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(-3.99) * Expected Volume: exp(0.00) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|********************************************* * | +1.0e-10
Z=-inf(0.00%) | Like=-4169.53..-62.95 [-4169.5281..-351.3616] | it/evals=0/301 eff=0.0000% N=300
Z=-546.2(0.00%) | Like=-536.32..-62.95 [-4169.5281..-351.3616] | it/evals=22/324 eff=91.6667% N=300
Z=-534.4(0.00%) | Like=-528.15..-62.95 [-4169.5281..-351.3616] | it/evals=30/333 eff=90.9091% N=300
Z=-503.2(0.00%) | Like=-495.22..-62.95 [-4169.5281..-351.3616] | it/evals=50/357 eff=87.7193% N=300
Z=-490.3(0.00%) | Like=-484.01..-62.95 [-4169.5281..-351.3616] | it/evals=60/370 eff=85.7143% N=300
Mono-modal Volume: ~exp(-3.99) Expected Volume: exp(-0.22) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|***************************** ******** ****** * | +1.0e-10
Z=-474.2(0.00%) | Like=-467.93..-61.57 [-4169.5281..-351.3616] | it/evals=78/392 eff=84.7826% N=300
Z=-463.9(0.00%) | Like=-458.07..-61.57 [-4169.5281..-351.3616] | it/evals=90/408 eff=83.3333% N=300
Z=-445.2(0.00%) | Like=-437.90..-61.57 [-4169.5281..-351.3616] | it/evals=109/432 eff=82.5758% N=300
Z=-437.7(0.00%) | Like=-432.32..-58.87 [-4169.5281..-351.3616] | it/evals=120/445 eff=82.7586% N=300
Mono-modal Volume: ~exp(-4.21) * Expected Volume: exp(-0.45) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|***************************** *************** **| +1.0e-10
Z=-428.2(0.00%) | Like=-422.65..-58.87 [-4169.5281..-351.3616] | it/evals=134/463 eff=82.2086% N=300
Z=-414.9(0.00%) | Like=-409.69..-58.87 [-4169.5281..-351.3616] | it/evals=150/482 eff=82.4176% N=300
Z=-391.6(0.00%) | Like=-385.20..-58.87 [-4169.5281..-351.3616] | it/evals=169/506 eff=82.0388% N=300
Z=-379.2(0.00%) | Like=-371.45..-58.87 [-4169.5281..-351.3616] | it/evals=180/519 eff=82.1918% N=300
Mono-modal Volume: ~exp(-4.21) Expected Volume: exp(-0.67) Quality: ok
index : +1.0| *********************************************| +5.0
amplitude: +1.0e-12| **************************** *************** **| +1.0e-10
Z=-357.8(0.00%) | Like=-350.62..-58.87 [-350.6204..-194.7231] | it/evals=201/543 eff=82.7160% N=300
Z=-347.1(0.00%) | Like=-339.08..-58.87 [-350.6204..-194.7231] | it/evals=210/553 eff=83.0040% N=300
Z=-328.3(0.00%) | Like=-321.95..-58.87 [-350.6204..-194.7231] | it/evals=228/578 eff=82.0144% N=300
Z=-312.3(0.00%) | Like=-305.12..-58.87 [-350.6204..-194.7231] | it/evals=240/594 eff=81.6327% N=300
Z=-295.8(0.00%) | Like=-289.61..-58.87 [-350.6204..-194.7231] | it/evals=256/618 eff=80.5031% N=300
Mono-modal Volume: ~exp(-4.63) * Expected Volume: exp(-0.89) Quality: ok
index : +1.0| ********************************************| +5.0
amplitude: +1.0e-12| ***************************************** * **| +1.0e-10
Z=-287.4(0.00%) | Like=-281.14..-58.87 [-350.6204..-194.7231] | it/evals=268/636 eff=79.7619% N=300
Z=-286.1(0.00%) | Like=-279.15..-58.87 [-350.6204..-194.7231] | it/evals=270/639 eff=79.6460% N=300
Z=-275.6(0.00%) | Like=-269.78..-58.87 [-350.6204..-194.7231] | it/evals=285/663 eff=78.5124% N=300
Z=-263.1(0.00%) | Like=-256.59..-58.87 [-350.6204..-194.7231] | it/evals=300/682 eff=78.5340% N=300
Z=-250.3(0.00%) | Like=-243.25..-58.87 [-350.6204..-194.7231] | it/evals=317/706 eff=78.0788% N=300
Z=-242.3(0.00%) | Like=-235.96..-58.87 [-350.6204..-194.7231] | it/evals=330/728 eff=77.1028% N=300
Mono-modal Volume: ~exp(-5.01) * Expected Volume: exp(-1.12) Quality: ok
index : +1.0| *******************************************| +5.0
amplitude: +1.0e-12| ******************************************* *| +1.0e-10
Z=-239.1(0.00%) | Like=-232.40..-58.87 [-350.6204..-194.7231] | it/evals=335/733 eff=77.3672% N=300
Z=-229.8(0.00%) | Like=-223.90..-58.87 [-350.6204..-194.7231] | it/evals=351/757 eff=76.8053% N=300
Z=-225.6(0.00%) | Like=-219.22..-58.87 [-350.6204..-194.7231] | it/evals=360/772 eff=76.2712% N=300
Z=-219.1(0.00%) | Like=-212.64..-58.87 [-350.6204..-194.7231] | it/evals=374/797 eff=75.2515% N=300
Z=-209.5(0.00%) | Like=-203.41..-58.87 [-350.6204..-194.7231] | it/evals=390/816 eff=75.5814% N=300
Mono-modal Volume: ~exp(-5.30) * Expected Volume: exp(-1.34) Quality: ok
index : +1.0| *****************************************| +5.0
amplitude: +1.0e-12| *************************************** ** | +1.0e-10
Z=-204.5(0.00%) | Like=-197.71..-58.87 [-350.6204..-194.7231] | it/evals=402/833 eff=75.4221% N=300
Z=-197.8(0.00%) | Like=-191.86..-58.87 [-194.6205..-126.8600] | it/evals=418/857 eff=75.0449% N=300
Z=-197.3(0.00%) | Like=-190.88..-58.87 [-194.6205..-126.8600] | it/evals=420/859 eff=75.1342% N=300
Z=-188.3(0.00%) | Like=-181.87..-58.87 [-194.6205..-126.8600] | it/evals=439/884 eff=75.1712% N=300
Z=-182.2(0.00%) | Like=-175.48..-58.87 [-194.6205..-126.8600] | it/evals=450/902 eff=74.7508% N=300
Z=-177.7(0.00%) | Like=-171.82..-58.87 [-194.6205..-126.8600] | it/evals=467/926 eff=74.6006% N=300
Mono-modal Volume: ~exp(-5.30) Expected Volume: exp(-1.56) Quality: ok
index : +1.0| ************************************ | +5.0
amplitude: +1.0e-12| ************************************* ** | +1.0e-10
Z=-173.0(0.00%) | Like=-166.57..-58.87 [-194.6205..-126.8600] | it/evals=480/943 eff=74.6501% N=300
Z=-167.2(0.00%) | Like=-160.69..-58.87 [-194.6205..-126.8600] | it/evals=494/968 eff=73.9521% N=300
Z=-162.0(0.00%) | Like=-155.73..-58.87 [-194.6205..-126.8600] | it/evals=510/987 eff=74.2358% N=300
Z=-156.9(0.00%) | Like=-151.47..-58.87 [-194.6205..-126.8600] | it/evals=531/1011 eff=74.6835% N=300
Mono-modal Volume: ~exp(-5.92) * Expected Volume: exp(-1.79) Quality: ok
index : +1.0| ******************************* | +5.0
amplitude: +1.0e-12| ************************************* ** | +1.0e-10
Z=-155.8(0.00%) | Like=-149.37..-58.87 [-194.6205..-126.8600] | it/evals=536/1016 eff=74.8603% N=300
Z=-154.5(0.00%) | Like=-148.24..-58.87 [-194.6205..-126.8600] | it/evals=540/1021 eff=74.8960% N=300
Z=-148.8(0.00%) | Like=-142.55..-58.87 [-194.6205..-126.8600] | it/evals=560/1045 eff=75.1678% N=300
Z=-146.3(0.00%) | Like=-139.74..-58.87 [-194.6205..-126.8600] | it/evals=570/1055 eff=75.4967% N=300
Z=-141.0(0.00%) | Like=-134.19..-58.87 [-194.6205..-126.8600] | it/evals=588/1078 eff=75.5784% N=300
Z=-137.6(0.00%) | Like=-131.73..-58.87 [-194.6205..-126.8600] | it/evals=600/1098 eff=75.1880% N=300
Mono-modal Volume: ~exp(-6.15) * Expected Volume: exp(-2.01) Quality: ok
index : +1.0| *************************** +4.0 | +5.0
amplitude: +1.0e-12| *********************************** | +1.0e-10
Z=-137.2(0.00%) | Like=-131.35..-58.87 [-194.6205..-126.8600] | it/evals=603/1103 eff=75.0934% N=300
Z=-134.3(0.00%) | Like=-127.96..-58.87 [-194.6205..-126.8600] | it/evals=621/1127 eff=75.0907% N=300
Z=-132.6(0.00%) | Like=-126.49..-58.87 [-126.8288..-94.5837] | it/evals=630/1139 eff=75.0894% N=300
Z=-128.4(0.00%) | Like=-122.07..-58.87 [-126.8288..-94.5837] | it/evals=650/1163 eff=75.3187% N=300
Z=-126.4(0.00%) | Like=-120.25..-58.87 [-126.8288..-94.5837] | it/evals=660/1175 eff=75.4286% N=300
Mono-modal Volume: ~exp(-6.15) Expected Volume: exp(-2.23) Quality: ok
index : +1.0| +1.9 ************************ +3.8 | +5.0
amplitude: +1.0e-12| ********************************* | +1.0e-10
Z=-122.9(0.00%) | Like=-116.58..-58.87 [-126.8288..-94.5837] | it/evals=676/1197 eff=75.3623% N=300
Z=-120.2(0.00%) | Like=-114.11..-58.87 [-126.8288..-94.5837] | it/evals=690/1216 eff=75.3275% N=300
Z=-117.7(0.00%) | Like=-111.24..-58.87 [-126.8288..-94.5837] | it/evals=707/1240 eff=75.2128% N=300
Z=-115.1(0.00%) | Like=-108.88..-58.87 [-126.8288..-94.5837] | it/evals=720/1258 eff=75.1566% N=300
Z=-113.1(0.00%) | Like=-106.52..-58.80 [-126.8288..-94.5837] | it/evals=734/1285 eff=74.5178% N=300
Mono-modal Volume: ~exp(-6.80) * Expected Volume: exp(-2.46) Quality: ok
index : +1.0| +2.0 ******************** +3.6 | +5.0
amplitude: +1.0e-12| *************************** +7.7e-11 | +1.0e-10
Z=-112.5(0.00%) | Like=-106.09..-58.80 [-126.8288..-94.5837] | it/evals=737/1288 eff=74.5951% N=300
Z=-110.6(0.00%) | Like=-104.55..-58.80 [-126.8288..-94.5837] | it/evals=750/1305 eff=74.6269% N=300
Z=-107.5(0.00%) | Like=-101.24..-58.80 [-126.8288..-94.5837] | it/evals=766/1330 eff=74.3689% N=300
Z=-106.1(0.00%) | Like=-100.36..-58.80 [-126.8288..-94.5837] | it/evals=780/1354 eff=74.0038% N=300
Z=-103.6(0.00%) | Like=-97.22..-58.80 [-126.8288..-94.5837] | it/evals=800/1378 eff=74.2115% N=300
Mono-modal Volume: ~exp(-6.80) Expected Volume: exp(-2.68) Quality: ok
index : +1.0| +2.1 ******************* +3.5 | +5.0
amplitude: +1.0e-12| ************************** +7.5e-11 | +1.0e-10
Z=-102.5(0.00%) | Like=-96.01..-58.80 [-126.8288..-94.5837] | it/evals=810/1391 eff=74.2438% N=300
Z=-100.8(0.00%) | Like=-94.87..-58.80 [-126.8288..-94.5837] | it/evals=830/1416 eff=74.3728% N=300
Z=-100.0(0.00%) | Like=-94.03..-58.80 [-94.3779..-80.1934] | it/evals=840/1431 eff=74.2706% N=300
Z=-98.5(0.00%) | Like=-92.42..-58.80 [-94.3779..-80.1934] | it/evals=856/1456 eff=74.0484% N=300
Z=-97.2(0.00%) | Like=-90.84..-58.80 [-94.3779..-80.1934] | it/evals=870/1478 eff=73.8540% N=300
Mono-modal Volume: ~exp(-6.89) * Expected Volume: exp(-2.90) Quality: ok
index : +1.0| +2.1 ***************** +3.5 | +5.0
amplitude: +1.0e-12| +2.4e-11 ************************ +7.2e-11 | +1.0e-10
Z=-97.1(0.00%) | Like=-90.81..-58.80 [-94.3779..-80.1934] | it/evals=871/1479 eff=73.8762% N=300
Z=-95.6(0.00%) | Like=-89.57..-58.80 [-94.3779..-80.1934] | it/evals=889/1503 eff=73.8986% N=300
Z=-94.8(0.00%) | Like=-88.70..-58.80 [-94.3779..-80.1934] | it/evals=900/1522 eff=73.6498% N=300
Z=-93.4(0.00%) | Like=-87.04..-58.80 [-94.3779..-80.1934] | it/evals=920/1547 eff=73.7771% N=300
Z=-92.5(0.00%) | Like=-86.33..-58.80 [-94.3779..-80.1934] | it/evals=930/1563 eff=73.6342% N=300
Mono-modal Volume: ~exp(-7.28) * Expected Volume: exp(-3.13) Quality: ok
index : +1.0| +2.1 **************** +3.4 | +5.0
amplitude: +1.0e-12| +2.6e-11 ********************* +6.8e-11 | +1.0e-10
Z=-92.0(0.00%) | Like=-85.70..-58.80 [-94.3779..-80.1934] | it/evals=938/1572 eff=73.7421% N=300
Z=-90.7(0.00%) | Like=-84.48..-58.80 [-94.3779..-80.1934] | it/evals=955/1596 eff=73.6883% N=300
Z=-90.4(0.00%) | Like=-84.07..-58.80 [-94.3779..-80.1934] | it/evals=960/1607 eff=73.4507% N=300
Z=-89.2(0.00%) | Like=-83.27..-58.80 [-94.3779..-80.1934] | it/evals=980/1631 eff=73.6289% N=300
Z=-88.6(0.00%) | Like=-82.42..-58.80 [-94.3779..-80.1934] | it/evals=990/1646 eff=73.5513% N=300
Mono-modal Volume: ~exp(-7.34) * Expected Volume: exp(-3.35) Quality: ok
index : +1.0| +2.2 ************** +3.3 | +5.0
amplitude: +1.0e-12| +2.8e-11 ********************* +6.8e-11 | +1.0e-10
Z=-87.6(0.00%) | Like=-81.29..-58.80 [-94.3779..-80.1934] | it/evals=1005/1666 eff=73.5725% N=300
Z=-86.5(0.00%) | Like=-80.09..-58.80 [-80.1261..-70.4926] | it/evals=1020/1682 eff=73.8061% N=300
Z=-85.1(0.00%) | Like=-78.65..-58.78 [-80.1261..-70.4926] | it/evals=1038/1708 eff=73.7216% N=300
Z=-84.5(0.00%) | Like=-78.19..-58.78 [-80.1261..-70.4926] | it/evals=1050/1724 eff=73.7360% N=300
Mono-modal Volume: ~exp(-7.61) * Expected Volume: exp(-3.57) Quality: ok
index : +1.0| +2.2 ************* +3.2 | +5.0
amplitude: +1.0e-12| +2.9e-11 ******************* +6.5e-11 | +1.0e-10
Z=-83.4(0.00%) | Like=-77.19..-58.78 [-80.1261..-70.4926] | it/evals=1072/1747 eff=74.0843% N=300
Z=-83.1(0.00%) | Like=-76.72..-58.78 [-80.1261..-70.4926] | it/evals=1080/1755 eff=74.2268% N=300
Z=-82.1(0.00%) | Like=-75.64..-58.78 [-80.1261..-70.4926] | it/evals=1098/1779 eff=74.2394% N=300
Z=-81.4(0.00%) | Like=-74.81..-58.78 [-80.1261..-70.4926] | it/evals=1110/1799 eff=74.0494% N=300
Z=-80.3(0.00%) | Like=-73.99..-58.77 [-80.1261..-70.4926] | it/evals=1131/1822 eff=74.3101% N=300
Mono-modal Volume: ~exp(-7.61) Expected Volume: exp(-3.80) Quality: ok
index : +1.0| +2.3 *********** +3.2 | +5.0
amplitude: +1.0e-12| +3.0e-11 ***************** +6.3e-11 | +1.0e-10
Z=-80.0(0.00%) | Like=-73.82..-58.77 [-80.1261..-70.4926] | it/evals=1140/1835 eff=74.2671% N=300
Z=-79.3(0.00%) | Like=-72.89..-58.77 [-80.1261..-70.4926] | it/evals=1159/1859 eff=74.3425% N=300
Z=-78.8(0.00%) | Like=-72.28..-58.77 [-80.1261..-70.4926] | it/evals=1170/1875 eff=74.2857% N=300
Z=-78.1(0.00%) | Like=-71.70..-58.77 [-80.1261..-70.4926] | it/evals=1187/1900 eff=74.1875% N=300
Z=-77.6(0.00%) | Like=-71.16..-58.77 [-80.1261..-70.4926] | it/evals=1200/1920 eff=74.0741% N=300
Mono-modal Volume: ~exp(-7.61) Expected Volume: exp(-4.02) Quality: ok
index : +1.0| +2.3 *********** +3.1 | +5.0
amplitude: +1.0e-12| +3.1e-11 *************** +6.0e-11 | +1.0e-10
Z=-77.0(0.00%) | Like=-70.51..-58.77 [-80.1261..-70.4926] | it/evals=1217/1943 eff=74.0718% N=300
Z=-76.5(0.00%) | Like=-70.10..-58.77 [-70.4889..-65.6202] | it/evals=1230/1963 eff=73.9627% N=300
Z=-76.2(0.00%) | Like=-69.76..-58.77 [-70.4889..-65.6202] | it/evals=1241/1987 eff=73.5625% N=300
Z=-75.7(0.00%) | Like=-69.23..-58.77 [-70.4889..-65.6202] | it/evals=1256/2013 eff=73.3217% N=300
Z=-75.6(0.00%) | Like=-69.14..-58.77 [-70.4889..-65.6202] | it/evals=1260/2020 eff=73.2558% N=300
Mono-modal Volume: ~exp(-8.22) * Expected Volume: exp(-4.24) Quality: ok
index : +1.0| +2.3 ********* +3.1 | +5.0
amplitude: +1.0e-12| +3.3e-11 ************* +5.9e-11 | +1.0e-10
Z=-75.2(0.00%) | Like=-68.80..-58.77 [-70.4889..-65.6202] | it/evals=1273/2043 eff=73.0350% N=300
Z=-74.8(0.01%) | Like=-68.36..-58.77 [-70.4889..-65.6202] | it/evals=1288/2068 eff=72.8507% N=300
Z=-74.7(0.01%) | Like=-68.29..-58.77 [-70.4889..-65.6202] | it/evals=1290/2071 eff=72.8402% N=300
Z=-74.3(0.01%) | Like=-67.87..-58.77 [-70.4889..-65.6202] | it/evals=1305/2095 eff=72.7019% N=300
Z=-73.9(0.02%) | Like=-67.51..-58.77 [-70.4889..-65.6202] | it/evals=1319/2119 eff=72.5124% N=300
Z=-73.9(0.02%) | Like=-67.49..-58.77 [-70.4889..-65.6202] | it/evals=1320/2120 eff=72.5275% N=300
Z=-73.5(0.03%) | Like=-67.03..-58.77 [-70.4889..-65.6202] | it/evals=1337/2144 eff=72.5054% N=300
Mono-modal Volume: ~exp(-8.29) * Expected Volume: exp(-4.47) Quality: ok
index : +1.0| +2.4 ********* +3.0 | +5.0
amplitude: +1.0e-12| +3.4e-11 ************* +5.7e-11 | +1.0e-10
Z=-73.4(0.03%) | Like=-66.84..-58.77 [-70.4889..-65.6202] | it/evals=1340/2147 eff=72.5501% N=300
Z=-73.2(0.04%) | Like=-66.59..-58.77 [-70.4889..-65.6202] | it/evals=1350/2161 eff=72.5416% N=300
Z=-72.7(0.06%) | Like=-66.17..-58.77 [-70.4889..-65.6202] | it/evals=1369/2185 eff=72.6260% N=300
Z=-72.4(0.08%) | Like=-65.98..-58.77 [-70.4889..-65.6202] | it/evals=1380/2201 eff=72.5934% N=300
Z=-72.1(0.11%) | Like=-65.58..-58.77 [-65.5769..-65.1485] | it/evals=1395/2225 eff=72.4675% N=300
Mono-modal Volume: ~exp(-8.69) * Expected Volume: exp(-4.69) 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=-71.9(0.14%) | Like=-65.33..-58.77 [-65.5769..-65.1485] | it/evals=1407/2241 eff=72.4884% N=300
Z=-71.8(0.15%) | Like=-65.30..-58.77 [-65.5769..-65.1485] | it/evals=1410/2244 eff=72.5309% N=300
Z=-71.5(0.21%) | Like=-65.04..-58.77 [-65.0355..-65.0124] | it/evals=1429/2267 eff=72.6487% N=300
Z=-71.3(0.26%) | Like=-64.83..-58.77 [-64.8488..-64.8304] | it/evals=1440/2283 eff=72.6172% N=300
Z=-71.0(0.35%) | Like=-64.59..-58.77 [-64.6069..-64.5858] | it/evals=1458/2306 eff=72.6820% N=300
Z=-70.8(0.42%) | Like=-64.37..-58.77 [-64.3876..-64.3736] | it/evals=1470/2319 eff=72.8083% N=300
Mono-modal Volume: ~exp(-8.69) Expected Volume: exp(-4.91) 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=-70.6(0.52%) | Like=-63.96..-58.77 [-63.9576..-63.9119] | it/evals=1484/2341 eff=72.7095% N=300
Z=-70.3(0.69%) | Like=-63.73..-58.77 [-63.7298..-63.7287]*| it/evals=1500/2362 eff=72.7449% N=300
Z=-70.0(0.94%) | Like=-63.33..-58.77 [-63.3334..-63.3051] | it/evals=1519/2385 eff=72.8537% N=300
Z=-69.8(1.15%) | Like=-63.10..-58.77 [-63.0974..-63.0954]*| it/evals=1530/2404 eff=72.7186% N=300
Mono-modal Volume: ~exp(-9.14) * 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=-69.6(1.39%) | Like=-62.92..-58.77 [-62.9452..-62.9208] | it/evals=1541/2422 eff=72.6202% N=300
Z=-69.3(1.80%) | Like=-62.66..-58.77 [-62.6635..-62.6611]*| it/evals=1560/2445 eff=72.7273% N=300
Z=-69.1(2.38%) | Like=-62.42..-58.77 [-62.4198..-62.4186]*| it/evals=1580/2470 eff=72.8111% N=300
Z=-68.9(2.65%) | Like=-62.27..-58.75 [-62.2677..-62.2622]*| it/evals=1590/2482 eff=72.8689% N=300
Z=-68.8(3.16%) | Like=-62.09..-58.75 [-62.0943..-62.0925]*| it/evals=1604/2506 eff=72.7108% N=300
Mono-modal Volume: ~exp(-9.25) * Expected Volume: exp(-5.36) Quality: ok
index : +1.0| +2.5 ****** +2.9 | +5.0
amplitude: +1.0e-12| +3.7e-11 ******** +5.2e-11 | +1.0e-10
Z=-68.7(3.30%) | Like=-62.05..-58.75 [-62.0530..-62.0233] | it/evals=1608/2513 eff=72.6615% N=300
Z=-68.6(3.82%) | Like=-61.89..-58.75 [-61.8938..-61.8724] | it/evals=1620/2531 eff=72.6132% N=300
Z=-68.4(4.65%) | Like=-61.64..-58.75 [-61.6441..-61.6332] | it/evals=1636/2555 eff=72.5499% N=300
Z=-68.2(5.39%) | Like=-61.53..-58.75 [-61.5289..-61.5227]*| it/evals=1650/2576 eff=72.4956% N=300
Z=-68.1(6.30%) | Like=-61.36..-58.75 [-61.3769..-61.3594] | it/evals=1665/2601 eff=72.3598% N=300
Mono-modal Volume: ~exp(-9.69) * Expected Volume: exp(-5.58) 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=-68.0(7.03%) | Like=-61.27..-58.75 [-61.2658..-61.2628]*| it/evals=1675/2616 eff=72.3230% N=300
Z=-67.9(7.36%) | Like=-61.20..-58.75 [-61.2167..-61.2017] | it/evals=1680/2622 eff=72.3514% N=300
Z=-67.7(8.61%) | Like=-61.07..-58.75 [-61.0739..-61.0661]*| it/evals=1698/2646 eff=72.3785% N=300
Z=-67.6(9.57%) | Like=-60.98..-58.75 [-60.9810..-60.9781]*| it/evals=1710/2660 eff=72.4576% N=300
Z=-67.5(10.85%) | Like=-60.86..-58.75 [-60.8710..-60.8561] | it/evals=1728/2683 eff=72.5136% N=300
Z=-67.4(11.97%) | Like=-60.80..-58.75 [-60.7982..-60.7966]*| it/evals=1740/2698 eff=72.5605% N=300
Mono-modal Volume: ~exp(-9.69) Expected Volume: exp(-5.81) Quality: ok
index : +1.0| +2.5 ***** +2.8 | +5.0
amplitude: +1.0e-12| +3.9e-11 ******* +5.1e-11 | +1.0e-10
Z=-67.3(13.50%) | Like=-60.71..-58.75 [-60.7104..-60.7082]*| it/evals=1759/2719 eff=72.7160% N=300
Z=-67.2(14.48%) | Like=-60.65..-58.75 [-60.6750..-60.6491] | it/evals=1770/2732 eff=72.7796% N=300
Z=-67.1(16.11%) | Like=-60.53..-58.75 [-60.5287..-60.5249]*| it/evals=1786/2758 eff=72.6607% N=300
Z=-67.0(17.16%) | Like=-60.45..-58.75 [-60.4480..-60.4453]*| it/evals=1796/2779 eff=72.4486% N=300
Z=-67.0(17.54%) | Like=-60.43..-58.75 [-60.4304..-60.4193] | it/evals=1800/2785 eff=72.4346% N=300
Mono-modal Volume: ~exp(-9.81) * Expected Volume: exp(-6.03) Quality: ok
index : +1.0| +2.5 **** +2.8 | +5.0
amplitude: +1.0e-12| +3.9e-11 ****** +5.0e-11 | +1.0e-10
Z=-67.0(18.37%) | Like=-60.37..-58.75 [-60.3667..-60.3662]*| it/evals=1809/2799 eff=72.3890% N=300
Z=-66.9(20.21%) | Like=-60.25..-58.75 [-60.2505..-60.2439]*| it/evals=1826/2822 eff=72.4029% N=300
Z=-66.8(20.73%) | Like=-60.23..-58.75 [-60.2317..-60.2307]*| it/evals=1830/2829 eff=72.3606% N=300
Z=-66.7(23.09%) | Like=-60.14..-58.75 [-60.1383..-60.1377]*| it/evals=1849/2854 eff=72.3962% N=300
Z=-66.7(24.53%) | Like=-60.09..-58.75 [-60.0893..-60.0877]*| it/evals=1860/2871 eff=72.3454% N=300
Mono-modal Volume: ~exp(-10.10) * 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=-66.6(26.53%) | Like=-60.02..-58.75 [-60.0359..-60.0216] | it/evals=1876/2891 eff=72.4045% N=300
Z=-66.5(28.28%) | Like=-59.97..-58.75 [-59.9738..-59.9722]*| it/evals=1890/2912 eff=72.3583% N=300
Z=-66.5(30.20%) | Like=-59.92..-58.75 [-59.9154..-59.9132]*| it/evals=1906/2937 eff=72.2791% N=300
Z=-66.4(31.83%) | Like=-59.87..-58.75 [-59.8675..-59.8625]*| it/evals=1920/2958 eff=72.2348% N=300
Z=-66.3(34.44%) | Like=-59.81..-58.75 [-59.8301..-59.8148] | it/evals=1940/2981 eff=72.3611% N=300
Mono-modal Volume: ~exp(-10.32) * Expected Volume: exp(-6.48) Quality: ok
index : +1.0| +2.5 **** +2.8 | +5.0
amplitude: +1.0e-12| +4.0e-11 ***** +4.9e-11 | +1.0e-10
Z=-66.3(34.78%) | Like=-59.80..-58.75 [-59.8043..-59.8035]*| it/evals=1943/2984 eff=72.3920% N=300
Z=-66.3(35.62%) | Like=-59.79..-58.75 [-59.7882..-59.7856]*| it/evals=1950/2996 eff=72.3294% N=300
Z=-66.3(37.61%) | Like=-59.73..-58.75 [-59.7309..-59.7284]*| it/evals=1967/3019 eff=72.3428% N=300
Z=-66.2(39.21%) | Like=-59.70..-58.75 [-59.6967..-59.6892]*| it/evals=1980/3033 eff=72.4479% N=300
Z=-66.1(41.61%) | Like=-59.63..-58.75 [-59.6325..-59.6320]*| it/evals=2001/3058 eff=72.5526% N=300
Mono-modal Volume: ~exp(-10.64) * 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=-66.1(42.61%) | Like=-59.60..-58.75 [-59.5985..-59.5983]*| it/evals=2010/3068 eff=72.6156% N=300
Z=-66.1(45.10%) | Like=-59.53..-58.75 [-59.5339..-59.5335]*| it/evals=2030/3092 eff=72.7077% N=300
Z=-66.0(46.23%) | Like=-59.52..-58.75 [-59.5153..-59.5139]*| it/evals=2040/3106 eff=72.7014% N=300
Z=-66.0(48.22%) | Like=-59.47..-58.75 [-59.4657..-59.4645]*| it/evals=2057/3130 eff=72.6855% N=300
Z=-66.0(49.80%) | Like=-59.43..-58.75 [-59.4307..-59.4290]*| it/evals=2070/3147 eff=72.7081% N=300
Mono-modal Volume: ~exp(-11.05) * Expected Volume: exp(-6.92) Quality: ok
index : +1.0| +2.6 **** +2.8 | +5.0
amplitude: +1.0e-12| +4.1e-11 **** +4.8e-11 | +1.0e-10
Z=-66.0(50.63%) | Like=-59.41..-58.75 [-59.4069..-59.4027]*| it/evals=2077/3156 eff=72.7241% N=300
Z=-65.9(52.72%) | Like=-59.38..-58.75 [-59.3789..-59.3782]*| it/evals=2095/3179 eff=72.7683% N=300
Z=-65.9(53.25%) | Like=-59.37..-58.75 [-59.3702..-59.3683]*| it/evals=2100/3187 eff=72.7399% N=300
Z=-65.9(55.28%) | Like=-59.33..-58.75 [-59.3251..-59.3236]*| it/evals=2119/3210 eff=72.8179% N=300
Z=-65.8(56.50%) | Like=-59.32..-58.75 [-59.3164..-59.3122]*| it/evals=2130/3225 eff=72.8205% N=300
Mono-modal Volume: ~exp(-11.14) * 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=-65.8(58.01%) | Like=-59.28..-58.75 [-59.2809..-59.2798]*| it/evals=2144/3244 eff=72.8261% N=300
Z=-65.8(59.59%) | Like=-59.24..-58.75 [-59.2383..-59.2363]*| it/evals=2159/3268 eff=72.7426% N=300
Z=-65.8(59.70%) | Like=-59.24..-58.75 [-59.2363..-59.2327]*| it/evals=2160/3269 eff=72.7518% N=300
Z=-65.8(61.86%) | Like=-59.19..-58.75 [-59.1920..-59.1847]*| it/evals=2181/3296 eff=72.7971% N=300
Z=-65.7(62.76%) | Like=-59.18..-58.75 [-59.1792..-59.1784]*| it/evals=2190/3310 eff=72.7575% N=300
Z=-65.7(64.56%) | Like=-59.16..-58.75 [-59.1585..-59.1585]*| it/evals=2209/3335 eff=72.7842% N=300
Mono-modal Volume: ~exp(-11.48) * 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=-65.7(64.76%) | Like=-59.16..-58.75 [-59.1570..-59.1569]*| it/evals=2211/3339 eff=72.7542% N=300
Z=-65.7(65.64%) | Like=-59.15..-58.75 [-59.1514..-59.1513]*| it/evals=2220/3348 eff=72.8346% N=300
Z=-65.7(67.51%) | Like=-59.13..-58.75 [-59.1256..-59.1246]*| it/evals=2241/3373 eff=72.9255% N=300
Z=-65.7(68.32%) | Like=-59.12..-58.75 [-59.1184..-59.1181]*| it/evals=2250/3383 eff=72.9809% N=300
Z=-65.6(69.83%) | Like=-59.10..-58.75 [-59.0957..-59.0942]*| it/evals=2268/3406 eff=73.0200% N=300
[ultranest] Explored until L=-6e+01
[ultranest] Likelihood function evaluations: 3407
[ultranest] logZ = -65.31 +- 0.09183
[ultranest] Effective samples strategy satisfied (ESS = 986.8, 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: 0.14 bs:0.09 tail:0.26 total:0.28 required:<0.50
[ultranest] done iterating.
logZ = -65.273 +- 0.282
single instance: logZ = -65.273 +- 0.136
bootstrapped : logZ = -65.311 +- 0.104
tail : logZ = +- 0.262
insert order U test : converged: True correlation: inf iterations
index : 2.367 │ ▁▁ ▁▁▁▁▁▂▂▃▂▄▆▆▇▆▆▇▅▅▄▄▃▃▂▁▁▁▁▁▁▁ ▁▁▁ │3.006 2.668 +- 0.082
amplitude : 0.0000000000324│ ▁ ▁▁▁▁▂▂▂▃▄▅▅▇▅▇▅▆▆▅▄▃▃▂▂▁▁▁▁▁▁ ▁ ▁ │0.0000000000590 0.0000000000444 +- 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 +/- 0.08
amplitude : 4.44e-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': [2.6677405904809914, 4.444334504360902e-11], 'stdev': [0.08211061411540356, 3.0330655754923534e-12], 'median': [2.664262654511589, 4.4398501926440816e-11], 'errlo': [2.5917868470844168, 4.1357398633402765e-11], 'errup': [2.752507569560234, 4.746520829699537e-11], 'information_gain_bits': [2.7152755575134018, 3.096187457776693]}
Besides mean, errors, etc, an interesting value is the
information gain which estimates how much the posterior
distribution has shrunk with respect to the prior (i.e. how much
we’ve learned). A value < 1 means that the parameter is poorly
constrained within the prior range (we haven’t learned much with respect to our prior assumption).
For a physical interpretation of the information gain see this
example.
The SamplerResult dictionary contains also other interesting
information :
print(result_joint.sampler_results.keys())
dict_keys(['niter', 'logz', 'logzerr', 'logz_bs', 'logz_single', 'logzerr_tail', 'logzerr_bs', 'ess', 'H', 'Herr', 'posterior', 'weighted_samples', 'samples', 'maximum_likelihood', 'ncall', 'paramnames', 'logzerr_single', 'insertion_order_MWW_test'])
Of particular interest, the samples used in the process to approximate the posterior distribution can be accessed via :
for i, n in enumerate(model.parameters.free_parameters.names):
s = result_joint.samples[:, i]
fig, ax = plt.subplots()
ax.hist(s, bins=30)
ax.axvline(np.mean(s), ls="--", color="red")
ax.set_xlabel(n)
plt.show()
While the above plots are interesting, the real strength of the Bayesian analysis is to visualise all parameters correlations which is usually done using “corner plots”. Ultranest corner plot function is a wrapper around the corner package. See the above link for optional keywords. Other packages exist for corner plots, like chainconsumer which is discussed later in this tutorial.
from ultranest.plot import cornerplot
cornerplot(
result_joint.sampler_results,
plot_datapoints=True,
plot_density=True,
bins=20,
title_fmt=".2e",
smooth=False,
)
plt.show()
Spectral model error band from samples#
To compute the spectral error band (“butterfly plots”), we will directly use the samples of the posterior distribution. This is more robust as compared to the traditional method of using the covariance matrix of the parameters which implicitly assumes Gaussian errors while for the posterior distribution there is no shape assumed. This difference can become significant when the parameter errors are non-Gaussian. For this we will need to convert the list of samples back to the spectral model parameters with the relevant units (e.g. normalisation units).
def get_samples_from_posterior(spectral_model, results):
"""
Create a list of spectral parameters with correct units
from the unitless parameters returned by the sampler.
"""
n_samples = results.samples.shape[0]
samples = []
for p in spectral_model.parameters:
try:
idx = spectral_model.parameters.free_unique_parameters.index(p)
samples.append(results.samples[:, idx] * p.unit)
except ValueError:
samples.append(np.ones(n_samples) * p.quantity)
return samples
samples = get_samples_from_posterior(datasets.models[0].spectral_model, result_joint)
Next we can provide these samples to the plot_error
method.
Individual run analysis#
Now we’ll analyse several Crab runs individually so that we can compare them.
result_0 = sampler.run(datasets[0])
result_1 = sampler.run(datasets[1])
result_2 = sampler.run(datasets[2])
[ultranest] Sampling 300 live points from prior ...
Mono-modal Volume: ~exp(-3.91) * Expected Volume: exp(0.00) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|**************************** ******** ******** *| +1.0e-10
Z=-inf(0.00%) | Like=-1470.64..-21.30 [-1470.6417..-104.8247] | it/evals=0/301 eff=0.0000% N=300
Z=-178.6(0.00%) | Like=-171.70..-21.30 [-1470.6417..-104.8247] | it/evals=30/331 eff=96.7742% N=300
Z=-164.4(0.00%) | Like=-159.54..-21.30 [-1470.6417..-104.8247] | it/evals=60/366 eff=90.9091% 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=-161.3(0.00%) | Like=-156.34..-21.30 [-1470.6417..-104.8247] | it/evals=67/374 eff=90.5405% N=300
Z=-155.4(0.00%) | Like=-149.55..-21.30 [-1470.6417..-104.8247] | it/evals=90/401 eff=89.1089% N=300
Z=-141.6(0.00%) | Like=-136.63..-21.30 [-1470.6417..-104.8247] | it/evals=120/436 eff=88.2353% N=300
Mono-modal Volume: ~exp(-4.38) * Expected Volume: exp(-0.45) Quality: ok
index : +1.0| **********************************************| +5.0
amplitude: +1.0e-12|**************************** ****** ******* * | +1.0e-10
Z=-135.2(0.00%) | Like=-129.47..-21.30 [-1470.6417..-104.8247] | it/evals=134/452 eff=88.1579% N=300
Z=-129.1(0.00%) | Like=-124.53..-21.30 [-1470.6417..-104.8247] | it/evals=150/475 eff=85.7143% N=300
Z=-118.8(0.00%) | Like=-114.23..-20.75 [-1470.6417..-104.8247] | it/evals=180/516 eff=83.3333% 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=-111.5(0.00%) | Like=-106.45..-20.75 [-1470.6417..-104.8247] | it/evals=201/539 eff=84.1004% N=300
Z=-109.3(0.00%) | Like=-103.82..-20.75 [-104.3863..-65.7999] | it/evals=210/548 eff=84.6774% N=300
Z=-96.5(0.00%) | Like=-90.75..-20.75 [-104.3863..-65.7999] | it/evals=240/590 eff=82.7586% 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=-90.5(0.00%) | Like=-85.58..-20.54 [-104.3863..-65.7999] | it/evals=268/626 eff=82.2086% N=300
Z=-90.1(0.00%) | Like=-85.03..-20.54 [-104.3863..-65.7999] | it/evals=270/628 eff=82.3171% N=300
Z=-84.7(0.00%) | Like=-79.65..-20.54 [-104.3863..-65.7999] | it/evals=300/669 eff=81.3008% N=300
Z=-80.0(0.00%) | Like=-75.41..-20.54 [-104.3863..-65.7999] | it/evals=330/710 eff=80.4878% N=300
Mono-modal Volume: ~exp(-5.20) * Expected Volume: exp(-1.12) Quality: ok
index : +1.0| ******************************************| +5.0
amplitude: +1.0e-12| **************************** ********** | +1.0e-10
Z=-79.3(0.00%) | Like=-74.38..-20.54 [-104.3863..-65.7999] | it/evals=335/716 eff=80.5288% N=300
Z=-76.2(0.00%) | Like=-71.77..-20.54 [-104.3863..-65.7999] | it/evals=360/756 eff=78.9474% N=300
Z=-73.3(0.00%) | Like=-68.58..-20.54 [-104.3863..-65.7999] | it/evals=390/795 eff=78.7879% N=300
Mono-modal Volume: ~exp(-5.58) * Expected Volume: exp(-1.34) Quality: ok
index : +1.0| ************************************** | +5.0
amplitude: +1.0e-12| **************************** ******** | +1.0e-10
Z=-72.1(0.00%) | Like=-67.46..-20.54 [-104.3863..-65.7999] | it/evals=402/816 eff=77.9070% N=300
Z=-70.4(0.00%) | Like=-65.43..-20.54 [-65.7393..-46.0622] | it/evals=420/840 eff=77.7778% N=300
Z=-67.2(0.00%) | Like=-62.47..-20.54 [-65.7393..-46.0622] | it/evals=448/882 eff=76.9759% N=300
Z=-67.1(0.00%) | Like=-62.22..-20.54 [-65.7393..-46.0622] | it/evals=450/885 eff=76.9231% N=300
Mono-modal Volume: ~exp(-5.58) Expected Volume: exp(-1.56) Quality: ok
index : +1.0| ******************************** | +5.0
amplitude: +1.0e-12| ***************************** *** +7.6e-11 | +1.0e-10
Z=-65.1(0.00%) | Like=-60.13..-20.54 [-65.7393..-46.0622] | it/evals=473/924 eff=75.8013% N=300
Z=-64.4(0.00%) | Like=-59.36..-20.54 [-65.7393..-46.0622] | it/evals=480/932 eff=75.9494% N=300
Z=-61.3(0.00%) | Like=-56.34..-20.54 [-65.7393..-46.0622] | it/evals=510/976 eff=75.4438% N=300
Mono-modal Volume: ~exp(-5.85) * Expected Volume: exp(-1.79) Quality: ok
index : +1.0| ***************************** +4.0 | +5.0
amplitude: +1.0e-12| **************************** ** +7.4e-11 | +1.0e-10
Z=-59.0(0.00%) | Like=-53.48..-20.54 [-65.7393..-46.0622] | it/evals=536/1018 eff=74.6518% N=300
Z=-58.5(0.00%) | Like=-52.99..-20.54 [-65.7393..-46.0622] | it/evals=540/1025 eff=74.4828% N=300
Z=-55.9(0.00%) | Like=-50.54..-20.54 [-65.7393..-46.0622] | it/evals=569/1070 eff=73.8961% N=300
Z=-55.8(0.00%) | Like=-50.53..-20.54 [-65.7393..-46.0622] | it/evals=570/1071 eff=73.9300% N=300
Z=-53.2(0.00%) | Like=-47.86..-20.54 [-65.7393..-46.0622] | it/evals=597/1114 eff=73.3415% N=300
Z=-53.0(0.00%) | Like=-47.80..-20.54 [-65.7393..-46.0622] | it/evals=600/1117 eff=73.4394% N=300
Mono-modal Volume: ~exp(-5.85) Expected Volume: exp(-2.01) Quality: ok
index : +1.0| ************************ +3.8 | +5.0
amplitude: +1.0e-12| **************************** +6.7e-11 | +1.0e-10
Z=-50.9(0.00%) | Like=-45.69..-20.54 [-46.0586..-33.9898] | it/evals=625/1156 eff=73.0140% N=300
Z=-50.5(0.00%) | Like=-45.02..-20.54 [-46.0586..-33.9898] | it/evals=630/1162 eff=73.0858% N=300
Z=-48.4(0.00%) | Like=-43.13..-20.54 [-46.0586..-33.9898] | it/evals=659/1204 eff=72.8982% N=300
Z=-48.3(0.00%) | Like=-42.75..-20.54 [-46.0586..-33.9898] | it/evals=660/1205 eff=72.9282% N=300
Mono-modal Volume: ~exp(-6.56) * Expected Volume: exp(-2.23) Quality: ok
index : +1.0| ********************** +3.6 | +5.0
amplitude: +1.0e-12| ************************ +6.2e-11 | +1.0e-10
Z=-47.6(0.00%) | Like=-42.26..-20.54 [-46.0586..-33.9898] | it/evals=670/1219 eff=72.9053% N=300
Z=-46.2(0.00%) | Like=-40.95..-20.54 [-46.0586..-33.9898] | it/evals=690/1250 eff=72.6316% N=300
Z=-44.7(0.00%) | Like=-39.48..-20.54 [-46.0586..-33.9898] | it/evals=718/1293 eff=72.3061% N=300
Z=-44.6(0.00%) | Like=-39.31..-20.54 [-46.0586..-33.9898] | it/evals=720/1295 eff=72.3618% N=300
Mono-modal Volume: ~exp(-6.73) * Expected Volume: exp(-2.46) Quality: ok
index : +1.0| +1.9 ******************* +3.4 | +5.0
amplitude: +1.0e-12| ********************* +5.8e-11 | +1.0e-10
Z=-43.7(0.00%) | Like=-38.25..-20.54 [-46.0586..-33.9898] | it/evals=737/1321 eff=72.1841% N=300
Z=-42.8(0.00%) | Like=-37.19..-20.54 [-46.0586..-33.9898] | it/evals=750/1337 eff=72.3240% N=300
Z=-41.3(0.00%) | Like=-36.06..-20.54 [-46.0586..-33.9898] | it/evals=774/1378 eff=71.7996% N=300
Z=-41.0(0.00%) | Like=-35.72..-20.46 [-46.0586..-33.9898] | it/evals=780/1384 eff=71.9557% N=300
Mono-modal Volume: ~exp(-6.73) Expected Volume: exp(-2.68) Quality: ok
index : +1.0| +2.0 ****************** +3.3 | +5.0
amplitude: +1.0e-12| ********************* +5.8e-11 | +1.0e-10
Z=-39.8(0.00%) | Like=-34.55..-20.46 [-46.0586..-33.9898] | it/evals=807/1424 eff=71.7972% N=300
Z=-39.7(0.00%) | Like=-34.42..-20.46 [-46.0586..-33.9898] | it/evals=810/1427 eff=71.8722% N=300
Z=-38.5(0.00%) | Like=-33.25..-20.46 [-33.9877..-27.6278] | it/evals=840/1468 eff=71.9178% N=300
Z=-37.3(0.00%) | Like=-31.97..-20.46 [-33.9877..-27.6278] | it/evals=870/1509 eff=71.9603% N=300
Mono-modal Volume: ~exp(-6.75) * Expected Volume: exp(-2.90) Quality: ok
index : +1.0| +2.0 *************** +3.2 | +5.0
amplitude: +1.0e-12| ****************** +5.4e-11 | +1.0e-10
Z=-37.3(0.00%) | Like=-31.96..-20.46 [-33.9877..-27.6278] | it/evals=871/1511 eff=71.9240% N=300
Z=-36.2(0.00%) | Like=-30.87..-20.46 [-33.9877..-27.6278] | it/evals=900/1546 eff=72.2311% N=300
Z=-35.4(0.01%) | Like=-29.92..-20.46 [-33.9877..-27.6278] | it/evals=928/1589 eff=71.9938% N=300
Z=-35.3(0.01%) | Like=-29.87..-20.46 [-33.9877..-27.6278] | it/evals=930/1591 eff=72.0372% N=300
Mono-modal Volume: ~exp(-7.23) * Expected Volume: exp(-3.13) Quality: ok
index : +1.0| +2.1 ************* +3.2 | +5.0
amplitude: +1.0e-12| **************** +5.2e-11 | +1.0e-10
Z=-35.0(0.01%) | Like=-29.43..-20.46 [-33.9877..-27.6278] | it/evals=938/1601 eff=72.0984% N=300
Z=-34.2(0.02%) | Like=-28.72..-20.46 [-33.9877..-27.6278] | it/evals=960/1629 eff=72.2348% N=300
Z=-33.4(0.05%) | Like=-27.90..-20.46 [-33.9877..-27.6278] | it/evals=986/1671 eff=71.9183% N=300
Z=-33.2(0.06%) | Like=-27.83..-20.46 [-33.9877..-27.6278] | it/evals=990/1678 eff=71.8433% N=300
Mono-modal Volume: ~exp(-7.68) * Expected Volume: exp(-3.35) Quality: ok
index : +1.0| +2.1 ************ +3.1 | +5.0
amplitude: +1.0e-12| ************** +4.9e-11 | +1.0e-10
Z=-32.9(0.08%) | Like=-27.48..-20.46 [-27.6094..-26.9503] | it/evals=1005/1698 eff=71.8884% N=300
Z=-32.5(0.12%) | Like=-27.22..-20.46 [-27.6094..-26.9503] | it/evals=1020/1724 eff=71.6292% N=300
Z=-31.8(0.22%) | Like=-26.50..-20.46 [-26.5042..-26.4541] | it/evals=1049/1765 eff=71.6041% N=300
Z=-31.8(0.22%) | Like=-26.45..-20.46 [-26.5042..-26.4541] | it/evals=1050/1766 eff=71.6235% N=300
Mono-modal Volume: ~exp(-7.79) * Expected Volume: exp(-3.57) Quality: ok
index : +1.0| +2.2 ********** +3.0 | +5.0
amplitude: +1.0e-12| ************* +4.8e-11 | +1.0e-10
Z=-31.4(0.34%) | Like=-26.03..-20.46 [-26.0304..-26.0122] | it/evals=1072/1795 eff=71.7057% N=300
Z=-31.2(0.40%) | Like=-25.90..-20.46 [-25.8963..-25.8776] | it/evals=1080/1804 eff=71.8085% N=300
Z=-30.7(0.67%) | Like=-25.49..-20.46 [-25.4914..-25.4882]*| it/evals=1109/1848 eff=71.6408% N=300
Z=-30.7(0.68%) | Like=-25.49..-20.46 [-25.4882..-25.4477] | it/evals=1110/1849 eff=71.6591% N=300
Mono-modal Volume: ~exp(-7.83) * Expected Volume: exp(-3.80) Quality: ok
index : +1.0| +2.2 ********** +3.0 | +5.0
amplitude: +1.0e-12| +2.4e-11 *********** +4.6e-11 | +1.0e-10
Z=-30.3(1.03%) | Like=-24.98..-20.46 [-24.9834..-24.9810]*| it/evals=1139/1890 eff=71.6352% N=300
Z=-30.3(1.05%) | Like=-24.98..-20.46 [-24.9810..-24.9591] | it/evals=1140/1891 eff=71.6530% N=300
Z=-29.9(1.55%) | Like=-24.48..-20.46 [-24.4995..-24.4821] | it/evals=1167/1938 eff=71.2454% N=300
Z=-29.8(1.60%) | Like=-24.45..-20.46 [-24.4542..-24.4512]*| it/evals=1170/1943 eff=71.2112% N=300
Z=-29.4(2.36%) | Like=-24.06..-20.46 [-24.0579..-24.0371] | it/evals=1199/1986 eff=71.1151% N=300
Z=-29.4(2.39%) | Like=-24.04..-20.46 [-24.0579..-24.0371] | it/evals=1200/1987 eff=71.1322% N=300
Mono-modal Volume: ~exp(-8.02) * Expected Volume: exp(-4.02) Quality: ok
index : +1.0| +2.3 ******** +2.9 | +5.0
amplitude: +1.0e-12| +2.5e-11 *********** +4.5e-11 | +1.0e-10
Z=-29.3(2.60%) | Like=-23.95..-20.46 [-23.9468..-23.9441]*| it/evals=1206/1996 eff=71.1085% N=300
Z=-29.0(3.43%) | Like=-23.67..-20.46 [-23.7284..-23.6725] | it/evals=1230/2030 eff=71.0983% N=300
Z=-28.7(4.63%) | Like=-23.44..-20.46 [-23.4394..-23.4143] | it/evals=1258/2073 eff=70.9532% N=300
Z=-28.7(4.70%) | Like=-23.41..-20.46 [-23.4109..-23.4029]*| it/evals=1260/2076 eff=70.9459% N=300
Mono-modal Volume: ~exp(-8.51) * Expected Volume: exp(-4.24) 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=-28.6(5.39%) | Like=-23.31..-20.46 [-23.3273..-23.3079] | it/evals=1273/2095 eff=70.9192% N=300
Z=-28.4(6.31%) | Like=-23.17..-20.46 [-23.1682..-23.1679]*| it/evals=1290/2112 eff=71.1921% N=300
Z=-28.2(8.00%) | Like=-22.94..-20.46 [-22.9388..-22.9135] | it/evals=1319/2153 eff=71.1819% N=300
Z=-28.2(8.05%) | Like=-22.91..-20.46 [-22.9388..-22.9135] | it/evals=1320/2155 eff=71.1590% N=300
Mono-modal Volume: ~exp(-8.51) Expected Volume: exp(-4.47) Quality: ok
index : +1.0| +2.3 ******** +2.8 | +5.0
amplitude: +1.0e-12| +2.7e-11 ********* +4.2e-11 | +1.0e-10
Z=-28.0(9.75%) | Like=-22.71..-20.46 [-22.7080..-22.6792] | it/evals=1345/2194 eff=71.0137% N=300
Z=-28.0(10.16%) | Like=-22.66..-20.46 [-22.6590..-22.6573]*| it/evals=1350/2200 eff=71.0526% N=300
Z=-27.8(12.50%) | Like=-22.45..-20.46 [-22.4516..-22.4410] | it/evals=1379/2242 eff=71.0093% N=300
Z=-27.7(12.60%) | Like=-22.44..-20.46 [-22.4516..-22.4410] | it/evals=1380/2243 eff=71.0242% N=300
Mono-modal Volume: ~exp(-8.84) * Expected Volume: exp(-4.69) Quality: ok
index : +1.0| +2.3 ****** +2.8 | +5.0
amplitude: +1.0e-12| +2.7e-11 ******** +4.2e-11 | +1.0e-10
Z=-27.6(15.04%) | Like=-22.24..-20.46 [-22.2367..-22.2357]*| it/evals=1407/2277 eff=71.1684% N=300
Z=-27.6(15.38%) | Like=-22.21..-20.46 [-22.2351..-22.2108] | it/evals=1410/2280 eff=71.2121% N=300
Z=-27.4(18.64%) | Like=-22.02..-20.46 [-22.0218..-22.0190]*| it/evals=1440/2320 eff=71.2871% N=300
Z=-27.2(21.87%) | Like=-21.86..-20.46 [-21.8569..-21.8543]*| it/evals=1468/2362 eff=71.1930% N=300
Z=-27.2(22.08%) | Like=-21.85..-20.46 [-21.8452..-21.8419]*| it/evals=1470/2366 eff=71.1520% N=300
Mono-modal Volume: ~exp(-8.86) * Expected Volume: exp(-4.91) 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=-27.2(22.56%) | Like=-21.82..-20.46 [-21.8241..-21.8080] | it/evals=1474/2373 eff=71.1047% N=300
Z=-27.1(25.93%) | Like=-21.67..-20.46 [-21.6724..-21.6720]*| it/evals=1500/2413 eff=70.9891% N=300
Z=-26.9(29.65%) | Like=-21.58..-20.46 [-21.5760..-21.5735]*| it/evals=1530/2452 eff=71.0967% N=300
Mono-modal Volume: ~exp(-9.44) * Expected Volume: exp(-5.14) 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=-26.9(31.08%) | Like=-21.54..-20.46 [-21.5364..-21.5346]*| it/evals=1541/2468 eff=71.0793% N=300
Z=-26.8(33.54%) | Like=-21.48..-20.46 [-21.4775..-21.4771]*| it/evals=1560/2495 eff=71.0706% N=300
Z=-26.7(37.31%) | Like=-21.35..-20.46 [-21.3509..-21.3492]*| it/evals=1590/2531 eff=71.2685% N=300
Mono-modal Volume: ~exp(-9.74) * Expected Volume: exp(-5.36) 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(39.62%) | Like=-21.30..-20.46 [-21.3024..-21.2962]*| it/evals=1608/2556 eff=71.2766% N=300
Z=-26.6(41.06%) | Like=-21.26..-20.46 [-21.2639..-21.2579]*| it/evals=1620/2571 eff=71.3342% N=300
Z=-26.5(44.21%) | Like=-21.19..-20.46 [-21.1860..-21.1830]*| it/evals=1644/2602 eff=71.4162% N=300
Z=-26.5(45.02%) | Like=-21.18..-20.46 [-21.1763..-21.1742]*| it/evals=1650/2609 eff=71.4595% N=300
Z=-26.4(48.05%) | Like=-21.14..-20.46 [-21.1382..-21.1380]*| it/evals=1674/2647 eff=71.3251% N=300
Mono-modal Volume: ~exp(-9.96) * Expected Volume: exp(-5.58) Quality: ok
index : +1.0| +2.4 **** +2.7 | +5.0
amplitude: +1.0e-12| +3.0e-11 ***** +3.9e-11 | +1.0e-10
Z=-26.4(48.17%) | Like=-21.14..-20.46 [-21.1380..-21.1336]*| it/evals=1675/2649 eff=71.3069% N=300
Z=-26.4(48.83%) | Like=-21.12..-20.46 [-21.1217..-21.1188]*| it/evals=1680/2654 eff=71.3679% N=300
Z=-26.4(52.40%) | Like=-21.06..-20.46 [-21.0615..-21.0602]*| it/evals=1710/2696 eff=71.3689% N=300
Z=-26.3(54.94%) | Like=-21.00..-20.46 [-20.9967..-20.9966]*| it/evals=1731/2726 eff=71.3520% N=300
Z=-26.3(55.90%) | Like=-20.98..-20.46 [-20.9849..-20.9827]*| it/evals=1740/2742 eff=71.2531% N=300
Mono-modal Volume: ~exp(-10.01) * Expected Volume: exp(-5.81) 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=-26.3(56.15%) | Like=-20.98..-20.46 [-20.9796..-20.9791]*| it/evals=1742/2745 eff=71.2474% N=300
Z=-26.2(59.20%) | Like=-20.94..-20.46 [-20.9393..-20.9375]*| it/evals=1770/2783 eff=71.2847% N=300
Z=-26.2(61.66%) | Like=-20.90..-20.46 [-20.9031..-20.8999]*| it/evals=1794/2825 eff=71.0495% N=300
Z=-26.2(62.29%) | Like=-20.89..-20.46 [-20.8894..-20.8850]*| it/evals=1800/2831 eff=71.1181% N=300
Mono-modal Volume: ~exp(-10.54) * Expected Volume: exp(-6.03) Quality: ok
index : +1.0| +2.5 **** +2.7 | +5.0
amplitude: +1.0e-12| +3.1e-11 **** +3.8e-11 | +1.0e-10
Z=-26.2(63.20%) | Like=-20.88..-20.46 [-20.8762..-20.8746]*| it/evals=1809/2842 eff=71.1644% N=300
Z=-26.1(65.27%) | Like=-20.85..-20.46 [-20.8474..-20.8473]*| it/evals=1830/2873 eff=71.1232% N=300
Z=-26.1(67.93%) | Like=-20.82..-20.46 [-20.8237..-20.8236]*| it/evals=1858/2914 eff=71.0788% N=300
Z=-26.1(68.10%) | Like=-20.82..-20.46 [-20.8221..-20.8209]*| it/evals=1860/2918 eff=71.0466% N=300
Mono-modal Volume: ~exp(-10.79) * Expected Volume: exp(-6.25) 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=-26.1(69.48%) | Like=-20.81..-20.46 [-20.8087..-20.8086]*| it/evals=1876/2948 eff=70.8459% N=300
[ultranest] Explored until L=-2e+01
[ultranest] Likelihood function evaluations: 2953
[ultranest] logZ = -25.68 +- 0.1003
[ultranest] Effective samples strategy satisfied (ESS = 971.6, 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:0.10 tail:0.26 total:0.28 required:<0.50
[ultranest] done iterating.
logZ = -25.707 +- 0.332
single instance: logZ = -25.707 +- 0.120
bootstrapped : logZ = -25.682 +- 0.203
tail : logZ = +- 0.262
insert order U test : converged: True correlation: inf iterations
index : 2.15 │ ▁▁▁▁▁▁▁▂▃▂▄▅▆▆▆▆▇▇▇▆▆▆▃▃▃▂▁▁▁▁▁▁▁▁ ▁▁ │3.08 2.57 +- 0.13
amplitude : 0.0000000000220│ ▁▁▁▁▁▂▂▃▄▆▆▇▇▇▇▇▆▅▅▃▃▂▂▂▁▁▁▁▁▁▁ ▁ ▁ │0.0000000000530 0.0000000000339 +- 0.0000000000037
[ultranest] Sampling 300 live points from prior ...
Mono-modal Volume: ~exp(-3.53) * Expected Volume: exp(0.00) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|******************************* ***** **********| +1.0e-10
Z=-inf(0.00%) | Like=-1046.22..-19.21 [-1046.2206..-131.2756] | it/evals=0/301 eff=0.0000% N=300
Z=-215.9(0.00%) | Like=-210.52..-19.21 [-1046.2206..-131.2756] | it/evals=30/333 eff=90.9091% N=300
Z=-201.9(0.00%) | Like=-197.33..-19.21 [-1046.2206..-131.2756] | it/evals=60/366 eff=90.9091% N=300
Mono-modal Volume: ~exp(-4.20) * Expected Volume: exp(-0.22) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|************************************* **********| +1.0e-10
Z=-198.0(0.00%) | Like=-192.10..-19.21 [-1046.2206..-131.2756] | it/evals=67/375 eff=89.3333% N=300
Z=-186.6(0.00%) | Like=-182.09..-19.21 [-1046.2206..-131.2756] | it/evals=90/406 eff=84.9057% N=300
Z=-167.2(0.00%) | Like=-161.78..-19.21 [-1046.2206..-131.2756] | it/evals=120/440 eff=85.7143% 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=-161.8(0.00%) | Like=-156.51..-19.21 [-1046.2206..-131.2756] | it/evals=134/458 eff=84.8101% N=300
Z=-154.0(0.00%) | Like=-148.67..-19.21 [-1046.2206..-131.2756] | it/evals=150/481 eff=82.8729% N=300
Z=-144.9(0.00%) | Like=-139.74..-19.21 [-1046.2206..-131.2756] | it/evals=180/523 eff=80.7175% N=300
Mono-modal Volume: ~exp(-4.43) Expected Volume: exp(-0.67) Quality: ok
index : +1.0| **********************************************| +5.0
amplitude: +1.0e-12| ********************************** * **********| +1.0e-10
Z=-131.8(0.00%) | Like=-125.38..-19.21 [-131.2311..-65.3805] | it/evals=210/559 eff=81.0811% N=300
Z=-116.1(0.00%) | Like=-109.87..-19.21 [-131.2311..-65.3805] | it/evals=240/601 eff=79.7342% N=300
Z=-107.3(0.00%) | Like=-102.37..-19.21 [-131.2311..-65.3805] | it/evals=266/643 eff=77.5510% N=300
Mono-modal Volume: ~exp(-4.48) * Expected Volume: exp(-0.89) Quality: ok
index : +1.0| *******************************************| +5.0
amplitude: +1.0e-12| ********************************** **********| +1.0e-10
Z=-107.0(0.00%) | Like=-102.30..-19.21 [-131.2311..-65.3805] | it/evals=268/646 eff=77.4566% N=300
Z=-106.7(0.00%) | Like=-101.78..-19.21 [-131.2311..-65.3805] | it/evals=270/649 eff=77.3639% N=300
Z=-97.9(0.00%) | Like=-92.44..-19.21 [-131.2311..-65.3805] | it/evals=299/690 eff=76.6667% N=300
Z=-97.6(0.00%) | Like=-92.15..-19.21 [-131.2311..-65.3805] | it/evals=300/692 eff=76.5306% N=300
Z=-90.3(0.00%) | Like=-84.59..-19.21 [-131.2311..-65.3805] | it/evals=330/730 eff=76.7442% N=300
Mono-modal Volume: ~exp(-4.93) * Expected Volume: exp(-1.12) Quality: ok
index : +1.0| ******************************************| +5.0
amplitude: +1.0e-12| ********************************* ***** ****| +1.0e-10
Z=-89.0(0.00%) | Like=-83.48..-19.21 [-131.2311..-65.3805] | it/evals=335/735 eff=77.0115% N=300
Z=-81.7(0.00%) | Like=-76.27..-19.21 [-131.2311..-65.3805] | it/evals=360/770 eff=76.5957% N=300
Z=-73.9(0.00%) | Like=-68.56..-19.21 [-131.2311..-65.3805] | it/evals=390/801 eff=77.8443% N=300
Mono-modal Volume: ~exp(-5.19) * Expected Volume: exp(-1.34) Quality: ok
index : +1.0| ************************************** *| +5.0
amplitude: +1.0e-12| ************************************* ****| +1.0e-10
Z=-70.5(0.00%) | Like=-64.68..-19.21 [-65.3645..-43.1420] | it/evals=402/823 eff=76.8642% N=300
Z=-66.6(0.00%) | Like=-61.37..-19.21 [-65.3645..-43.1420] | it/evals=420/846 eff=76.9231% N=300
Z=-61.3(0.00%) | Like=-56.01..-19.21 [-65.3645..-43.1420] | it/evals=450/885 eff=76.9231% N=300
Mono-modal Volume: ~exp(-5.56) * Expected Volume: exp(-1.56) Quality: ok
index : +1.0| ************************************* *| +5.0
amplitude: +1.0e-12| *********************************** ****| +1.0e-10
Z=-59.5(0.00%) | Like=-54.87..-19.21 [-65.3645..-43.1420] | it/evals=469/910 eff=76.8852% N=300
Z=-58.6(0.00%) | Like=-53.68..-19.21 [-65.3645..-43.1420] | it/evals=480/928 eff=76.4331% N=300
Z=-55.9(0.00%) | Like=-50.86..-19.21 [-65.3645..-43.1420] | it/evals=510/964 eff=76.8072% N=300
Mono-modal Volume: ~exp(-5.82) * Expected Volume: exp(-1.79) Quality: ok
index : +1.0| ********************************** | +5.0
amplitude: +1.0e-12| ********************************* ****| +1.0e-10
Z=-53.4(0.00%) | Like=-48.02..-19.21 [-65.3645..-43.1420] | it/evals=536/1000 eff=76.5714% N=300
Z=-53.0(0.00%) | Like=-47.50..-19.21 [-65.3645..-43.1420] | it/evals=540/1004 eff=76.7045% N=300
Z=-49.8(0.00%) | Like=-44.88..-19.21 [-65.3645..-43.1420] | it/evals=570/1045 eff=76.5101% N=300
Z=-47.9(0.00%) | Like=-43.00..-19.21 [-43.1165..-31.5092] | it/evals=600/1082 eff=76.7263% N=300
Mono-modal Volume: ~exp(-5.92) * Expected Volume: exp(-2.01) Quality: ok
index : +1.0| +2.0 ***************************** | +5.0
amplitude: +1.0e-12| +2.4e-11 ************************************ | +1.0e-10
Z=-47.7(0.00%) | Like=-42.81..-19.21 [-43.1165..-31.5092] | it/evals=603/1086 eff=76.7176% N=300
Z=-46.0(0.00%) | Like=-41.14..-19.21 [-43.1165..-31.5092] | it/evals=630/1127 eff=76.1790% N=300
Z=-44.6(0.00%) | Like=-39.51..-19.21 [-43.1165..-31.5092] | it/evals=660/1163 eff=76.4774% N=300
Mono-modal Volume: ~exp(-6.27) * Expected Volume: exp(-2.23) Quality: ok
index : +1.0| +2.0 *************************** | +5.0
amplitude: +1.0e-12| +2.7e-11 *********************************** | +1.0e-10
Z=-44.0(0.00%) | Like=-39.03..-19.21 [-43.1165..-31.5092] | it/evals=670/1179 eff=76.2230% N=300
Z=-42.7(0.00%) | Like=-37.26..-19.21 [-43.1165..-31.5092] | it/evals=690/1205 eff=76.2431% N=300
Z=-41.1(0.00%) | Like=-35.98..-19.21 [-43.1165..-31.5092] | it/evals=718/1249 eff=75.6586% N=300
Z=-41.0(0.00%) | Like=-35.94..-19.21 [-43.1165..-31.5092] | it/evals=720/1251 eff=75.7098% N=300
Mono-modal Volume: ~exp(-6.27) Expected Volume: exp(-2.46) Quality: ok
index : +1.0| +2.1 ************************ +4.0 | +5.0
amplitude: +1.0e-12| +2.8e-11 ********************************** | +1.0e-10
Z=-39.8(0.00%) | Like=-34.80..-19.21 [-43.1165..-31.5092] | it/evals=747/1290 eff=75.4545% N=300
Z=-39.7(0.00%) | Like=-34.72..-19.21 [-43.1165..-31.5092] | it/evals=750/1294 eff=75.4527% N=300
Z=-38.1(0.00%) | Like=-32.51..-19.21 [-43.1165..-31.5092] | it/evals=779/1336 eff=75.1931% N=300
Z=-38.0(0.00%) | Like=-32.48..-19.21 [-43.1165..-31.5092] | it/evals=780/1338 eff=75.1445% N=300
Mono-modal Volume: ~exp(-6.57) * 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=-31.61..-19.21 [-43.1165..-31.5092] | it/evals=804/1379 eff=74.5134% N=300
Z=-36.6(0.00%) | Like=-31.50..-19.21 [-31.5066..-26.0807] | it/evals=810/1391 eff=74.2438% N=300
Z=-35.6(0.00%) | Like=-30.50..-19.21 [-31.5066..-26.0807] | it/evals=835/1433 eff=73.6981% N=300
Z=-35.5(0.00%) | Like=-30.41..-19.21 [-31.5066..-26.0807] | it/evals=840/1439 eff=73.7489% N=300
Z=-34.7(0.00%) | Like=-29.56..-19.21 [-31.5066..-26.0807] | it/evals=864/1483 eff=73.0347% N=300
Z=-34.5(0.00%) | Like=-29.30..-19.21 [-31.5066..-26.0807] | it/evals=870/1497 eff=72.6817% N=300
Mono-modal Volume: ~exp(-6.71) * 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=-34.5(0.00%) | Like=-29.25..-19.21 [-31.5066..-26.0807] | it/evals=871/1498 eff=72.7045% N=300
Z=-33.7(0.01%) | Like=-28.58..-19.21 [-31.5066..-26.0807] | it/evals=900/1533 eff=72.9927% N=300
Z=-33.0(0.02%) | Like=-27.96..-19.21 [-31.5066..-26.0807] | it/evals=926/1575 eff=72.6275% N=300
Z=-32.9(0.02%) | Like=-27.90..-19.21 [-31.5066..-26.0807] | it/evals=930/1579 eff=72.7131% N=300
Mono-modal Volume: ~exp(-7.07) * 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=-32.7(0.02%) | Like=-27.49..-19.21 [-31.5066..-26.0807] | it/evals=938/1589 eff=72.7696% N=300
Z=-32.1(0.04%) | Like=-26.70..-19.21 [-31.5066..-26.0807] | it/evals=960/1620 eff=72.7273% N=300
Z=-31.3(0.10%) | Like=-25.96..-19.21 [-26.0772..-25.6012] | it/evals=990/1663 eff=72.6339% N=300
Mono-modal Volume: ~exp(-7.07) Expected Volume: exp(-3.35) Quality: ok
index : +1.0| +2.3 *************** +3.5 | +5.0
amplitude: +1.0e-12| +3.6e-11 ********************* +7.8e-11| +1.0e-10
Z=-30.7(0.18%) | Like=-25.47..-19.21 [-25.4706..-25.4678]*| it/evals=1014/1703 eff=72.2737% N=300
Z=-30.6(0.21%) | Like=-25.44..-19.21 [-25.4368..-25.4058] | it/evals=1020/1714 eff=72.1358% N=300
Z=-30.1(0.36%) | Like=-24.90..-19.21 [-24.8995..-24.8850] | it/evals=1049/1757 eff=71.9973% N=300
Z=-30.1(0.37%) | Like=-24.89..-19.21 [-24.8995..-24.8850] | it/evals=1050/1758 eff=72.0165% N=300
Mono-modal Volume: ~exp(-7.63) * Expected Volume: exp(-3.57) Quality: ok
index : +1.0| +2.4 ************* +3.4 | +5.0
amplitude: +1.0e-12| +3.8e-11 ******************** +7.6e-11 | +1.0e-10
Z=-29.7(0.54%) | Like=-24.48..-19.21 [-24.4756..-24.4376] | it/evals=1072/1794 eff=71.7537% N=300
Z=-29.5(0.61%) | Like=-24.33..-19.21 [-24.3398..-24.3264] | it/evals=1080/1803 eff=71.8563% N=300
Z=-29.1(0.93%) | Like=-23.95..-19.21 [-23.9707..-23.9467] | it/evals=1107/1844 eff=71.6969% N=300
Z=-29.1(0.98%) | Like=-23.92..-19.21 [-23.9282..-23.9152] | it/evals=1110/1851 eff=71.5667% N=300
Mono-modal Volume: ~exp(-7.99) * Expected Volume: exp(-3.80) Quality: ok
index : +1.0| +2.4 ************* +3.3 | +5.0
amplitude: +1.0e-12| +3.9e-11 ****************** +7.3e-11 | +1.0e-10
Z=-28.7(1.47%) | Like=-23.46..-19.21 [-23.4784..-23.4565] | it/evals=1139/1883 eff=71.9520% N=300
Z=-28.7(1.49%) | Like=-23.44..-19.21 [-23.4429..-23.3929] | it/evals=1140/1884 eff=71.9697% N=300
Z=-28.3(2.20%) | Like=-23.05..-19.17 [-23.0451..-23.0201] | it/evals=1170/1923 eff=72.0887% N=300
Z=-27.9(3.07%) | Like=-22.67..-19.17 [-22.6708..-22.6682]*| it/evals=1197/1966 eff=71.8487% N=300
Z=-27.9(3.20%) | Like=-22.62..-19.17 [-22.6197..-22.6151]*| it/evals=1200/1969 eff=71.8993% N=300
Mono-modal Volume: ~exp(-7.99) Expected Volume: exp(-4.02) Quality: ok
index : +1.0| +2.4 *********** +3.3 | +5.0
amplitude: +1.0e-12| +4.1e-11 **************** +7.2e-11 | +1.0e-10
Z=-27.6(4.24%) | Like=-22.38..-19.17 [-22.3792..-22.3758]*| it/evals=1226/2007 eff=71.8219% N=300
Z=-27.6(4.42%) | Like=-22.37..-19.17 [-22.3689..-22.3607]*| it/evals=1230/2014 eff=71.7620% N=300
Z=-27.3(5.72%) | Like=-22.08..-19.17 [-22.0753..-22.0662]*| it/evals=1257/2056 eff=71.5831% N=300
Z=-27.3(5.87%) | Like=-22.05..-19.17 [-22.0484..-22.0345] | it/evals=1260/2059 eff=71.6316% N=300
Mono-modal Volume: ~exp(-8.18) * Expected Volume: exp(-4.24) Quality: ok
index : +1.0| +2.5 ********** +3.2 | +5.0
amplitude: +1.0e-12| +4.3e-11 ************** +6.9e-11 | +1.0e-10
Z=-27.2(6.57%) | Like=-21.96..-19.17 [-21.9612..-21.9542]*| it/evals=1273/2077 eff=71.6376% N=300
Z=-27.0(7.53%) | Like=-21.85..-19.17 [-21.8478..-21.8407]*| it/evals=1290/2096 eff=71.8263% N=300
Z=-26.8(9.31%) | Like=-21.61..-19.17 [-21.6116..-21.6112]*| it/evals=1318/2137 eff=71.7474% N=300
Z=-26.8(9.40%) | Like=-21.61..-19.17 [-21.6058..-21.6018]*| it/evals=1320/2139 eff=71.7781% N=300
Mono-modal Volume: ~exp(-8.26) * Expected Volume: exp(-4.47) Quality: ok
index : +1.0| +2.5 ********* +3.2 | +5.0
amplitude: +1.0e-12| +4.4e-11 ************* +6.9e-11 | +1.0e-10
Z=-26.6(10.97%) | Like=-21.41..-19.17 [-21.4125..-21.4058]*| it/evals=1340/2164 eff=71.8884% N=300
Z=-26.6(11.74%) | Like=-21.33..-19.17 [-21.3301..-21.3190] | it/evals=1350/2176 eff=71.9616% N=300
Z=-26.4(14.09%) | Like=-21.11..-19.17 [-21.1134..-21.1079]*| it/evals=1377/2218 eff=71.7935% N=300
Z=-26.4(14.37%) | Like=-21.10..-19.17 [-21.1027..-21.0908] | it/evals=1380/2223 eff=71.7629% N=300
Mono-modal Volume: ~exp(-8.66) * Expected Volume: exp(-4.69) Quality: ok
index : +1.0| +2.5 ******** +3.1 | +5.0
amplitude: +1.0e-12| +4.5e-11 *********** +6.7e-11 | +1.0e-10
Z=-26.2(16.99%) | Like=-20.90..-19.17 [-20.8958..-20.8830] | it/evals=1407/2263 eff=71.6760% N=300
Z=-26.2(17.29%) | Like=-20.88..-19.17 [-20.8807..-20.8787]*| it/evals=1410/2266 eff=71.7192% N=300
Z=-26.0(20.46%) | Like=-20.75..-19.17 [-20.7506..-20.7305] | it/evals=1440/2307 eff=71.7489% N=300
Z=-25.9(23.80%) | Like=-20.60..-19.17 [-20.6011..-20.6003]*| it/evals=1470/2346 eff=71.8475% N=300
Mono-modal Volume: ~exp(-8.76) * Expected Volume: exp(-4.91) Quality: ok
index : +1.0| +2.5 ******** +3.1 | +5.0
amplitude: +1.0e-12| +4.6e-11 *********** +6.5e-11 | +1.0e-10
Z=-25.8(24.24%) | Like=-20.59..-19.17 [-20.5928..-20.5925]*| it/evals=1474/2351 eff=71.8674% N=300
Z=-25.7(27.22%) | Like=-20.48..-19.17 [-20.4800..-20.4757]*| it/evals=1500/2382 eff=72.0461% N=300
Z=-25.6(30.24%) | Like=-20.35..-19.16 [-20.3534..-20.3484]*| it/evals=1528/2424 eff=71.9397% N=300
Z=-25.6(30.47%) | Like=-20.35..-19.16 [-20.3475..-20.3450]*| it/evals=1530/2427 eff=71.9323% N=300
Mono-modal Volume: ~exp(-9.04) * 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=-25.6(31.90%) | Like=-20.31..-19.16 [-20.3131..-20.3066]*| it/evals=1541/2440 eff=72.0093% N=300
Z=-25.5(34.07%) | Like=-20.25..-19.16 [-20.2537..-20.2531]*| it/evals=1560/2467 eff=71.9889% N=300
Z=-25.4(37.56%) | Like=-20.15..-19.16 [-20.1537..-20.1525]*| it/evals=1590/2505 eff=72.1088% N=300
Mono-modal Volume: ~exp(-9.09) * 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=-25.3(39.78%) | Like=-20.05..-19.16 [-20.0532..-20.0507]*| it/evals=1608/2531 eff=72.0753% N=300
Z=-25.3(41.34%) | Like=-20.01..-19.16 [-20.0129..-20.0104]*| it/evals=1620/2544 eff=72.1925% N=300
Z=-25.2(44.80%) | Like=-19.93..-19.16 [-19.9283..-19.9266]*| it/evals=1648/2585 eff=72.1225% N=300
Z=-25.2(45.04%) | Like=-19.92..-19.16 [-19.9222..-19.9220]*| it/evals=1650/2587 eff=72.1469% N=300
Mono-modal Volume: ~exp(-9.67) * 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=-25.2(48.11%) | Like=-19.86..-19.16 [-19.8593..-19.8588]*| it/evals=1675/2616 eff=72.3230% N=300
Z=-25.1(48.63%) | Like=-19.85..-19.16 [-19.8486..-19.8482]*| it/evals=1680/2621 eff=72.3826% N=300
Z=-25.1(52.08%) | Like=-19.80..-19.16 [-19.7976..-19.7966]*| it/evals=1708/2663 eff=72.2810% N=300
Z=-25.1(52.30%) | Like=-19.79..-19.16 [-19.7932..-19.7932]*| it/evals=1710/2666 eff=72.2739% N=300
Z=-25.0(55.70%) | Like=-19.72..-19.16 [-19.7203..-19.7184]*| it/evals=1740/2707 eff=72.2892% N=300
Mono-modal Volume: ~exp(-9.92) * 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=-25.0(55.94%) | Like=-19.72..-19.16 [-19.7181..-19.7178]*| it/evals=1742/2709 eff=72.3122% N=300
Z=-25.0(59.11%) | Like=-19.68..-19.16 [-19.6848..-19.6845]*| it/evals=1770/2743 eff=72.4519% N=300
Z=-24.9(62.00%) | Like=-19.63..-19.16 [-19.6318..-19.6307]*| it/evals=1799/2784 eff=72.4235% N=300
Z=-24.9(62.09%) | Like=-19.63..-19.16 [-19.6307..-19.6303]*| it/evals=1800/2785 eff=72.4346% N=300
Mono-modal Volume: ~exp(-10.25) * 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=-24.9(62.94%) | Like=-19.61..-19.16 [-19.6187..-19.6085] | it/evals=1809/2802 eff=72.3022% N=300
Z=-24.9(64.94%) | Like=-19.58..-19.16 [-19.5827..-19.5794]*| it/evals=1830/2834 eff=72.2178% N=300
Z=-24.8(67.71%) | Like=-19.55..-19.16 [-19.5532..-19.5531]*| it/evals=1860/2870 eff=72.3735% N=300
Mono-modal Volume: ~exp(-10.25) Expected Volume: exp(-6.25) Quality: ok
index : +1.0| +2.7 ***** +2.9 | +5.0
amplitude: +1.0e-12| +5.0e-11 ****** +6.0e-11 | +1.0e-10
[ultranest] Explored until L=-2e+01
[ultranest] Likelihood function evaluations: 2907
[ultranest] logZ = -24.39 +- 0.07202
[ultranest] Effective samples strategy satisfied (ESS = 997.9, 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=0.27, need <0.5)
[ultranest] logZ error budget: single: 0.12 bs:0.07 tail:0.26 total:0.27 required:<0.50
[ultranest] done iterating.
logZ = -24.424 +- 0.305
single instance: logZ = -24.424 +- 0.119
bootstrapped : logZ = -24.392 +- 0.156
tail : logZ = +- 0.262
insert order U test : converged: True correlation: inf iterations
index : 2.27 │ ▁ ▁▁▁▁▂▂▃▃▄▆▆▆▇▇▆▇▆▅▄▄▃▄▃▂▁▁▁▁▁▁▁▁ ▁▁ │3.55 2.83 +- 0.17
amplitude : 0.0000000000363│ ▁▁▁▁▁▂▂▃▄▇▅▆▆▇▇▆▇▆▄▄▃▂▂▁▂▁▁▁ ▁ ▁ ▁ │0.0000000000856 0.0000000000552 +- 0.0000000000060
[ultranest] Sampling 300 live points from prior ...
Mono-modal Volume: ~exp(-4.02) * Expected Volume: exp(0.00) Quality: ok
index : +1.0|*************************************** ********| +5.0
amplitude: +1.0e-12|**************************** ******* ** * ***| +1.0e-10
Z=-inf(0.00%) | Like=-652.50..-13.61 [-652.4991..-91.9415] | it/evals=0/301 eff=0.0000% N=300
Z=-153.6(0.00%) | Like=-149.26..-13.61 [-652.4991..-91.9415] | it/evals=30/331 eff=96.7742% N=300
Z=-145.6(0.00%) | Like=-141.30..-13.61 [-652.4991..-91.9415] | it/evals=60/362 eff=96.7742% N=300
Mono-modal Volume: ~exp(-4.07) * Expected Volume: exp(-0.22) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|**************************** ******* ** ** **| +1.0e-10
Z=-143.4(0.00%) | Like=-137.90..-13.61 [-652.4991..-91.9415] | it/evals=67/369 eff=97.1014% N=300
Z=-135.0(0.00%) | Like=-130.31..-13.61 [-652.4991..-91.9415] | it/evals=90/396 eff=93.7500% N=300
Z=-124.0(0.00%) | Like=-118.68..-13.61 [-652.4991..-91.9415] | it/evals=120/431 eff=91.6031% N=300
Mono-modal Volume: ~exp(-4.58) * Expected Volume: exp(-0.45) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|**************************** *************** **| +1.0e-10
Z=-119.0(0.00%) | Like=-113.67..-13.61 [-652.4991..-91.9415] | it/evals=134/447 eff=91.1565% N=300
Z=-112.2(0.00%) | Like=-107.16..-13.61 [-652.4991..-91.9415] | it/evals=150/466 eff=90.3614% N=300
Z=-103.8(0.00%) | Like=-98.77..-13.61 [-652.4991..-91.9415] | it/evals=180/505 eff=87.8049% N=300
Mono-modal Volume: ~exp(-4.58) Expected Volume: exp(-0.67) Quality: ok
index : +1.0| ****************************************** ***| +5.0
amplitude: +1.0e-12| *************************** **************** **| +1.0e-10
Z=-95.6(0.00%) | Like=-90.40..-13.39 [-91.7012..-47.5616] | it/evals=209/543 eff=86.0082% N=300
Z=-95.3(0.00%) | Like=-89.84..-13.39 [-91.7012..-47.5616] | it/evals=210/544 eff=86.0656% N=300
Z=-86.0(0.00%) | Like=-80.33..-13.39 [-91.7012..-47.5616] | it/evals=240/582 eff=85.1064% N=300
Mono-modal Volume: ~exp(-4.58) Expected Volume: exp(-0.89) Quality: ok
index : +1.0| ********************************************| +5.0
amplitude: +1.0e-12| **********************************************| +1.0e-10
Z=-76.2(0.00%) | Like=-70.66..-13.39 [-91.7012..-47.5616] | it/evals=268/621 eff=83.4891% N=300
Z=-75.7(0.00%) | Like=-70.41..-13.39 [-91.7012..-47.5616] | it/evals=270/623 eff=83.5913% N=300
Z=-69.5(0.00%) | Like=-64.35..-13.39 [-91.7012..-47.5616] | it/evals=300/664 eff=82.4176% N=300
Z=-63.7(0.00%) | Like=-58.55..-13.39 [-91.7012..-47.5616] | it/evals=330/701 eff=82.2943% N=300
Mono-modal Volume: ~exp(-4.93) * Expected Volume: exp(-1.12) Quality: ok
index : +1.0| ******************************************| +5.0
amplitude: +1.0e-12| *********************************************| +1.0e-10
Z=-62.8(0.00%) | Like=-57.28..-13.39 [-91.7012..-47.5616] | it/evals=335/710 eff=81.7073% N=300
Z=-58.5(0.00%) | Like=-52.84..-13.39 [-91.7012..-47.5616] | it/evals=360/745 eff=80.8989% N=300
Z=-54.0(0.00%) | Like=-48.93..-13.39 [-91.7012..-47.5616] | it/evals=390/780 eff=81.2500% N=300
Mono-modal Volume: ~exp(-5.53) * Expected Volume: exp(-1.34) Quality: ok
index : +1.0| ****************************************| +5.0
amplitude: +1.0e-12| *******************************************| +1.0e-10
Z=-52.3(0.00%) | Like=-47.48..-13.39 [-47.5477..-29.9747] | it/evals=402/798 eff=80.7229% N=300
Z=-50.8(0.00%) | Like=-46.24..-13.39 [-47.5477..-29.9747] | it/evals=420/824 eff=80.1527% N=300
Z=-47.6(0.00%) | Like=-42.32..-13.39 [-47.5477..-29.9747] | it/evals=450/863 eff=79.9290% N=300
Mono-modal Volume: ~exp(-5.53) Expected Volume: exp(-1.56) Quality: ok
index : +1.0| *********************************** | +5.0
amplitude: +1.0e-12| ******************************************| +1.0e-10
Z=-44.8(0.00%) | Like=-39.50..-13.39 [-47.5477..-29.9747] | it/evals=476/902 eff=79.0698% N=300
Z=-44.3(0.00%) | Like=-38.88..-13.39 [-47.5477..-29.9747] | it/evals=480/908 eff=78.9474% N=300
Z=-41.7(0.00%) | Like=-36.93..-13.39 [-47.5477..-29.9747] | it/evals=509/951 eff=78.1874% N=300
Z=-41.7(0.00%) | Like=-36.90..-13.39 [-47.5477..-29.9747] | it/evals=510/952 eff=78.2209% N=300
Mono-modal Volume: ~exp(-6.17) * Expected Volume: exp(-1.79) Quality: ok
index : +1.0| ****************************** | +5.0
amplitude: +1.0e-12| *****************************************| +1.0e-10
Z=-39.6(0.00%) | Like=-34.47..-13.39 [-47.5477..-29.9747] | it/evals=536/989 eff=77.7939% N=300
Z=-39.3(0.00%) | Like=-34.40..-13.39 [-47.5477..-29.9747] | it/evals=540/994 eff=77.8098% N=300
Z=-37.1(0.00%) | Like=-32.29..-13.39 [-47.5477..-29.9747] | it/evals=569/1036 eff=77.3098% N=300
Z=-37.0(0.00%) | Like=-32.28..-13.39 [-47.5477..-29.9747] | it/evals=570/1038 eff=77.2358% N=300
Z=-35.7(0.00%) | Like=-30.97..-13.39 [-47.5477..-29.9747] | it/evals=596/1082 eff=76.2148% N=300
Z=-35.5(0.00%) | Like=-30.64..-13.39 [-47.5477..-29.9747] | it/evals=600/1087 eff=76.2389% N=300
Mono-modal Volume: ~exp(-6.19) * Expected Volume: exp(-2.01) Quality: ok
index : +1.0| +1.9 ************************** +4.0 | +5.0
amplitude: +1.0e-12| ***************************************| +1.0e-10
Z=-35.3(0.00%) | Like=-30.28..-13.39 [-47.5477..-29.9747] | it/evals=603/1091 eff=76.2326% N=300
Z=-33.5(0.00%) | Like=-28.46..-13.33 [-29.9448..-21.0222] | it/evals=630/1133 eff=75.6303% N=300
Z=-31.9(0.00%) | Like=-26.95..-13.32 [-29.9448..-21.0222] | it/evals=660/1167 eff=76.1246% N=300
Mono-modal Volume: ~exp(-6.19) Expected Volume: exp(-2.23) Quality: ok
index : +1.0| +2.0 ********************** +3.8 | +5.0
amplitude: +1.0e-12| **************************************| +1.0e-10
Z=-30.5(0.00%) | Like=-25.56..-13.32 [-29.9448..-21.0222] | it/evals=687/1205 eff=75.9116% N=300
Z=-30.4(0.00%) | Like=-25.49..-13.32 [-29.9448..-21.0222] | it/evals=690/1208 eff=75.9912% N=300
Z=-29.2(0.00%) | Like=-24.11..-13.32 [-29.9448..-21.0222] | it/evals=718/1251 eff=75.4995% N=300
Z=-29.1(0.00%) | Like=-24.08..-13.32 [-29.9448..-21.0222] | it/evals=720/1254 eff=75.4717% N=300
Mono-modal Volume: ~exp(-6.63) * Expected Volume: exp(-2.46) Quality: ok
index : +1.0| +2.1 ******************** +3.7 | +5.0
amplitude: +1.0e-12| +2.6e-11 *********************************** | +1.0e-10
Z=-28.4(0.00%) | Like=-23.63..-13.32 [-29.9448..-21.0222] | it/evals=737/1280 eff=75.2041% N=300
Z=-28.0(0.01%) | Like=-23.03..-13.32 [-29.9448..-21.0222] | it/evals=750/1295 eff=75.3769% N=300
Z=-27.0(0.02%) | Like=-22.18..-13.32 [-29.9448..-21.0222] | it/evals=780/1331 eff=75.6547% N=300
Mono-modal Volume: ~exp(-6.84) * Expected Volume: exp(-2.68) Quality: ok
index : +1.0| +2.1 ****************** +3.5 | +5.0
amplitude: +1.0e-12| +2.8e-11 ******************************** | +1.0e-10
Z=-26.3(0.03%) | Like=-21.33..-13.32 [-29.9448..-21.0222] | it/evals=804/1368 eff=75.2809% N=300
Z=-26.1(0.04%) | Like=-21.20..-13.32 [-29.9448..-21.0222] | it/evals=810/1375 eff=75.3488% N=300
Z=-25.3(0.08%) | Like=-20.72..-13.32 [-21.0091..-19.9056] | it/evals=840/1415 eff=75.3363% N=300
Z=-24.8(0.13%) | Like=-20.30..-13.32 [-21.0091..-19.9056] | it/evals=870/1456 eff=75.2595% N=300
Mono-modal Volume: ~exp(-7.11) * Expected Volume: exp(-2.90) Quality: ok
index : +1.0| +2.2 **************** +3.5 | +5.0
amplitude: +1.0e-12| +3.0e-11 ***************************** | +1.0e-10
Z=-24.8(0.13%) | Like=-20.27..-13.32 [-21.0091..-19.9056] | it/evals=871/1457 eff=75.2809% N=300
Z=-24.3(0.21%) | Like=-19.73..-13.32 [-19.9035..-19.6670] | it/evals=900/1499 eff=75.0626% N=300
Z=-23.9(0.31%) | Like=-19.30..-13.32 [-19.3192..-19.2997] | it/evals=925/1541 eff=74.5367% N=300
Z=-23.8(0.33%) | Like=-19.25..-13.32 [-19.2536..-19.2245] | it/evals=930/1548 eff=74.5192% N=300
Mono-modal Volume: ~exp(-7.34) * Expected Volume: exp(-3.13) Quality: ok
index : +1.0| +2.2 *************** +3.4 | +5.0
amplitude: +1.0e-12| +3.1e-11 ************************** | +1.0e-10
Z=-23.7(0.38%) | Like=-19.06..-13.32 [-19.0625..-19.0567]*| it/evals=938/1559 eff=74.5036% N=300
Z=-23.4(0.53%) | Like=-18.69..-13.32 [-18.6858..-18.6257] | it/evals=960/1588 eff=74.5342% N=300
Z=-23.0(0.82%) | Like=-18.17..-13.32 [-18.1743..-18.1608] | it/evals=986/1630 eff=74.1353% N=300
Z=-22.9(0.87%) | Like=-18.07..-13.32 [-18.0710..-18.0587] | it/evals=990/1635 eff=74.1573% N=300
Mono-modal Volume: ~exp(-7.34) Expected Volume: exp(-3.35) Quality: ok
index : +1.0| +2.3 ************* +3.3 | +5.0
amplitude: +1.0e-12| +3.3e-11 *********************** +7.9e-11| +1.0e-10
Z=-22.5(1.32%) | Like=-17.64..-13.32 [-17.6395..-17.6322]*| it/evals=1019/1672 eff=74.2711% N=300
Z=-22.5(1.34%) | Like=-17.63..-13.32 [-17.6322..-17.6179] | it/evals=1020/1673 eff=74.2899% N=300
Z=-22.1(2.00%) | Like=-17.30..-13.32 [-17.3022..-17.2962]*| it/evals=1049/1717 eff=74.0296% N=300
Z=-22.1(2.02%) | Like=-17.30..-13.32 [-17.2962..-17.2872]*| it/evals=1050/1718 eff=74.0480% N=300
Mono-modal Volume: ~exp(-7.34) Expected Volume: exp(-3.57) Quality: ok
index : +1.0| +2.3 *********** +3.2 | +5.0
amplitude: +1.0e-12| +3.5e-11 ********************* +7.7e-11 | +1.0e-10
Z=-21.8(2.72%) | Like=-17.01..-13.32 [-17.0144..-16.9954] | it/evals=1075/1756 eff=73.8324% N=300
Z=-21.7(2.86%) | Like=-16.96..-13.32 [-16.9563..-16.9532]*| it/evals=1080/1765 eff=73.7201% N=300
Z=-21.4(3.88%) | Like=-16.58..-13.32 [-16.5813..-16.5742]*| it/evals=1110/1806 eff=73.7052% N=300
Mono-modal Volume: ~exp(-7.92) * Expected Volume: exp(-3.80) Quality: ok
index : +1.0| +2.4 *********** +3.2 | +5.0
amplitude: +1.0e-12| +3.8e-11 ******************* +7.4e-11 | +1.0e-10
Z=-21.1(5.00%) | Like=-16.28..-13.31 [-16.3688..-16.2831] | it/evals=1139/1848 eff=73.5788% N=300
Z=-21.1(5.06%) | Like=-16.28..-13.31 [-16.2815..-16.2704] | it/evals=1140/1849 eff=73.5959% N=300
Z=-20.9(6.51%) | Like=-15.95..-13.31 [-15.9768..-15.9533] | it/evals=1166/1892 eff=73.2412% N=300
Z=-20.9(6.76%) | Like=-15.94..-13.31 [-15.9379..-15.9263] | it/evals=1170/1901 eff=73.0793% N=300
Z=-20.6(8.61%) | Like=-15.65..-13.31 [-15.6543..-15.6516]*| it/evals=1198/1942 eff=72.9598% N=300
Z=-20.6(8.79%) | Like=-15.65..-13.31 [-15.6477..-15.6472]*| it/evals=1200/1946 eff=72.9040% N=300
Mono-modal Volume: ~exp(-7.92) Expected Volume: exp(-4.02) 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=-20.4(10.65%) | Like=-15.46..-13.31 [-15.4595..-15.4585]*| it/evals=1224/1987 eff=72.5548% N=300
Z=-20.3(11.17%) | Like=-15.41..-13.31 [-15.4075..-15.3910] | it/evals=1230/1996 eff=72.5236% N=300
Z=-20.1(13.89%) | Like=-15.22..-13.31 [-15.2328..-15.2198] | it/evals=1259/2037 eff=72.4813% N=300
Z=-20.1(14.00%) | Like=-15.22..-13.31 [-15.2193..-15.2128]*| it/evals=1260/2038 eff=72.4971% N=300
Mono-modal Volume: ~exp(-8.49) * Expected Volume: exp(-4.24) Quality: ok
index : +1.0| +2.4 ******** +3.1 | +5.0
amplitude: +1.0e-12| +4.0e-11 *************** +6.9e-11 | +1.0e-10
Z=-20.0(15.24%) | Like=-15.16..-13.31 [-15.1594..-15.1501]*| it/evals=1273/2055 eff=72.5356% N=300
Z=-19.9(16.95%) | Like=-15.10..-13.31 [-15.1024..-15.0949]*| it/evals=1290/2077 eff=72.5943% N=300
Z=-19.8(19.90%) | Like=-14.92..-13.31 [-14.9203..-14.9187]*| it/evals=1320/2116 eff=72.6872% N=300
Mono-modal Volume: ~exp(-8.49) Expected Volume: exp(-4.47) Quality: ok
index : +1.0| +2.5 ******** +3.0 | +5.0
amplitude: +1.0e-12| +4.1e-11 ************** +6.8e-11 | +1.0e-10
Z=-19.7(22.30%) | Like=-14.81..-13.31 [-14.8075..-14.7945] | it/evals=1344/2154 eff=72.4919% N=300
Z=-19.6(23.00%) | Like=-14.76..-13.31 [-14.7745..-14.7583] | it/evals=1350/2162 eff=72.5027% N=300
Z=-19.5(25.67%) | Like=-14.61..-13.31 [-14.6302..-14.6112] | it/evals=1374/2205 eff=72.1260% N=300
Z=-19.5(26.32%) | Like=-14.58..-13.31 [-14.5844..-14.5814]*| it/evals=1380/2214 eff=72.1003% N=300
Mono-modal Volume: ~exp(-8.83) * Expected Volume: exp(-4.69) Quality: ok
index : +1.0| +2.5 ******** +3.0 | +5.0
amplitude: +1.0e-12| +4.2e-11 ************ +6.4e-11 | +1.0e-10
Z=-19.4(29.61%) | Like=-14.51..-13.31 [-14.5062..-14.5031]*| it/evals=1407/2250 eff=72.1538% N=300
Z=-19.4(29.99%) | Like=-14.50..-13.31 [-14.5018..-14.5014]*| it/evals=1410/2253 eff=72.1966% N=300
Z=-19.3(33.44%) | Like=-14.41..-13.31 [-14.4123..-14.4081]*| it/evals=1439/2296 eff=72.0942% N=300
Z=-19.3(33.59%) | Like=-14.41..-13.31 [-14.4081..-14.4074]*| it/evals=1440/2297 eff=72.1082% N=300
Z=-19.2(37.13%) | Like=-14.28..-13.31 [-14.2831..-14.2806]*| it/evals=1469/2338 eff=72.0805% N=300
Z=-19.2(37.24%) | Like=-14.28..-13.31 [-14.2806..-14.2755]*| it/evals=1470/2339 eff=72.0942% N=300
Mono-modal Volume: ~exp(-8.93) * Expected Volume: exp(-4.91) Quality: ok
index : +1.0| +2.5 ****** +3.0 | +5.0
amplitude: +1.0e-12| +4.3e-11 *********** +6.4e-11 | +1.0e-10
Z=-19.1(37.79%) | Like=-14.27..-13.31 [-14.2699..-14.2659]*| it/evals=1474/2347 eff=72.0078% N=300
Z=-19.1(40.93%) | Like=-14.16..-13.30 [-14.1642..-14.1611]*| it/evals=1500/2387 eff=71.8735% N=300
Z=-19.0(44.15%) | Like=-14.09..-13.30 [-14.0939..-14.0924]*| it/evals=1527/2429 eff=71.7238% N=300
Z=-19.0(44.51%) | Like=-14.09..-13.30 [-14.0894..-14.0829]*| it/evals=1530/2433 eff=71.7300% N=300
Mono-modal Volume: ~exp(-9.53) * Expected Volume: exp(-5.14) Quality: ok
index : +1.0| +2.5 ****** +2.9 | +5.0
amplitude: +1.0e-12| +4.4e-11 ********** +6.3e-11 | +1.0e-10
Z=-18.9(45.82%) | Like=-14.06..-13.30 [-14.0619..-14.0576]*| it/evals=1541/2455 eff=71.5081% N=300
Z=-18.9(48.21%) | Like=-13.99..-13.30 [-13.9909..-13.9907]*| it/evals=1560/2475 eff=71.7241% N=300
Z=-18.8(51.81%) | Like=-13.93..-13.30 [-13.9317..-13.9265]*| it/evals=1590/2512 eff=71.8807% N=300
Mono-modal Volume: ~exp(-9.57) * Expected Volume: exp(-5.36) Quality: ok
index : +1.0| +2.6 ***** +2.9 | +5.0
amplitude: +1.0e-12| +4.5e-11 ********* +6.2e-11 | +1.0e-10
Z=-18.8(53.83%) | Like=-13.89..-13.30 [-13.8932..-13.8924]*| it/evals=1608/2533 eff=72.0107% N=300
Z=-18.8(55.21%) | Like=-13.86..-13.30 [-13.8625..-13.8616]*| it/evals=1620/2546 eff=72.1282% N=300
Z=-18.7(58.41%) | Like=-13.81..-13.30 [-13.8123..-13.8068]*| it/evals=1649/2587 eff=72.1032% N=300
Z=-18.7(58.53%) | Like=-13.81..-13.30 [-13.8068..-13.8060]*| it/evals=1650/2588 eff=72.1154% 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 ******** +6.1e-11 | +1.0e-10
Z=-18.7(61.09%) | Like=-13.77..-13.30 [-13.7694..-13.7669]*| it/evals=1675/2625 eff=72.0430% N=300
Z=-18.7(61.60%) | Like=-13.76..-13.30 [-13.7610..-13.7605]*| it/evals=1680/2632 eff=72.0412% N=300
Z=-18.6(64.50%) | Like=-13.72..-13.30 [-13.7220..-13.7201]*| it/evals=1710/2674 eff=72.0303% N=300
Z=-18.6(67.29%) | Like=-13.68..-13.30 [-13.6799..-13.6797]*| it/evals=1740/2714 eff=72.0795% N=300
Mono-modal Volume: ~exp(-9.78) Expected Volume: exp(-5.81) Quality: ok
index : +1.0| +2.6 **** +2.9 | +5.0
amplitude: +1.0e-12| +4.7e-11 ******* +6.0e-11 | +1.0e-10
Z=-18.5(69.61%) | Like=-13.64..-13.30 [-13.6371..-13.6356]*| it/evals=1767/2754 eff=72.0049% N=300
Z=-18.5(69.88%) | Like=-13.63..-13.30 [-13.6327..-13.6322]*| it/evals=1770/2757 eff=72.0391% N=300
[ultranest] Explored until L=-1e+01
[ultranest] Likelihood function evaluations: 2760
[ultranest] logZ = -18.17 +- 0.08884
[ultranest] Effective samples strategy satisfied (ESS = 997.3, need >400)
[ultranest] Posterior uncertainty strategy is satisfied (KL: 0.46+-0.08 nat, need <0.50 nat)
[ultranest] Evidency uncertainty strategy is satisfied (dlogz=0.28, need <0.5)
[ultranest] logZ error budget: single: 0.11 bs:0.09 tail:0.26 total:0.28 required:<0.50
[ultranest] done iterating.
logZ = -18.167 +- 0.304
single instance: logZ = -18.167 +- 0.114
bootstrapped : logZ = -18.169 +- 0.154
tail : logZ = +- 0.262
insert order U test : converged: True correlation: inf iterations
index : 2.11 │ ▁▁ ▁▁▁▁▁▂▂▂▃▄▅▆▆▇▆▆▇▆▅▄▄▃▃▂▂▂▁▁▁▁▁▁▁▁ │3.38 2.74 +- 0.17
amplitude : 0.0000000000271│ ▁▁ ▁▁▁▂▃▃▄▅▇▇▆▇▆▇▆▆▄▃▃▂▁▁▁▁▁▁▁▁ ▁ ▁ │0.0000000000919 0.0000000000533 +- 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).
Highest density intervals#
Given the samples, one can also compute the highest density interval (HDI) which is also known as the smallest credible interval (SCI). See more details here. This is the smallest interval in which a given probability (e.g. 68%) is contained.
For unimodal distributions, the HDI is a single continuous interval containing the mode whereas for multimodal distributions, the HDI can be a set of disconnected intervals. The HDI can be particularly helpful with multimodal distributions as opposed to the mean and quantiles approaches which will not report the important information. Here, we showcase the HDI using the Arviz package. Check out the many possibilities offered by Arviz, a package to analyze the samples posterior distributions.
from arviz import hdi
import scipy.stats as stats
# Multi-modal samples example
weight = 0.3
n_samples = 10000
mu1 = 5.5e-11
sigma1 = 0.7e-11
mu2 = 3.5e-11
sigma2 = 0.3e-11
weight = 0.7
rng = np.random.default_rng(42)
component_mask = rng.uniform(size=n_samples) < weight
samples = np.empty(n_samples)
samples[component_mask] = rng.normal(mu1, sigma1, component_mask.sum())
samples[~component_mask] = rng.normal(mu2, sigma2, (~component_mask).sum())
fig, (ax1, ax2) = plt.subplots(
2, 1, sharex=True, figsize=(9, 7), gridspec_kw={"height_ratios": [5, 2]}
)
# Highest density intervals
hdis = hdi(samples, hdi_prob=0.68, multimodal=True)
ax1.hist(
samples,
bins=50,
histtype="step",
color="k",
alpha=0.5,
)
yl = ax1.get_ylim()
for k in range(hdis.shape[0]):
label = "68% HDI" if k == 0 else None
ax2.hlines(
1 + 3 * 0.015, hdis[k, 0], hdis[k, 1], lw=15, color="k", alpha=0.5, label=label
)
# Percentile
percentile = np.percentile(samples, q=[16, 84])
ax2.hlines(
1 + 2 * 0.015,
percentile[0],
percentile[1],
lw=15,
color="y",
alpha=0.5,
label="16-84% percentile",
)
# Mean and standard deviation
mean = np.mean(samples)
std = np.std(samples)
ax1.plot([mean, mean], yl, label="mean", color="r", ls="--")
ax2.hlines(
1 + 1 * 0.015,
mean - std,
mean + std,
lw=15,
color="r",
alpha=0.5,
label=r"mean $\pm$ std",
)
# Median and median absolute deviation
median = np.median(samples)
mad = stats.median_abs_deviation(samples)
ax1.plot([median, median], yl, label="median", color="b", ls="--")
ax2.hlines(
1,
median - mad,
median + mad,
lw=15,
color="b",
alpha=0.5,
label=r"median $\pm$ mad",
)
ax2.legend(loc=6)
ax1.legend(loc="upper left")
ax1.set_xlim(1e-11, 8e-11)
ax2.set_ylim(0.98, 1.06)
ax2.set_xlabel("Amplitude")
ax2.tick_params(left=False, labelleft=False)
plt.show()
Total running time of the script: (0 minutes 42.974 seconds)
