like the
amplitude parameter.
A UniformPrior means that the samples will be drawn with uniform
probability between the (min,max) values in the linear or log space
in the case of a LogUniformPrior.
model.spectral_model.amplitude.prior = LogUniformPrior(min=1e-12, max=1e-10)
model.spectral_model.index.prior = UniformPrior(min=1, max=5)
datasets.models = [model]
print(datasets.models)
DatasetModels
Component 0: SkyModel
Name : crab
Datasets names : None
Spectral model type : PowerLawSpectralModel
Spatial model type :
Temporal model type :
Parameters:
index : 2.000 +/- 0.00
amplitude : 1.00e-12 +/- 0.0e+00 1 / (TeV s cm2)
reference (frozen): 1.000 TeV
Defining the sampler and options#
As for the Fit object, the Sampler object can receive
different backend (although just one is available for now).
The Sampler comes with “reasonable” default parameters, but you can
change them via the sampler_opts dictionary.
Here is a short description of the most relevant parameters that you
could change :
live_points: minimum number of live points throughout the run. More points allow to discover multiple peaks if existing, but is slower. To test the Prior boundaries and for debugging, a lower number (~100) can be used before a production run with more points (~400 or more).frac_remain: the cut-off condition for the integration, set by the maximum allowed fraction of posterior mass left in the live points vs the dead points. High values (e.g., 0.5) are faster and can be used if the posterior distribution is a relatively simple shape. A low value (1e-1, 1e-2) is optimal for finding peaks, but slower.log_dir: directory where the output files will be stored. If set to None, no files will be written. If set to a string, a directory will be created containing the ongoing status of the run and final results. For time consuming analysis, it is highly recommended to use that option to monitor the run and restart it in case of a crash (withresume=True).
Important note: unlike the MCMC method, you don’t need to define the number of steps for which the sampler will run. The algorithm will automatically stop once a convergence criteria has been reached.
sampler_opts = {
"live_points": 300,
"frac_remain": 0.3,
"log_dir": None,
}
sampler = Sampler(backend="ultranest", sampler_opts=sampler_opts)
Next we can run the sampler on a given dataset. No options are accepted in the run method.
[ultranest] Sampling 300 live points from prior ...
Mono-modal Volume: ~exp(-4.09) * Expected Volume: exp(0.00) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|************** ********** **** ******** *** * *| +1.0e-10
Z=-inf(0.00%) | Like=-2983.33..-59.10 [-2983.3335..-328.6061] | it/evals=0/301 eff=0.0000% N=300
Z=-546.4(0.00%) | Like=-539.97..-59.10 [-2983.3335..-328.6061] | it/evals=22/324 eff=91.6667% N=300
Z=-537.0(0.00%) | Like=-531.82..-59.10 [-2983.3335..-328.6061] | it/evals=30/333 eff=90.9091% N=300
Z=-508.7(0.00%) | Like=-502.35..-59.10 [-2983.3335..-328.6061] | it/evals=51/356 eff=91.0714% N=300
Z=-500.9(0.00%) | Like=-494.17..-59.10 [-2983.3335..-328.6061] | it/evals=60/365 eff=92.3077% N=300
Mono-modal Volume: ~exp(-4.44) * Expected Volume: exp(-0.22) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|************** ********** **** *************** | +1.0e-10
Z=-489.1(0.00%) | Like=-476.60..-59.10 [-2983.3335..-328.6061] | it/evals=67/373 eff=91.7808% N=300
Z=-453.5(0.00%) | Like=-446.78..-59.10 [-2983.3335..-328.6061] | it/evals=88/397 eff=90.7216% N=300
Z=-451.2(0.00%) | Like=-445.41..-58.80 [-2983.3335..-328.6061] | it/evals=90/399 eff=90.9091% N=300
Z=-428.4(0.00%) | Like=-420.71..-58.80 [-2983.3335..-328.6061] | it/evals=110/422 eff=90.1639% N=300
Z=-417.0(0.00%) | Like=-410.69..-58.80 [-2983.3335..-328.6061] | it/evals=120/432 eff=90.9091% N=300
Mono-modal Volume: ~exp(-4.51) * Expected Volume: exp(-0.45) Quality: ok
index : +1.0| ***********************************************| +5.0
amplitude: +1.0e-12|****************************** *************** | +1.0e-10
Z=-404.1(0.00%) | Like=-398.79..-58.80 [-2983.3335..-328.6061] | it/evals=134/451 eff=88.7417% N=300
Z=-388.2(0.00%) | Like=-380.38..-58.80 [-2983.3335..-328.6061] | it/evals=150/471 eff=87.7193% N=300
Z=-361.9(0.00%) | Like=-355.88..-58.80 [-2983.3335..-328.6061] | it/evals=171/496 eff=87.2449% N=300
Z=-348.8(0.00%) | Like=-341.63..-58.80 [-2983.3335..-328.6061] | it/evals=180/506 eff=87.3786% N=300
Z=-333.7(0.00%) | Like=-327.52..-58.80 [-328.4588..-185.7884] | it/evals=198/530 eff=86.0870% N=300
Mono-modal Volume: ~exp(-4.51) Expected Volume: exp(-0.67) Quality: ok
index : +1.0| **********************************************| +5.0
amplitude: +1.0e-12| ***************************** **************** | +1.0e-10
Z=-323.6(0.00%) | Like=-317.45..-58.80 [-328.4588..-185.7884] | it/evals=210/546 eff=85.3659% N=300
Z=-304.8(0.00%) | Like=-298.58..-58.80 [-328.4588..-185.7884] | it/evals=228/569 eff=84.7584% N=300
Z=-291.7(0.00%) | Like=-285.34..-58.80 [-328.4588..-185.7884] | it/evals=240/585 eff=84.2105% N=300
Z=-281.9(0.00%) | Like=-276.26..-58.80 [-328.4588..-185.7884] | it/evals=256/608 eff=83.1169% 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=-271.9(0.00%) | Like=-265.71..-58.80 [-328.4588..-185.7884] | it/evals=268/624 eff=82.7160% N=300
Z=-270.9(0.00%) | Like=-264.91..-58.80 [-328.4588..-185.7884] | it/evals=270/626 eff=82.8221% N=300
Z=-250.0(0.00%) | Like=-243.15..-58.80 [-328.4588..-185.7884] | it/evals=292/649 eff=83.6676% N=300
Z=-240.9(0.00%) | Like=-234.30..-58.80 [-328.4588..-185.7884] | it/evals=300/657 eff=84.0336% N=300
Z=-224.4(0.00%) | Like=-218.14..-58.80 [-328.4588..-185.7884] | it/evals=319/680 eff=83.9474% N=300
Z=-220.1(0.00%) | Like=-214.42..-58.80 [-328.4588..-185.7884] | it/evals=330/694 eff=83.7563% 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=-217.6(0.00%) | Like=-211.31..-58.80 [-328.4588..-185.7884] | it/evals=335/700 eff=83.7500% N=300
Z=-209.9(0.00%) | Like=-203.69..-58.80 [-328.4588..-185.7884] | it/evals=354/723 eff=83.6879% N=300
Z=-207.8(0.00%) | Like=-202.25..-58.80 [-328.4588..-185.7884] | it/evals=360/730 eff=83.7209% N=300
Z=-199.3(0.00%) | Like=-192.99..-58.80 [-328.4588..-185.7884] | it/evals=378/753 eff=83.4437% N=300
Z=-194.1(0.00%) | Like=-188.81..-58.80 [-328.4588..-185.7884] | it/evals=390/768 eff=83.3333% N=300
Mono-modal Volume: ~exp(-5.69) * Expected Volume: exp(-1.34) Quality: ok
index : +1.0| *************************************** | +5.0
amplitude: +1.0e-12| ****************************************** | +1.0e-10
Z=-190.8(0.00%) | Like=-185.06..-58.80 [-185.6223..-134.0982] | it/evals=402/789 eff=82.2086% N=300
Z=-185.3(0.00%) | Like=-179.75..-58.80 [-185.6223..-134.0982] | it/evals=420/810 eff=82.3529% N=300
Z=-180.8(0.00%) | Like=-174.92..-58.80 [-185.6223..-134.0982] | it/evals=440/835 eff=82.2430% N=300
Z=-178.6(0.00%) | Like=-172.70..-58.80 [-185.6223..-134.0982] | it/evals=450/848 eff=82.1168% N=300
Z=-172.6(0.00%) | Like=-166.99..-58.80 [-185.6223..-134.0982] | it/evals=467/873 eff=81.5009% N=300
Mono-modal Volume: ~exp(-5.69) Expected Volume: exp(-1.56) Quality: ok
index : +1.0| *********************************** | +5.0
amplitude: +1.0e-12| ****************************************** | +1.0e-10
Z=-170.5(0.00%) | Like=-164.43..-58.80 [-185.6223..-134.0982] | it/evals=480/889 eff=81.4941% N=300
Z=-167.2(0.00%) | Like=-161.90..-58.80 [-185.6223..-134.0982] | it/evals=499/913 eff=81.4029% N=300
Z=-165.7(0.00%) | Like=-160.30..-58.80 [-185.6223..-134.0982] | it/evals=510/929 eff=81.0811% N=300
Z=-163.4(0.00%) | Like=-157.59..-58.80 [-185.6223..-134.0982] | it/evals=525/952 eff=80.5215% N=300
Mono-modal Volume: ~exp(-5.69) Expected Volume: exp(-1.79) Quality: ok
index : +1.0| ********************************* | +5.0
amplitude: +1.0e-12| ************************************ *** | +1.0e-10
Z=-159.7(0.00%) | Like=-153.70..-58.80 [-185.6223..-134.0982] | it/evals=540/969 eff=80.7175% N=300
Z=-155.5(0.00%) | Like=-148.79..-58.80 [-185.6223..-134.0982] | it/evals=556/993 eff=80.2309% N=300
Z=-151.1(0.00%) | Like=-144.36..-58.80 [-185.6223..-134.0982] | it/evals=569/1016 eff=79.4693% N=300
Z=-150.7(0.00%) | Like=-144.15..-58.80 [-185.6223..-134.0982] | it/evals=570/1017 eff=79.4979% N=300
Z=-147.5(0.00%) | Like=-141.70..-58.80 [-185.6223..-134.0982] | it/evals=585/1040 eff=79.0541% N=300
Z=-143.4(0.00%) | Like=-136.52..-58.80 [-185.6223..-134.0982] | it/evals=600/1058 eff=79.1557% N=300
Mono-modal Volume: ~exp(-5.75) * Expected Volume: exp(-2.01) Quality: ok
index : +1.0| **************************** +4.1 | +5.0
amplitude: +1.0e-12| *********************************** * | +1.0e-10
Z=-142.6(0.00%) | Like=-136.42..-58.80 [-185.6223..-134.0982] | it/evals=603/1062 eff=79.1339% N=300
Z=-138.8(0.00%) | Like=-132.43..-58.80 [-134.0587..-97.4002] | it/evals=620/1085 eff=78.9809% N=300
Z=-136.7(0.00%) | Like=-130.17..-58.80 [-134.0587..-97.4002] | it/evals=630/1095 eff=79.2453% N=300
Z=-132.3(0.00%) | Like=-125.77..-58.80 [-134.0587..-97.4002] | it/evals=648/1118 eff=79.2176% N=300
Z=-129.7(0.00%) | Like=-123.67..-58.80 [-134.0587..-97.4002] | it/evals=660/1134 eff=79.1367% N=300
Mono-modal Volume: ~exp(-6.42) * Expected Volume: exp(-2.23) Quality: ok
index : +1.0| ************************* +3.9 | +5.0
amplitude: +1.0e-12| ********************************** | +1.0e-10
Z=-127.3(0.00%) | Like=-120.17..-58.80 [-134.0587..-97.4002] | it/evals=670/1149 eff=78.9164% N=300
Z=-123.3(0.00%) | Like=-117.24..-58.80 [-134.0587..-97.4002] | it/evals=688/1173 eff=78.8087% N=300
Z=-123.1(0.00%) | Like=-116.58..-58.80 [-134.0587..-97.4002] | it/evals=690/1176 eff=78.7671% N=300
Z=-118.9(0.00%) | Like=-112.44..-58.80 [-134.0587..-97.4002] | it/evals=710/1199 eff=78.9766% N=300
Z=-117.3(0.00%) | Like=-111.00..-58.80 [-134.0587..-97.4002] | it/evals=720/1211 eff=79.0340% N=300
Z=-115.5(0.00%) | Like=-109.30..-58.80 [-134.0587..-97.4002] | it/evals=734/1234 eff=78.5867% N=300
Mono-modal Volume: ~exp(-6.50) * Expected Volume: exp(-2.46) Quality: ok
index : +1.0| +2.0 ********************** +3.7 | +5.0
amplitude: +1.0e-12| ****************************** | +1.0e-10
Z=-115.2(0.00%) | Like=-108.49..-58.80 [-134.0587..-97.4002] | it/evals=737/1239 eff=78.4878% N=300
Z=-112.7(0.00%) | Like=-106.48..-58.80 [-134.0587..-97.4002] | it/evals=750/1257 eff=78.3699% N=300
Z=-110.5(0.00%) | Like=-104.19..-58.80 [-134.0587..-97.4002] | it/evals=768/1280 eff=78.3673% N=300
Z=-108.9(0.00%) | Like=-102.12..-58.80 [-134.0587..-97.4002] | it/evals=780/1294 eff=78.4708% N=300
Z=-106.6(0.00%) | Like=-100.36..-58.80 [-134.0587..-97.4002] | it/evals=798/1319 eff=78.3121% N=300
Mono-modal Volume: ~exp(-6.77) * Expected Volume: exp(-2.68) Quality: ok
index : +1.0| +2.0 ******************** +3.6 | +5.0
amplitude: +1.0e-12| *************************** +7.6e-11 | +1.0e-10
Z=-106.0(0.00%) | Like=-100.00..-58.80 [-134.0587..-97.4002] | it/evals=804/1330 eff=78.0583% N=300
Z=-105.4(0.00%) | Like=-98.95..-58.80 [-134.0587..-97.4002] | it/evals=810/1342 eff=77.7351% N=300
Z=-103.7(0.00%) | Like=-97.86..-58.80 [-134.0587..-97.4002] | it/evals=827/1365 eff=77.6526% N=300
Z=-102.1(0.00%) | Like=-95.23..-58.80 [-97.3452..-78.5180] | it/evals=840/1381 eff=77.7058% N=300
Z=-100.4(0.00%) | Like=-94.21..-58.80 [-97.3452..-78.5180] | it/evals=854/1404 eff=77.3551% N=300
Z=-98.7(0.00%) | Like=-92.28..-58.80 [-97.3452..-78.5180] | it/evals=870/1427 eff=77.1961% N=300
Mono-modal Volume: ~exp(-6.94) * Expected Volume: exp(-2.90) Quality: ok
index : +1.0| +2.1 ***************** +3.5 | +5.0
amplitude: +1.0e-12| +2.4e-11 ************************* +7.4e-11 | +1.0e-10
Z=-98.6(0.00%) | Like=-92.08..-58.80 [-97.3452..-78.5180] | it/evals=871/1429 eff=77.1479% N=300
Z=-97.3(0.00%) | Like=-91.07..-58.80 [-97.3452..-78.5180] | it/evals=884/1453 eff=76.6696% N=300
Z=-96.0(0.00%) | Like=-89.70..-58.80 [-97.3452..-78.5180] | it/evals=899/1476 eff=76.4456% N=300
Z=-95.9(0.00%) | Like=-89.67..-58.80 [-97.3452..-78.5180] | it/evals=900/1477 eff=76.4656% N=300
Z=-94.2(0.00%) | Like=-87.54..-58.80 [-97.3452..-78.5180] | it/evals=916/1501 eff=76.2698% N=300
Z=-92.7(0.00%) | Like=-86.11..-58.80 [-97.3452..-78.5180] | it/evals=930/1522 eff=76.1047% N=300
Mono-modal Volume: ~exp(-7.30) * Expected Volume: exp(-3.13) Quality: ok
index : +1.0| +2.2 **************** +3.4 | +5.0
amplitude: +1.0e-12| +2.6e-11 ********************* +6.8e-11 | +1.0e-10
Z=-91.9(0.00%) | Like=-85.40..-58.80 [-97.3452..-78.5180] | it/evals=938/1534 eff=76.0130% N=300
Z=-90.2(0.00%) | Like=-83.85..-58.80 [-97.3452..-78.5180] | it/evals=958/1558 eff=76.1526% N=300
Z=-90.0(0.00%) | Like=-83.68..-58.80 [-97.3452..-78.5180] | it/evals=960/1560 eff=76.1905% N=300
Z=-88.4(0.00%) | Like=-82.06..-58.80 [-97.3452..-78.5180] | it/evals=979/1583 eff=76.3055% N=300
Z=-87.8(0.00%) | Like=-81.74..-58.80 [-97.3452..-78.5180] | it/evals=990/1595 eff=76.4479% N=300
Mono-modal Volume: ~exp(-7.30) Expected Volume: exp(-3.35) Quality: ok
index : +1.0| +2.2 ************** +3.3 | +5.0
amplitude: +1.0e-12| +2.8e-11 ******************** +6.7e-11 | +1.0e-10
Z=-87.0(0.00%) | Like=-80.96..-58.80 [-97.3452..-78.5180] | it/evals=1005/1618 eff=76.2519% N=300
Z=-86.3(0.00%) | Like=-79.95..-58.80 [-97.3452..-78.5180] | it/evals=1020/1638 eff=76.2332% N=300
Z=-84.9(0.00%) | Like=-78.49..-58.80 [-78.4958..-68.3673] | it/evals=1041/1660 eff=76.5441% N=300
Z=-84.3(0.00%) | Like=-77.76..-58.80 [-78.4958..-68.3673] | it/evals=1050/1676 eff=76.3081% N=300
Z=-83.5(0.00%) | Like=-77.17..-58.80 [-78.4958..-68.3673] | it/evals=1065/1702 eff=75.9629% N=300
Mono-modal Volume: ~exp(-7.33) * Expected Volume: exp(-3.57) Quality: ok
index : +1.0| +2.2 ************* +3.2 | +5.0
amplitude: +1.0e-12| +2.9e-11 ****************** +6.4e-11 | +1.0e-10
Z=-83.1(0.00%) | Like=-76.79..-58.80 [-78.4958..-68.3673] | it/evals=1072/1712 eff=75.9207% N=300
Z=-82.7(0.00%) | Like=-76.41..-58.80 [-78.4958..-68.3673] | it/evals=1080/1726 eff=75.7363% N=300
Z=-81.7(0.00%) | Like=-75.56..-58.80 [-78.4958..-68.3673] | it/evals=1102/1748 eff=76.1050% N=300
Z=-81.4(0.00%) | Like=-75.30..-58.80 [-78.4958..-68.3673] | it/evals=1110/1765 eff=75.7679% N=300
Z=-81.0(0.00%) | Like=-74.87..-58.80 [-78.4958..-68.3673] | it/evals=1121/1788 eff=75.3360% N=300
Mono-modal Volume: ~exp(-7.75) * Expected Volume: exp(-3.80) Quality: ok
index : +1.0| +2.3 *********** +3.2 | +5.0
amplitude: +1.0e-12| +3.0e-11 ***************** +6.2e-11 | +1.0e-10
Z=-80.2(0.00%) | Like=-73.52..-58.80 [-78.4958..-68.3673] | it/evals=1139/1812 eff=75.3307% N=300
Z=-80.1(0.00%) | Like=-73.49..-58.80 [-78.4958..-68.3673] | it/evals=1140/1814 eff=75.2972% N=300
Z=-79.2(0.00%) | Like=-72.38..-58.80 [-78.4958..-68.3673] | it/evals=1156/1837 eff=75.2115% N=300
Z=-78.4(0.00%) | Like=-71.77..-58.80 [-78.4958..-68.3673] | it/evals=1170/1854 eff=75.2896% N=300
Z=-77.8(0.00%) | Like=-71.31..-58.80 [-78.4958..-68.3673] | it/evals=1183/1878 eff=74.9683% N=300
Z=-77.0(0.00%) | Like=-70.30..-58.80 [-78.4958..-68.3673] | it/evals=1200/1899 eff=75.0469% N=300
Mono-modal Volume: ~exp(-8.15) * 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=-76.7(0.00%) | Like=-70.09..-58.80 [-78.4958..-68.3673] | it/evals=1206/1906 eff=75.0934% N=300
Z=-75.9(0.00%) | Like=-69.29..-58.80 [-78.4958..-68.3673] | it/evals=1225/1929 eff=75.1995% N=300
Z=-75.7(0.00%) | Like=-69.05..-58.80 [-78.4958..-68.3673] | it/evals=1230/1934 eff=75.2754% N=300
Z=-74.9(0.01%) | Like=-68.41..-58.80 [-78.4958..-68.3673] | it/evals=1249/1958 eff=75.3317% N=300
Z=-74.5(0.01%) | Like=-67.95..-58.80 [-68.3204..-65.4513] | it/evals=1260/1972 eff=75.3589% N=300
Mono-modal Volume: ~exp(-8.15) Expected Volume: exp(-4.24) Quality: ok
index : +1.0| +2.4 ********* +3.0 | +5.0
amplitude: +1.0e-12| +3.3e-11 ************* +5.8e-11 | +1.0e-10
Z=-74.0(0.01%) | Like=-67.56..-58.80 [-68.3204..-65.4513] | it/evals=1276/1993 eff=75.3692% N=300
Z=-73.6(0.02%) | Like=-67.10..-58.76 [-68.3204..-65.4513] | it/evals=1290/2009 eff=75.4827% N=300
Z=-73.0(0.04%) | Like=-66.40..-58.76 [-68.3204..-65.4513] | it/evals=1309/2032 eff=75.5774% N=300
Z=-72.7(0.05%) | Like=-66.00..-58.76 [-68.3204..-65.4513] | it/evals=1320/2045 eff=75.6447% N=300
Z=-72.3(0.08%) | Like=-65.71..-58.76 [-68.3204..-65.4513] | it/evals=1334/2068 eff=75.4525% N=300
Mono-modal Volume: ~exp(-8.30) * 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=-65.56..-58.76 [-68.3204..-65.4513] | it/evals=1340/2075 eff=75.4930% N=300
Z=-71.8(0.12%) | Like=-65.34..-58.76 [-65.4442..-65.1405] | it/evals=1350/2086 eff=75.5879% N=300
Z=-71.4(0.19%) | Like=-64.82..-58.76 [-64.9084..-64.8221] | it/evals=1369/2109 eff=75.6772% N=300
Z=-71.1(0.25%) | Like=-64.57..-58.76 [-64.6452..-64.5670] | it/evals=1380/2126 eff=75.5750% N=300
Z=-70.8(0.34%) | Like=-64.34..-58.76 [-64.3556..-64.3353] | it/evals=1394/2149 eff=75.3921% N=300
Mono-modal Volume: ~exp(-8.60) * Expected Volume: exp(-4.69) Quality: ok
index : +1.0| +2.4 ******** +3.0 | +5.0
amplitude: +1.0e-12| +3.6e-11 *********** +5.5e-11 | +1.0e-10
Z=-70.6(0.46%) | Like=-64.12..-58.76 [-64.1160..-64.0800] | it/evals=1407/2170 eff=75.2406% N=300
Z=-70.5(0.48%) | Like=-64.07..-58.76 [-64.0663..-64.0631]*| it/evals=1410/2174 eff=75.2401% N=300
Z=-70.2(0.67%) | Like=-63.78..-58.76 [-63.7750..-63.7514] | it/evals=1429/2197 eff=75.3295% N=300
Z=-70.0(0.80%) | Like=-63.65..-58.76 [-63.6940..-63.6549] | it/evals=1440/2215 eff=75.1958% N=300
Z=-69.8(1.01%) | Like=-63.47..-58.76 [-63.4857..-63.4719] | it/evals=1455/2238 eff=75.0774% N=300
Z=-69.6(1.28%) | Like=-63.22..-58.76 [-63.2220..-63.2099] | it/evals=1470/2259 eff=75.0383% N=300
Mono-modal Volume: ~exp(-8.79) * 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=-69.5(1.33%) | Like=-63.14..-58.76 [-63.1396..-63.1304]*| it/evals=1474/2263 eff=75.0891% N=300
Z=-69.3(1.64%) | Like=-62.88..-58.76 [-62.8798..-62.8294] | it/evals=1491/2285 eff=75.1134% N=300
Z=-69.1(1.90%) | Like=-62.68..-58.76 [-62.6956..-62.6825] | it/evals=1500/2298 eff=75.0751% N=300
Z=-69.0(2.27%) | Like=-62.54..-58.76 [-62.5742..-62.5404] | it/evals=1512/2324 eff=74.7036% N=300
Z=-68.8(2.71%) | Like=-62.31..-58.76 [-62.3341..-62.3100] | it/evals=1525/2347 eff=74.4993% N=300
Z=-68.7(2.90%) | Like=-62.27..-58.76 [-62.2838..-62.2686] | it/evals=1530/2353 eff=74.5251% N=300
Mono-modal Volume: ~exp(-8.79) 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=-68.5(3.53%) | Like=-62.12..-58.76 [-62.1242..-62.1242]*| it/evals=1545/2375 eff=74.4578% N=300
Z=-68.4(4.13%) | Like=-61.97..-58.76 [-61.9682..-61.9679]*| it/evals=1560/2394 eff=74.4986% N=300
Z=-68.2(4.93%) | Like=-61.75..-58.76 [-61.7664..-61.7515] | it/evals=1578/2416 eff=74.5747% N=300
Z=-68.1(5.55%) | Like=-61.65..-58.76 [-61.6498..-61.6473]*| it/evals=1590/2434 eff=74.5080% N=300
Z=-67.9(6.41%) | Like=-61.52..-58.76 [-61.5210..-61.5194]*| it/evals=1606/2458 eff=74.4208% N=300
Mono-modal Volume: ~exp(-9.13) * 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=-67.9(6.51%) | Like=-61.51..-58.76 [-61.5121..-61.5101]*| it/evals=1608/2462 eff=74.3756% N=300
Z=-67.8(7.21%) | Like=-61.36..-58.76 [-61.3724..-61.3613] | it/evals=1620/2475 eff=74.4828% N=300
Z=-67.6(8.58%) | Like=-61.20..-58.76 [-61.1984..-61.1907]*| it/evals=1639/2498 eff=74.5678% N=300
Z=-67.5(9.31%) | Like=-61.09..-58.76 [-61.0908..-61.0688] | it/evals=1650/2514 eff=74.5257% N=300
Z=-67.4(10.83%) | Like=-60.92..-58.76 [-60.9220..-60.9158]*| it/evals=1669/2536 eff=74.6422% N=300
Mono-modal Volume: ~exp(-10.05) * 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=-67.3(11.33%) | Like=-60.88..-58.76 [-60.8764..-60.8731]*| it/evals=1675/2547 eff=74.5438% N=300
Z=-67.3(11.78%) | Like=-60.85..-58.76 [-60.8481..-60.8442]*| it/evals=1680/2552 eff=74.6004% N=300
Z=-67.2(13.34%) | Like=-60.69..-58.76 [-60.6939..-60.6929]*| it/evals=1696/2574 eff=74.5822% N=300
Z=-67.1(14.63%) | Like=-60.65..-58.76 [-60.6463..-60.6266] | it/evals=1710/2591 eff=74.6399% N=300
Z=-66.9(16.22%) | Like=-60.54..-58.76 [-60.5436..-60.5313] | it/evals=1726/2616 eff=74.5250% N=300
Z=-66.9(17.39%) | Like=-60.46..-58.76 [-60.4552..-60.4545]*| it/evals=1740/2634 eff=74.5501% N=300
Mono-modal Volume: ~exp(-10.05) 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=-66.8(19.08%) | Like=-60.37..-58.76 [-60.3929..-60.3742] | it/evals=1757/2654 eff=74.6389% N=300
Z=-66.7(20.26%) | Like=-60.33..-58.75 [-60.3255..-60.3224]*| it/evals=1770/2672 eff=74.6206% N=300
Z=-66.6(21.79%) | Like=-60.22..-58.75 [-60.2166..-60.2101]*| it/evals=1785/2696 eff=74.4992% N=300
Z=-66.5(23.57%) | Like=-60.13..-58.75 [-60.1332..-60.1281]*| it/evals=1800/2717 eff=74.4725% N=300
Mono-modal Volume: ~exp(-10.17) * Expected Volume: exp(-6.03) Quality: ok
index : +1.0| +2.5 **** +2.8 | +5.0
amplitude: +1.0e-12| +4.0e-11 ****** +5.0e-11 | +1.0e-10
Z=-66.5(24.68%) | Like=-60.11..-58.75 [-60.1082..-60.1040]*| it/evals=1809/2731 eff=74.4138% N=300
Z=-66.4(27.34%) | Like=-60.04..-58.75 [-60.0432..-60.0414]*| it/evals=1830/2753 eff=74.6025% N=300
Z=-66.3(29.20%) | Like=-59.99..-58.75 [-59.9939..-59.9916]*| it/evals=1847/2775 eff=74.6263% N=300
Z=-66.3(30.70%) | Like=-59.93..-58.75 [-59.9321..-59.9305]*| it/evals=1860/2789 eff=74.7288% N=300
Mono-modal Volume: ~exp(-10.35) * 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=-66.2(32.73%) | Like=-59.88..-58.75 [-59.8802..-59.8777]*| it/evals=1876/2812 eff=74.6815% N=300
Z=-66.2(34.34%) | Like=-59.81..-58.75 [-59.8127..-59.8114]*| it/evals=1890/2830 eff=74.7036% N=300
Z=-66.1(36.68%) | Like=-59.75..-58.75 [-59.7548..-59.7459]*| it/evals=1909/2852 eff=74.8041% N=300
Z=-66.1(38.03%) | Like=-59.71..-58.75 [-59.7150..-59.7095]*| it/evals=1920/2870 eff=74.7082% N=300
Z=-66.0(39.79%) | Like=-59.68..-58.75 [-59.6763..-59.6758]*| it/evals=1934/2893 eff=74.5854% N=300
Mono-modal Volume: ~exp(-10.82) * 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=-66.0(40.92%) | Like=-59.63..-58.75 [-59.6347..-59.6297]*| it/evals=1943/2903 eff=74.6446% N=300
Z=-66.0(41.78%) | Like=-59.62..-58.75 [-59.6163..-59.6113]*| it/evals=1950/2911 eff=74.6840% N=300
Z=-65.9(44.13%) | Like=-59.55..-58.75 [-59.5540..-59.5516]*| it/evals=1969/2935 eff=74.7249% N=300
Z=-65.9(45.52%) | Like=-59.52..-58.75 [-59.5179..-59.5151]*| it/evals=1980/2951 eff=74.6888% N=300
Z=-65.8(47.82%) | Like=-59.48..-58.75 [-59.4778..-59.4707]*| it/evals=1999/2974 eff=74.7569% N=300
Mono-modal Volume: ~exp(-10.94) * Expected Volume: exp(-6.70) Quality: ok
index : +1.0| +2.6 **** +2.8 | +5.0
amplitude: +1.0e-12| +4.1e-11 **** +4.8e-11 | +1.0e-10
Z=-65.8(49.16%) | Like=-59.45..-58.75 [-59.4500..-59.4450]*| it/evals=2010/2989 eff=74.7490% N=300
Z=-65.8(51.51%) | Like=-59.41..-58.75 [-59.4149..-59.4139]*| it/evals=2030/3011 eff=74.8801% N=300
Z=-65.8(52.61%) | Like=-59.40..-58.75 [-59.4046..-59.4014]*| it/evals=2040/3026 eff=74.8349% N=300
Z=-65.7(54.66%) | Like=-59.36..-58.75 [-59.3637..-59.3635]*| it/evals=2058/3049 eff=74.8636% N=300
Z=-65.7(56.03%) | Like=-59.34..-58.75 [-59.3370..-59.3351]*| it/evals=2070/3062 eff=74.9457% N=300
Mono-modal Volume: ~exp(-10.94) 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=-65.7(57.72%) | Like=-59.31..-58.75 [-59.3079..-59.3016]*| it/evals=2086/3082 eff=74.9820% N=300
Z=-65.6(59.26%) | Like=-59.28..-58.75 [-59.2819..-59.2803]*| it/evals=2100/3100 eff=75.0000% N=300
Z=-65.6(60.65%) | Like=-59.27..-58.75 [-59.2652..-59.2639]*| it/evals=2114/3123 eff=74.8849% N=300
Z=-65.6(62.22%) | Like=-59.23..-58.75 [-59.2315..-59.2304]*| it/evals=2130/3146 eff=74.8419% N=300
Mono-modal Volume: ~exp(-10.94) 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=-65.6(63.93%) | Like=-59.20..-58.75 [-59.2009..-59.1998]*| it/evals=2148/3168 eff=74.8954% N=300
Z=-65.5(65.07%) | Like=-59.18..-58.75 [-59.1823..-59.1823]*| it/evals=2160/3186 eff=74.8441% N=300
Z=-65.5(66.56%) | Like=-59.16..-58.75 [-59.1586..-59.1569]*| it/evals=2176/3209 eff=74.8023% N=300
Z=-65.5(67.60%) | Like=-59.14..-58.75 [-59.1418..-59.1413]*| it/evals=2188/3232 eff=74.6248% N=300
Z=-65.5(67.77%) | Like=-59.14..-58.75 [-59.1405..-59.1391]*| it/evals=2190/3237 eff=74.5659% N=300
Z=-65.5(68.91%) | Like=-59.13..-58.75 [-59.1281..-59.1250]*| it/evals=2203/3260 eff=74.4257% N=300
Mono-modal Volume: ~exp(-11.62) * 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=-65.5(69.58%) | Like=-59.12..-58.75 [-59.1159..-59.1159]*| it/evals=2211/3277 eff=74.2694% N=300
[ultranest] Explored until L=-6e+01
[ultranest] Likelihood function evaluations: 3286
[ultranest] logZ = -65.09 +- 0.1192
[ultranest] Effective samples strategy satisfied (ESS = 991.4, need >400)
[ultranest] Posterior uncertainty strategy is satisfied (KL: 0.46+-0.06 nat, need <0.50 nat)
[ultranest] Evidency uncertainty strategy is satisfied (dlogz=0.29, 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 = -65.110 +- 0.445
single instance: logZ = -65.110 +- 0.134
bootstrapped : logZ = -65.088 +- 0.359
tail : logZ = +- 0.262
insert order U test : converged: True correlation: inf iterations
index : 2.365 │ ▁ ▁▁▁▁▁▂▂▃▃▄▄▅▄▇▆▅▇▆▅▄▄▂▂▂▁▁▁▁▁▁▁▁ ▁ │3.043 2.668 +- 0.085
amplitude : 0.0000000000337│ ▁▁▁▁▁▁▁▁▂▃▃▃▄▅▅▇▇▅▆▆▆▄▄▃▂▃▂▁▁▁▁▁▁▁▁▁▁ │0.0000000000562 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 : 2.668 +/- 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 dictionary.
The most “standard” information about the posterior distributions can
be found in :
print(result_joint.sampler_results["posterior"])
{'mean': [2.6678145620190823, 4.433451524494846e-11], 'stdev': [0.0850830143047406, 3.0448889024953076e-12], 'median': [2.6691791568620538, 4.4170433592618776e-11], 'errlo': [2.5794384899868144, 4.1267619825642644e-11], 'errup': [2.7508985883148744, 4.743464688155637e-11], 'information_gain_bits': [2.710800512445947, 3.0992023377781215]}
Besides mean, errors, etc, an interesting value is the
information gain which estimates how much the posterior
distribution has shrunk with respect to the prior (i.e. how much
we’ve learned). A value < 1 means that the parameter is poorly
constrained within the prior range (we haven’t learned much with respect to our prior assumption).
For a physical 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(-4.06) * Expected Volume: exp(0.00) Quality: ok
index : +1.0|***************** ******************************| +5.0
amplitude: +1.0e-12|***************************** ****** ******* **| +1.0e-10
Z=-inf(0.00%) | Like=-1157.36..-21.89 [-1157.3590..-113.8723] | it/evals=0/301 eff=0.0000% N=300
Z=-176.3(0.00%) | Like=-171.52..-21.89 [-1157.3590..-113.8723] | it/evals=30/333 eff=90.9091% N=300
Z=-165.6(0.00%) | Like=-160.35..-21.89 [-1157.3590..-113.8723] | it/evals=60/370 eff=85.7143% N=300
Mono-modal Volume: ~exp(-4.06) Expected Volume: exp(-0.22) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|***************************** **** * **** ** **| +1.0e-10
Z=-153.4(0.00%) | Like=-149.04..-21.89 [-1157.3590..-113.8723] | it/evals=87/407 eff=81.3084% N=300
Z=-152.8(0.00%) | Like=-148.14..-21.89 [-1157.3590..-113.8723] | it/evals=90/410 eff=81.8182% N=300
Z=-141.6(0.00%) | Like=-136.83..-21.89 [-1157.3590..-113.8723] | it/evals=120/444 eff=83.3333% N=300
Mono-modal Volume: ~exp(-4.06) Expected Volume: exp(-0.45) Quality: ok
index : +1.0| ***********************************************| +5.0
amplitude: +1.0e-12|********************************** * *** *** **| +1.0e-10
Z=-130.7(0.00%) | Like=-125.41..-21.89 [-1157.3590..-113.8723] | it/evals=150/478 eff=84.2697% N=300
Z=-123.4(0.00%) | Like=-118.49..-21.89 [-1157.3590..-113.8723] | it/evals=175/522 eff=78.8288% N=300
Z=-122.5(0.00%) | Like=-117.53..-21.31 [-1157.3590..-113.8723] | it/evals=180/528 eff=78.9474% N=300
Mono-modal Volume: ~exp(-4.87) * Expected Volume: exp(-0.67) Quality: ok
index : +1.0| **********************************************| +5.0
amplitude: +1.0e-12| *********************************** *** *** **| +1.0e-10
Z=-117.9(0.00%) | Like=-112.69..-21.31 [-113.7174..-72.3857] | it/evals=201/553 eff=79.4466% N=300
Z=-115.2(0.00%) | Like=-109.81..-21.31 [-113.7174..-72.3857] | it/evals=210/566 eff=78.9474% N=300
Z=-104.0(0.00%) | Like=-98.40..-21.31 [-113.7174..-72.3857] | it/evals=240/605 eff=78.6885% N=300
Mono-modal Volume: ~exp(-4.93) * Expected Volume: exp(-0.89) Quality: ok
index : +1.0| ********************************************| +5.0
amplitude: +1.0e-12| ********************************** **** *** | +1.0e-10
Z=-96.9(0.00%) | Like=-91.93..-21.31 [-113.7174..-72.3857] | it/evals=268/643 eff=78.1341% N=300
Z=-96.6(0.00%) | Like=-91.71..-21.31 [-113.7174..-72.3857] | it/evals=270/645 eff=78.2609% N=300
Z=-90.6(0.00%) | Like=-84.90..-21.31 [-113.7174..-72.3857] | it/evals=300/685 eff=77.9221% N=300
Z=-84.5(0.00%) | Like=-79.52..-20.50 [-113.7174..-72.3857] | it/evals=330/722 eff=78.1991% N=300
Mono-modal Volume: ~exp(-5.08) * Expected Volume: exp(-1.12) Quality: ok
index : +1.0| *******************************************| +5.0
amplitude: +1.0e-12| ******************************* * **** * | +1.0e-10
Z=-83.8(0.00%) | Like=-78.87..-20.50 [-113.7174..-72.3857] | it/evals=335/727 eff=78.4543% N=300
Z=-79.8(0.00%) | Like=-74.45..-20.50 [-113.7174..-72.3857] | it/evals=360/762 eff=77.9221% N=300
Z=-77.0(0.00%) | Like=-72.38..-20.50 [-72.3813..-50.5750] | it/evals=381/805 eff=75.4455% N=300
Z=-75.8(0.00%) | Like=-70.83..-20.50 [-72.3813..-50.5750] | it/evals=390/816 eff=75.5814% N=300
Mono-modal Volume: ~exp(-5.34) * Expected Volume: exp(-1.34) Quality: ok
index : +1.0| *************************************** | +5.0
amplitude: +1.0e-12| ******************************* **** | +1.0e-10
Z=-73.5(0.00%) | Like=-68.40..-20.50 [-72.3813..-50.5750] | it/evals=402/835 eff=75.1402% N=300
Z=-71.4(0.00%) | Like=-66.67..-20.50 [-72.3813..-50.5750] | it/evals=420/858 eff=75.2688% N=300
Z=-68.9(0.00%) | Like=-63.96..-20.50 [-72.3813..-50.5750] | it/evals=447/900 eff=74.5000% N=300
Z=-68.6(0.00%) | Like=-63.67..-20.50 [-72.3813..-50.5750] | it/evals=450/906 eff=74.2574% N=300
Mono-modal Volume: ~exp(-5.83) * Expected Volume: exp(-1.56) Quality: ok
index : +1.0| ********************************* | +5.0
amplitude: +1.0e-12| ********************************** +7.9e-11| +1.0e-10
Z=-66.4(0.00%) | Like=-61.10..-20.50 [-72.3813..-50.5750] | it/evals=469/935 eff=73.8583% N=300
Z=-65.1(0.00%) | Like=-59.80..-20.50 [-72.3813..-50.5750] | it/evals=480/950 eff=73.8462% N=300
Z=-62.1(0.00%) | Like=-57.10..-20.50 [-72.3813..-50.5750] | it/evals=509/992 eff=73.5549% N=300
Z=-62.0(0.00%) | Like=-57.09..-20.50 [-72.3813..-50.5750] | it/evals=510/993 eff=73.5931% N=300
Mono-modal Volume: ~exp(-5.83) Expected Volume: exp(-1.79) Quality: ok
index : +1.0| ****************************** +4.1 | +5.0
amplitude: +1.0e-12| ******************************* +7.1e-11 | +1.0e-10
Z=-60.0(0.00%) | Like=-55.11..-20.50 [-72.3813..-50.5750] | it/evals=536/1034 eff=73.0245% N=300
Z=-59.7(0.00%) | Like=-54.69..-20.50 [-72.3813..-50.5750] | it/evals=540/1038 eff=73.1707% N=300
Z=-57.7(0.00%) | Like=-52.76..-20.50 [-72.3813..-50.5750] | it/evals=567/1080 eff=72.6923% N=300
Z=-57.5(0.00%) | Like=-52.57..-20.50 [-72.3813..-50.5750] | it/evals=570/1084 eff=72.7041% N=300
Z=-55.7(0.00%) | Like=-50.29..-20.50 [-50.5614..-38.5327] | it/evals=599/1126 eff=72.5182% N=300
Z=-55.6(0.00%) | Like=-50.18..-20.50 [-50.5614..-38.5327] | it/evals=600/1127 eff=72.5514% N=300
Mono-modal Volume: ~exp(-6.08) * Expected Volume: exp(-2.01) Quality: ok
index : +1.0| *************************** +3.9 | +5.0
amplitude: +1.0e-12| ***************************** +7.0e-11 | +1.0e-10
Z=-55.3(0.00%) | Like=-50.11..-20.50 [-50.5614..-38.5327] | it/evals=603/1130 eff=72.6506% N=300
Z=-53.4(0.00%) | Like=-48.26..-20.50 [-50.5614..-38.5327] | it/evals=630/1167 eff=72.6644% N=300
Z=-51.2(0.00%) | Like=-45.70..-20.50 [-50.5614..-38.5327] | it/evals=657/1211 eff=72.1186% N=300
Z=-50.9(0.00%) | Like=-45.40..-20.50 [-50.5614..-38.5327] | it/evals=660/1214 eff=72.2101% N=300
Mono-modal Volume: ~exp(-6.43) * Expected Volume: exp(-2.23) Quality: ok
index : +1.0| ************************ +3.7 | +5.0
amplitude: +1.0e-12| ************************** +6.6e-11 | +1.0e-10
Z=-50.0(0.00%) | Like=-44.44..-20.50 [-50.5614..-38.5327] | it/evals=670/1225 eff=72.4324% N=300
Z=-48.5(0.00%) | Like=-43.06..-20.50 [-50.5614..-38.5327] | it/evals=690/1256 eff=72.1757% N=300
Z=-46.8(0.00%) | Like=-41.57..-20.50 [-50.5614..-38.5327] | it/evals=720/1299 eff=72.0721% N=300
Mono-modal Volume: ~exp(-6.57) * Expected Volume: exp(-2.46) Quality: ok
index : +1.0| ********************* +3.5 | +5.0
amplitude: +1.0e-12| *********************** * +6.3e-11 | +1.0e-10
Z=-45.9(0.00%) | Like=-40.86..-20.50 [-50.5614..-38.5327] | it/evals=737/1328 eff=71.6926% N=300
Z=-45.4(0.00%) | Like=-40.22..-20.50 [-50.5614..-38.5327] | it/evals=750/1346 eff=71.7017% N=300
Z=-44.1(0.00%) | Like=-39.00..-20.50 [-50.5614..-38.5327] | it/evals=780/1385 eff=71.8894% N=300
Mono-modal Volume: ~exp(-6.57) Expected Volume: exp(-2.68) Quality: ok
index : +1.0| +2.0 ******************* +3.5 | +5.0
amplitude: +1.0e-12| *********************** +6.1e-11 | +1.0e-10
Z=-42.8(0.00%) | Like=-37.29..-20.50 [-38.5050..-29.7963] | it/evals=810/1419 eff=72.3861% N=300
Z=-41.3(0.00%) | Like=-35.74..-20.50 [-38.5050..-29.7963] | it/evals=839/1462 eff=72.2031% N=300
Z=-41.2(0.00%) | Like=-35.66..-20.50 [-38.5050..-29.7963] | it/evals=840/1463 eff=72.2270% N=300
Z=-40.1(0.00%) | Like=-34.62..-20.50 [-38.5050..-29.7963] | it/evals=863/1505 eff=71.6183% N=300
Z=-39.8(0.00%) | Like=-34.28..-20.50 [-38.5050..-29.7963] | it/evals=870/1516 eff=71.5461% N=300
Mono-modal Volume: ~exp(-6.57) 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=-38.6(0.00%) | Like=-32.92..-20.50 [-38.5050..-29.7963] | it/evals=895/1554 eff=71.3716% N=300
Z=-38.4(0.00%) | Like=-32.77..-20.50 [-38.5050..-29.7963] | it/evals=900/1559 eff=71.4853% N=300
Z=-37.3(0.00%) | Like=-32.06..-20.50 [-38.5050..-29.7963] | it/evals=928/1600 eff=71.3846% N=300
Z=-37.3(0.00%) | Like=-31.98..-20.50 [-38.5050..-29.7963] | it/evals=930/1602 eff=71.4286% N=300
Mono-modal Volume: ~exp(-6.57) Expected Volume: exp(-3.13) Quality: ok
index : +1.0| +2.0 *************** +3.2 | +5.0
amplitude: +1.0e-12| ****************** +5.4e-11 | +1.0e-10
Z=-36.6(0.00%) | Like=-30.97..-20.50 [-38.5050..-29.7963] | it/evals=949/1640 eff=70.8209% N=300
Z=-36.2(0.00%) | Like=-30.84..-20.50 [-38.5050..-29.7963] | it/evals=960/1655 eff=70.8487% N=300
Z=-35.6(0.01%) | Like=-29.99..-20.50 [-38.5050..-29.7963] | it/evals=985/1696 eff=70.5587% N=300
Z=-35.4(0.01%) | Like=-29.81..-20.50 [-38.5050..-29.7963] | it/evals=990/1706 eff=70.4125% N=300
Mono-modal Volume: ~exp(-7.23) * Expected Volume: exp(-3.35) Quality: ok
index : +1.0| +2.1 ************** +3.2 | +5.0
amplitude: +1.0e-12| **************** +5.2e-11 | +1.0e-10
Z=-34.9(0.01%) | Like=-29.40..-20.50 [-29.7811..-27.3383] | it/evals=1005/1731 eff=70.2306% N=300
Z=-34.5(0.02%) | Like=-28.93..-20.50 [-29.7811..-27.3383] | it/evals=1020/1746 eff=70.5394% N=300
Z=-33.8(0.05%) | Like=-28.23..-20.50 [-29.7811..-27.3383] | it/evals=1047/1787 eff=70.4102% N=300
Z=-33.7(0.05%) | Like=-28.04..-20.50 [-29.7811..-27.3383] | it/evals=1050/1791 eff=70.4225% N=300
Mono-modal Volume: ~exp(-7.62) * Expected Volume: exp(-3.57) Quality: ok
index : +1.0| +2.1 ************ +3.1 | +5.0
amplitude: +1.0e-12| *************** +5.1e-11 | +1.0e-10
Z=-33.2(0.09%) | Like=-27.64..-20.50 [-29.7811..-27.3383] | it/evals=1072/1820 eff=70.5263% N=300
Z=-33.0(0.11%) | Like=-27.50..-20.50 [-29.7811..-27.3383] | it/evals=1080/1828 eff=70.6806% N=300
Z=-32.4(0.21%) | Like=-26.95..-20.50 [-26.9470..-26.9344] | it/evals=1110/1862 eff=71.0627% N=300
Mono-modal Volume: ~exp(-7.95) * Expected Volume: exp(-3.80) Quality: ok
index : +1.0| +2.2 *********** +3.0 | +5.0
amplitude: +1.0e-12| ************** +4.9e-11 | +1.0e-10
Z=-31.8(0.34%) | Like=-26.32..-20.50 [-26.3168..-26.3096]*| it/evals=1139/1899 eff=71.2320% N=300
Z=-31.8(0.35%) | Like=-26.31..-20.50 [-26.3096..-26.3017]*| it/evals=1140/1900 eff=71.2500% N=300
Z=-31.3(0.59%) | Like=-25.89..-20.50 [-25.9089..-25.8916] | it/evals=1170/1938 eff=71.4286% N=300
Z=-30.9(0.84%) | Like=-25.49..-20.50 [-25.5428..-25.4932] | it/evals=1199/1978 eff=71.4541% N=300
Z=-30.9(0.85%) | Like=-25.45..-20.50 [-25.4469..-25.4467]*| it/evals=1200/1981 eff=71.3861% N=300
Mono-modal Volume: ~exp(-8.01) * Expected Volume: exp(-4.02) Quality: ok
index : +1.0| +2.2 ********** +3.0 | +5.0
amplitude: +1.0e-12| +2.4e-11 ************ +4.7e-11 | +1.0e-10
Z=-30.8(0.91%) | Like=-25.31..-20.50 [-25.3248..-25.3132] | it/evals=1206/1988 eff=71.4455% N=300
Z=-30.5(1.31%) | Like=-24.95..-20.50 [-24.9486..-24.9311] | it/evals=1230/2015 eff=71.7201% N=300
Z=-30.1(2.01%) | Like=-24.45..-20.50 [-24.4645..-24.4504] | it/evals=1260/2056 eff=71.7540% N=300
Mono-modal Volume: ~exp(-8.37) * Expected Volume: exp(-4.24) Quality: ok
index : +1.0| +2.2 ********** +2.9 | +5.0
amplitude: +1.0e-12| +2.5e-11 *********** +4.6e-11 | +1.0e-10
Z=-29.9(2.35%) | Like=-24.33..-20.50 [-24.3326..-24.3289]*| it/evals=1273/2079 eff=71.5571% N=300
Z=-29.7(2.89%) | Like=-24.11..-20.50 [-24.1120..-24.1085]*| it/evals=1290/2104 eff=71.5078% N=300
Z=-29.4(4.00%) | Like=-23.81..-20.50 [-23.8098..-23.8071]*| it/evals=1319/2145 eff=71.4905% N=300
Z=-29.4(4.01%) | Like=-23.81..-20.50 [-23.8071..-23.8020]*| it/evals=1320/2147 eff=71.4672% N=300
Mono-modal Volume: ~exp(-8.37) Expected Volume: exp(-4.47) Quality: ok
index : +1.0| +2.3 ******** +2.9 | +5.0
amplitude: +1.0e-12| +2.5e-11 ********** +4.4e-11 | +1.0e-10
Z=-29.1(5.19%) | Like=-23.52..-20.50 [-23.5173..-23.4991] | it/evals=1349/2183 eff=71.6410% N=300
Z=-29.1(5.25%) | Like=-23.50..-20.50 [-23.5173..-23.4991] | it/evals=1350/2184 eff=71.6561% N=300
Z=-28.8(6.80%) | Like=-23.25..-20.50 [-23.2644..-23.2473] | it/evals=1378/2226 eff=71.5472% N=300
Z=-28.8(6.89%) | Like=-23.22..-20.50 [-23.2246..-23.2209]*| it/evals=1380/2229 eff=71.5397% N=300
Z=-28.6(8.49%) | Like=-23.01..-20.49 [-23.0093..-23.0084]*| it/evals=1406/2270 eff=71.3706% N=300
Mono-modal Volume: ~exp(-8.37) Expected Volume: exp(-4.69) Quality: ok
index : +1.0| +2.3 ******** +2.9 | +5.0
amplitude: +1.0e-12| +2.6e-11 ********* +4.3e-11 | +1.0e-10
Z=-28.5(8.76%) | Like=-23.00..-20.49 [-23.0007..-22.9941]*| it/evals=1410/2274 eff=71.4286% N=300
Z=-28.3(10.53%) | Like=-22.81..-20.49 [-22.8133..-22.8080]*| it/evals=1437/2318 eff=71.2091% N=300
Z=-28.3(10.76%) | Like=-22.79..-20.49 [-22.7923..-22.7904]*| it/evals=1440/2321 eff=71.2519% N=300
Z=-28.1(12.87%) | Like=-22.57..-20.49 [-22.5713..-22.5626]*| it/evals=1469/2365 eff=71.1380% N=300
Z=-28.1(12.97%) | Like=-22.56..-20.49 [-22.5626..-22.5618]*| it/evals=1470/2366 eff=71.1520% N=300
Mono-modal Volume: ~exp(-8.95) * Expected Volume: exp(-4.91) Quality: ok
index : +1.0| +2.3 ******* +2.8 | +5.0
amplitude: +1.0e-12| +2.7e-11 ******** +4.2e-11 | +1.0e-10
Z=-28.1(13.31%) | Like=-22.55..-20.49 [-22.5537..-22.5354] | it/evals=1474/2376 eff=71.0019% N=300
Z=-27.9(15.59%) | Like=-22.34..-20.47 [-22.3442..-22.3335] | it/evals=1500/2409 eff=71.1238% N=300
Z=-27.8(18.54%) | Like=-22.15..-20.47 [-22.1507..-22.1433]*| it/evals=1530/2451 eff=71.1297% N=300
Mono-modal Volume: ~exp(-8.95) Expected Volume: exp(-5.14) Quality: ok
index : +1.0| +2.3 ****** +2.8 | +5.0
amplitude: +1.0e-12| +2.8e-11 ******** +4.1e-11 | +1.0e-10
Z=-27.6(21.48%) | Like=-22.00..-20.47 [-22.0043..-21.9907] | it/evals=1557/2488 eff=71.1609% N=300
Z=-27.6(21.79%) | Like=-21.99..-20.47 [-21.9855..-21.9846]*| it/evals=1560/2492 eff=71.1679% N=300
Z=-27.5(24.87%) | Like=-21.86..-20.46 [-21.8623..-21.8574]*| it/evals=1589/2533 eff=71.1599% N=300
Z=-27.5(25.01%) | Like=-21.86..-20.46 [-21.8574..-21.8495]*| it/evals=1590/2535 eff=71.1409% N=300
Mono-modal Volume: ~exp(-9.04) * Expected Volume: exp(-5.36) Quality: ok
index : +1.0| +2.4 ****** +2.8 | +5.0
amplitude: +1.0e-12| +2.8e-11 ******* +4.0e-11 | +1.0e-10
Z=-27.4(27.14%) | Like=-21.77..-20.46 [-21.7686..-21.7659]*| it/evals=1608/2558 eff=71.2135% N=300
Z=-27.3(28.33%) | Like=-21.70..-20.46 [-21.7017..-21.6977]*| it/evals=1620/2572 eff=71.3028% N=300
Z=-27.2(31.79%) | Like=-21.58..-20.46 [-21.5781..-21.5692]*| it/evals=1650/2609 eff=71.4595% N=300
Mono-modal Volume: ~exp(-9.72) * 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=-27.1(34.81%) | Like=-21.45..-20.46 [-21.4507..-21.4491]*| it/evals=1675/2642 eff=71.5201% N=300
Z=-27.1(35.47%) | Like=-21.43..-20.46 [-21.4299..-21.4286]*| it/evals=1680/2647 eff=71.5807% N=300
Z=-27.0(39.23%) | Like=-21.33..-20.46 [-21.3271..-21.3251]*| it/evals=1710/2683 eff=71.7583% N=300
Z=-26.9(42.62%) | Like=-21.26..-20.46 [-21.2554..-21.2538]*| it/evals=1738/2724 eff=71.6997% N=300
Z=-26.9(42.88%) | Like=-21.25..-20.46 [-21.2536..-21.2534]*| it/evals=1740/2726 eff=71.7230% N=300
Mono-modal Volume: ~exp(-9.78) * 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=-26.9(43.16%) | Like=-21.25..-20.46 [-21.2489..-21.2456]*| it/evals=1742/2728 eff=71.7463% N=300
Z=-26.8(46.46%) | Like=-21.17..-20.46 [-21.1673..-21.1635]*| it/evals=1770/2768 eff=71.7180% N=300
Z=-26.8(49.97%) | Like=-21.12..-20.46 [-21.1215..-21.1138]*| it/evals=1800/2805 eff=71.8563% N=300
Mono-modal Volume: ~exp(-9.78) 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=-26.7(52.86%) | Like=-21.06..-20.46 [-21.0568..-21.0544]*| it/evals=1825/2842 eff=71.7939% N=300
Z=-26.7(53.39%) | Like=-21.05..-20.46 [-21.0466..-21.0391]*| it/evals=1830/2849 eff=71.7929% N=300
Z=-26.6(56.62%) | Like=-20.98..-20.46 [-20.9789..-20.9775]*| it/evals=1859/2889 eff=71.8038% N=300
Z=-26.6(56.73%) | Like=-20.98..-20.46 [-20.9775..-20.9770]*| it/evals=1860/2890 eff=71.8147% N=300
Mono-modal Volume: ~exp(-9.96) * 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=-26.6(58.50%) | Like=-20.95..-20.46 [-20.9510..-20.9469]*| it/evals=1876/2916 eff=71.7125% N=300
Z=-26.6(60.03%) | Like=-20.92..-20.46 [-20.9239..-20.9222]*| it/evals=1890/2932 eff=71.8085% N=300
Z=-26.5(63.05%) | Like=-20.87..-20.46 [-20.8725..-20.8695]*| it/evals=1920/2968 eff=71.9640% N=300
Mono-modal Volume: ~exp(-10.36) * 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=-26.5(65.30%) | Like=-20.85..-20.46 [-20.8484..-20.8477]*| it/evals=1943/3003 eff=71.8831% N=300
Z=-26.5(65.97%) | Like=-20.83..-20.46 [-20.8298..-20.8292]*| it/evals=1950/3010 eff=71.9557% N=300
Z=-26.4(68.75%) | Like=-20.79..-20.46 [-20.7868..-20.7862]*| it/evals=1980/3050 eff=72.0000% N=300
[ultranest] Explored until L=-2e+01
[ultranest] Likelihood function evaluations: 3069
[ultranest] logZ = -26.07 +- 0.08965
[ultranest] Effective samples strategy satisfied (ESS = 999.7, need >400)
[ultranest] Posterior uncertainty strategy is satisfied (KL: 0.46+-0.10 nat, need <0.50 nat)
[ultranest] Evidency uncertainty strategy is satisfied (dlogz=0.28, need <0.5)
[ultranest] logZ error budget: single: 0.12 bs:0.09 tail:0.26 total:0.28 required:<0.50
[ultranest] done iterating.
logZ = -26.061 +- 0.284
single instance: logZ = -26.061 +- 0.124
bootstrapped : logZ = -26.071 +- 0.111
tail : logZ = +- 0.262
insert order U test : converged: True correlation: inf iterations
index : 2.14 │ ▁▁ ▁▁▁▁▂▂▂▃▄▄▅▇▆▇▄▆▅▄▄▃▃▂▂▂▁▁▁▁▁▁▁▁▁▁ │3.09 2.58 +- 0.13
amplitude : 0.0000000000217│ ▁▁▁▁▁▂▂▃▃▄▅▅▆▆▇▆▇▆▃▃▃▁▂▁▁▁▁▁▁▁ ▁ │0.0000000000533 0.0000000000341 +- 0.0000000000037
[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=-836.34..-19.51 [-836.3384..-141.7705] | it/evals=0/301 eff=0.0000% N=300
Z=-218.6(0.00%) | Like=-213.12..-19.51 [-836.3384..-141.7705] | it/evals=30/332 eff=93.7500% N=300
Z=-204.7(0.00%) | Like=-198.69..-19.51 [-836.3384..-141.7705] | it/evals=60/364 eff=93.7500% N=300
Mono-modal Volume: ~exp(-4.33) Expected Volume: exp(-0.22) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|************************ ********* *** *** *** | +1.0e-10
Z=-192.2(0.00%) | Like=-186.88..-19.51 [-836.3384..-141.7705] | it/evals=90/401 eff=89.1089% N=300
Z=-181.6(0.00%) | Like=-175.99..-19.51 [-836.3384..-141.7705] | it/evals=120/438 eff=86.9565% 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=-169.4(0.00%) | Like=-164.40..-19.51 [-836.3384..-141.7705] | it/evals=149/476 eff=84.6591% N=300
Z=-169.2(0.00%) | Like=-164.26..-19.51 [-836.3384..-141.7705] | it/evals=150/479 eff=83.7989% N=300
Z=-160.6(0.00%) | Like=-155.17..-19.51 [-836.3384..-141.7705] | it/evals=180/517 eff=82.9493% N=300
Mono-modal Volume: ~exp(-4.48) * Expected Volume: exp(-0.67) Quality: ok
index : +1.0| ***********************************************| +5.0
amplitude: +1.0e-12| ********************************* ******* * ** | +1.0e-10
Z=-152.9(0.00%) | Like=-146.51..-19.51 [-836.3384..-141.7705] | it/evals=201/542 eff=83.0579% N=300
Z=-147.0(0.00%) | Like=-139.91..-19.51 [-141.7232..-69.6643] | it/evals=210/551 eff=83.6653% N=300
Z=-130.1(0.00%) | Like=-124.60..-19.51 [-141.7232..-69.6643] | it/evals=240/587 eff=83.6237% N=300
Mono-modal Volume: ~exp(-4.74) * Expected Volume: exp(-0.89) Quality: ok
index : +1.0| *******************************************| +5.0
amplitude: +1.0e-12| *********************************** *** **** | +1.0e-10
Z=-117.8(0.00%) | Like=-111.85..-19.51 [-141.7232..-69.6643] | it/evals=268/625 eff=82.4615% N=300
Z=-116.9(0.00%) | Like=-111.31..-19.51 [-141.7232..-69.6643] | it/evals=270/629 eff=82.0669% N=300
Z=-104.7(0.00%) | Like=-98.96..-19.51 [-141.7232..-69.6643] | it/evals=300/665 eff=82.1918% N=300
Z=-96.4(0.00%) | Like=-90.92..-19.51 [-141.7232..-69.6643] | it/evals=329/706 eff=81.0345% N=300
Z=-96.1(0.00%) | Like=-90.43..-19.51 [-141.7232..-69.6643] | it/evals=330/707 eff=81.0811% N=300
Mono-modal Volume: ~exp(-4.74) Expected Volume: exp(-1.12) Quality: ok
index : +1.0| *******************************************| +5.0
amplitude: +1.0e-12| ********************************** *** *****| +1.0e-10
Z=-89.3(0.00%) | Like=-83.69..-19.50 [-141.7232..-69.6643] | it/evals=359/744 eff=80.8559% N=300
Z=-89.0(0.00%) | Like=-83.49..-19.50 [-141.7232..-69.6643] | it/evals=360/745 eff=80.8989% N=300
Z=-79.6(0.00%) | Like=-74.12..-19.50 [-141.7232..-69.6643] | it/evals=390/783 eff=80.7453% N=300
Mono-modal Volume: ~exp(-5.18) * Expected Volume: exp(-1.34) Quality: ok
index : +1.0| *****************************************| +5.0
amplitude: +1.0e-12| ******************************** *** * ***| +1.0e-10
Z=-77.2(0.00%) | Like=-71.81..-19.50 [-141.7232..-69.6643] | it/evals=402/799 eff=80.5611% N=300
Z=-73.7(0.00%) | Like=-68.44..-19.50 [-69.6347..-41.1540] | it/evals=420/823 eff=80.3059% N=300
Z=-67.8(0.00%) | Like=-62.36..-19.50 [-69.6347..-41.1540] | it/evals=450/863 eff=79.9290% N=300
Mono-modal Volume: ~exp(-5.18) Expected Volume: exp(-1.56) Quality: ok
index : +1.0| ****************************************| +5.0
amplitude: +1.0e-12| ****************************** ***** ***| +1.0e-10
Z=-63.3(0.00%) | Like=-58.01..-19.50 [-69.6347..-41.1540] | it/evals=476/902 eff=79.0698% N=300
Z=-62.4(0.00%) | Like=-56.44..-19.50 [-69.6347..-41.1540] | it/evals=480/906 eff=79.2079% N=300
Z=-59.0(0.00%) | Like=-53.90..-19.50 [-69.6347..-41.1540] | it/evals=506/948 eff=78.0864% N=300
Z=-58.6(0.00%) | Like=-53.31..-19.50 [-69.6347..-41.1540] | it/evals=510/955 eff=77.8626% N=300
Mono-modal Volume: ~exp(-5.52) * Expected Volume: exp(-1.79) Quality: ok
index : +1.0| ************************************ | +5.0
amplitude: +1.0e-12| ***************************** ***** ***| +1.0e-10
Z=-56.3(0.00%) | Like=-51.03..-19.47 [-69.6347..-41.1540] | it/evals=536/996 eff=77.0115% N=300
Z=-55.9(0.00%) | Like=-50.71..-19.47 [-69.6347..-41.1540] | it/evals=540/1000 eff=77.1429% N=300
Z=-53.4(0.00%) | Like=-48.32..-19.47 [-69.6347..-41.1540] | it/evals=570/1036 eff=77.4457% N=300
Z=-50.2(0.00%) | Like=-44.61..-19.47 [-69.6347..-41.1540] | it/evals=600/1078 eff=77.1208% N=300
Mono-modal Volume: ~exp(-5.73) * Expected Volume: exp(-2.01) Quality: ok
index : +1.0| +1.9 ******************************* | +5.0
amplitude: +1.0e-12| +2.4e-11 *************************** ******* *| +1.0e-10
Z=-49.9(0.00%) | Like=-44.35..-19.47 [-69.6347..-41.1540] | it/evals=603/1081 eff=77.2087% N=300
Z=-46.8(0.00%) | Like=-41.23..-19.25 [-69.6347..-41.1540] | it/evals=630/1118 eff=77.0171% N=300
Z=-44.5(0.00%) | Like=-38.92..-19.25 [-41.0774..-30.2208] | it/evals=660/1161 eff=76.6551% N=300
Mono-modal Volume: ~exp(-6.43) * 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=-43.6(0.00%) | Like=-38.26..-19.25 [-41.0774..-30.2208] | it/evals=670/1172 eff=76.8349% N=300
Z=-42.4(0.00%) | Like=-37.29..-19.25 [-41.0774..-30.2208] | it/evals=690/1201 eff=76.5816% N=300
Z=-40.9(0.00%) | Like=-35.77..-19.25 [-41.0774..-30.2208] | it/evals=720/1242 eff=76.4331% N=300
Mono-modal Volume: ~exp(-6.53) * 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=-40.1(0.00%) | Like=-35.03..-19.25 [-41.0774..-30.2208] | it/evals=737/1270 eff=75.9794% N=300
Z=-39.5(0.00%) | Like=-34.31..-19.25 [-41.0774..-30.2208] | it/evals=750/1288 eff=75.9109% N=300
Z=-37.9(0.00%) | Like=-32.58..-19.25 [-41.0774..-30.2208] | it/evals=780/1326 eff=76.0234% N=300
Mono-modal Volume: ~exp(-6.53) Expected Volume: exp(-2.68) Quality: ok
index : +1.0| +2.2 ********************* +3.8 | +5.0
amplitude: +1.0e-12| +3.0e-11 ***************************** | +1.0e-10
Z=-36.8(0.00%) | Like=-31.49..-19.25 [-41.0774..-30.2208] | it/evals=804/1366 eff=75.4221% N=300
Z=-36.6(0.00%) | Like=-31.33..-19.25 [-41.0774..-30.2208] | it/evals=810/1373 eff=75.4893% N=300
Z=-35.5(0.00%) | Like=-30.36..-19.22 [-41.0774..-30.2208] | it/evals=838/1415 eff=75.1570% N=300
Z=-35.4(0.00%) | Like=-30.28..-19.22 [-41.0774..-30.2208] | it/evals=840/1417 eff=75.2014% N=300
Z=-34.5(0.00%) | Like=-29.15..-19.22 [-30.2173..-26.0896] | it/evals=866/1462 eff=74.5267% N=300
Z=-34.3(0.01%) | Like=-28.78..-19.22 [-30.2173..-26.0896] | it/evals=870/1469 eff=74.4226% N=300
Mono-modal Volume: ~exp(-6.66) * Expected Volume: exp(-2.90) Quality: ok
index : +1.0| +2.2 ****************** +3.6 | +5.0
amplitude: +1.0e-12| +3.3e-11 ************************** | +1.0e-10
Z=-34.3(0.01%) | Like=-28.78..-19.22 [-30.2173..-26.0896] | it/evals=871/1470 eff=74.4444% N=300
Z=-33.3(0.01%) | Like=-28.02..-19.22 [-30.2173..-26.0896] | it/evals=900/1510 eff=74.3802% N=300
Z=-32.5(0.03%) | Like=-27.26..-19.22 [-30.2173..-26.0896] | it/evals=929/1551 eff=74.2606% N=300
Z=-32.5(0.03%) | Like=-27.20..-19.22 [-30.2173..-26.0896] | it/evals=930/1553 eff=74.2219% N=300
Mono-modal Volume: ~exp(-7.27) * 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=-26.96..-19.22 [-30.2173..-26.0896] | it/evals=938/1565 eff=74.1502% N=300
Z=-31.6(0.08%) | Like=-26.30..-19.22 [-30.2173..-26.0896] | it/evals=960/1598 eff=73.9599% N=300
Z=-30.8(0.16%) | Like=-25.49..-19.22 [-25.5080..-25.4867] | it/evals=990/1636 eff=74.1018% N=300
Mono-modal Volume: ~exp(-7.54) * Expected Volume: exp(-3.35) Quality: ok
index : +1.0| +2.3 *************** +3.5 | +5.0
amplitude: +1.0e-12| +3.8e-11 ******************** +7.7e-11 | +1.0e-10
Z=-30.5(0.22%) | Like=-25.22..-19.22 [-25.2241..-25.2205]*| it/evals=1005/1664 eff=73.6804% N=300
Z=-30.2(0.30%) | Like=-24.96..-19.22 [-24.9635..-24.9534] | it/evals=1020/1681 eff=73.8595% N=300
Z=-29.6(0.50%) | Like=-24.41..-19.22 [-24.4078..-24.4063]*| it/evals=1050/1719 eff=73.9958% N=300
Mono-modal Volume: ~exp(-7.74) * Expected Volume: exp(-3.57) Quality: ok
index : +1.0| +2.4 ************* +3.4 | +5.0
amplitude: +1.0e-12| +3.9e-11 ****************** +7.5e-11 | +1.0e-10
Z=-29.2(0.75%) | Like=-24.01..-19.22 [-24.0113..-23.9814] | it/evals=1072/1751 eff=73.8801% N=300
Z=-29.1(0.87%) | Like=-23.89..-19.22 [-23.8936..-23.8677] | it/evals=1080/1760 eff=73.9726% N=300
Z=-28.6(1.38%) | Like=-23.51..-19.22 [-23.5057..-23.5010]*| it/evals=1110/1793 eff=74.3470% N=300
Mono-modal Volume: ~exp(-8.14) * Expected Volume: exp(-3.80) Quality: ok
index : +1.0| +2.4 ************ +3.3 | +5.0
amplitude: +1.0e-12| +4.1e-11 **************** +7.3e-11 | +1.0e-10
Z=-28.3(2.02%) | Like=-23.08..-19.22 [-23.1025..-23.0823] | it/evals=1139/1831 eff=74.3958% N=300
Z=-28.2(2.05%) | Like=-23.07..-19.22 [-23.0736..-23.0690]*| it/evals=1140/1833 eff=74.3640% N=300
Z=-27.9(2.95%) | Like=-22.82..-19.22 [-22.8190..-22.8183]*| it/evals=1170/1866 eff=74.7126% N=300
Z=-27.6(4.06%) | Like=-22.57..-19.22 [-22.5851..-22.5731] | it/evals=1200/1903 eff=74.8596% N=300
Mono-modal Volume: ~exp(-8.14) 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=-27.3(5.16%) | Like=-22.22..-19.22 [-22.2180..-22.2143]*| it/evals=1227/1940 eff=74.8171% N=300
Z=-27.3(5.30%) | Like=-22.20..-19.22 [-22.2111..-22.2004] | it/evals=1230/1944 eff=74.8175% N=300
Z=-27.1(6.83%) | Like=-21.94..-19.20 [-21.9386..-21.9188] | it/evals=1260/1984 eff=74.8219% N=300
Mono-modal Volume: ~exp(-8.17) * Expected Volume: exp(-4.24) Quality: ok
index : +1.0| +2.5 ********** +3.2 | +5.0
amplitude: +1.0e-12| +4.3e-11 ************** +7.0e-11 | +1.0e-10
Z=-26.9(7.63%) | Like=-21.84..-19.17 [-21.8352..-21.8334]*| it/evals=1273/2001 eff=74.8383% N=300
Z=-26.8(8.71%) | Like=-21.62..-19.17 [-21.6233..-21.6226]*| it/evals=1290/2022 eff=74.9129% N=300
Z=-26.6(10.71%) | Like=-21.41..-19.17 [-21.4087..-21.4005]*| it/evals=1319/2063 eff=74.8157% N=300
Z=-26.6(10.80%) | Like=-21.40..-19.17 [-21.4005..-21.3930]*| it/evals=1320/2064 eff=74.8299% N=300
Mono-modal Volume: ~exp(-8.39) * Expected Volume: exp(-4.47) Quality: ok
index : +1.0| +2.5 ******** +3.2 | +5.0
amplitude: +1.0e-12| +4.4e-11 ************* +6.8e-11 | +1.0e-10
Z=-26.4(12.50%) | Like=-21.29..-19.17 [-21.2916..-21.2856]*| it/evals=1340/2092 eff=74.7768% N=300
Z=-26.4(13.45%) | Like=-21.17..-19.17 [-21.1664..-21.1617]*| it/evals=1350/2108 eff=74.6681% N=300
Z=-26.2(16.07%) | Like=-20.98..-19.17 [-20.9896..-20.9779] | it/evals=1378/2149 eff=74.5268% N=300
Z=-26.2(16.24%) | Like=-20.96..-19.17 [-20.9603..-20.9580]*| it/evals=1380/2151 eff=74.5543% N=300
Z=-26.0(18.78%) | Like=-20.81..-19.17 [-20.8149..-20.8094]*| it/evals=1406/2192 eff=74.3129% N=300
Mono-modal Volume: ~exp(-9.11) * 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(18.90%) | Like=-20.81..-19.17 [-20.8094..-20.8005]*| it/evals=1407/2193 eff=74.3265% N=300
Z=-26.0(19.26%) | Like=-20.80..-19.17 [-20.7966..-20.7962]*| it/evals=1410/2199 eff=74.2496% N=300
Z=-25.8(22.48%) | Like=-20.60..-19.17 [-20.6209..-20.6038] | it/evals=1440/2235 eff=74.4186% N=300
Z=-25.7(25.92%) | Like=-20.48..-19.17 [-20.4811..-20.4796]*| it/evals=1469/2276 eff=74.3421% N=300
Z=-25.7(26.02%) | Like=-20.48..-19.17 [-20.4796..-20.4795]*| it/evals=1470/2277 eff=74.3551% N=300
Mono-modal Volume: ~exp(-9.18) * 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(26.48%) | Like=-20.47..-19.17 [-20.4702..-20.4533] | it/evals=1474/2283 eff=74.3318% N=300
Z=-25.6(29.68%) | Like=-20.35..-19.17 [-20.3491..-20.3480]*| it/evals=1500/2315 eff=74.4417% N=300
Z=-25.5(33.29%) | Like=-20.26..-19.17 [-20.2631..-20.2603]*| it/evals=1529/2355 eff=74.4039% N=300
Z=-25.5(33.44%) | Like=-20.26..-19.17 [-20.2603..-20.2597]*| it/evals=1530/2356 eff=74.4163% N=300
Mono-modal Volume: ~exp(-9.23) * 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(34.79%) | Like=-20.21..-19.17 [-20.2076..-20.2051]*| it/evals=1541/2369 eff=74.4804% N=300
Z=-25.3(37.06%) | Like=-20.14..-19.17 [-20.1416..-20.1375]*| it/evals=1560/2395 eff=74.4630% N=300
Z=-25.3(40.72%) | Like=-20.05..-19.17 [-20.0467..-20.0463]*| it/evals=1590/2431 eff=74.6129% N=300
Mono-modal Volume: ~exp(-9.63) * 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(42.97%) | Like=-19.98..-19.17 [-19.9779..-19.9731]*| it/evals=1608/2455 eff=74.6172% N=300
Z=-25.2(44.36%) | Like=-19.94..-19.17 [-19.9406..-19.9275] | it/evals=1620/2473 eff=74.5513% N=300
Z=-25.1(47.76%) | Like=-19.87..-19.17 [-19.8733..-19.8728]*| it/evals=1650/2510 eff=74.6606% N=300
Mono-modal Volume: ~exp(-9.63) 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=-25.0(50.84%) | Like=-19.79..-19.17 [-19.7919..-19.7842]*| it/evals=1676/2547 eff=74.5883% N=300
Z=-25.0(51.33%) | Like=-19.78..-19.17 [-19.7790..-19.7777]*| it/evals=1680/2552 eff=74.6004% N=300
Z=-24.9(54.68%) | Like=-19.72..-19.16 [-19.7158..-19.7153]*| it/evals=1709/2592 eff=74.5637% N=300
Z=-24.9(54.82%) | Like=-19.72..-19.16 [-19.7153..-19.7141]*| it/evals=1710/2594 eff=74.5423% N=300
Z=-24.9(57.70%) | Like=-19.69..-19.16 [-19.6873..-19.6861]*| it/evals=1736/2637 eff=74.2833% N=300
Z=-24.9(58.16%) | Like=-19.68..-19.16 [-19.6848..-19.6844]*| it/evals=1740/2642 eff=74.2955% N=300
Mono-modal Volume: ~exp(-10.04) * 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(58.41%) | Like=-19.68..-19.16 [-19.6835..-19.6813]*| it/evals=1742/2644 eff=74.3174% N=300
Z=-24.8(61.28%) | Like=-19.65..-19.16 [-19.6460..-19.6457]*| it/evals=1770/2676 eff=74.4949% N=300
Z=-24.8(64.15%) | Like=-19.60..-19.16 [-19.6040..-19.5985]*| it/evals=1800/2710 eff=74.6888% N=300
Mono-modal Volume: ~exp(-10.17) * 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(64.98%) | Like=-19.58..-19.16 [-19.5824..-19.5820]*| it/evals=1809/2719 eff=74.7830% N=300
Z=-24.7(66.86%) | Like=-19.55..-19.16 [-19.5475..-19.5466]*| it/evals=1830/2746 eff=74.8160% N=300
Z=-24.7(69.50%) | Like=-19.51..-19.16 [-19.5094..-19.5090]*| it/evals=1860/2783 eff=74.9094% N=300
[ultranest] Explored until L=-2e+01
[ultranest] Likelihood function evaluations: 2789
[ultranest] logZ = -24.36 +- 0.09076
[ultranest] Effective samples strategy satisfied (ESS = 993.7, 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:0.09 tail:0.26 total:0.28 required:<0.50
[ultranest] done iterating.
logZ = -24.345 +- 0.297
single instance: logZ = -24.345 +- 0.118
bootstrapped : logZ = -24.356 +- 0.139
tail : logZ = +- 0.262
insert order U test : converged: True correlation: inf iterations
index : 2.27 │ ▁▁▁▁▁▁▁▁▃▃▄▅▆▆▇▇▆▇▆▆▆▄▅▃▃▂▂▁▁▁▁▁▁▁▁▁▁ │3.50 2.83 +- 0.17
amplitude : 0.0000000000338│ ▁▁▁▁▁▁▁▁▂▃▄▄▄▅▅▅▇▅▆▅▄▃▄▂▂▂▁▁▁▁▁▁▁ ▁ ▁ │0.0000000000815 0.0000000000553 +- 0.0000000000060
[ultranest] Sampling 300 live points from prior ...
Mono-modal Volume: ~exp(-4.23) * Expected Volume: exp(0.00) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|**************************** ******* ********* *| +1.0e-10
Z=-inf(0.00%) | Like=-1313.20..-13.43 [-1313.2039..-86.4104] | it/evals=0/301 eff=0.0000% N=300
Z=-152.7(0.00%) | Like=-147.73..-13.43 [-1313.2039..-86.4104] | it/evals=30/331 eff=96.7742% N=300
Z=-141.3(0.00%) | Like=-136.26..-13.43 [-1313.2039..-86.4104] | it/evals=60/362 eff=96.7742% N=300
Mono-modal Volume: ~exp(-4.23) Expected Volume: exp(-0.22) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|****************************************** ** * | +1.0e-10
Z=-128.4(0.00%) | Like=-123.11..-13.43 [-1313.2039..-86.4104] | it/evals=90/398 eff=91.8367% N=300
Z=-118.0(0.00%) | Like=-112.92..-13.43 [-1313.2039..-86.4104] | it/evals=120/435 eff=88.8889% N=300
Mono-modal Volume: ~exp(-4.43) * Expected Volume: exp(-0.45) Quality: ok
index : +1.0| ***********************************************| +5.0
amplitude: +1.0e-12|********************************************* * | +1.0e-10
Z=-114.0(0.00%) | Like=-108.52..-13.43 [-1313.2039..-86.4104] | it/evals=134/451 eff=88.7417% N=300
Z=-107.0(0.00%) | Like=-101.83..-13.43 [-1313.2039..-86.4104] | it/evals=150/470 eff=88.2353% N=300
Z=-95.7(0.00%) | Like=-89.90..-13.43 [-1313.2039..-86.4104] | it/evals=180/506 eff=87.3786% N=300
Mono-modal Volume: ~exp(-5.02) * Expected Volume: exp(-0.67) Quality: ok
index : +1.0| *********************************************| +5.0
amplitude: +1.0e-12| ******************************************** **| +1.0e-10
Z=-90.4(0.00%) | Like=-84.82..-13.40 [-86.4091..-44.7106] | it/evals=201/533 eff=86.2661% N=300
Z=-87.9(0.00%) | Like=-82.62..-13.40 [-86.4091..-44.7106] | it/evals=210/542 eff=86.7769% N=300
Z=-79.3(0.00%) | Like=-73.35..-13.40 [-86.4091..-44.7106] | it/evals=240/581 eff=85.4093% N=300
Mono-modal Volume: ~exp(-5.08) * Expected Volume: exp(-0.89) Quality: ok
index : +1.0| ********************************************| +5.0
amplitude: +1.0e-12| ****************************************** **| +1.0e-10
Z=-72.7(0.00%) | Like=-67.59..-13.40 [-86.4091..-44.7106] | it/evals=268/621 eff=83.4891% N=300
Z=-72.3(0.00%) | Like=-67.29..-13.40 [-86.4091..-44.7106] | it/evals=270/623 eff=83.5913% N=300
Z=-67.0(0.00%) | Like=-62.29..-13.40 [-86.4091..-44.7106] | it/evals=298/664 eff=81.8681% N=300
Z=-66.7(0.00%) | Like=-62.14..-13.40 [-86.4091..-44.7106] | it/evals=300/666 eff=81.9672% N=300
Z=-61.0(0.00%) | Like=-55.85..-13.40 [-86.4091..-44.7106] | it/evals=330/708 eff=80.8824% N=300
Mono-modal Volume: ~exp(-5.08) Expected Volume: exp(-1.12) Quality: ok
index : +1.0| ******************************************| +5.0
amplitude: +1.0e-12| ***************************************** **| +1.0e-10
Z=-56.3(0.00%) | Like=-51.36..-13.40 [-86.4091..-44.7106] | it/evals=357/746 eff=80.0448% N=300
Z=-55.8(0.00%) | Like=-50.93..-13.40 [-86.4091..-44.7106] | it/evals=360/753 eff=79.4702% N=300
Z=-51.0(0.00%) | Like=-46.00..-13.40 [-86.4091..-44.7106] | it/evals=389/794 eff=78.7449% N=300
Z=-50.9(0.00%) | Like=-45.99..-13.40 [-86.4091..-44.7106] | it/evals=390/795 eff=78.7879% N=300
Mono-modal Volume: ~exp(-5.52) * Expected Volume: exp(-1.34) Quality: ok
index : +1.0| *************************************** | +5.0
amplitude: +1.0e-12| ******************************************| +1.0e-10
Z=-49.3(0.00%) | Like=-44.38..-13.40 [-44.6783..-29.2908] | it/evals=402/810 eff=78.8235% N=300
Z=-47.3(0.00%) | Like=-42.38..-13.40 [-44.6783..-29.2908] | it/evals=420/837 eff=78.2123% N=300
Z=-44.5(0.00%) | Like=-39.68..-13.40 [-44.6783..-29.2908] | it/evals=448/880 eff=77.2414% N=300
Z=-44.3(0.00%) | Like=-39.67..-13.40 [-44.6783..-29.2908] | it/evals=450/883 eff=77.1870% N=300
Mono-modal Volume: ~exp(-6.20) * Expected Volume: exp(-1.56) Quality: ok
index : +1.0| ********************************* | +5.0
amplitude: +1.0e-12| ******************************************| +1.0e-10
Z=-42.9(0.00%) | Like=-38.10..-13.40 [-44.6783..-29.2908] | it/evals=469/908 eff=77.1382% N=300
Z=-42.0(0.00%) | Like=-37.06..-13.40 [-44.6783..-29.2908] | it/evals=480/924 eff=76.9231% N=300
Z=-40.1(0.00%) | Like=-35.65..-13.40 [-44.6783..-29.2908] | it/evals=509/965 eff=76.5414% N=300
Z=-40.0(0.00%) | Like=-35.32..-13.40 [-44.6783..-29.2908] | it/evals=510/966 eff=76.5766% N=300
Mono-modal Volume: ~exp(-6.20) Expected Volume: exp(-1.79) Quality: ok
index : +1.0| ***************************** | +5.0
amplitude: +1.0e-12| ************************************* *| +1.0e-10
Z=-38.3(0.00%) | Like=-33.65..-13.40 [-44.6783..-29.2908] | it/evals=540/1001 eff=77.0328% N=300
Z=-36.6(0.00%) | Like=-31.94..-13.40 [-44.6783..-29.2908] | it/evals=570/1040 eff=77.0270% N=300
Z=-35.0(0.00%) | Like=-30.04..-13.40 [-44.6783..-29.2908] | it/evals=600/1074 eff=77.5194% N=300
Mono-modal Volume: ~exp(-6.20) Expected Volume: exp(-2.01) Quality: ok
index : +1.0| +1.9 ************************** +4.0 | +5.0
amplitude: +1.0e-12| ************************************ **| +1.0e-10
Z=-33.6(0.00%) | Like=-28.63..-13.32 [-29.2783..-20.8093] | it/evals=624/1113 eff=76.7528% N=300
Z=-33.3(0.00%) | Like=-28.43..-13.32 [-29.2783..-20.8093] | it/evals=630/1119 eff=76.9231% N=300
Z=-31.9(0.00%) | Like=-27.14..-13.32 [-29.2783..-20.8093] | it/evals=658/1161 eff=76.4228% N=300
Z=-31.9(0.00%) | Like=-27.07..-13.32 [-29.2783..-20.8093] | it/evals=660/1163 eff=76.4774% N=300
Mono-modal Volume: ~exp(-6.20) Expected Volume: exp(-2.23) Quality: ok
index : +1.0| +2.0 ********************** +3.8 | +5.0
amplitude: +1.0e-12| *********************************** *| +1.0e-10
Z=-30.8(0.00%) | Like=-25.61..-13.32 [-29.2783..-20.8093] | it/evals=682/1201 eff=75.6937% N=300
Z=-30.4(0.00%) | Like=-25.34..-13.32 [-29.2783..-20.8093] | it/evals=690/1210 eff=75.8242% N=300
Z=-29.1(0.00%) | Like=-24.09..-13.32 [-29.2783..-20.8093] | it/evals=718/1254 eff=75.2621% N=300
Z=-29.0(0.00%) | Like=-23.95..-13.32 [-29.2783..-20.8093] | it/evals=720/1256 eff=75.3138% N=300
Mono-modal Volume: ~exp(-6.77) * Expected Volume: exp(-2.46) Quality: ok
index : +1.0| +2.1 ******************* +3.6 | +5.0
amplitude: +1.0e-12| +2.6e-11 ********************************* | +1.0e-10
Z=-28.1(0.00%) | Like=-23.20..-13.32 [-29.2783..-20.8093] | it/evals=737/1280 eff=75.2041% N=300
Z=-27.6(0.01%) | Like=-22.65..-13.32 [-29.2783..-20.8093] | it/evals=750/1297 eff=75.2257% N=300
Z=-26.7(0.02%) | Like=-21.88..-13.32 [-29.2783..-20.8093] | it/evals=780/1333 eff=75.5082% N=300
Mono-modal Volume: ~exp(-6.77) Expected Volume: exp(-2.68) Quality: ok
index : +1.0| +2.2 ***************** +3.5 | +5.0
amplitude: +1.0e-12| +2.8e-11 ****************************** | +1.0e-10
Z=-25.8(0.04%) | Like=-20.95..-13.32 [-29.2783..-20.8093] | it/evals=808/1371 eff=75.4435% N=300
Z=-25.8(0.04%) | Like=-20.89..-13.32 [-29.2783..-20.8093] | it/evals=810/1374 eff=75.4190% N=300
Z=-25.1(0.09%) | Like=-20.32..-13.32 [-20.8033..-19.8553] | it/evals=835/1416 eff=74.8208% N=300
Z=-25.0(0.10%) | Like=-20.24..-13.32 [-20.8033..-19.8553] | it/evals=840/1423 eff=74.7996% N=300
Z=-24.4(0.16%) | Like=-19.72..-13.32 [-19.8516..-19.6582] | it/evals=865/1464 eff=74.3127% N=300
Z=-24.3(0.18%) | Like=-19.55..-13.32 [-19.5498..-19.5434]*| it/evals=870/1470 eff=74.3590% N=300
Mono-modal Volume: ~exp(-7.26) * Expected Volume: exp(-2.90) Quality: ok
index : +1.0| +2.2 *************** +3.4 | +5.0
amplitude: +1.0e-12| +3.1e-11 *************************** | +1.0e-10
Z=-24.3(0.19%) | Like=-19.54..-13.32 [-19.5434..-19.5413]*| it/evals=871/1471 eff=74.3809% N=300
Z=-23.7(0.33%) | Like=-19.02..-13.32 [-19.0407..-19.0164] | it/evals=900/1508 eff=74.5033% N=300
Z=-23.2(0.58%) | Like=-18.50..-13.32 [-18.5261..-18.5046] | it/evals=930/1545 eff=74.6988% N=300
Mono-modal Volume: ~exp(-7.59) * Expected Volume: exp(-3.13) Quality: ok
index : +1.0| +2.3 ************* +3.3 | +5.0
amplitude: +1.0e-12| +3.3e-11 ************************ | +1.0e-10
Z=-23.1(0.66%) | Like=-18.28..-13.32 [-18.3114..-18.2802] | it/evals=938/1554 eff=74.8006% N=300
Z=-22.7(0.96%) | Like=-17.85..-13.32 [-17.8513..-17.8475]*| it/evals=960/1583 eff=74.8246% N=300
Z=-22.2(1.49%) | Like=-17.49..-13.32 [-17.4874..-17.4557] | it/evals=990/1618 eff=75.1138% N=300
Mono-modal Volume: ~exp(-7.59) Expected Volume: exp(-3.35) Quality: ok
index : +1.0| +2.3 ************* +3.3 | +5.0
amplitude: +1.0e-12| +3.4e-11 ********************** +7.8e-11| +1.0e-10
Z=-21.9(2.16%) | Like=-17.08..-13.32 [-17.0789..-17.0700]*| it/evals=1016/1656 eff=74.9263% N=300
Z=-21.8(2.30%) | Like=-17.02..-13.32 [-17.0162..-17.0044] | it/evals=1020/1662 eff=74.8899% N=300
Z=-21.5(3.24%) | Like=-16.68..-13.32 [-16.6785..-16.6776]*| it/evals=1047/1703 eff=74.6258% N=300
Z=-21.4(3.36%) | Like=-16.65..-13.32 [-16.6755..-16.6538] | it/evals=1050/1706 eff=74.6799% N=300
Mono-modal Volume: ~exp(-7.96) * Expected Volume: exp(-3.57) Quality: ok
index : +1.0| +2.4 *********** +3.2 | +5.0
amplitude: +1.0e-12| +3.7e-11 ******************* +7.4e-11 | +1.0e-10
Z=-21.2(4.31%) | Like=-16.46..-13.32 [-16.4762..-16.4579] | it/evals=1072/1741 eff=74.3928% N=300
Z=-21.1(4.70%) | Like=-16.43..-13.32 [-16.4394..-16.4265] | it/evals=1080/1752 eff=74.3802% N=300
Z=-20.8(6.09%) | Like=-16.12..-13.32 [-16.1166..-16.1152]*| it/evals=1110/1790 eff=74.4966% N=300
Mono-modal Volume: ~exp(-8.03) * Expected Volume: exp(-3.80) 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=-20.6(7.99%) | Like=-15.88..-13.32 [-15.8816..-15.8808]*| it/evals=1139/1829 eff=74.4931% N=300
Z=-20.6(8.07%) | Like=-15.88..-13.32 [-15.8808..-15.8593] | it/evals=1140/1830 eff=74.5098% N=300
Z=-20.4(10.04%) | Like=-15.64..-13.32 [-15.6423..-15.6409]*| it/evals=1166/1871 eff=74.2202% N=300
Z=-20.3(10.34%) | Like=-15.62..-13.32 [-15.6219..-15.6171]*| it/evals=1170/1876 eff=74.2386% N=300
Z=-20.1(12.67%) | Like=-15.44..-13.32 [-15.4429..-15.4409]*| it/evals=1198/1917 eff=74.0878% N=300
Z=-20.1(12.87%) | Like=-15.43..-13.32 [-15.4334..-15.4302]*| it/evals=1200/1919 eff=74.1198% N=300
Mono-modal Volume: ~exp(-8.39) * 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=-20.1(13.31%) | Like=-15.40..-13.32 [-15.3968..-15.3853] | it/evals=1206/1929 eff=74.0331% N=300
Z=-19.9(15.36%) | Like=-15.26..-13.31 [-15.2606..-15.2553]*| it/evals=1230/1968 eff=73.7410% N=300
Z=-19.8(17.83%) | Like=-15.11..-13.31 [-15.1117..-15.1057]*| it/evals=1256/2010 eff=73.4503% N=300
Z=-19.8(18.28%) | Like=-15.10..-13.31 [-15.1029..-15.1023]*| it/evals=1260/2016 eff=73.4266% N=300
Mono-modal Volume: ~exp(-8.39) Expected Volume: exp(-4.24) Quality: ok
index : +1.0| +2.5 ******** +3.1 | +5.0
amplitude: +1.0e-12| +4.0e-11 *************** +6.9e-11 | +1.0e-10
Z=-19.6(20.92%) | Like=-14.96..-13.31 [-14.9581..-14.9574]*| it/evals=1286/2053 eff=73.3600% N=300
Z=-19.6(21.29%) | Like=-14.93..-13.31 [-14.9323..-14.9304]*| it/evals=1290/2060 eff=73.2955% N=300
Z=-19.5(24.14%) | Like=-14.83..-13.31 [-14.8265..-14.8221]*| it/evals=1317/2102 eff=73.0855% N=300
Z=-19.5(24.48%) | Like=-14.81..-13.31 [-14.8138..-14.8041]*| it/evals=1320/2106 eff=73.0897% N=300
Mono-modal Volume: ~exp(-8.63) * Expected Volume: exp(-4.47) 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=-19.4(26.66%) | Like=-14.72..-13.31 [-14.7207..-14.7157]*| it/evals=1340/2134 eff=73.0643% N=300
Z=-19.4(27.64%) | Like=-14.69..-13.31 [-14.6902..-14.6835]*| it/evals=1350/2146 eff=73.1311% N=300
Z=-19.2(30.92%) | Like=-14.57..-13.31 [-14.5668..-14.5608]*| it/evals=1380/2182 eff=73.3262% N=300
Mono-modal Volume: ~exp(-9.00) * 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=-19.1(34.18%) | Like=-14.48..-13.31 [-14.4805..-14.4719]*| it/evals=1407/2223 eff=73.1669% N=300
Z=-19.1(34.60%) | Like=-14.44..-13.31 [-14.4361..-14.4311]*| it/evals=1410/2227 eff=73.1707% N=300
Z=-19.0(38.18%) | Like=-14.32..-13.31 [-14.3195..-14.3185]*| it/evals=1440/2268 eff=73.1707% N=300
Z=-19.0(41.71%) | Like=-14.23..-13.31 [-14.2337..-14.2219] | it/evals=1470/2305 eff=73.3167% N=300
Mono-modal Volume: ~exp(-9.23) * Expected Volume: exp(-4.91) Quality: ok
index : +1.0| +2.5 ****** +3.0 | +5.0
amplitude: +1.0e-12| +4.3e-11 *********** +6.4e-11 | +1.0e-10
Z=-18.9(42.17%) | Like=-14.21..-13.31 [-14.2147..-14.2096]*| it/evals=1474/2311 eff=73.2969% N=300
Z=-18.9(45.11%) | Like=-14.10..-13.31 [-14.1032..-14.1028]*| it/evals=1500/2347 eff=73.2780% N=300
Z=-18.8(48.43%) | Like=-14.03..-13.30 [-14.0335..-14.0298]*| it/evals=1530/2386 eff=73.3461% N=300
Mono-modal Volume: ~exp(-9.78) * 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(49.78%) | Like=-14.00..-13.30 [-14.0003..-14.0001]*| it/evals=1541/2399 eff=73.4159% N=300
Z=-18.7(51.92%) | Like=-13.95..-13.30 [-13.9517..-13.9502]*| it/evals=1560/2423 eff=73.4809% N=300
Z=-18.7(55.30%) | Like=-13.89..-13.30 [-13.8937..-13.8923]*| it/evals=1590/2462 eff=73.5430% 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(58.08%) | Like=-13.84..-13.30 [-13.8363..-13.8360]*| it/evals=1616/2498 eff=73.5214% N=300
Z=-18.6(58.49%) | Like=-13.83..-13.30 [-13.8310..-13.8287]*| it/evals=1620/2502 eff=73.5695% N=300
Z=-18.6(61.48%) | Like=-13.78..-13.30 [-13.7829..-13.7823]*| it/evals=1649/2542 eff=73.5504% N=300
Z=-18.6(61.57%) | Like=-13.78..-13.30 [-13.7823..-13.7816]*| it/evals=1650/2543 eff=73.5622% N=300
Z=-18.5(63.88%) | Like=-13.75..-13.30 [-13.7489..-13.7479]*| it/evals=1674/2584 eff=73.2925% N=300
Mono-modal Volume: ~exp(-9.78) Expected Volume: exp(-5.58) 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=-18.5(64.46%) | Like=-13.74..-13.30 [-13.7432..-13.7430]*| it/evals=1680/2591 eff=73.3304% N=300
Z=-18.5(67.16%) | Like=-13.70..-13.30 [-13.6992..-13.6880] | it/evals=1709/2633 eff=73.2533% N=300
Z=-18.5(67.25%) | Like=-13.69..-13.30 [-13.6992..-13.6880] | it/evals=1710/2634 eff=73.2648% N=300
Z=-18.4(69.50%) | Like=-13.66..-13.30 [-13.6553..-13.6543]*| it/evals=1737/2675 eff=73.1368% N=300
Z=-18.4(69.73%) | Like=-13.65..-13.30 [-13.6514..-13.6513]*| it/evals=1740/2678 eff=73.1707% N=300
Mono-modal Volume: ~exp(-10.23) * Expected Volume: exp(-5.81) 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=-18.4(69.89%) | Like=-13.65..-13.30 [-13.6511..-13.6501]*| it/evals=1742/2682 eff=73.1318% N=300
[ultranest] Explored until L=-1e+01
[ultranest] Likelihood function evaluations: 2683
[ultranest] logZ = -18.06 +- 0.07516
[ultranest] Effective samples strategy satisfied (ESS = 1022.0, need >400)
[ultranest] Posterior uncertainty strategy is satisfied (KL: 0.46+-0.06 nat, need <0.50 nat)
[ultranest] Evidency uncertainty strategy is satisfied (dlogz=0.27, 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 = -18.072 +- 0.302
single instance: logZ = -18.072 +- 0.111
bootstrapped : logZ = -18.063 +- 0.151
tail : logZ = +- 0.262
insert order U test : converged: True correlation: inf iterations
index : 2.25 │ ▁▁▁▁▂▂▂▃▃▅▅▅▇▆▇▆▆▆▅▅▆▄▄▂▃▂▁▁▁▁▁▁▁▁▁▁▁ │3.41 2.75 +- 0.18
amplitude : 0.0000000000283│ ▁ ▁▁▁▁▂▂▃▃▃▄▅▆▆▇▇▆▄▄▄▄▃▃▂▂▂▁▁▁▁▁▁▁ ▁▁ │0.0000000000839 0.0000000000530 +- 0.0000000000076
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 38.920 seconds)
