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(-4.06) * Expected Volume: exp(0.00) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|********************************* ************ | +1.0e-10
Z=-inf(0.00%) | Like=-2940.74..-62.63 [-2940.7396..-324.0275] | it/evals=0/301 eff=0.0000% N=300
Z=-587.7(0.00%) | Like=-553.18..-62.63 [-2940.7396..-324.0275] | it/evals=22/323 eff=95.6522% N=300
Z=-542.9(0.00%) | Like=-535.45..-62.63 [-2940.7396..-324.0275] | it/evals=30/331 eff=96.7742% N=300
Z=-508.4(0.00%) | Like=-502.65..-62.63 [-2940.7396..-324.0275] | it/evals=52/353 eff=98.1132% N=300
Z=-503.0(0.00%) | Like=-497.20..-62.63 [-2940.7396..-324.0275] | it/evals=60/361 eff=98.3607% 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=-498.9(0.00%) | Like=-492.82..-62.63 [-2940.7396..-324.0275] | it/evals=67/370 eff=95.7143% N=300
Z=-473.9(0.00%) | Like=-465.53..-62.63 [-2940.7396..-324.0275] | it/evals=88/392 eff=95.6522% N=300
Z=-466.9(0.00%) | Like=-458.73..-62.63 [-2940.7396..-324.0275] | it/evals=90/394 eff=95.7447% N=300
Z=-442.8(0.00%) | Like=-435.94..-62.63 [-2940.7396..-324.0275] | it/evals=112/416 eff=96.5517% N=300
Z=-435.4(0.00%) | Like=-429.50..-62.63 [-2940.7396..-324.0275] | it/evals=120/426 eff=95.2381% 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=-420.0(0.00%) | Like=-412.56..-62.63 [-2940.7396..-324.0275] | it/evals=136/447 eff=92.5170% N=300
Z=-398.2(0.00%) | Like=-390.56..-62.63 [-2940.7396..-324.0275] | it/evals=150/464 eff=91.4634% N=300
Z=-371.3(0.00%) | Like=-365.10..-60.52 [-2940.7396..-324.0275] | it/evals=171/486 eff=91.9355% N=300
Z=-358.4(0.00%) | Like=-349.24..-60.52 [-2940.7396..-324.0275] | it/evals=180/499 eff=90.4523% N=300
Z=-336.2(0.00%) | Like=-330.44..-60.52 [-2940.7396..-324.0275] | it/evals=199/522 eff=89.6396% N=300
Mono-modal Volume: ~exp(-4.18) * Expected Volume: exp(-0.67) Quality: ok
index : +1.0| **********************************************| +5.0
amplitude: +1.0e-12| ******************************** **************| +1.0e-10
Z=-334.4(0.00%) | Like=-325.21..-60.52 [-2940.7396..-324.0275] | it/evals=201/525 eff=89.3333% N=300
Z=-324.9(0.00%) | Like=-318.37..-60.52 [-323.7946..-178.4779] | it/evals=210/536 eff=88.9831% N=300
Z=-309.6(0.00%) | Like=-302.17..-60.52 [-323.7946..-178.4779] | it/evals=228/558 eff=88.3721% N=300
Z=-300.3(0.00%) | Like=-292.86..-60.52 [-323.7946..-178.4779] | it/evals=240/575 eff=87.2727% N=300
Z=-288.9(0.00%) | Like=-280.89..-60.25 [-323.7946..-178.4779] | it/evals=255/598 eff=85.5705% N=300
Mono-modal Volume: ~exp(-4.88) * Expected Volume: exp(-0.89) Quality: ok
index : +1.0| ********************************************| +5.0
amplitude: +1.0e-12| ******************************* **************| +1.0e-10
Z=-274.1(0.00%) | Like=-266.36..-60.25 [-323.7946..-178.4779] | it/evals=268/613 eff=85.6230% N=300
Z=-271.9(0.00%) | Like=-263.01..-60.25 [-323.7946..-178.4779] | it/evals=270/615 eff=85.7143% N=300
Z=-252.8(0.00%) | Like=-246.58..-60.25 [-323.7946..-178.4779] | it/evals=289/637 eff=85.7567% N=300
Z=-244.5(0.00%) | Like=-237.18..-60.25 [-323.7946..-178.4779] | it/evals=300/649 eff=85.9599% N=300
Z=-232.1(0.00%) | Like=-223.71..-60.25 [-323.7946..-178.4779] | it/evals=314/673 eff=84.1823% N=300
Z=-220.4(0.00%) | Like=-214.18..-60.25 [-323.7946..-178.4779] | it/evals=330/697 eff=83.1234% N=300
Mono-modal Volume: ~exp(-5.21) * Expected Volume: exp(-1.12) Quality: ok
index : +1.0| ******************************************| +5.0
amplitude: +1.0e-12| ***************************** **************| +1.0e-10
Z=-216.1(0.00%) | Like=-209.06..-60.25 [-323.7946..-178.4779] | it/evals=335/703 eff=83.1266% N=300
Z=-210.4(0.00%) | Like=-204.13..-60.25 [-323.7946..-178.4779] | it/evals=351/725 eff=82.5882% N=300
Z=-207.1(0.00%) | Like=-199.92..-60.25 [-323.7946..-178.4779] | it/evals=360/741 eff=81.6327% N=300
Z=-201.2(0.00%) | Like=-195.56..-60.25 [-323.7946..-178.4779] | it/evals=378/764 eff=81.4655% N=300
Z=-197.2(0.00%) | Like=-191.00..-60.25 [-323.7946..-178.4779] | it/evals=390/778 eff=81.5900% N=300
Mono-modal Volume: ~exp(-5.42) * Expected Volume: exp(-1.34) Quality: ok
index : +1.0| *************************************** | +5.0
amplitude: +1.0e-12| *******************************************| +1.0e-10
Z=-192.5(0.00%) | Like=-186.26..-60.25 [-323.7946..-178.4779] | it/evals=402/796 eff=81.0484% N=300
Z=-186.9(0.00%) | Like=-181.03..-60.25 [-323.7946..-178.4779] | it/evals=420/817 eff=81.2379% N=300
Z=-180.4(0.00%) | Like=-173.78..-60.25 [-178.2311..-130.8566] | it/evals=440/839 eff=81.6327% N=300
Z=-177.5(0.00%) | Like=-171.58..-59.37 [-178.2311..-130.8566] | it/evals=450/852 eff=81.5217% N=300
Z=-171.1(0.00%) | Like=-164.37..-59.37 [-178.2311..-130.8566] | it/evals=468/875 eff=81.3913% 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=-170.7(0.00%) | Like=-163.63..-59.37 [-178.2311..-130.8566] | it/evals=469/878 eff=81.1419% N=300
Z=-166.1(0.00%) | Like=-160.16..-59.37 [-178.2311..-130.8566] | it/evals=480/893 eff=80.9444% N=300
Z=-162.5(0.00%) | Like=-156.59..-59.37 [-178.2311..-130.8566] | it/evals=496/914 eff=80.7818% N=300
Z=-160.4(0.00%) | Like=-154.76..-59.37 [-178.2311..-130.8566] | it/evals=509/937 eff=79.9058% N=300
Z=-160.2(0.00%) | Like=-154.54..-59.37 [-178.2311..-130.8566] | it/evals=510/938 eff=79.9373% N=300
Z=-157.0(0.00%) | Like=-151.08..-59.37 [-178.2311..-130.8566] | it/evals=525/961 eff=79.4251% N=300
Mono-modal Volume: ~exp(-6.09) * Expected Volume: exp(-1.79) Quality: ok
index : +1.0| ******************************* | +5.0
amplitude: +1.0e-12| ************************************** * | +1.0e-10
Z=-154.4(0.00%) | Like=-148.74..-59.37 [-178.2311..-130.8566] | it/evals=536/973 eff=79.6434% N=300
Z=-153.9(0.00%) | Like=-148.04..-59.37 [-178.2311..-130.8566] | it/evals=540/978 eff=79.6460% N=300
Z=-149.9(0.00%) | Like=-143.97..-59.37 [-178.2311..-130.8566] | it/evals=558/1000 eff=79.7143% N=300
Z=-148.1(0.00%) | Like=-142.32..-59.37 [-178.2311..-130.8566] | it/evals=570/1019 eff=79.2768% N=300
Z=-145.2(0.00%) | Like=-139.34..-59.26 [-178.2311..-130.8566] | it/evals=588/1041 eff=79.3522% N=300
Z=-143.1(0.00%) | Like=-137.12..-59.26 [-178.2311..-130.8566] | it/evals=600/1057 eff=79.2602% N=300
Mono-modal Volume: ~exp(-6.09) Expected Volume: exp(-2.01) Quality: ok
index : +1.0| **************************** +4.1 | +5.0
amplitude: +1.0e-12| *********************************** * | +1.0e-10
Z=-140.9(0.00%) | Like=-135.14..-59.26 [-178.2311..-130.8566] | it/evals=615/1079 eff=78.9474% N=300
Z=-138.8(0.00%) | Like=-132.62..-59.26 [-178.2311..-130.8566] | it/evals=630/1097 eff=79.0464% N=300
Z=-135.3(0.00%) | Like=-128.35..-59.26 [-130.7431..-95.4654] | it/evals=646/1121 eff=78.6845% N=300
Z=-132.5(0.00%) | Like=-126.22..-58.88 [-130.7431..-95.4654] | it/evals=660/1141 eff=78.4780% N=300
Mono-modal Volume: ~exp(-6.09) Expected Volume: exp(-2.23) Quality: ok
index : +1.0| ************************** +3.9 | +5.0
amplitude: +1.0e-12| ********************************** | +1.0e-10
Z=-129.2(0.00%) | Like=-122.77..-58.88 [-130.7431..-95.4654] | it/evals=674/1162 eff=78.1903% N=300
Z=-127.3(0.00%) | Like=-121.12..-58.88 [-130.7431..-95.4654] | it/evals=687/1184 eff=77.7149% N=300
Z=-126.9(0.00%) | Like=-120.85..-58.88 [-130.7431..-95.4654] | it/evals=690/1188 eff=77.7027% N=300
Z=-124.7(0.00%) | Like=-117.96..-58.88 [-130.7431..-95.4654] | it/evals=704/1213 eff=77.1084% N=300
Z=-121.4(0.00%) | Like=-115.08..-58.88 [-130.7431..-95.4654] | it/evals=718/1237 eff=76.6275% N=300
Z=-121.1(0.00%) | Like=-115.00..-58.88 [-130.7431..-95.4654] | it/evals=720/1239 eff=76.6773% N=300
Z=-119.4(0.00%) | Like=-113.35..-58.88 [-130.7431..-95.4654] | it/evals=735/1262 eff=76.4033% N=300
Mono-modal Volume: ~exp(-6.40) * Expected Volume: exp(-2.46) Quality: ok
index : +1.0| +1.9 *********************** +3.8 | +5.0
amplitude: +1.0e-12| ****************************** | +1.0e-10
Z=-119.2(0.00%) | Like=-112.96..-58.88 [-130.7431..-95.4654] | it/evals=737/1264 eff=76.4523% N=300
Z=-117.5(0.00%) | Like=-111.34..-58.88 [-130.7431..-95.4654] | it/evals=750/1279 eff=76.6088% N=300
Z=-114.2(0.00%) | Like=-106.94..-58.88 [-130.7431..-95.4654] | it/evals=768/1301 eff=76.7233% N=300
Z=-111.9(0.00%) | Like=-105.67..-58.88 [-130.7431..-95.4654] | it/evals=780/1315 eff=76.8473% N=300
Z=-109.5(0.00%) | Like=-103.15..-58.88 [-130.7431..-95.4654] | it/evals=794/1338 eff=76.4933% N=300
Mono-modal Volume: ~exp(-6.88) * 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=-108.3(0.00%) | Like=-101.95..-58.88 [-130.7431..-95.4654] | it/evals=804/1354 eff=76.2808% N=300
Z=-107.5(0.00%) | Like=-100.99..-58.88 [-130.7431..-95.4654] | it/evals=810/1360 eff=76.4151% N=300
Z=-104.8(0.00%) | Like=-98.16..-58.88 [-130.7431..-95.4654] | it/evals=829/1383 eff=76.5466% N=300
Z=-103.2(0.00%) | Like=-96.76..-58.88 [-130.7431..-95.4654] | it/evals=840/1398 eff=76.5027% N=300
Z=-101.1(0.00%) | Like=-94.89..-58.82 [-95.4245..-77.3572] | it/evals=858/1420 eff=76.6071% N=300
Z=-99.6(0.00%) | Like=-92.63..-58.82 [-95.4245..-77.3572] | it/evals=870/1435 eff=76.6520% N=300
Mono-modal Volume: ~exp(-6.88) Expected Volume: exp(-2.90) Quality: ok
index : +1.0| +2.1 ***************** +3.5 | +5.0
amplitude: +1.0e-12| +2.5e-11 ************************ +7.3e-11 | +1.0e-10
Z=-98.3(0.00%) | Like=-92.01..-58.82 [-95.4245..-77.3572] | it/evals=881/1457 eff=76.1452% N=300
Z=-97.0(0.00%) | Like=-90.71..-58.82 [-95.4245..-77.3572] | it/evals=896/1480 eff=75.9322% N=300
Z=-96.7(0.00%) | Like=-90.03..-58.82 [-95.4245..-77.3572] | it/evals=900/1487 eff=75.8214% N=300
Z=-95.1(0.00%) | Like=-88.47..-58.82 [-95.4245..-77.3572] | it/evals=916/1511 eff=75.6400% N=300
Z=-93.6(0.00%) | Like=-86.94..-58.82 [-95.4245..-77.3572] | it/evals=929/1534 eff=75.2836% N=300
Z=-93.5(0.00%) | Like=-86.85..-58.82 [-95.4245..-77.3572] | it/evals=930/1535 eff=75.3036% N=300
Mono-modal Volume: ~exp(-6.96) * Expected Volume: exp(-3.13) Quality: ok
index : +1.0| +2.1 **************** +3.4 | +5.0
amplitude: +1.0e-12| +2.6e-11 *********************** +7.0e-11 | +1.0e-10
Z=-92.6(0.00%) | Like=-85.92..-58.82 [-95.4245..-77.3572] | it/evals=938/1549 eff=75.1001% N=300
Z=-90.8(0.00%) | Like=-84.56..-58.82 [-95.4245..-77.3572] | it/evals=956/1572 eff=75.1572% N=300
Z=-90.6(0.00%) | Like=-84.42..-58.82 [-95.4245..-77.3572] | it/evals=960/1576 eff=75.2351% N=300
Z=-89.2(0.00%) | Like=-82.59..-58.82 [-95.4245..-77.3572] | it/evals=976/1598 eff=75.1926% N=300
Z=-87.8(0.00%) | Like=-81.00..-58.82 [-95.4245..-77.3572] | it/evals=990/1617 eff=75.1708% N=300
Mono-modal Volume: ~exp(-7.10) * Expected Volume: exp(-3.35) Quality: ok
index : +1.0| +2.2 ************** +3.3 | +5.0
amplitude: +1.0e-12| +2.7e-11 ******************** +6.6e-11 | +1.0e-10
Z=-86.6(0.00%) | Like=-80.13..-58.82 [-95.4245..-77.3572] | it/evals=1005/1639 eff=75.0560% N=300
Z=-85.6(0.00%) | Like=-79.13..-58.82 [-95.4245..-77.3572] | it/evals=1020/1658 eff=75.1105% N=300
Z=-84.3(0.00%) | Like=-77.61..-58.82 [-95.4245..-77.3572] | it/evals=1038/1681 eff=75.1629% N=300
Z=-83.4(0.00%) | Like=-76.81..-58.82 [-77.3440..-68.3715] | it/evals=1050/1700 eff=75.0000% N=300
Z=-82.4(0.00%) | Like=-76.01..-58.82 [-77.3440..-68.3715] | it/evals=1066/1724 eff=74.8596% 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 ****************** +6.4e-11 | +1.0e-10
Z=-82.1(0.00%) | Like=-75.89..-58.82 [-77.3440..-68.3715] | it/evals=1072/1731 eff=74.9126% N=300
Z=-81.7(0.00%) | Like=-75.51..-58.82 [-77.3440..-68.3715] | it/evals=1080/1739 eff=75.0521% N=300
Z=-81.0(0.00%) | Like=-74.57..-58.82 [-77.3440..-68.3715] | it/evals=1095/1761 eff=74.9487% N=300
Z=-80.3(0.00%) | Like=-73.78..-58.82 [-77.3440..-68.3715] | it/evals=1110/1782 eff=74.8988% N=300
Z=-79.4(0.00%) | Like=-72.78..-58.82 [-77.3440..-68.3715] | it/evals=1127/1804 eff=74.9335% N=300
Mono-modal Volume: ~exp(-7.94) * Expected Volume: exp(-3.80) Quality: ok
index : +1.0| +2.3 *********** +3.1 | +5.0
amplitude: +1.0e-12| +3.1e-11 **************** +6.1e-11 | +1.0e-10
Z=-78.8(0.00%) | Like=-72.24..-58.82 [-77.3440..-68.3715] | it/evals=1139/1827 eff=74.5907% N=300
Z=-78.7(0.00%) | Like=-72.22..-58.82 [-77.3440..-68.3715] | it/evals=1140/1828 eff=74.6073% N=300
Z=-78.0(0.00%) | Like=-71.62..-58.82 [-77.3440..-68.3715] | it/evals=1157/1850 eff=74.6452% N=300
Z=-77.5(0.00%) | Like=-71.26..-58.82 [-77.3440..-68.3715] | it/evals=1170/1868 eff=74.6173% N=300
Z=-77.0(0.00%) | Like=-70.76..-58.82 [-77.3440..-68.3715] | it/evals=1185/1891 eff=74.4815% N=300
Z=-76.5(0.00%) | Like=-70.20..-58.82 [-77.3440..-68.3715] | it/evals=1200/1911 eff=74.4879% N=300
Mono-modal Volume: ~exp(-7.99) * Expected Volume: exp(-4.02) Quality: ok
index : +1.0| +2.3 *********** +3.1 | +5.0
amplitude: +1.0e-12| +3.2e-11 *************** +6.0e-11 | +1.0e-10
Z=-76.3(0.00%) | Like=-69.90..-58.82 [-77.3440..-68.3715] | it/evals=1206/1922 eff=74.3527% N=300
Z=-75.6(0.00%) | Like=-69.27..-58.82 [-77.3440..-68.3715] | it/evals=1226/1946 eff=74.4836% N=300
Z=-75.5(0.00%) | Like=-69.11..-58.82 [-77.3440..-68.3715] | it/evals=1230/1950 eff=74.5455% N=300
Z=-75.0(0.01%) | Like=-68.56..-58.82 [-77.3440..-68.3715] | it/evals=1246/1973 eff=74.4770% N=300
Z=-74.6(0.01%) | Like=-68.23..-58.82 [-68.3408..-65.7898] | it/evals=1260/1993 eff=74.4241% N=300
Mono-modal Volume: ~exp(-8.21) * Expected Volume: exp(-4.24) Quality: ok
index : +1.0| +2.3 ********* +3.0 | +5.0
amplitude: +1.0e-12| +3.3e-11 ************* +5.9e-11 | +1.0e-10
Z=-74.2(0.02%) | Like=-67.90..-58.82 [-68.3408..-65.7898] | it/evals=1273/2010 eff=74.4444% N=300
Z=-73.8(0.02%) | Like=-67.55..-58.82 [-68.3408..-65.7898] | it/evals=1290/2031 eff=74.5234% N=300
Z=-73.4(0.03%) | Like=-67.01..-58.76 [-68.3408..-65.7898] | it/evals=1308/2053 eff=74.6149% N=300
Z=-73.1(0.04%) | Like=-66.77..-58.76 [-68.3408..-65.7898] | it/evals=1320/2066 eff=74.7452% N=300
Z=-72.7(0.06%) | Like=-66.38..-58.76 [-68.3408..-65.7898] | it/evals=1336/2090 eff=74.6369% N=300
Mono-modal Volume: ~exp(-8.41) * 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=-72.6(0.07%) | Like=-66.15..-58.76 [-68.3408..-65.7898] | it/evals=1340/2096 eff=74.6102% N=300
Z=-72.4(0.09%) | Like=-65.97..-58.76 [-68.3408..-65.7898] | it/evals=1350/2110 eff=74.5856% N=300
Z=-72.0(0.13%) | Like=-65.44..-58.76 [-65.7415..-65.3326] | it/evals=1367/2133 eff=74.5772% N=300
Z=-71.7(0.17%) | Like=-65.24..-58.76 [-65.2602..-65.2416] | it/evals=1380/2153 eff=74.4738% N=300
Z=-71.4(0.23%) | Like=-65.02..-58.76 [-65.0352..-65.0225] | it/evals=1395/2175 eff=74.4000% N=300
Mono-modal Volume: ~exp(-8.41) Expected Volume: exp(-4.69) Quality: ok
index : +1.0| +2.4 ******** +3.0 | +5.0
amplitude: +1.0e-12| +3.5e-11 *********** +5.6e-11 | +1.0e-10
Z=-71.1(0.31%) | Like=-64.63..-58.75 [-64.6253..-64.6248]*| it/evals=1410/2195 eff=74.4063% N=300
Z=-70.8(0.41%) | Like=-64.44..-58.75 [-64.4712..-64.4419] | it/evals=1425/2218 eff=74.2961% N=300
Z=-70.6(0.50%) | Like=-64.11..-58.75 [-64.1345..-64.1150] | it/evals=1438/2241 eff=74.0855% N=300
Z=-70.6(0.52%) | Like=-64.10..-58.75 [-64.0997..-64.0792] | it/evals=1440/2244 eff=74.0741% N=300
Z=-70.3(0.70%) | Like=-63.79..-58.75 [-63.8177..-63.7939] | it/evals=1457/2266 eff=74.1099% N=300
Z=-70.1(0.86%) | Like=-63.62..-58.75 [-63.6196..-63.6170]*| it/evals=1470/2286 eff=74.0181% N=300
Mono-modal Volume: ~exp(-8.83) * 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=-70.0(0.90%) | Like=-63.55..-58.75 [-63.5814..-63.5498] | it/evals=1474/2290 eff=74.0704% N=300
Z=-69.8(1.07%) | Like=-63.40..-58.75 [-63.4177..-63.4027] | it/evals=1486/2313 eff=73.8202% N=300
Z=-69.6(1.28%) | Like=-63.23..-58.75 [-63.2268..-63.1974] | it/evals=1500/2335 eff=73.7101% N=300
Z=-69.4(1.64%) | Like=-62.89..-58.75 [-62.9029..-62.8871] | it/evals=1518/2358 eff=73.7609% N=300
Z=-69.2(1.96%) | Like=-62.75..-58.75 [-62.7468..-62.7423]*| it/evals=1530/2379 eff=73.5931% N=300
Mono-modal Volume: ~exp(-9.13) * 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=-69.1(2.24%) | Like=-62.62..-58.75 [-62.6212..-62.6070] | it/evals=1541/2398 eff=73.4509% N=300
Z=-68.9(2.64%) | Like=-62.32..-58.75 [-62.3415..-62.3200] | it/evals=1557/2420 eff=73.4434% N=300
Z=-68.8(2.77%) | Like=-62.30..-58.75 [-62.2982..-62.2864] | it/evals=1560/2424 eff=73.4463% N=300
Z=-68.7(3.28%) | Like=-62.15..-58.75 [-62.1561..-62.1454] | it/evals=1575/2446 eff=73.3924% N=300
Z=-68.5(3.81%) | Like=-62.03..-58.75 [-62.0270..-62.0240]*| it/evals=1589/2468 eff=73.2934% N=300
Z=-68.5(3.85%) | Like=-62.02..-58.75 [-62.0240..-62.0236]*| it/evals=1590/2470 eff=73.2719% 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| +3.7e-11 ******** +5.2e-11 | +1.0e-10
Z=-68.3(4.54%) | Like=-61.84..-58.75 [-61.8407..-61.8275] | it/evals=1608/2492 eff=73.3577% N=300
Z=-68.2(5.10%) | Like=-61.76..-58.75 [-61.7595..-61.7520]*| it/evals=1620/2510 eff=73.3032% N=300
Z=-68.1(5.71%) | Like=-61.66..-58.75 [-61.6588..-61.6421] | it/evals=1634/2533 eff=73.1751% N=300
Z=-67.9(6.49%) | Like=-61.50..-58.75 [-61.5043..-61.4961]*| it/evals=1648/2557 eff=73.0173% N=300
Z=-67.9(6.59%) | Like=-61.49..-58.75 [-61.4921..-61.4847]*| it/evals=1650/2559 eff=73.0412% N=300
Z=-67.8(7.46%) | Like=-61.33..-58.75 [-61.3430..-61.3316] | it/evals=1665/2581 eff=72.9943% N=300
Mono-modal Volume: ~exp(-9.52) * 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=-67.7(8.10%) | Like=-61.24..-58.75 [-61.2403..-61.2387]*| it/evals=1675/2596 eff=72.9530% N=300
Z=-67.7(8.40%) | Like=-61.21..-58.75 [-61.2248..-61.2113] | it/evals=1680/2602 eff=72.9800% N=300
Z=-67.5(9.67%) | Like=-61.05..-58.75 [-61.0455..-61.0450]*| it/evals=1698/2624 eff=73.0637% N=300
Z=-67.4(10.64%) | Like=-60.96..-58.75 [-60.9647..-60.9623]*| it/evals=1710/2643 eff=72.9834% N=300
Z=-67.3(12.41%) | Like=-60.83..-58.75 [-60.8285..-60.8102] | it/evals=1730/2665 eff=73.1501% N=300
Z=-67.2(13.20%) | Like=-60.76..-58.75 [-60.7778..-60.7602] | it/evals=1740/2679 eff=73.1400% N=300
Mono-modal Volume: ~exp(-9.58) * 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=-67.2(13.38%) | Like=-60.75..-58.75 [-60.7478..-60.7468]*| it/evals=1742/2682 eff=73.1318% N=300
Z=-67.1(14.76%) | Like=-60.67..-58.75 [-60.6729..-60.6463] | it/evals=1757/2704 eff=73.0865% N=300
Z=-67.1(16.08%) | Like=-60.56..-58.75 [-60.5602..-60.5555]*| it/evals=1770/2725 eff=72.9897% N=300
Z=-67.0(17.73%) | Like=-60.49..-58.75 [-60.4890..-60.4748] | it/evals=1787/2747 eff=73.0282% N=300
Z=-66.9(18.97%) | Like=-60.39..-58.75 [-60.3943..-60.3941]*| it/evals=1800/2762 eff=73.1113% N=300
Mono-modal Volume: ~exp(-9.70) * Expected Volume: exp(-6.03) Quality: ok
index : +1.0| +2.5 **** +2.8 | +5.0
amplitude: +1.0e-12| +3.9e-11 ****** +5.0e-11 | +1.0e-10
Z=-66.8(19.85%) | Like=-60.32..-58.75 [-60.3189..-60.3184]*| it/evals=1809/2772 eff=73.1796% N=300
Z=-66.7(21.84%) | Like=-60.22..-58.75 [-60.2532..-60.2203] | it/evals=1826/2794 eff=73.2157% N=300
Z=-66.7(22.26%) | Like=-60.22..-58.75 [-60.2180..-60.2117]*| it/evals=1830/2798 eff=73.2586% N=300
Z=-66.6(24.08%) | Like=-60.14..-58.75 [-60.1420..-60.1403]*| it/evals=1846/2820 eff=73.2540% N=300
Z=-66.6(25.79%) | Like=-60.08..-58.75 [-60.0768..-60.0759]*| it/evals=1860/2838 eff=73.2861% N=300
Mono-modal Volume: ~exp(-10.09) * 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=-66.5(27.42%) | Like=-60.03..-58.75 [-60.0313..-60.0304]*| it/evals=1876/2857 eff=73.3672% N=300
Z=-66.5(28.96%) | Like=-59.99..-58.75 [-59.9898..-59.9855]*| it/evals=1890/2874 eff=73.4266% N=300
Z=-66.4(31.23%) | Like=-59.91..-58.75 [-59.9076..-59.9037]*| it/evals=1909/2896 eff=73.5362% N=300
Z=-66.3(32.43%) | Like=-59.84..-58.75 [-59.8367..-59.8367]*| it/evals=1920/2910 eff=73.5632% N=300
Z=-66.3(34.41%) | Like=-59.79..-58.75 [-59.7912..-59.7827]*| it/evals=1935/2932 eff=73.5182% N=300
Mono-modal Volume: ~exp(-10.42) * 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(35.57%) | Like=-59.76..-58.75 [-59.7631..-59.7598]*| it/evals=1943/2946 eff=73.4316% N=300
Z=-66.2(36.49%) | Like=-59.74..-58.75 [-59.7434..-59.7412]*| it/evals=1950/2953 eff=73.5017% N=300
Z=-66.2(38.43%) | Like=-59.70..-58.75 [-59.6958..-59.6903]*| it/evals=1965/2975 eff=73.4579% N=300
Z=-66.1(40.29%) | Like=-59.64..-58.75 [-59.6381..-59.6378]*| it/evals=1980/2995 eff=73.4694% N=300
Z=-66.1(42.39%) | Like=-59.58..-58.75 [-59.5805..-59.5783]*| it/evals=1997/3017 eff=73.5002% N=300
Mono-modal Volume: ~exp(-10.56) * 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(43.99%) | Like=-59.56..-58.75 [-59.5554..-59.5541]*| it/evals=2010/3038 eff=73.4112% N=300
Z=-66.0(45.84%) | Like=-59.52..-58.75 [-59.5242..-59.5233]*| it/evals=2026/3060 eff=73.4058% N=300
Z=-66.0(47.40%) | Like=-59.49..-58.75 [-59.4897..-59.4841]*| it/evals=2040/3077 eff=73.4606% N=300
Z=-65.9(49.49%) | Like=-59.44..-58.75 [-59.4435..-59.4383]*| it/evals=2058/3099 eff=73.5263% N=300
Z=-65.9(50.81%) | Like=-59.41..-58.75 [-59.4105..-59.4099]*| it/evals=2070/3118 eff=73.4564% N=300
Mono-modal Volume: ~exp(-10.99) * 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=-65.9(51.60%) | Like=-59.38..-58.75 [-59.3841..-59.3841]*| it/evals=2077/3128 eff=73.4441% N=300
Z=-65.9(53.72%) | Like=-59.35..-58.75 [-59.3514..-59.3470]*| it/evals=2096/3151 eff=73.5181% N=300
Z=-65.8(54.14%) | Like=-59.35..-58.75 [-59.3455..-59.3444]*| it/evals=2100/3157 eff=73.5037% N=300
Z=-65.8(55.87%) | Like=-59.31..-58.75 [-59.3140..-59.3131]*| it/evals=2116/3181 eff=73.4467% N=300
Z=-65.8(57.43%) | Like=-59.29..-58.75 [-59.2895..-59.2799]*| it/evals=2130/3200 eff=73.4483% N=300
Mono-modal Volume: ~exp(-10.99) 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(58.96%) | Like=-59.26..-58.75 [-59.2616..-59.2585]*| it/evals=2144/3225 eff=73.2991% N=300
Z=-65.7(60.67%) | Like=-59.23..-58.75 [-59.2280..-59.2250]*| it/evals=2160/3256 eff=73.0717% N=300
Z=-65.7(62.26%) | Like=-59.21..-58.75 [-59.2054..-59.2050]*| it/evals=2176/3279 eff=73.0446% N=300
Z=-65.7(63.59%) | Like=-59.18..-58.75 [-59.1815..-59.1813]*| it/evals=2190/3301 eff=72.9757% N=300
Z=-65.7(65.03%) | Like=-59.16..-58.75 [-59.1560..-59.1554]*| it/evals=2205/3325 eff=72.8926% N=300
Mono-modal Volume: ~exp(-11.30) * 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(65.57%) | Like=-59.15..-58.75 [-59.1459..-59.1449]*| it/evals=2211/3334 eff=72.8741% N=300
Z=-65.6(66.42%) | Like=-59.13..-58.75 [-59.1266..-59.1232]*| it/evals=2220/3344 eff=72.9304% N=300
Z=-65.6(68.15%) | Like=-59.10..-58.75 [-59.0982..-59.0973]*| it/evals=2240/3366 eff=73.0594% N=300
Z=-65.6(69.02%) | Like=-59.08..-58.75 [-59.0825..-59.0824]*| it/evals=2250/3381 eff=73.0282% N=300
[ultranest] Explored until L=-6e+01
[ultranest] Likelihood function evaluations: 3397
[ultranest] logZ = -65.23 +- 0.1231
[ultranest] Effective samples strategy satisfied (ESS = 1004.8, need >400)
[ultranest] Posterior uncertainty strategy is satisfied (KL: 0.46+-0.07 nat, need <0.50 nat)
[ultranest] Evidency uncertainty strategy is satisfied (dlogz=0.29, need <0.5)
[ultranest] logZ error budget: single: 0.14 bs:0.12 tail:0.26 total:0.29 required:<0.50
[ultranest] done iterating.
logZ = -65.235 +- 0.341
single instance: logZ = -65.235 +- 0.135
bootstrapped : logZ = -65.227 +- 0.218
tail : logZ = +- 0.262
insert order U test : converged: True correlation: inf iterations
index : 2.379 │ ▁▁▁▁▁▁▂▂▃▃▅▄▅▆▆▇▇▆▆▅▄▃▃▂▂▁▁▁▁▁▁▁ ▁ ▁ │3.056 2.672 +- 0.085
amplitude : 0.0000000000344│ ▁▁▁▁▁▁▁▂▂▂▃▄▄▅▇▆▅▆▆▄▄▃▄▂▂▂▁▁▁▁▁▁▁▁▁▁▁ │0.0000000000566 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 : 2.672 +/- 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': [2.6720183583673704, 4.438276546278543e-11], 'stdev': [0.08483789184562701, 3.0163071911927864e-12], 'median': [2.671939732786746, 4.43117582885524e-11], 'errlo': [2.583993378476902, 4.1471704423419637e-11], 'errup': [2.7538857752290973, 4.742430775366671e-11], 'information_gain_bits': [2.6812735173932456, 3.0941507172641978]}
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(-3.87) * Expected Volume: exp(0.00) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|******************** ********* *********** * ***| +1.0e-10
Z=-inf(0.00%) | Like=-2230.38..-22.11 [-2230.3802..-107.3581] | it/evals=0/301 eff=0.0000% N=300
Z=-177.0(0.00%) | Like=-171.84..-20.59 [-2230.3802..-107.3581] | it/evals=30/333 eff=90.9091% N=300
Z=-162.8(0.00%) | Like=-157.93..-20.59 [-2230.3802..-107.3581] | it/evals=60/374 eff=81.0811% N=300
Mono-modal Volume: ~exp(-4.39) * Expected Volume: exp(-0.22) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|****************************** *********** * **| +1.0e-10
Z=-160.5(0.00%) | Like=-155.60..-20.59 [-2230.3802..-107.3581] | it/evals=67/381 eff=82.7160% N=300
Z=-154.2(0.00%) | Like=-149.06..-20.59 [-2230.3802..-107.3581] | it/evals=90/410 eff=81.8182% N=300
Z=-145.2(0.00%) | Like=-140.13..-20.57 [-2230.3802..-107.3581] | it/evals=120/445 eff=82.7586% N=300
Mono-modal Volume: ~exp(-4.39) Expected Volume: exp(-0.45) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|************************************** *** ** **| +1.0e-10
Z=-135.4(0.00%) | Like=-130.29..-20.57 [-2230.3802..-107.3581] | it/evals=150/479 eff=83.7989% N=300
Z=-122.1(0.00%) | Like=-117.17..-20.57 [-2230.3802..-107.3581] | it/evals=179/521 eff=80.9955% N=300
Z=-121.9(0.00%) | Like=-117.00..-20.57 [-2230.3802..-107.3581] | it/evals=180/523 eff=80.7175% N=300
Mono-modal Volume: ~exp(-5.12) * Expected Volume: exp(-0.67) Quality: ok
index : +1.0| **********************************************| +5.0
amplitude: +1.0e-12| ************************ **************** ** **| +1.0e-10
Z=-114.9(0.00%) | Like=-108.87..-20.57 [-2230.3802..-107.3581] | it/evals=201/549 eff=80.7229% N=300
Z=-112.2(0.00%) | Like=-107.05..-20.57 [-107.0545..-70.1641] | it/evals=210/564 eff=79.5455% N=300
Z=-104.3(0.00%) | Like=-99.56..-20.57 [-107.0545..-70.1641] | it/evals=240/599 eff=80.2676% N=300
Mono-modal Volume: ~exp(-5.12) Expected Volume: exp(-0.89) Quality: ok
index : +1.0| ********************************************| +5.0
amplitude: +1.0e-12| ************************ ****** ***** *** ****| +1.0e-10
Z=-98.9(0.00%) | Like=-93.56..-20.57 [-107.0545..-70.1641] | it/evals=270/632 eff=81.3253% N=300
Z=-93.2(0.00%) | Like=-88.86..-20.57 [-107.0545..-70.1641] | it/evals=300/670 eff=81.0811% N=300
Z=-86.9(0.00%) | Like=-81.95..-20.57 [-107.0545..-70.1641] | it/evals=330/709 eff=80.6846% N=300
Mono-modal Volume: ~exp(-5.12) Expected Volume: exp(-1.12) Quality: ok
index : +1.0| *******************************************| +5.0
amplitude: +1.0e-12| ****************************** ****** ** | +1.0e-10
Z=-81.1(0.00%) | Like=-75.91..-20.57 [-107.0545..-70.1641] | it/evals=360/745 eff=80.8989% N=300
Z=-78.0(0.00%) | Like=-73.42..-20.57 [-107.0545..-70.1641] | it/evals=386/787 eff=79.2608% N=300
Z=-77.6(0.00%) | Like=-73.10..-20.57 [-107.0545..-70.1641] | it/evals=390/792 eff=79.2683% N=300
Mono-modal Volume: ~exp(-5.12) Expected Volume: exp(-1.34) Quality: ok
index : +1.0| ***************************************** | +5.0
amplitude: +1.0e-12| ********************************** * * | +1.0e-10
Z=-74.7(0.00%) | Like=-70.16..-20.57 [-70.1637..-47.9113] | it/evals=418/828 eff=79.1667% N=300
Z=-74.6(0.00%) | Like=-70.07..-20.57 [-70.1637..-47.9113] | it/evals=420/832 eff=78.9474% N=300
Z=-71.2(0.00%) | Like=-66.12..-20.57 [-70.1637..-47.9113] | it/evals=450/870 eff=78.9474% N=300
Mono-modal Volume: ~exp(-5.56) * Expected Volume: exp(-1.56) Quality: ok
index : +1.0| ********************************** | +5.0
amplitude: +1.0e-12| ********************************* +7.6e-11 | +1.0e-10
Z=-68.9(0.00%) | Like=-63.14..-20.57 [-70.1637..-47.9113] | it/evals=469/912 eff=76.6340% N=300
Z=-67.2(0.00%) | Like=-61.83..-20.57 [-70.1637..-47.9113] | it/evals=480/931 eff=76.0697% N=300
Z=-64.8(0.00%) | Like=-60.04..-20.57 [-70.1637..-47.9113] | it/evals=503/973 eff=74.7400% N=300
Z=-64.4(0.00%) | Like=-59.70..-20.57 [-70.1637..-47.9113] | it/evals=510/985 eff=74.4526% N=300
Z=-62.4(0.00%) | Like=-57.52..-20.57 [-70.1637..-47.9113] | it/evals=534/1027 eff=73.4525% N=300
Mono-modal Volume: ~exp(-5.99) * Expected Volume: exp(-1.79) Quality: ok
index : +1.0| ****************************** +4.2 | +5.0
amplitude: +1.0e-12| ******************************* +7.2e-11 | +1.0e-10
Z=-62.2(0.00%) | Like=-57.50..-20.57 [-70.1637..-47.9113] | it/evals=536/1030 eff=73.4247% N=300
Z=-62.0(0.00%) | Like=-56.95..-20.57 [-70.1637..-47.9113] | it/evals=540/1035 eff=73.4694% N=300
Z=-59.2(0.00%) | Like=-53.94..-20.57 [-70.1637..-47.9113] | it/evals=568/1078 eff=73.0077% N=300
Z=-59.0(0.00%) | Like=-53.81..-20.57 [-70.1637..-47.9113] | it/evals=570/1081 eff=72.9834% N=300
Z=-56.7(0.00%) | Like=-51.63..-20.57 [-70.1637..-47.9113] | it/evals=600/1120 eff=73.1707% N=300
Mono-modal Volume: ~exp(-5.99) Expected Volume: exp(-2.01) Quality: ok
index : +1.0| **************************** +3.9 | +5.0
amplitude: +1.0e-12| ****************************** +7.2e-11 | +1.0e-10
Z=-54.8(0.00%) | Like=-49.49..-20.57 [-70.1637..-47.9113] | it/evals=627/1157 eff=73.1622% N=300
Z=-54.5(0.00%) | Like=-49.09..-20.57 [-70.1637..-47.9113] | it/evals=630/1160 eff=73.2558% N=300
Z=-52.1(0.00%) | Like=-46.89..-20.57 [-47.7776..-36.3649] | it/evals=656/1203 eff=72.6467% N=300
Z=-51.8(0.00%) | Like=-46.49..-20.57 [-47.7776..-36.3649] | it/evals=660/1210 eff=72.5275% N=300
Mono-modal Volume: ~exp(-6.12) * Expected Volume: exp(-2.23) Quality: ok
index : +1.0| ************************ +3.7 | +5.0
amplitude: +1.0e-12| ************************* +6.4e-11 | +1.0e-10
Z=-51.1(0.00%) | Like=-45.75..-20.57 [-47.7776..-36.3649] | it/evals=670/1225 eff=72.4324% N=300
Z=-49.9(0.00%) | Like=-44.96..-20.57 [-47.7776..-36.3649] | it/evals=690/1251 eff=72.5552% N=300
Z=-48.3(0.00%) | Like=-42.86..-20.57 [-47.7776..-36.3649] | it/evals=716/1292 eff=72.1774% N=300
Z=-48.1(0.00%) | Like=-42.64..-20.57 [-47.7776..-36.3649] | it/evals=720/1299 eff=72.0721% N=300
Mono-modal Volume: ~exp(-6.12) Expected Volume: exp(-2.46) Quality: ok
index : +1.0| ********************* +3.5 | +5.0
amplitude: +1.0e-12| ************************* +6.4e-11 | +1.0e-10
Z=-46.6(0.00%) | Like=-41.19..-20.57 [-47.7776..-36.3649] | it/evals=746/1336 eff=72.0077% N=300
Z=-46.4(0.00%) | Like=-41.07..-20.57 [-47.7776..-36.3649] | it/evals=750/1341 eff=72.0461% N=300
Z=-45.2(0.00%) | Like=-39.67..-20.57 [-47.7776..-36.3649] | it/evals=770/1384 eff=71.0332% N=300
Z=-44.6(0.00%) | Like=-39.04..-20.57 [-47.7776..-36.3649] | it/evals=780/1396 eff=71.1679% N=300
Mono-modal Volume: ~exp(-6.61) * Expected Volume: exp(-2.68) Quality: ok
index : +1.0| +1.9 ******************* +3.5 | +5.0
amplitude: +1.0e-12| ********************** +6.0e-11 | +1.0e-10
Z=-43.2(0.00%) | Like=-37.86..-20.46 [-47.7776..-36.3649] | it/evals=804/1429 eff=71.2135% N=300
Z=-42.9(0.00%) | Like=-37.46..-20.46 [-47.7776..-36.3649] | it/evals=810/1439 eff=71.1150% N=300
Z=-41.6(0.00%) | Like=-36.34..-20.46 [-36.3549..-28.3334] | it/evals=839/1481 eff=71.0415% N=300
Z=-41.6(0.00%) | Like=-36.33..-20.46 [-36.3549..-28.3334] | it/evals=840/1482 eff=71.0660% N=300
Z=-40.7(0.00%) | Like=-35.26..-20.46 [-36.3549..-28.3334] | it/evals=863/1525 eff=70.4490% N=300
Z=-40.4(0.00%) | Like=-35.05..-20.46 [-36.3549..-28.3334] | it/evals=870/1533 eff=70.5596% N=300
Mono-modal Volume: ~exp(-6.61) Expected Volume: exp(-2.90) Quality: ok
index : +1.0| +2.0 ****************** +3.3 | +5.0
amplitude: +1.0e-12| ********************* +5.8e-11 | +1.0e-10
Z=-39.2(0.00%) | Like=-33.61..-20.46 [-36.3549..-28.3334] | it/evals=895/1570 eff=70.4724% N=300
Z=-39.0(0.00%) | Like=-33.51..-20.46 [-36.3549..-28.3334] | it/evals=900/1575 eff=70.5882% N=300
Z=-37.9(0.00%) | Like=-32.37..-20.46 [-36.3549..-28.3334] | it/evals=927/1616 eff=70.4407% N=300
Z=-37.8(0.00%) | Like=-32.34..-20.46 [-36.3549..-28.3334] | it/evals=930/1619 eff=70.5080% N=300
Mono-modal Volume: ~exp(-7.03) * Expected Volume: exp(-3.13) Quality: ok
index : +1.0| +2.0 *************** +3.2 | +5.0
amplitude: +1.0e-12| ******************* +5.5e-11 | +1.0e-10
Z=-37.5(0.00%) | Like=-31.91..-20.46 [-36.3549..-28.3334] | it/evals=938/1629 eff=70.5794% N=300
Z=-36.7(0.00%) | Like=-30.96..-20.46 [-36.3549..-28.3334] | it/evals=960/1655 eff=70.8487% N=300
Z=-35.8(0.01%) | Like=-30.39..-20.46 [-36.3549..-28.3334] | it/evals=988/1696 eff=70.7736% N=300
Z=-35.7(0.01%) | Like=-30.33..-20.46 [-36.3549..-28.3334] | it/evals=990/1698 eff=70.8155% N=300
Mono-modal Volume: ~exp(-7.63) * 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=-35.3(0.01%) | Like=-29.63..-20.46 [-36.3549..-28.3334] | it/evals=1005/1722 eff=70.6751% N=300
Z=-34.8(0.02%) | Like=-29.19..-20.46 [-36.3549..-28.3334] | it/evals=1020/1737 eff=70.9812% N=300
Z=-34.0(0.05%) | Like=-28.40..-20.46 [-36.3549..-28.3334] | it/evals=1050/1777 eff=71.0900% N=300
Mono-modal Volume: ~exp(-7.63) Expected Volume: exp(-3.57) Quality: ok
index : +1.0| +2.1 ************* +3.1 | +5.0
amplitude: +1.0e-12| ************** +5.0e-11 | +1.0e-10
Z=-33.4(0.08%) | Like=-27.89..-20.46 [-28.2662..-27.3611] | it/evals=1073/1814 eff=70.8719% N=300
Z=-33.2(0.10%) | Like=-27.74..-20.46 [-28.2662..-27.3611] | it/evals=1080/1825 eff=70.8197% N=300
Z=-32.6(0.16%) | Like=-27.13..-20.46 [-27.3219..-27.0620] | it/evals=1110/1864 eff=70.9719% N=300
Mono-modal Volume: ~exp(-7.91) * 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=-32.1(0.27%) | Like=-26.63..-20.46 [-26.6540..-26.6293] | it/evals=1139/1903 eff=71.0543% N=300
Z=-32.1(0.28%) | Like=-26.62..-20.46 [-26.6180..-26.6108]*| it/evals=1140/1904 eff=71.0723% N=300
Z=-31.7(0.39%) | Like=-26.26..-20.46 [-26.2580..-26.2532]*| it/evals=1164/1946 eff=70.7169% N=300
Z=-31.6(0.43%) | Like=-26.22..-20.46 [-26.2171..-26.1721] | it/evals=1170/1953 eff=70.7804% N=300
Z=-31.2(0.62%) | Like=-25.75..-20.46 [-25.7506..-25.7363] | it/evals=1196/1992 eff=70.6856% N=300
Z=-31.1(0.65%) | Like=-25.71..-20.46 [-25.7086..-25.6547] | it/evals=1200/1999 eff=70.6298% N=300
Mono-modal Volume: ~exp(-7.95) * Expected Volume: exp(-4.02) Quality: ok
index : +1.0| +2.2 ********** +3.0 | +5.0
amplitude: +1.0e-12| ************* +4.7e-11 | +1.0e-10
Z=-31.1(0.70%) | Like=-25.55..-20.46 [-25.5707..-25.5491] | it/evals=1206/2008 eff=70.6089% N=300
Z=-30.7(1.03%) | Like=-25.21..-20.46 [-25.2131..-25.1974] | it/evals=1230/2041 eff=70.6491% N=300
Z=-30.4(1.45%) | Like=-24.79..-20.46 [-24.7937..-24.7341] | it/evals=1257/2082 eff=70.5387% N=300
Z=-30.3(1.52%) | Like=-24.73..-20.46 [-24.7284..-24.7089] | it/evals=1260/2086 eff=70.5487% N=300
Mono-modal Volume: ~exp(-8.27) * Expected Volume: exp(-4.24) Quality: ok
index : +1.0| +2.2 ********** +3.0 | +5.0
amplitude: +1.0e-12| +2.5e-11 *********** +4.5e-11 | +1.0e-10
Z=-30.1(1.81%) | Like=-24.59..-20.46 [-24.5942..-24.5937]*| it/evals=1273/2101 eff=70.6830% N=300
Z=-29.9(2.22%) | Like=-24.36..-20.46 [-24.3595..-24.3249] | it/evals=1290/2122 eff=70.8013% N=300
Z=-29.6(3.00%) | Like=-24.09..-20.46 [-24.0934..-24.0685] | it/evals=1316/2161 eff=70.7147% N=300
Z=-29.6(3.11%) | Like=-24.04..-20.46 [-24.0373..-24.0354]*| it/evals=1320/2169 eff=70.6260% N=300
Mono-modal Volume: ~exp(-8.27) Expected Volume: exp(-4.47) 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(4.04%) | Like=-23.72..-20.46 [-23.7375..-23.7218] | it/evals=1346/2205 eff=70.6562% N=300
Z=-29.3(4.18%) | Like=-23.66..-20.46 [-23.7051..-23.6572] | it/evals=1350/2211 eff=70.6436% N=300
Z=-29.0(5.53%) | Like=-23.43..-20.46 [-23.4305..-23.4158] | it/evals=1377/2251 eff=70.5792% N=300
Z=-29.0(5.72%) | Like=-23.41..-20.46 [-23.4090..-23.4084]*| it/evals=1380/2255 eff=70.5882% N=300
Z=-28.8(6.84%) | Like=-23.23..-20.46 [-23.2272..-23.2083] | it/evals=1400/2296 eff=70.1403% N=300
Mono-modal Volume: ~exp(-8.30) * Expected Volume: exp(-4.69) 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=-28.8(7.30%) | Like=-23.17..-20.46 [-23.1726..-23.1723]*| it/evals=1407/2305 eff=70.1746% N=300
Z=-28.8(7.47%) | Like=-23.16..-20.46 [-23.1624..-23.1524]*| it/evals=1410/2308 eff=70.2191% N=300
Z=-28.5(9.54%) | Like=-22.94..-20.46 [-22.9383..-22.9173] | it/evals=1440/2347 eff=70.3468% N=300
Z=-28.3(11.53%) | Like=-22.74..-20.46 [-22.7401..-22.7253] | it/evals=1470/2384 eff=70.5374% N=300
Mono-modal Volume: ~exp(-8.61) * Expected Volume: exp(-4.91) Quality: ok
index : +1.0| +2.3 ******* +2.8 | +5.0
amplitude: +1.0e-12| +2.7e-11 ********* +4.3e-11 | +1.0e-10
Z=-28.3(11.84%) | Like=-22.70..-20.46 [-22.7048..-22.6922] | it/evals=1474/2388 eff=70.5939% N=300
Z=-28.1(13.88%) | Like=-22.49..-20.46 [-22.4893..-22.4721] | it/evals=1500/2423 eff=70.6547% N=300
Z=-28.0(16.40%) | Like=-22.28..-20.46 [-22.2926..-22.2811] | it/evals=1527/2463 eff=70.5964% N=300
Z=-27.9(16.72%) | Like=-22.24..-20.46 [-22.2423..-22.2138] | it/evals=1530/2466 eff=70.6371% N=300
Mono-modal Volume: ~exp(-8.79) * Expected Volume: exp(-5.14) Quality: ok
index : +1.0| +2.4 ****** +2.8 | +5.0
amplitude: +1.0e-12| +2.8e-11 ******** +4.2e-11 | +1.0e-10
Z=-27.9(17.75%) | Like=-22.17..-20.46 [-22.1744..-22.1668]*| it/evals=1541/2484 eff=70.5586% N=300
Z=-27.8(19.88%) | Like=-22.07..-20.46 [-22.0739..-22.0683]*| it/evals=1560/2509 eff=70.6202% N=300
Z=-27.6(23.05%) | Like=-21.94..-20.46 [-21.9577..-21.9398] | it/evals=1589/2549 eff=70.6536% N=300
Z=-27.6(23.18%) | Like=-21.93..-20.46 [-21.9318..-21.9213] | it/evals=1590/2550 eff=70.6667% N=300
Mono-modal Volume: ~exp(-8.94) * 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=-27.5(25.16%) | Like=-21.85..-20.46 [-21.8511..-21.8494]*| it/evals=1608/2581 eff=70.4954% N=300
Z=-27.5(26.61%) | Like=-21.81..-20.46 [-21.8130..-21.8122]*| it/evals=1620/2595 eff=70.5882% N=300
Z=-27.4(30.03%) | Like=-21.70..-20.46 [-21.7046..-21.6976]*| it/evals=1650/2635 eff=70.6638% N=300
Mono-modal Volume: ~exp(-9.13) * Expected Volume: exp(-5.58) Quality: ok
index : +1.0| +2.4 ***** +2.7 | +5.0
amplitude: +1.0e-12| +2.9e-11 ****** +4.0e-11 | +1.0e-10
Z=-27.3(33.10%) | Like=-21.62..-20.46 [-21.6248..-21.6062] | it/evals=1675/2668 eff=70.7348% N=300
Z=-27.2(33.66%) | Like=-21.59..-20.46 [-21.5900..-21.5889]*| it/evals=1680/2673 eff=70.7965% N=300
Z=-27.1(37.38%) | Like=-21.46..-20.46 [-21.4642..-21.4628]*| it/evals=1710/2711 eff=70.9249% N=300
Z=-27.1(40.46%) | Like=-21.38..-20.46 [-21.3799..-21.3793]*| it/evals=1737/2751 eff=70.8690% N=300
Z=-27.0(40.80%) | Like=-21.37..-20.46 [-21.3733..-21.3591] | it/evals=1740/2755 eff=70.8758% N=300
Mono-modal Volume: ~exp(-9.45) * Expected Volume: exp(-5.81) 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=-27.0(41.00%) | Like=-21.36..-20.46 [-21.3552..-21.3527]*| it/evals=1742/2757 eff=70.8995% N=300
Z=-27.0(44.28%) | Like=-21.26..-20.46 [-21.2555..-21.2553]*| it/evals=1770/2794 eff=70.9703% N=300
Z=-26.9(47.67%) | Like=-21.20..-20.46 [-21.2010..-21.1979]*| it/evals=1798/2834 eff=70.9550% N=300
Z=-26.9(47.89%) | Like=-21.20..-20.46 [-21.1959..-21.1945]*| it/evals=1800/2837 eff=70.9499% N=300
Mono-modal Volume: ~exp(-9.57) * Expected Volume: exp(-6.03) 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(48.98%) | Like=-21.18..-20.46 [-21.1827..-21.1780]*| it/evals=1809/2850 eff=70.9412% N=300
Z=-26.8(51.48%) | Like=-21.12..-20.46 [-21.1236..-21.1152]*| it/evals=1830/2882 eff=70.8753% N=300
Z=-26.8(54.80%) | Like=-21.07..-20.46 [-21.0732..-21.0715]*| it/evals=1860/2918 eff=71.0466% N=300
Mono-modal Volume: ~exp(-10.10) * Expected Volume: exp(-6.25) 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(56.55%) | Like=-21.04..-20.46 [-21.0398..-21.0395]*| it/evals=1876/2941 eff=71.0337% N=300
Z=-26.7(58.14%) | Like=-21.02..-20.46 [-21.0218..-21.0176]*| it/evals=1890/2957 eff=71.1329% N=300
Z=-26.6(61.12%) | Like=-20.97..-20.46 [-20.9657..-20.9640]*| it/evals=1920/2996 eff=71.2166% N=300
Mono-modal Volume: ~exp(-10.42) * Expected Volume: exp(-6.48) 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(63.35%) | Like=-20.92..-20.46 [-20.9296..-20.9190] | it/evals=1943/3026 eff=71.2766% N=300
Z=-26.6(64.05%) | Like=-20.91..-20.46 [-20.9061..-20.9028]*| it/evals=1950/3035 eff=71.2980% N=300
Z=-26.6(66.87%) | Like=-20.85..-20.46 [-20.8526..-20.8521]*| it/evals=1980/3069 eff=71.5060% N=300
Mono-modal Volume: ~exp(-10.42) Expected Volume: exp(-6.70) 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(69.52%) | Like=-20.83..-20.46 [-20.8328..-20.8292]*| it/evals=2010/3108 eff=71.5812% N=300
[ultranest] Explored until L=-2e+01
[ultranest] Likelihood function evaluations: 3113
[ultranest] logZ = -26.16 +- 0.08736
[ultranest] Effective samples strategy satisfied (ESS = 1007.6, need >400)
[ultranest] Posterior uncertainty strategy is satisfied (KL: 0.46+-0.09 nat, need <0.50 nat)
[ultranest] Evidency uncertainty strategy is satisfied (dlogz=0.28, need <0.5)
[ultranest] logZ error budget: single: 0.12 bs:0.09 tail:0.26 total:0.28 required:<0.50
[ultranest] done iterating.
logZ = -26.155 +- 0.303
single instance: logZ = -26.155 +- 0.125
bootstrapped : logZ = -26.159 +- 0.151
tail : logZ = +- 0.262
insert order U test : converged: True correlation: inf iterations
index : 2.12 │ ▁ ▁▁▁▁▁▂▂▄▄▄▅▇▆▇▇▇▆▅▅▄▃▂▂▂▁▁▁▁▁▁▁ ▁ ▁ │3.15 2.57 +- 0.13
amplitude : 0.0000000000211│ ▁ ▁▁▁▁▁▁▂▂▂▄▄▅▆▆▆▅▇▅▄▄▃▃▂▂▂▂▁▁▁▁▁▁ ▁▁ │0.0000000000484 0.0000000000339 +- 0.0000000000037
[ultranest] Sampling 300 live points from prior ...
Mono-modal Volume: ~exp(-3.89) * Expected Volume: exp(0.00) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|************* ************** * ***** * *** *** | +1.0e-10
Z=-inf(0.00%) | Like=-954.59..-19.84 [-954.5903..-134.5042] | it/evals=0/301 eff=0.0000% N=300
Z=-219.8(0.00%) | Like=-214.13..-19.84 [-954.5903..-134.5042] | it/evals=30/332 eff=93.7500% N=300
Z=-200.6(0.00%) | Like=-194.24..-19.84 [-954.5903..-134.5042] | it/evals=60/368 eff=88.2353% N=300
Mono-modal Volume: ~exp(-4.35) * Expected Volume: exp(-0.22) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|************* *********************** * ******* | +1.0e-10
Z=-196.1(0.00%) | Like=-190.53..-19.84 [-954.5903..-134.5042] | it/evals=67/378 eff=85.8974% N=300
Z=-185.4(0.00%) | Like=-179.88..-19.84 [-954.5903..-134.5042] | it/evals=90/405 eff=85.7143% N=300
Z=-173.4(0.00%) | Like=-168.50..-19.84 [-954.5903..-134.5042] | it/evals=120/438 eff=86.9565% N=300
Mono-modal Volume: ~exp(-4.35) Expected Volume: exp(-0.45) Quality: ok
index : +1.0| ***********************************************| +5.0
amplitude: +1.0e-12|************************************* * ********| +1.0e-10
Z=-160.6(0.00%) | Like=-155.34..-19.84 [-954.5903..-134.5042] | it/evals=148/475 eff=84.5714% N=300
Z=-159.8(0.00%) | Like=-154.22..-19.84 [-954.5903..-134.5042] | it/evals=150/477 eff=84.7458% N=300
Z=-145.6(0.00%) | Like=-140.02..-19.52 [-954.5903..-134.5042] | it/evals=180/512 eff=84.9057% N=300
Mono-modal Volume: ~exp(-4.37) * Expected Volume: exp(-0.67) Quality: ok
index : +1.0| *********************************************| +5.0
amplitude: +1.0e-12| *********************************** * ********| +1.0e-10
Z=-136.5(0.00%) | Like=-130.65..-19.19 [-134.3154..-65.6762] | it/evals=201/537 eff=84.8101% N=300
Z=-130.7(0.00%) | Like=-125.07..-19.19 [-134.3154..-65.6762] | it/evals=210/548 eff=84.6774% N=300
Z=-117.4(0.00%) | Like=-111.25..-19.19 [-134.3154..-65.6762] | it/evals=236/590 eff=81.3793% N=300
Z=-115.5(0.00%) | Like=-109.57..-19.19 [-134.3154..-65.6762] | it/evals=240/595 eff=81.3559% N=300
Mono-modal Volume: ~exp(-4.61) * Expected Volume: exp(-0.89) Quality: ok
index : +1.0| ********************************************| +5.0
amplitude: +1.0e-12| ********************************** * ********| +1.0e-10
Z=-106.0(0.00%) | Like=-99.95..-19.19 [-134.3154..-65.6762] | it/evals=268/627 eff=81.9572% N=300
Z=-104.9(0.00%) | Like=-99.01..-19.19 [-134.3154..-65.6762] | it/evals=270/630 eff=81.8182% N=300
Z=-91.4(0.00%) | Like=-85.23..-19.19 [-134.3154..-65.6762] | it/evals=300/667 eff=81.7439% N=300
Z=-85.1(0.00%) | Like=-80.01..-19.19 [-134.3154..-65.6762] | it/evals=325/709 eff=79.4621% N=300
Z=-84.4(0.00%) | Like=-79.56..-19.19 [-134.3154..-65.6762] | it/evals=330/714 eff=79.7101% N=300
Mono-modal Volume: ~exp(-5.22) * Expected Volume: exp(-1.12) Quality: ok
index : +1.0| *****************************************| +5.0
amplitude: +1.0e-12| ******************************** * ********| +1.0e-10
Z=-83.6(0.00%) | Like=-78.29..-19.19 [-134.3154..-65.6762] | it/evals=335/719 eff=79.9523% N=300
Z=-76.7(0.00%) | Like=-70.65..-19.19 [-134.3154..-65.6762] | it/evals=360/746 eff=80.7175% N=300
Z=-70.5(0.00%) | Like=-64.63..-19.19 [-65.6220..-39.5909] | it/evals=390/787 eff=80.0821% N=300
Mono-modal Volume: ~exp(-5.50) * Expected Volume: exp(-1.34) Quality: ok
index : +1.0| ****************************************| +5.0
amplitude: +1.0e-12| ******************************** ********| +1.0e-10
Z=-68.4(0.00%) | Like=-63.29..-19.19 [-65.6220..-39.5909] | it/evals=402/800 eff=80.4000% N=300
Z=-65.7(0.00%) | Like=-60.41..-19.19 [-65.6220..-39.5909] | it/evals=420/824 eff=80.1527% N=300
Z=-60.4(0.00%) | Like=-54.58..-19.19 [-65.6220..-39.5909] | it/evals=450/861 eff=80.2139% N=300
Mono-modal Volume: ~exp(-5.50) Expected Volume: exp(-1.56) Quality: ok
index : +1.0| ************************************** | +5.0
amplitude: +1.0e-12| ****************************** ******* | +1.0e-10
Z=-57.4(0.00%) | Like=-52.71..-19.19 [-65.6220..-39.5909] | it/evals=476/899 eff=79.4658% N=300
Z=-57.1(0.00%) | Like=-52.46..-19.19 [-65.6220..-39.5909] | it/evals=480/907 eff=79.0774% N=300
Z=-54.6(0.00%) | Like=-49.43..-19.19 [-65.6220..-39.5909] | it/evals=510/945 eff=79.0698% N=300
Mono-modal Volume: ~exp(-5.50) Expected Volume: exp(-1.79) Quality: ok
index : +1.0| ******************************** * | +5.0
amplitude: +1.0e-12| ****************************** ********| +1.0e-10
Z=-52.1(0.00%) | Like=-47.17..-19.19 [-65.6220..-39.5909] | it/evals=536/983 eff=78.4773% N=300
Z=-51.8(0.00%) | Like=-46.81..-19.19 [-65.6220..-39.5909] | it/evals=540/990 eff=78.2609% N=300
Z=-48.6(0.00%) | Like=-43.31..-19.19 [-65.6220..-39.5909] | it/evals=570/1031 eff=77.9754% N=300
Z=-45.9(0.00%) | Like=-40.56..-19.19 [-65.6220..-39.5909] | it/evals=597/1074 eff=77.1318% N=300
Z=-45.6(0.00%) | Like=-40.45..-19.19 [-65.6220..-39.5909] | it/evals=600/1082 eff=76.7263% N=300
Mono-modal Volume: ~exp(-5.96) * Expected Volume: exp(-2.01) Quality: ok
index : +1.0| +2.0 *************************** | +5.0
amplitude: +1.0e-12| +2.6e-11 *********************** *************| +1.0e-10
Z=-45.4(0.00%) | Like=-40.19..-19.19 [-65.6220..-39.5909] | it/evals=603/1086 eff=76.7176% N=300
Z=-43.4(0.00%) | Like=-38.58..-19.19 [-39.5675..-30.9446] | it/evals=630/1124 eff=76.4563% N=300
Z=-42.1(0.00%) | Like=-37.06..-19.19 [-39.5675..-30.9446] | it/evals=657/1166 eff=75.8661% N=300
Z=-41.9(0.00%) | Like=-36.92..-19.19 [-39.5675..-30.9446] | it/evals=660/1169 eff=75.9494% N=300
Mono-modal Volume: ~exp(-6.17) * Expected Volume: exp(-2.23) Quality: ok
index : +1.0| +2.1 ************************* +4.1 | +5.0
amplitude: +1.0e-12| +2.7e-11 ********************** ************ | +1.0e-10
Z=-41.4(0.00%) | Like=-36.39..-19.19 [-39.5675..-30.9446] | it/evals=670/1187 eff=75.5355% N=300
Z=-40.5(0.00%) | Like=-35.70..-19.19 [-39.5675..-30.9446] | it/evals=690/1211 eff=75.7409% N=300
Z=-39.4(0.00%) | Like=-34.56..-19.19 [-39.5675..-30.9446] | it/evals=720/1252 eff=75.6303% N=300
Mono-modal Volume: ~exp(-6.56) * Expected Volume: exp(-2.46) Quality: ok
index : +1.0| +2.1 ********************** +3.9 | +5.0
amplitude: +1.0e-12| +2.9e-11 ****************************** | +1.0e-10
Z=-38.8(0.00%) | Like=-33.76..-19.19 [-39.5675..-30.9446] | it/evals=737/1277 eff=75.4350% N=300
Z=-38.2(0.00%) | Like=-33.13..-19.19 [-39.5675..-30.9446] | it/evals=750/1299 eff=75.0751% N=300
Z=-37.1(0.00%) | Like=-32.21..-19.19 [-39.5675..-30.9446] | it/evals=780/1337 eff=75.2170% N=300
Mono-modal Volume: ~exp(-6.63) * Expected Volume: exp(-2.68) Quality: ok
index : +1.0| +2.2 ********************* +3.8 | +5.0
amplitude: +1.0e-12| +3.1e-11 ***************************** | +1.0e-10
Z=-36.3(0.00%) | Like=-31.26..-19.19 [-39.5675..-30.9446] | it/evals=804/1373 eff=74.9301% N=300
Z=-36.1(0.00%) | Like=-31.03..-19.19 [-39.5675..-30.9446] | it/evals=810/1384 eff=74.7232% N=300
Z=-35.1(0.00%) | Like=-29.93..-19.19 [-30.9406..-26.2755] | it/evals=839/1425 eff=74.5778% N=300
Z=-35.1(0.00%) | Like=-29.91..-19.19 [-30.9406..-26.2755] | it/evals=840/1427 eff=74.5342% N=300
Z=-34.2(0.01%) | Like=-28.98..-19.19 [-30.9406..-26.2755] | it/evals=864/1467 eff=74.0360% N=300
Z=-34.0(0.01%) | Like=-28.84..-19.19 [-30.9406..-26.2755] | it/evals=870/1474 eff=74.1056% N=300
Mono-modal Volume: ~exp(-7.08) * Expected Volume: exp(-2.90) Quality: ok
index : +1.0| +2.2 ****************** +3.7 | +5.0
amplitude: +1.0e-12| +3.4e-11 ************************* | +1.0e-10
Z=-34.0(0.01%) | Like=-28.76..-19.19 [-30.9406..-26.2755] | it/evals=871/1475 eff=74.1277% N=300
Z=-33.1(0.02%) | Like=-27.85..-19.19 [-30.9406..-26.2755] | it/evals=900/1509 eff=74.4417% N=300
Z=-32.3(0.04%) | Like=-27.31..-19.19 [-30.9406..-26.2755] | it/evals=929/1549 eff=74.3795% N=300
Z=-32.3(0.04%) | Like=-27.31..-19.19 [-30.9406..-26.2755] | it/evals=930/1550 eff=74.4000% N=300
Mono-modal Volume: ~exp(-7.08) 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=-31.8(0.06%) | Like=-26.71..-19.19 [-30.9406..-26.2755] | it/evals=952/1588 eff=73.9130% N=300
Z=-31.6(0.07%) | Like=-26.54..-19.19 [-30.9406..-26.2755] | it/evals=960/1599 eff=73.9030% N=300
Z=-31.0(0.13%) | Like=-25.92..-19.19 [-26.2469..-25.6643] | it/evals=989/1642 eff=73.6960% N=300
Z=-31.0(0.13%) | Like=-25.91..-19.19 [-26.2469..-25.6643] | it/evals=990/1643 eff=73.7156% N=300
Mono-modal Volume: ~exp(-7.14) * Expected Volume: exp(-3.35) Quality: ok
index : +1.0| +2.3 *************** +3.5 | +5.0
amplitude: +1.0e-12| +3.7e-11 ********************* +7.9e-11| +1.0e-10
Z=-30.7(0.18%) | Like=-25.55..-19.19 [-25.6103..-25.5167] | it/evals=1005/1667 eff=73.5187% N=300
Z=-30.4(0.24%) | Like=-25.09..-19.19 [-25.0868..-25.0610] | it/evals=1020/1687 eff=73.5400% N=300
Z=-29.8(0.43%) | Like=-24.65..-19.18 [-24.6484..-24.6343] | it/evals=1048/1728 eff=73.3894% N=300
Z=-29.8(0.45%) | Like=-24.62..-19.18 [-24.6225..-24.6146]*| it/evals=1050/1733 eff=73.2729% N=300
Mono-modal Volume: ~exp(-7.81) * Expected Volume: exp(-3.57) Quality: ok
index : +1.0| +2.4 ************* +3.4 | +5.0
amplitude: +1.0e-12| +3.9e-11 ****************** +7.4e-11 | +1.0e-10
Z=-29.4(0.66%) | Like=-24.24..-19.18 [-24.2440..-24.2385]*| it/evals=1072/1770 eff=72.9252% N=300
Z=-29.3(0.77%) | Like=-24.08..-19.18 [-24.1036..-24.0753] | it/evals=1080/1782 eff=72.8745% N=300
Z=-28.8(1.25%) | Like=-23.59..-19.18 [-23.5908..-23.5763] | it/evals=1110/1818 eff=73.1225% 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| +4.0e-11 ****************** +7.3e-11 | +1.0e-10
Z=-28.4(1.84%) | Like=-23.26..-19.18 [-23.2634..-23.2488] | it/evals=1139/1858 eff=73.1065% N=300
Z=-28.4(1.87%) | Like=-23.25..-19.18 [-23.2634..-23.2488] | it/evals=1140/1859 eff=73.1238% N=300
Z=-28.1(2.57%) | Like=-22.92..-19.18 [-22.9204..-22.9085] | it/evals=1168/1899 eff=73.0457% N=300
Z=-28.0(2.64%) | Like=-22.89..-19.18 [-22.8932..-22.8881]*| it/evals=1170/1901 eff=73.0793% N=300
Z=-27.8(3.51%) | Like=-22.64..-19.18 [-22.6357..-22.6320]*| it/evals=1196/1942 eff=72.8380% N=300
Z=-27.7(3.69%) | Like=-22.62..-19.18 [-22.6247..-22.6223]*| it/evals=1200/1948 eff=72.8155% 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| +4.2e-11 **************** +7.1e-11 | +1.0e-10
Z=-27.4(4.84%) | Like=-22.29..-19.16 [-22.2910..-22.2802] | it/evals=1230/1984 eff=73.0404% N=300
Z=-27.2(6.44%) | Like=-21.98..-19.16 [-22.0069..-21.9817] | it/evals=1259/2024 eff=73.0278% N=300
Z=-27.2(6.51%) | Like=-21.98..-19.16 [-21.9800..-21.9563] | it/evals=1260/2025 eff=73.0435% N=300
Mono-modal Volume: ~exp(-8.50) * Expected Volume: exp(-4.24) Quality: ok
index : +1.0| +2.5 ********** +3.2 | +5.0
amplitude: +1.0e-12| +4.3e-11 ************** +6.9e-11 | +1.0e-10
Z=-27.0(7.30%) | Like=-21.88..-19.16 [-21.8827..-21.8485] | it/evals=1273/2043 eff=73.0350% N=300
Z=-26.9(8.32%) | Like=-21.75..-19.16 [-21.7679..-21.7490] | it/evals=1290/2068 eff=72.9638% N=300
Z=-26.7(10.28%) | Like=-21.49..-19.16 [-21.4927..-21.4675] | it/evals=1317/2107 eff=72.8832% N=300
Z=-26.7(10.53%) | Like=-21.46..-19.16 [-21.4637..-21.4595]*| it/evals=1320/2118 eff=72.6073% N=300
Mono-modal Volume: ~exp(-8.50) 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=-26.5(12.57%) | Like=-21.25..-19.16 [-21.2456..-21.2439]*| it/evals=1345/2155 eff=72.5067% N=300
Z=-26.5(13.00%) | Like=-21.23..-19.16 [-21.2305..-21.2239]*| it/evals=1350/2161 eff=72.5416% N=300
Z=-26.3(15.58%) | Like=-21.10..-19.16 [-21.1034..-21.0983]*| it/evals=1378/2202 eff=72.4501% N=300
Z=-26.3(15.79%) | Like=-21.09..-19.16 [-21.0913..-21.0907]*| it/evals=1380/2204 eff=72.4790% N=300
Mono-modal Volume: ~exp(-8.84) * 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=-26.1(18.48%) | Like=-20.96..-19.16 [-20.9618..-20.9614]*| it/evals=1407/2241 eff=72.4884% N=300
Z=-26.1(18.77%) | Like=-20.96..-19.16 [-20.9559..-20.9523]*| it/evals=1410/2247 eff=72.4191% N=300
Z=-26.0(21.76%) | Like=-20.82..-19.16 [-20.8168..-20.8102]*| it/evals=1440/2286 eff=72.5076% N=300
Z=-25.8(24.74%) | Like=-20.69..-19.16 [-20.6939..-20.6914]*| it/evals=1470/2323 eff=72.6644% N=300
Mono-modal Volume: ~exp(-8.84) Expected Volume: exp(-4.91) Quality: ok
index : +1.0| +2.5 ******** +3.1 | +5.0
amplitude: +1.0e-12| +4.6e-11 *********** +6.6e-11 | +1.0e-10
Z=-25.7(27.58%) | Like=-20.54..-19.16 [-20.5519..-20.5375] | it/evals=1495/2360 eff=72.5728% N=300
Z=-25.7(28.18%) | Like=-20.52..-19.16 [-20.5211..-20.5209]*| it/evals=1500/2368 eff=72.5338% N=300
Z=-25.6(31.38%) | Like=-20.44..-19.16 [-20.4385..-20.4346]*| it/evals=1528/2408 eff=72.4858% N=300
Z=-25.6(31.64%) | Like=-20.42..-19.16 [-20.4346..-20.4189] | it/evals=1530/2410 eff=72.5118% N=300
Mono-modal Volume: ~exp(-8.84) Expected Volume: exp(-5.14) Quality: ok
index : +1.0| +2.6 ******* +3.1 | +5.0
amplitude: +1.0e-12| +4.7e-11 ********* +6.5e-11 | +1.0e-10
Z=-25.5(34.92%) | Like=-20.32..-19.16 [-20.3229..-20.3226]*| it/evals=1557/2447 eff=72.5198% N=300
Z=-25.5(35.25%) | Like=-20.31..-19.16 [-20.3198..-20.3097] | it/evals=1560/2451 eff=72.5244% N=300
Z=-25.4(38.23%) | Like=-20.22..-19.16 [-20.2171..-20.2135]*| it/evals=1585/2492 eff=72.3084% N=300
Z=-25.4(38.83%) | Like=-20.20..-19.16 [-20.1981..-20.1907]*| it/evals=1590/2503 eff=72.1743% N=300
Mono-modal Volume: ~exp(-9.02) * Expected Volume: exp(-5.36) Quality: ok
index : +1.0| +2.6 ****** +3.1 | +5.0
amplitude: +1.0e-12| +4.7e-11 ********* +6.3e-11 | +1.0e-10
Z=-25.3(41.05%) | Like=-20.15..-19.16 [-20.1462..-20.1393]*| it/evals=1608/2531 eff=72.0753% N=300
Z=-25.3(42.55%) | Like=-20.11..-19.16 [-20.1084..-20.1079]*| it/evals=1620/2546 eff=72.1282% N=300
Z=-25.2(45.96%) | Like=-20.03..-19.16 [-20.0333..-20.0324]*| it/evals=1649/2587 eff=72.1032% N=300
Z=-25.2(46.06%) | Like=-20.03..-19.16 [-20.0324..-20.0306]*| it/evals=1650/2588 eff=72.1154% N=300
Mono-modal Volume: ~exp(-9.70) * Expected Volume: exp(-5.58) 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(48.81%) | Like=-19.96..-19.16 [-19.9576..-19.9551]*| it/evals=1675/2619 eff=72.2294% N=300
Z=-25.2(49.41%) | Like=-19.95..-19.16 [-19.9457..-19.9433]*| it/evals=1680/2624 eff=72.2892% N=300
Z=-25.1(52.63%) | Like=-19.85..-19.16 [-19.8507..-19.8498]*| it/evals=1710/2661 eff=72.4269% N=300
Z=-25.0(55.82%) | Like=-19.79..-19.16 [-19.7924..-19.7896]*| it/evals=1740/2695 eff=72.6514% N=300
Mono-modal Volume: ~exp(-9.70) Expected Volume: exp(-5.81) 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(58.19%) | Like=-19.75..-19.16 [-19.7510..-19.7494]*| it/evals=1762/2732 eff=72.4507% N=300
Z=-25.0(58.96%) | Like=-19.74..-19.16 [-19.7388..-19.7369]*| it/evals=1770/2745 eff=72.3926% N=300
Z=-24.9(61.74%) | Like=-19.68..-19.16 [-19.6830..-19.6818]*| it/evals=1799/2785 eff=72.3944% N=300
Z=-24.9(61.82%) | Like=-19.68..-19.16 [-19.6818..-19.6810]*| it/evals=1800/2786 eff=72.4055% N=300
Mono-modal Volume: ~exp(-9.91) * Expected Volume: exp(-6.03) Quality: ok
index : +1.0| +2.7 ***** +3.0 | +5.0
amplitude: +1.0e-12| +4.9e-11 ******* +6.1e-11 | +1.0e-10
Z=-24.9(62.71%) | Like=-19.67..-19.16 [-19.6711..-19.6706]*| it/evals=1809/2799 eff=72.3890% N=300
Z=-24.9(64.63%) | Like=-19.63..-19.16 [-19.6346..-19.6337]*| it/evals=1830/2830 eff=72.3320% N=300
Z=-24.8(67.00%) | Like=-19.59..-19.16 [-19.5900..-19.5898]*| it/evals=1856/2870 eff=72.2179% N=300
Z=-24.8(67.35%) | Like=-19.59..-19.16 [-19.5876..-19.5868]*| it/evals=1860/2876 eff=72.2050% N=300
Mono-modal Volume: ~exp(-9.91) Expected Volume: exp(-6.25) 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(69.55%) | Like=-19.55..-19.16 [-19.5472..-19.5454]*| it/evals=1886/2913 eff=72.1776% N=300
Z=-24.8(69.88%) | Like=-19.54..-19.16 [-19.5413..-19.5398]*| it/evals=1890/2920 eff=72.1374% N=300
[ultranest] Explored until L=-2e+01
[ultranest] Likelihood function evaluations: 2921
[ultranest] logZ = -24.4 +- 0.06726
[ultranest] Effective samples strategy satisfied (ESS = 1026.8, need >400)
[ultranest] Posterior uncertainty strategy is satisfied (KL: 0.46+-0.07 nat, need <0.50 nat)
[ultranest] Evidency uncertainty strategy is satisfied (dlogz=0.27, 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 = -24.447 +- 0.319
single instance: logZ = -24.447 +- 0.118
bootstrapped : logZ = -24.402 +- 0.182
tail : logZ = +- 0.262
insert order U test : converged: True correlation: inf iterations
index : 2.25 │ ▁ ▁▁▁▁▂▃▃▃▅▅▅▇▇▇▇▆▆▆▅▄▂▄▂▂▂▁▁▁▁▁ ▁▁ ▁ │3.55 2.83 +- 0.18
amplitude : 0.0000000000339│ ▁ ▁▁▁▁▁▂▃▄▄▅▆▇▇▆▆▆▆▅▄▄▃▃▂▁▁▁▁▁▁▁ ▁ │0.0000000000806 0.0000000000550 +- 0.0000000000059
[ultranest] Sampling 300 live points from prior ...
Mono-modal Volume: ~exp(-3.98) * Expected Volume: exp(0.00) Quality: ok
index : +1.0|******************************************** ***| +5.0
amplitude: +1.0e-12|***************** ****** ********* *** ****** **| +1.0e-10
Z=-inf(0.00%) | Like=-1342.72..-13.33 [-1342.7177..-93.6927] | it/evals=0/301 eff=0.0000% N=300
Z=-149.1(0.00%) | Like=-144.54..-13.33 [-1342.7177..-93.6927] | it/evals=30/335 eff=85.7143% N=300
Z=-141.5(0.00%) | Like=-137.26..-13.33 [-1342.7177..-93.6927] | it/evals=60/370 eff=85.7143% N=300
Mono-modal Volume: ~exp(-3.98) Expected Volume: exp(-0.22) Quality: ok
index : +1.0|******************************************** ***| +5.0
amplitude: +1.0e-12|***************** ************** ***** *********| +1.0e-10
Z=-133.6(0.00%) | Like=-127.32..-13.33 [-1342.7177..-93.6927] | it/evals=90/403 eff=87.3786% N=300
Z=-123.6(0.00%) | Like=-118.53..-13.33 [-1342.7177..-93.6927] | it/evals=120/439 eff=86.3309% N=300
Mono-modal Volume: ~exp(-4.65) * Expected Volume: exp(-0.45) Quality: ok
index : +1.0| ***********************************************| +5.0
amplitude: +1.0e-12|************************************** *********| +1.0e-10
Z=-118.0(0.00%) | Like=-113.26..-13.33 [-1342.7177..-93.6927] | it/evals=134/460 eff=83.7500% N=300
Z=-115.0(0.00%) | Like=-110.53..-13.33 [-1342.7177..-93.6927] | it/evals=150/480 eff=83.3333% N=300
Z=-104.6(0.00%) | Like=-99.63..-13.33 [-1342.7177..-93.6927] | it/evals=180/515 eff=83.7209% N=300
Mono-modal Volume: ~exp(-4.68) * Expected Volume: exp(-0.67) Quality: ok
index : +1.0| *********************************************| +5.0
amplitude: +1.0e-12| ************************************* *********| +1.0e-10
Z=-98.5(0.00%) | Like=-92.88..-13.33 [-93.0462..-50.9192] | it/evals=201/542 eff=83.0579% N=300
Z=-95.9(0.00%) | Like=-90.72..-13.33 [-93.0462..-50.9192] | it/evals=210/553 eff=83.0040% N=300
Z=-89.4(0.00%) | Like=-84.35..-13.33 [-93.0462..-50.9192] | it/evals=240/591 eff=82.4742% N=300
Mono-modal Volume: ~exp(-4.68) Expected Volume: exp(-0.89) Quality: ok
index : +1.0| *********************************************| +5.0
amplitude: +1.0e-12| **********************************************| +1.0e-10
Z=-82.2(0.00%) | Like=-77.03..-13.33 [-93.0462..-50.9192] | it/evals=269/628 eff=82.0122% N=300
Z=-82.0(0.00%) | Like=-76.45..-13.33 [-93.0462..-50.9192] | it/evals=270/629 eff=82.0669% N=300
Z=-76.2(0.00%) | Like=-70.03..-13.33 [-93.0462..-50.9192] | it/evals=300/666 eff=81.9672% N=300
Z=-67.7(0.00%) | Like=-62.30..-13.33 [-93.0462..-50.9192] | it/evals=328/707 eff=80.5897% N=300
Z=-67.3(0.00%) | Like=-62.18..-13.33 [-93.0462..-50.9192] | it/evals=330/709 eff=80.6846% N=300
Mono-modal Volume: ~exp(-5.07) * Expected Volume: exp(-1.12) Quality: ok
index : +1.0| ******************************************| +5.0
amplitude: +1.0e-12| *********************************************| +1.0e-10
Z=-66.6(0.00%) | Like=-61.79..-13.33 [-93.0462..-50.9192] | it/evals=335/719 eff=79.9523% N=300
Z=-63.8(0.00%) | Like=-58.81..-13.33 [-93.0462..-50.9192] | it/evals=360/750 eff=80.0000% N=300
Z=-59.1(0.00%) | Like=-54.11..-13.33 [-93.0462..-50.9192] | it/evals=390/791 eff=79.4297% N=300
Mono-modal Volume: ~exp(-5.16) * Expected Volume: exp(-1.34) Quality: ok
index : +1.0| *****************************************| +5.0
amplitude: +1.0e-12| ********************************************| +1.0e-10
Z=-57.2(0.00%) | Like=-51.66..-13.33 [-93.0462..-50.9192] | it/evals=402/805 eff=79.6040% N=300
Z=-54.5(0.00%) | Like=-49.16..-13.33 [-50.8256..-31.8480] | it/evals=420/829 eff=79.3951% N=300
Z=-50.8(0.00%) | Like=-45.99..-13.33 [-50.8256..-31.8480] | it/evals=448/870 eff=78.5965% N=300
Z=-50.6(0.00%) | Like=-45.77..-13.33 [-50.8256..-31.8480] | it/evals=450/872 eff=78.6713% N=300
Mono-modal Volume: ~exp(-5.67) * Expected Volume: exp(-1.56) Quality: ok
index : +1.0| *************************************** | +5.0
amplitude: +1.0e-12| ******************************************| +1.0e-10
Z=-48.7(0.00%) | Like=-43.48..-13.33 [-50.8256..-31.8480] | it/evals=469/900 eff=78.1667% N=300
Z=-47.6(0.00%) | Like=-42.93..-13.33 [-50.8256..-31.8480] | it/evals=480/915 eff=78.0488% N=300
Z=-45.2(0.00%) | Like=-40.20..-13.32 [-50.8256..-31.8480] | it/evals=510/952 eff=78.2209% N=300
Mono-modal Volume: ~exp(-5.96) * Expected Volume: exp(-1.79) Quality: ok
index : +1.0| ********************************** | +5.0
amplitude: +1.0e-12| *****************************************| +1.0e-10
Z=-43.4(0.00%) | Like=-38.46..-13.32 [-50.8256..-31.8480] | it/evals=536/984 eff=78.3626% N=300
Z=-43.1(0.00%) | Like=-37.64..-13.32 [-50.8256..-31.8480] | it/evals=540/989 eff=78.3745% N=300
Z=-40.1(0.00%) | Like=-34.88..-13.32 [-50.8256..-31.8480] | it/evals=565/1031 eff=77.2914% N=300
Z=-39.7(0.00%) | Like=-34.77..-13.32 [-50.8256..-31.8480] | it/evals=570/1039 eff=77.1313% N=300
Z=-38.1(0.00%) | Like=-33.21..-13.31 [-50.8256..-31.8480] | it/evals=599/1081 eff=76.6965% N=300
Z=-38.0(0.00%) | Like=-33.17..-13.31 [-50.8256..-31.8480] | it/evals=600/1083 eff=76.6284% N=300
Mono-modal Volume: ~exp(-5.96) Expected Volume: exp(-2.01) Quality: ok
index : +1.0| ***************************** | +5.0
amplitude: +1.0e-12| ****************************************| +1.0e-10
Z=-36.4(0.00%) | Like=-31.50..-13.31 [-31.8182..-22.3331] | it/evals=629/1120 eff=76.7073% N=300
Z=-36.3(0.00%) | Like=-31.48..-13.31 [-31.8182..-22.3331] | it/evals=630/1121 eff=76.7357% N=300
Z=-35.0(0.00%) | Like=-30.07..-13.31 [-31.8182..-22.3331] | it/evals=655/1164 eff=75.8102% N=300
Z=-34.8(0.00%) | Like=-29.79..-13.31 [-31.8182..-22.3331] | it/evals=660/1169 eff=75.9494% N=300
Mono-modal Volume: ~exp(-6.08) * Expected Volume: exp(-2.23) Quality: ok
index : +1.0| +2.0 ************************* +3.9 | +5.0
amplitude: +1.0e-12| ***************************************| +1.0e-10
Z=-34.2(0.00%) | Like=-29.25..-13.31 [-31.8182..-22.3331] | it/evals=670/1181 eff=76.0499% N=300
Z=-33.2(0.00%) | Like=-28.31..-13.31 [-31.8182..-22.3331] | it/evals=690/1205 eff=76.2431% N=300
Z=-31.8(0.00%) | Like=-26.67..-13.31 [-31.8182..-22.3331] | it/evals=720/1245 eff=76.1905% N=300
Mono-modal Volume: ~exp(-6.56) * Expected Volume: exp(-2.46) Quality: ok
index : +1.0| +2.1 ********************** +3.8 | +5.0
amplitude: +1.0e-12| **************************************| +1.0e-10
Z=-30.9(0.00%) | Like=-25.83..-13.31 [-31.8182..-22.3331] | it/evals=737/1270 eff=75.9794% N=300
Z=-30.3(0.00%) | Like=-25.17..-13.31 [-31.8182..-22.3331] | it/evals=750/1289 eff=75.8342% N=300
Z=-29.1(0.00%) | Like=-23.99..-13.31 [-31.8182..-22.3331] | it/evals=780/1325 eff=76.0976% N=300
Mono-modal Volume: ~exp(-6.60) * Expected Volume: exp(-2.68) Quality: ok
index : +1.0| +2.1 ******************* +3.6 | +5.0
amplitude: +1.0e-12| +2.6e-11 ********************************** | +1.0e-10
Z=-28.2(0.00%) | Like=-23.26..-13.31 [-31.8182..-22.3331] | it/evals=804/1365 eff=75.4930% N=300
Z=-28.0(0.01%) | Like=-23.12..-13.31 [-31.8182..-22.3331] | it/evals=810/1371 eff=75.6303% N=300
Z=-27.1(0.01%) | Like=-22.08..-13.31 [-22.3185..-19.9673] | it/evals=838/1412 eff=75.3597% N=300
Z=-27.0(0.01%) | Like=-22.02..-13.31 [-22.3185..-19.9673] | it/evals=840/1414 eff=75.4039% N=300
Z=-26.3(0.03%) | Like=-21.32..-13.31 [-22.3185..-19.9673] | it/evals=869/1455 eff=75.2381% N=300
Z=-26.2(0.04%) | Like=-21.32..-13.31 [-22.3185..-19.9673] | it/evals=870/1456 eff=75.2595% N=300
Mono-modal Volume: ~exp(-6.83) * Expected Volume: exp(-2.90) Quality: ok
index : +1.0| +2.2 ****************** +3.5 | +5.0
amplitude: +1.0e-12| +2.8e-11 ****************************** | +1.0e-10
Z=-26.2(0.04%) | Like=-21.31..-13.31 [-22.3185..-19.9673] | it/evals=871/1457 eff=75.2809% N=300
Z=-25.5(0.07%) | Like=-20.53..-13.31 [-22.3185..-19.9673] | it/evals=899/1498 eff=75.0417% N=300
Z=-25.5(0.07%) | Like=-20.51..-13.31 [-22.3185..-19.9673] | it/evals=900/1499 eff=75.0626% N=300
Z=-24.8(0.13%) | Like=-19.74..-13.31 [-19.9327..-19.5706] | it/evals=930/1541 eff=74.9396% N=300
Mono-modal Volume: ~exp(-7.43) * Expected Volume: exp(-3.13) Quality: ok
index : +1.0| +2.2 *************** +3.4 | +5.0
amplitude: +1.0e-12| +3.0e-11 *************************** | +1.0e-10
Z=-24.6(0.15%) | Like=-19.45..-13.31 [-19.5111..-19.4546] | it/evals=938/1550 eff=75.0400% N=300
Z=-24.1(0.26%) | Like=-19.02..-13.31 [-19.0323..-19.0187] | it/evals=960/1579 eff=75.0586% N=300
Z=-23.5(0.48%) | Like=-18.33..-13.31 [-18.3307..-18.3272]*| it/evals=990/1615 eff=75.2852% N=300
Mono-modal Volume: ~exp(-7.43) Expected Volume: exp(-3.35) Quality: ok
index : +1.0| +2.3 ************* +3.3 | +5.0
amplitude: +1.0e-12| +3.3e-11 ************************ | +1.0e-10
Z=-23.0(0.78%) | Like=-18.10..-13.31 [-18.1011..-18.0979]*| it/evals=1017/1652 eff=75.2219% N=300
Z=-23.0(0.81%) | Like=-18.08..-13.31 [-18.0764..-18.0729]*| it/evals=1020/1656 eff=75.2212% N=300
Z=-22.5(1.30%) | Like=-17.49..-13.31 [-17.5460..-17.4922] | it/evals=1050/1693 eff=75.3769% N=300
Mono-modal Volume: ~exp(-7.78) * Expected Volume: exp(-3.57) Quality: ok
index : +1.0| +2.3 ************* +3.3 | +5.0
amplitude: +1.0e-12| +3.5e-11 ********************* +7.7e-11 | +1.0e-10
Z=-22.2(1.78%) | Like=-17.18..-13.31 [-17.1782..-17.1779]*| it/evals=1072/1723 eff=75.3338% N=300
Z=-22.1(1.99%) | Like=-17.11..-13.31 [-17.1137..-17.1116]*| it/evals=1080/1737 eff=75.1566% N=300
Z=-21.8(2.77%) | Like=-16.76..-13.31 [-16.7948..-16.7640] | it/evals=1107/1779 eff=74.8479% N=300
Z=-21.7(2.88%) | Like=-16.75..-13.31 [-16.7545..-16.7489]*| it/evals=1110/1785 eff=74.7475% N=300
Mono-modal Volume: ~exp(-7.78) Expected Volume: exp(-3.80) Quality: ok
index : +1.0| +2.4 *********** +3.2 | +5.0
amplitude: +1.0e-12| +3.6e-11 ******************** +7.5e-11 | +1.0e-10
Z=-21.4(3.92%) | Like=-16.47..-13.31 [-16.4708..-16.4660]*| it/evals=1140/1821 eff=74.9507% N=300
Z=-21.1(5.22%) | Like=-16.21..-13.31 [-16.2141..-16.2139]*| it/evals=1167/1861 eff=74.7598% N=300
Z=-21.1(5.36%) | Like=-16.18..-13.31 [-16.1810..-16.1756]*| it/evals=1170/1864 eff=74.8082% N=300
Z=-20.8(6.96%) | Like=-15.86..-13.31 [-15.8637..-15.8399] | it/evals=1200/1902 eff=74.9064% N=300
Mono-modal Volume: ~exp(-7.78) Expected Volume: exp(-4.02) Quality: ok
index : +1.0| +2.4 ********** +3.1 | +5.0
amplitude: +1.0e-12| +3.9e-11 ***************** +7.2e-11 | +1.0e-10
Z=-20.6(8.53%) | Like=-15.67..-13.31 [-15.6665..-15.6591]*| it/evals=1223/1940 eff=74.5732% N=300
Z=-20.6(9.08%) | Like=-15.59..-13.31 [-15.6029..-15.5920] | it/evals=1230/1949 eff=74.5907% N=300
Z=-20.4(11.31%) | Like=-15.35..-13.31 [-15.3495..-15.3461]*| it/evals=1257/1989 eff=74.4227% N=300
Z=-20.3(11.54%) | Like=-15.32..-13.31 [-15.3361..-15.3221] | it/evals=1260/1994 eff=74.3802% N=300
Mono-modal Volume: ~exp(-8.16) * Expected Volume: exp(-4.24) Quality: ok
index : +1.0| +2.4 ******** +3.1 | +5.0
amplitude: +1.0e-12| +3.9e-11 *************** +6.9e-11 | +1.0e-10
Z=-20.2(12.74%) | Like=-15.23..-13.31 [-15.2341..-15.2319]*| it/evals=1273/2013 eff=74.3141% N=300
Z=-20.1(14.34%) | Like=-15.16..-13.31 [-15.1749..-15.1594] | it/evals=1290/2038 eff=74.2232% N=300
Z=-20.0(17.30%) | Like=-15.00..-13.31 [-14.9987..-14.9959]*| it/evals=1317/2079 eff=74.0304% N=300
Z=-19.9(17.65%) | Like=-14.98..-13.31 [-14.9798..-14.9771]*| it/evals=1320/2082 eff=74.0741% N=300
Mono-modal Volume: ~exp(-8.80) * Expected Volume: exp(-4.47) Quality: ok
index : +1.0| +2.4 ******** +3.1 | +5.0
amplitude: +1.0e-12| +4.1e-11 ************** +6.8e-11 | +1.0e-10
Z=-19.8(19.64%) | Like=-14.90..-13.31 [-14.9023..-14.8967]*| it/evals=1340/2118 eff=73.7074% N=300
Z=-19.8(20.77%) | Like=-14.86..-13.31 [-14.8608..-14.8509]*| it/evals=1350/2133 eff=73.6498% N=300
Z=-19.6(23.75%) | Like=-14.71..-13.31 [-14.7123..-14.7094]*| it/evals=1379/2172 eff=73.6645% N=300
Z=-19.6(23.89%) | Like=-14.71..-13.31 [-14.7094..-14.6966] | it/evals=1380/2173 eff=73.6786% N=300
Mono-modal Volume: ~exp(-8.80) Expected Volume: exp(-4.69) Quality: ok
index : +1.0| +2.5 ******** +3.0 | +5.0
amplitude: +1.0e-12| +4.2e-11 ************* +6.6e-11 | +1.0e-10
Z=-19.5(27.11%) | Like=-14.58..-13.31 [-14.5783..-14.5779]*| it/evals=1407/2210 eff=73.6649% N=300
Z=-19.5(27.53%) | Like=-14.58..-13.31 [-14.5763..-14.5675]*| it/evals=1410/2213 eff=73.7062% N=300
Z=-19.4(31.29%) | Like=-14.47..-13.31 [-14.4664..-14.4631]*| it/evals=1440/2250 eff=73.8462% N=300
Z=-19.3(34.72%) | Like=-14.35..-13.31 [-14.3533..-14.3494]*| it/evals=1470/2284 eff=74.0927% N=300
Mono-modal Volume: ~exp(-8.80) Expected Volume: exp(-4.91) Quality: ok
index : +1.0| +2.5 ****** +3.0 | +5.0
amplitude: +1.0e-12| +4.3e-11 *********** +6.5e-11 | +1.0e-10
Z=-19.2(36.96%) | Like=-14.27..-13.31 [-14.2650..-14.2609]*| it/evals=1491/2322 eff=73.7389% N=300
Z=-19.2(38.08%) | Like=-14.23..-13.31 [-14.2293..-14.2277]*| it/evals=1500/2340 eff=73.5294% N=300
Z=-19.1(40.65%) | Like=-14.16..-13.31 [-14.1631..-14.1543]*| it/evals=1520/2382 eff=73.0067% N=300
Z=-19.1(41.84%) | Like=-14.13..-13.31 [-14.1269..-14.1240]*| it/evals=1530/2397 eff=72.9614% N=300
Mono-modal Volume: ~exp(-9.63) * Expected Volume: exp(-5.14) Quality: ok
index : +1.0| +2.5 ****** +3.0 | +5.0
amplitude: +1.0e-12| +4.4e-11 *********** +6.3e-11 | +1.0e-10
Z=-19.1(43.17%) | Like=-14.09..-13.31 [-14.0880..-14.0879]*| it/evals=1541/2412 eff=72.9640% N=300
Z=-19.0(45.49%) | Like=-14.02..-13.31 [-14.0206..-14.0191]*| it/evals=1560/2435 eff=73.0679% N=300
Z=-18.9(48.63%) | Like=-13.96..-13.30 [-13.9612..-13.9606]*| it/evals=1587/2476 eff=72.9320% N=300
Z=-18.9(48.95%) | Like=-13.96..-13.30 [-13.9566..-13.9554]*| it/evals=1590/2480 eff=72.9358% N=300
Mono-modal Volume: ~exp(-9.63) 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=-18.9(52.08%) | Like=-13.90..-13.30 [-13.9028..-13.9021]*| it/evals=1616/2518 eff=72.8584% N=300
Z=-18.9(52.50%) | Like=-13.89..-13.30 [-13.8899..-13.8896]*| it/evals=1620/2524 eff=72.8417% N=300
Z=-18.8(55.44%) | Like=-13.84..-13.30 [-13.8403..-13.8362]*| it/evals=1646/2565 eff=72.6711% N=300
Z=-18.8(55.90%) | Like=-13.83..-13.30 [-13.8255..-13.8241]*| it/evals=1650/2571 eff=72.6552% N=300
Mono-modal Volume: ~exp(-9.93) * 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=-18.8(58.53%) | Like=-13.78..-13.30 [-13.7779..-13.7775]*| it/evals=1675/2611 eff=72.4794% N=300
Z=-18.7(59.04%) | Like=-13.77..-13.30 [-13.7727..-13.7714]*| it/evals=1680/2619 eff=72.4450% N=300
Z=-18.7(62.11%) | Like=-13.72..-13.30 [-13.7201..-13.7166]*| it/evals=1709/2659 eff=72.4460% N=300
Z=-18.7(62.19%) | Like=-13.72..-13.30 [-13.7166..-13.7143]*| it/evals=1710/2660 eff=72.4576% N=300
Z=-18.6(65.07%) | Like=-13.67..-13.30 [-13.6694..-13.6673]*| it/evals=1739/2700 eff=72.4583% N=300
Z=-18.6(65.17%) | Like=-13.67..-13.30 [-13.6673..-13.6671]*| it/evals=1740/2701 eff=72.4698% N=300
Mono-modal Volume: ~exp(-9.93) 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=-18.6(67.83%) | Like=-13.62..-13.30 [-13.6227..-13.6223]*| it/evals=1769/2737 eff=72.5892% N=300
Z=-18.6(67.91%) | Like=-13.62..-13.30 [-13.6223..-13.6193]*| it/evals=1770/2738 eff=72.6005% N=300
[ultranest] Explored until L=-1e+01
[ultranest] Likelihood function evaluations: 2768
[ultranest] logZ = -18.23 +- 0.1044
[ultranest] Effective samples strategy satisfied (ESS = 997.1, 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: 0.11 bs:0.10 tail:0.26 total:0.28 required:<0.50
[ultranest] done iterating.
logZ = -18.218 +- 0.319
single instance: logZ = -18.218 +- 0.115
bootstrapped : logZ = -18.227 +- 0.182
tail : logZ = +- 0.262
insert order U test : converged: True correlation: inf iterations
index : 2.10 │ ▁ ▁▁▁▁▁▁▂▂▃▄▆▆▇▆▇▅▅▄▄▂▂▂▁▁▁▁▁▁▁ ▁ │3.64 2.76 +- 0.17
amplitude : 0.0000000000290│ ▁▁▁▁▁▁▁▃▃▃▄▆▇▇▇▇▇▅▅▅▅▃▂▂▂▂▁▁▁▁▁▁▁▁▁▁▁ │0.0000000000864 0.0000000000536 +- 0.0000000000075
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 45.534 seconds)
