Note
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Bayesian analysis with nested sampling#
A demonstration of a Bayesian analysis using the nested sampling technique.
Context#
1. Bayesian analysis#
Bayesian inference uses prior knowledge, in the form of a prior distribution, in order to estimate posterior probabilities which we traditionally visualise in the form of corner plots. These distributions contain more information than a maximum likelihood fit as they reveal not only the “best model” but provide a more accurate representation of errors and correlation between parameters. In particular, non-Gaussian degeneracies are complex to estimate with a maximum likelihood approach.
2. Limitations of the Markov Chain Monte Carlo approach#
A well-known approach to estimate this posterior distribution is the Markov Chain Monte Carlo (MCMC). This uses an ensemble of walkers to produce a chain of samples that after a convergence period will reach a stationary state. Once convergence is reached, the successive elements of the chain are samples of the target posterior distribution. However, the weakness of the MCMC approach lies in the “Once convergence” part. If the walkers are started far from the best likelihood region, the convergence time can be long or never reached if the walkers fall in a local minima. The choice of the initialisation point can become critical for complex models with a high number of dimensions and the ability of these walkers to escape a local minimum or to accurately describe a complex likelihood space is not guaranteed.
3. Nested sampling approach#
To overcome these issues, the nested sampling (NS) algorithm has gained traction in physics and astronomy. It is a Monte Carlo algorithm for computing an integral of the likelihood function over the prior model parameter space introduced in Skilling, 2004. The method performs this integral by evolving a collection of points through the parameter space (see recent reviews from Ashton et al., 2022, and Buchner, 2023). Without going into too many details, one important specificity of the NS method is that it starts from the entire parameter space and evolves a collection of live points to map all minima (including multiple modes if any), whereas Markov Chain Monte Carlo methods require an initialisation point and the walkers will explore the local likelihood. The ability of these walkers to escape a local minimum or to accurately describe a complex likelihood space is not guaranteed. This is a fundamental difference with MCMC or Minuit which will only ever probe the vicinity along their minimisation paths and do not have an overview of the global likelihood landscape. The analysis using the NS framework is more CPU time consuming than a standard classical fit, but it provides the full posterior distribution for all parameters, which is out of reach with traditional fitting techniques (N*(N-1)/2 contour plots to generate). In addition, it is more robust to the choice of initialisation, requires less human intervention and is therefore readily integrated in pipeline analysis. In Gammapy, we used the NS implementation of the UltraNest package (see here for more information), one of the leading package in Astronomy (already used in Cosmology and in X-rays). For a nice visualisation of the NS method see here : sampling visualisation. And for a tutorial of UltraNest applied to X-ray fitting with concrete examples and questions see : BXA Tutorial.
Note: please cite UltraNest if used for a paper
If you are using the “UltraNest” library for a paper, please follow its citation scheme: Cite UltraNest.
Proposed approach#
In this example, we will perform a Bayesian analysis with multiple 1D spectra of the Crab nebula data and investigate their posterior distributions.
Setup#
As usual, we’ll start with some setup …
import matplotlib.pyplot as plt
import numpy as np
import astropy.units as u
from gammapy.datasets import Datasets
from gammapy.datasets import SpectrumDatasetOnOff
from gammapy.modeling.models import (
SkyModel,
UniformPrior,
LogUniformPrior,
)
from gammapy.modeling.sampler import Sampler
Loading the spectral datasets#
Here we will load a few Crab 1D spectral data for which we will do a fit.
Model definition#
Now we want to define the spectral model that will be fitted to the data. The Crab spectra will be fitted here with a simple powerlaw for simplicity.
model = SkyModel.create(spectral_model="pl", name="crab")
Warning
Priors definition: Unlike a traditional fit where priors on the parameters are optional, here it is inherent to the Bayesian approach and are therefore mandatory.
In this case we will set (min,max) prior that will define the
boundaries in which the sampling will be performed.
Note that it is usually recommended to use a LogUniformPrior for
the parameters that have a large amplitude range like the
amplitude parameter.
A UniformPrior means that the samples will be drawn with uniform
probability between the (min,max) values in the linear or log space
in the case of a LogUniformPrior.
model.spectral_model.amplitude.prior = LogUniformPrior(min=1e-12, max=1e-10)
model.spectral_model.index.prior = UniformPrior(min=1, max=5)
datasets.models = [model]
print(datasets.models)
DatasetModels
Component 0: SkyModel
Name : crab
Datasets names : None
Spectral model type : PowerLawSpectralModel
Spatial model type :
Temporal model type :
Parameters:
index : 2.000 +/- 0.00
amplitude : 1.00e-12 +/- 0.0e+00 1 / (TeV s cm2)
reference (frozen): 1.000 TeV
Defining the sampler and options#
As for the Fit object, the Sampler object can receive
different backend (although just one is available for now).
The Sampler comes with “reasonable” default parameters, but you can
change them via the sampler_opts dictionary.
Here is a short description of the most relevant parameters that you
could change :
live_points: minimum number of live points throughout the run. More points allow to discover multiple peaks if existing, but is slower. To test the Prior boundaries and for debugging, a lower number (~100) can be used before a production run with more points (~400 or more).frac_remain: the cut-off condition for the integration, set by the maximum allowed fraction of posterior mass left in the live points vs the dead points. High values (e.g., 0.5) are faster and can be used if the posterior distribution is a relatively simple shape. A low value (1e-1, 1e-2) is optimal for finding peaks, but slower.log_dir: directory where the output files will be stored. If set to None, no files will be written. If set to a string, a directory will be created containing the ongoing status of the run and final results. For time consuming analysis, it is highly recommended to use that option to monitor the run and restart it in case of a crash (withresume=True).
Important note: unlike the MCMC method, you don’t need to define the number of steps for which the sampler will run. The algorithm will automatically stop once a convergence criteria has been reached.
sampler_opts = {
"live_points": 300,
"frac_remain": 0.3,
"log_dir": None,
}
sampler = Sampler(backend="ultranest", sampler_opts=sampler_opts)
Next we can run the sampler on a given dataset. No options are accepted in the run method.
[ultranest] Sampling 300 live points from prior ...
Mono-modal Volume: ~exp(-4.27) * Expected Volume: exp(0.00) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|***************** ************************ *****| +1.0e-10
Z=-inf(0.00%) | Like=-2343.14..-59.77 [-2343.1360..-343.3725] | it/evals=0/301 eff=0.0000% N=300
Z=-560.8(0.00%) | Like=-554.68..-59.77 [-2343.1360..-343.3725] | it/evals=19/323 eff=82.6087% N=300
Z=-540.0(0.00%) | Like=-534.72..-59.77 [-2343.1360..-343.3725] | it/evals=30/334 eff=88.2353% N=300
Z=-519.2(0.00%) | Like=-513.36..-59.77 [-2343.1360..-343.3725] | it/evals=50/356 eff=89.2857% N=300
Z=-506.0(0.00%) | Like=-498.11..-59.77 [-2343.1360..-343.3725] | it/evals=60/366 eff=90.9091% N=300
Mono-modal Volume: ~exp(-4.27) Expected Volume: exp(-0.22) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|*********************************************** | +1.0e-10
Z=-479.0(0.00%) | Like=-470.97..-59.77 [-2343.1360..-343.3725] | it/evals=77/386 eff=89.5349% N=300
Z=-460.2(0.00%) | Like=-454.03..-59.77 [-2343.1360..-343.3725] | it/evals=90/400 eff=90.0000% N=300
Z=-440.0(0.00%) | Like=-431.15..-59.77 [-2343.1360..-343.3725] | it/evals=108/422 eff=88.5246% N=300
Z=-425.6(0.00%) | Like=-419.42..-59.77 [-2343.1360..-343.3725] | it/evals=120/436 eff=88.2353% N=300
Mono-modal Volume: ~exp(-4.57) * Expected Volume: exp(-0.45) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|*********************************************** | +1.0e-10
Z=-412.8(0.00%) | Like=-407.21..-59.77 [-2343.1360..-343.3725] | it/evals=134/452 eff=88.1579% N=300
Z=-389.2(0.00%) | Like=-381.85..-59.77 [-2343.1360..-343.3725] | it/evals=150/474 eff=86.2069% N=300
Z=-369.9(0.00%) | Like=-362.41..-59.77 [-2343.1360..-343.3725] | it/evals=170/496 eff=86.7347% N=300
Z=-360.7(0.00%) | Like=-354.69..-59.77 [-2343.1360..-343.3725] | it/evals=180/506 eff=87.3786% N=300
Z=-350.2(0.00%) | Like=-344.40..-59.77 [-2343.1360..-343.3725] | it/evals=194/528 eff=85.0877% N=300
Mono-modal Volume: ~exp(-4.57) Expected Volume: exp(-0.67) Quality: ok
index : +1.0| * ********************************************| +5.0
amplitude: +1.0e-12| ********************************************** | +1.0e-10
Z=-335.9(0.00%) | Like=-329.35..-59.77 [-343.2535..-187.0530] | it/evals=209/548 eff=84.2742% N=300
Z=-335.1(0.00%) | Like=-328.06..-59.77 [-343.2535..-187.0530] | it/evals=210/549 eff=84.3373% N=300
Z=-315.0(0.00%) | Like=-308.27..-59.77 [-343.2535..-187.0530] | it/evals=228/571 eff=84.1328% N=300
Z=-304.0(0.00%) | Like=-297.80..-59.77 [-343.2535..-187.0530] | it/evals=240/586 eff=83.9161% N=300
Z=-279.8(0.00%) | Like=-270.67..-59.77 [-343.2535..-187.0530] | it/evals=255/608 eff=82.7922% N=300
Mono-modal Volume: ~exp(-4.57) Expected Volume: exp(-0.89) Quality: ok
index : +1.0| *******************************************| +5.0
amplitude: +1.0e-12| ******************************** *************| +1.0e-10
Z=-268.7(0.00%) | Like=-261.71..-59.77 [-343.2535..-187.0530] | it/evals=269/628 eff=82.0122% N=300
Z=-267.8(0.00%) | Like=-261.43..-59.77 [-343.2535..-187.0530] | it/evals=270/629 eff=82.0669% N=300
Z=-258.3(0.00%) | Like=-251.99..-59.77 [-343.2535..-187.0530] | it/evals=287/651 eff=81.7664% N=300
Z=-249.9(0.00%) | Like=-241.29..-58.76 [-343.2535..-187.0530] | it/evals=299/674 eff=79.9465% N=300
Z=-247.8(0.00%) | Like=-241.04..-58.76 [-343.2535..-187.0530] | it/evals=300/675 eff=80.0000% N=300
Z=-230.8(0.00%) | Like=-224.61..-58.76 [-343.2535..-187.0530] | it/evals=318/697 eff=80.1008% N=300
Z=-224.2(0.00%) | Like=-217.60..-58.76 [-343.2535..-187.0530] | it/evals=330/717 eff=79.1367% N=300
Mono-modal Volume: ~exp(-5.16) * Expected Volume: exp(-1.12) Quality: ok
index : +1.0| ******************************************| +5.0
amplitude: +1.0e-12| ******************************* *************| +1.0e-10
Z=-220.9(0.00%) | Like=-214.74..-58.76 [-343.2535..-187.0530] | it/evals=335/722 eff=79.3839% N=300
Z=-213.0(0.00%) | Like=-207.57..-58.76 [-343.2535..-187.0530] | it/evals=354/744 eff=79.7297% N=300
Z=-211.8(0.00%) | Like=-206.53..-58.76 [-343.2535..-187.0530] | it/evals=360/750 eff=80.0000% N=300
Z=-206.1(0.00%) | Like=-200.46..-58.76 [-343.2535..-187.0530] | it/evals=375/772 eff=79.4492% N=300
Z=-198.2(0.00%) | Like=-190.83..-58.76 [-343.2535..-187.0530] | it/evals=390/793 eff=79.1075% 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=-192.7(0.00%) | Like=-187.11..-58.76 [-343.2535..-187.0530] | it/evals=403/814 eff=78.4047% N=300
Z=-187.3(0.00%) | Like=-181.45..-58.75 [-186.9641..-125.9802] | it/evals=420/836 eff=78.3582% N=300
Z=-183.5(0.00%) | Like=-177.31..-58.75 [-186.9641..-125.9802] | it/evals=433/858 eff=77.5986% N=300
Z=-179.8(0.00%) | Like=-173.87..-58.75 [-186.9641..-125.9802] | it/evals=449/880 eff=77.4138% N=300
Z=-179.6(0.00%) | Like=-173.62..-58.75 [-186.9641..-125.9802] | it/evals=450/882 eff=77.3196% N=300
Z=-174.1(0.00%) | Like=-167.78..-58.75 [-186.9641..-125.9802] | it/evals=467/904 eff=77.3179% N=300
Mono-modal Volume: ~exp(-5.27) * Expected Volume: exp(-1.56) Quality: ok
index : +1.0| *********************************** | +5.0
amplitude: +1.0e-12| **************************************** | +1.0e-10
Z=-173.5(0.00%) | Like=-167.15..-58.75 [-186.9641..-125.9802] | it/evals=469/907 eff=77.2652% N=300
Z=-169.7(0.00%) | Like=-163.49..-58.75 [-186.9641..-125.9802] | it/evals=479/929 eff=76.1526% N=300
Z=-169.4(0.00%) | Like=-163.47..-58.75 [-186.9641..-125.9802] | it/evals=480/930 eff=76.1905% N=300
Z=-164.1(0.00%) | Like=-158.38..-58.75 [-186.9641..-125.9802] | it/evals=497/952 eff=76.2270% N=300
Z=-160.0(0.00%) | Like=-153.04..-58.75 [-186.9641..-125.9802] | it/evals=510/969 eff=76.2332% N=300
Z=-155.4(0.00%) | Like=-148.57..-58.75 [-186.9641..-125.9802] | it/evals=526/993 eff=75.9019% N=300
Mono-modal Volume: ~exp(-5.86) * Expected Volume: exp(-1.79) Quality: ok
index : +1.0| ******************************* | +5.0
amplitude: +1.0e-12| ************************************ ** | +1.0e-10
Z=-153.4(0.00%) | Like=-147.94..-58.75 [-186.9641..-125.9802] | it/evals=536/1014 eff=75.0700% N=300
Z=-152.9(0.00%) | Like=-147.29..-58.75 [-186.9641..-125.9802] | it/evals=540/1021 eff=74.8960% N=300
Z=-147.7(0.00%) | Like=-141.07..-58.75 [-186.9641..-125.9802] | it/evals=557/1045 eff=74.7651% N=300
Z=-142.7(0.00%) | Like=-136.00..-58.75 [-186.9641..-125.9802] | it/evals=570/1060 eff=75.0000% N=300
Z=-137.6(0.00%) | Like=-131.10..-58.75 [-186.9641..-125.9802] | it/evals=586/1082 eff=74.9361% N=300
Z=-133.5(0.00%) | Like=-127.51..-58.75 [-186.9641..-125.9802] | it/evals=600/1099 eff=75.0939% N=300
Mono-modal Volume: ~exp(-6.29) * Expected Volume: exp(-2.01) Quality: ok
index : +1.0| ************************** +4.0 | +5.0
amplitude: +1.0e-12| ********************************** | +1.0e-10
Z=-133.0(0.00%) | Like=-127.21..-58.75 [-186.9641..-125.9802] | it/evals=603/1102 eff=75.1870% N=300
Z=-130.0(0.00%) | Like=-123.42..-58.75 [-125.9703..-95.3216] | it/evals=620/1124 eff=75.2427% N=300
Z=-127.8(0.00%) | Like=-121.61..-58.75 [-125.9703..-95.3216] | it/evals=630/1137 eff=75.2688% N=300
Z=-125.9(0.00%) | Like=-120.05..-58.75 [-125.9703..-95.3216] | it/evals=643/1159 eff=74.8545% N=300
Z=-122.4(0.00%) | Like=-116.41..-58.75 [-125.9703..-95.3216] | it/evals=659/1183 eff=74.6319% N=300
Z=-122.3(0.00%) | Like=-115.84..-58.75 [-125.9703..-95.3216] | it/evals=660/1185 eff=74.5763% N=300
Mono-modal Volume: ~exp(-6.29) Expected Volume: exp(-2.23) Quality: ok
index : +1.0| +1.9 *********************** +3.8 | +5.0
amplitude: +1.0e-12| ******************************** | +1.0e-10
Z=-119.9(0.00%) | Like=-113.30..-58.75 [-125.9703..-95.3216] | it/evals=674/1206 eff=74.3929% N=300
Z=-117.4(0.00%) | Like=-111.21..-58.75 [-125.9703..-95.3216] | it/evals=689/1228 eff=74.2457% N=300
Z=-117.2(0.00%) | Like=-111.19..-58.75 [-125.9703..-95.3216] | it/evals=690/1229 eff=74.2734% N=300
Z=-113.8(0.00%) | Like=-107.48..-58.75 [-125.9703..-95.3216] | it/evals=707/1251 eff=74.3428% N=300
Z=-112.0(0.00%) | Like=-105.82..-58.75 [-125.9703..-95.3216] | it/evals=720/1271 eff=74.1504% N=300
Z=-110.0(0.00%) | Like=-104.04..-58.75 [-125.9703..-95.3216] | it/evals=734/1293 eff=73.9174% N=300
Mono-modal Volume: ~exp(-6.50) * Expected Volume: exp(-2.46) Quality: ok
index : +1.0| +2.0 ******************** +3.6 | +5.0
amplitude: +1.0e-12| **************************** +7.9e-11| +1.0e-10
Z=-109.7(0.00%) | Like=-103.63..-58.75 [-125.9703..-95.3216] | it/evals=737/1298 eff=73.8477% N=300
Z=-108.3(0.00%) | Like=-102.37..-58.75 [-125.9703..-95.3216] | it/evals=750/1316 eff=73.8189% N=300
Z=-106.8(0.00%) | Like=-100.50..-58.75 [-125.9703..-95.3216] | it/evals=768/1337 eff=74.0598% N=300
Z=-105.5(0.00%) | Like=-99.28..-58.75 [-125.9703..-95.3216] | it/evals=780/1351 eff=74.2150% N=300
Z=-103.1(0.00%) | Like=-97.03..-58.75 [-125.9703..-95.3216] | it/evals=796/1373 eff=74.1845% N=300
Mono-modal Volume: ~exp(-6.83) * Expected Volume: exp(-2.68) Quality: ok
index : +1.0| +2.1 ****************** +3.5 | +5.0
amplitude: +1.0e-12| *************************** +7.5e-11 | +1.0e-10
Z=-102.2(0.00%) | Like=-95.85..-58.75 [-125.9703..-95.3216] | it/evals=804/1385 eff=74.1014% N=300
Z=-101.5(0.00%) | Like=-95.38..-58.75 [-125.9703..-95.3216] | it/evals=810/1394 eff=74.0402% N=300
Z=-100.0(0.00%) | Like=-94.12..-58.75 [-95.2745..-75.4977] | it/evals=827/1416 eff=74.1039% N=300
Z=-98.8(0.00%) | Like=-92.50..-58.75 [-95.2745..-75.4977] | it/evals=840/1435 eff=74.0088% N=300
Z=-96.3(0.00%) | Like=-89.88..-58.75 [-95.2745..-75.4977] | it/evals=856/1457 eff=73.9844% N=300
Z=-94.9(0.00%) | Like=-88.50..-58.75 [-95.2745..-75.4977] | it/evals=870/1476 eff=73.9796% N=300
Mono-modal Volume: ~exp(-6.92) * Expected Volume: exp(-2.90) Quality: ok
index : +1.0| +2.1 ***************** +3.4 | +5.0
amplitude: +1.0e-12| +2.5e-11 *********************** +7.0e-11 | +1.0e-10
Z=-94.8(0.00%) | Like=-88.43..-58.75 [-95.2745..-75.4977] | it/evals=871/1477 eff=74.0017% N=300
Z=-93.5(0.00%) | Like=-87.19..-58.75 [-95.2745..-75.4977] | it/evals=886/1499 eff=73.8949% N=300
Z=-91.9(0.00%) | Like=-85.10..-58.75 [-95.2745..-75.4977] | it/evals=900/1519 eff=73.8310% N=300
Z=-90.3(0.00%) | Like=-83.95..-58.75 [-95.2745..-75.4977] | it/evals=915/1542 eff=73.6715% N=300
Z=-88.9(0.00%) | Like=-82.60..-58.75 [-95.2745..-75.4977] | it/evals=930/1561 eff=73.7510% N=300
Mono-modal Volume: ~exp(-7.44) * Expected Volume: exp(-3.13) Quality: ok
index : +1.0| +2.2 ************** +3.3 | +5.0
amplitude: +1.0e-12| +2.7e-11 ********************* +6.7e-11 | +1.0e-10
Z=-88.2(0.00%) | Like=-81.93..-58.75 [-95.2745..-75.4977] | it/evals=938/1577 eff=73.4534% N=300
Z=-87.2(0.00%) | Like=-80.95..-58.75 [-95.2745..-75.4977] | it/evals=952/1599 eff=73.2871% N=300
Z=-86.7(0.00%) | Like=-80.64..-58.75 [-95.2745..-75.4977] | it/evals=960/1610 eff=73.2824% N=300
Z=-85.7(0.00%) | Like=-79.09..-58.75 [-95.2745..-75.4977] | it/evals=977/1633 eff=73.2933% N=300
Z=-84.5(0.00%) | Like=-77.49..-58.75 [-95.2745..-75.4977] | it/evals=990/1648 eff=73.4421% N=300
Mono-modal Volume: ~exp(-7.44) Expected Volume: exp(-3.35) Quality: ok
index : +1.0| +2.2 ************* +3.2 | +5.0
amplitude: +1.0e-12| +2.9e-11 ****************** +6.5e-11 | +1.0e-10
Z=-83.2(0.00%) | Like=-76.85..-58.75 [-95.2745..-75.4977] | it/evals=1005/1668 eff=73.4649% N=300
Z=-82.2(0.00%) | Like=-75.98..-58.75 [-95.2745..-75.4977] | it/evals=1020/1689 eff=73.4341% N=300
Z=-81.3(0.00%) | Like=-75.15..-58.75 [-75.4543..-67.9574] | it/evals=1037/1711 eff=73.4940% N=300
Z=-80.7(0.00%) | Like=-74.63..-58.75 [-75.4543..-67.9574] | it/evals=1050/1731 eff=73.3753% N=300
Z=-80.2(0.00%) | Like=-74.30..-58.75 [-75.4543..-67.9574] | it/evals=1063/1753 eff=73.1590% N=300
Mono-modal Volume: ~exp(-7.51) * Expected Volume: exp(-3.57) Quality: ok
index : +1.0| +2.3 ************ +3.2 | +5.0
amplitude: +1.0e-12| +3.0e-11 ***************** +6.2e-11 | +1.0e-10
Z=-79.9(0.00%) | Like=-73.77..-58.75 [-75.4543..-67.9574] | it/evals=1072/1765 eff=73.1741% N=300
Z=-79.6(0.00%) | Like=-73.48..-58.75 [-75.4543..-67.9574] | it/evals=1080/1774 eff=73.2700% N=300
Z=-78.9(0.00%) | Like=-72.69..-58.75 [-75.4543..-67.9574] | it/evals=1095/1798 eff=73.0975% N=300
Z=-78.4(0.00%) | Like=-72.11..-58.75 [-75.4543..-67.9574] | it/evals=1108/1822 eff=72.7989% N=300
Z=-78.3(0.00%) | Like=-72.04..-58.75 [-75.4543..-67.9574] | it/evals=1110/1824 eff=72.8346% N=300
Z=-77.9(0.00%) | Like=-71.81..-58.75 [-75.4543..-67.9574] | it/evals=1121/1846 eff=72.5097% N=300
Z=-77.3(0.00%) | Like=-71.27..-58.75 [-75.4543..-67.9574] | it/evals=1138/1869 eff=72.5303% N=300
Mono-modal Volume: ~exp(-7.51) Expected Volume: exp(-3.80) Quality: ok
index : +1.0| +2.3 *********** +3.1 | +5.0
amplitude: +1.0e-12| +3.1e-11 **************** +6.1e-11 | +1.0e-10
Z=-77.2(0.00%) | Like=-71.19..-58.75 [-75.4543..-67.9574] | it/evals=1140/1871 eff=72.5652% N=300
Z=-76.8(0.00%) | Like=-70.65..-58.75 [-75.4543..-67.9574] | it/evals=1154/1893 eff=72.4419% N=300
Z=-76.4(0.00%) | Like=-70.15..-58.75 [-75.4543..-67.9574] | it/evals=1167/1915 eff=72.2601% N=300
Z=-76.3(0.00%) | Like=-70.10..-58.75 [-75.4543..-67.9574] | it/evals=1170/1919 eff=72.2668% N=300
Z=-76.0(0.00%) | Like=-69.79..-58.75 [-75.4543..-67.9574] | it/evals=1180/1942 eff=71.8636% N=300
Z=-75.6(0.00%) | Like=-69.42..-58.75 [-75.4543..-67.9574] | it/evals=1194/1967 eff=71.6257% N=300
Z=-75.4(0.00%) | Like=-69.28..-58.75 [-75.4543..-67.9574] | it/evals=1200/1976 eff=71.5990% N=300
Mono-modal Volume: ~exp(-7.78) * Expected Volume: exp(-4.02) Quality: ok
index : +1.0| +2.3 ********** +3.1 | +5.0
amplitude: +1.0e-12| +3.2e-11 *************** +5.9e-11 | +1.0e-10
Z=-75.2(0.00%) | Like=-69.09..-58.75 [-75.4543..-67.9574] | it/evals=1206/1987 eff=71.4878% N=300
Z=-74.8(0.01%) | Like=-68.52..-58.75 [-75.4543..-67.9574] | it/evals=1221/2009 eff=71.4453% N=300
Z=-74.5(0.01%) | Like=-68.25..-58.75 [-75.4543..-67.9574] | it/evals=1230/2021 eff=71.4701% N=300
Z=-74.0(0.01%) | Like=-67.76..-58.75 [-67.9030..-65.3940] | it/evals=1246/2043 eff=71.4859% N=300
Z=-73.7(0.02%) | Like=-67.43..-58.75 [-67.9030..-65.3940] | it/evals=1260/2063 eff=71.4691% N=300
Mono-modal Volume: ~exp(-7.98) * Expected Volume: exp(-4.24) Quality: ok
index : +1.0| +2.4 ********* +3.0 | +5.0
amplitude: +1.0e-12| +3.3e-11 ************* +5.8e-11 | +1.0e-10
Z=-73.4(0.02%) | Like=-67.19..-58.75 [-67.9030..-65.3940] | it/evals=1273/2084 eff=71.3565% N=300
Z=-72.9(0.03%) | Like=-66.61..-58.75 [-67.9030..-65.3940] | it/evals=1290/2105 eff=71.4681% N=300
Z=-72.5(0.05%) | Like=-66.15..-58.75 [-67.9030..-65.3940] | it/evals=1307/2127 eff=71.5380% N=300
Z=-72.2(0.07%) | Like=-65.86..-58.75 [-67.9030..-65.3940] | it/evals=1320/2146 eff=71.5060% N=300
Z=-71.9(0.10%) | Like=-65.47..-58.75 [-67.9030..-65.3940] | it/evals=1333/2168 eff=71.3597% N=300
Mono-modal Volume: ~exp(-8.38) * Expected Volume: exp(-4.47) Quality: ok
index : +1.0| +2.4 ******** +3.0 | +5.0
amplitude: +1.0e-12| +3.5e-11 *********** +5.6e-11 | +1.0e-10
Z=-71.7(0.12%) | Like=-65.35..-58.75 [-65.3809..-65.0231] | it/evals=1340/2178 eff=71.3525% N=300
Z=-71.5(0.15%) | Like=-65.15..-58.75 [-65.3809..-65.0231] | it/evals=1350/2190 eff=71.4286% N=300
Z=-71.1(0.22%) | Like=-64.75..-58.75 [-64.7451..-64.7235] | it/evals=1368/2211 eff=71.5856% N=300
Z=-70.9(0.28%) | Like=-64.56..-58.75 [-64.5610..-64.5257] | it/evals=1380/2227 eff=71.6139% N=300
Z=-70.6(0.38%) | Like=-64.23..-58.75 [-64.2332..-64.2179] | it/evals=1397/2249 eff=71.6778% N=300
Mono-modal Volume: ~exp(-8.82) * Expected Volume: exp(-4.69) Quality: ok
index : +1.0| +2.4 ******** +3.0 | +5.0
amplitude: +1.0e-12| +3.5e-11 ********** +5.4e-11 | +1.0e-10
Z=-70.4(0.45%) | Like=-64.02..-58.75 [-64.0184..-64.0081] | it/evals=1407/2262 eff=71.7125% N=300
Z=-70.3(0.47%) | Like=-63.95..-58.75 [-63.9674..-63.9468] | it/evals=1410/2265 eff=71.7557% N=300
Z=-70.0(0.64%) | Like=-63.63..-58.75 [-63.6261..-63.6248]*| it/evals=1427/2287 eff=71.8168% N=300
Z=-69.8(0.80%) | Like=-63.39..-58.75 [-63.4045..-63.3928] | it/evals=1440/2306 eff=71.7846% N=300
Z=-69.5(1.11%) | Like=-63.10..-58.75 [-63.1181..-63.1014] | it/evals=1458/2328 eff=71.8935% N=300
Z=-69.3(1.35%) | Like=-62.93..-58.75 [-62.9331..-62.9253]*| it/evals=1470/2345 eff=71.8826% N=300
Mono-modal Volume: ~exp(-9.23) * Expected Volume: exp(-4.91) Quality: ok
index : +1.0| +2.5 ******* +2.9 | +5.0
amplitude: +1.0e-12| +3.6e-11 ********* +5.3e-11 | +1.0e-10
Z=-69.3(1.43%) | Like=-62.88..-58.75 [-62.8847..-62.8710] | it/evals=1474/2353 eff=71.7974% N=300
Z=-69.0(1.89%) | Like=-62.60..-58.75 [-62.6021..-62.5839] | it/evals=1493/2374 eff=71.9865% N=300
Z=-68.9(2.08%) | Like=-62.48..-58.75 [-62.4932..-62.4826] | it/evals=1500/2382 eff=72.0461% N=300
Z=-68.7(2.65%) | Like=-62.20..-58.75 [-62.1968..-62.1849] | it/evals=1516/2403 eff=72.0875% N=300
Z=-68.5(3.15%) | Like=-62.07..-58.75 [-62.0714..-62.0625]*| it/evals=1530/2422 eff=72.1018% N=300
Mono-modal Volume: ~exp(-9.23) Expected Volume: exp(-5.14) 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(3.72%) | Like=-61.96..-58.75 [-61.9585..-61.9580]*| it/evals=1544/2441 eff=72.1158% N=300
Z=-68.2(4.39%) | Like=-61.79..-58.75 [-61.8489..-61.7921] | it/evals=1560/2461 eff=72.1888% N=300
Z=-68.0(5.27%) | Like=-61.57..-58.75 [-61.6042..-61.5740] | it/evals=1577/2483 eff=72.2400% N=300
Z=-67.8(6.03%) | Like=-61.40..-58.75 [-61.4021..-61.4005]*| it/evals=1590/2504 eff=72.1416% N=300
Z=-67.7(7.02%) | Like=-61.26..-58.75 [-61.2646..-61.2625]*| it/evals=1605/2526 eff=72.1024% N=300
Mono-modal Volume: ~exp(-9.37) * Expected Volume: exp(-5.36) Quality: ok
index : +1.0| +2.5 ***** +2.9 | +5.0
amplitude: +1.0e-12| +3.8e-11 ******* +5.1e-11 | +1.0e-10
Z=-67.7(7.25%) | Like=-61.26..-58.75 [-61.2573..-61.2541]*| it/evals=1608/2530 eff=72.1076% N=300
Z=-67.6(8.04%) | Like=-61.17..-58.75 [-61.1676..-61.1459] | it/evals=1620/2543 eff=72.2247% N=300
Z=-67.4(9.27%) | Like=-61.00..-58.75 [-61.0032..-61.0011]*| it/evals=1635/2567 eff=72.1217% N=300
Z=-67.3(10.51%) | Like=-60.89..-58.75 [-60.8906..-60.8859]*| it/evals=1650/2590 eff=72.0524% N=300
Z=-67.2(11.95%) | Like=-60.77..-58.75 [-60.7729..-60.7653]*| it/evals=1666/2613 eff=72.0277% N=300
Mono-modal Volume: ~exp(-9.88) * Expected Volume: exp(-5.58) 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=-67.1(12.79%) | Like=-60.73..-58.75 [-60.7345..-60.7256]*| it/evals=1675/2626 eff=72.0120% N=300
Z=-67.1(13.24%) | Like=-60.70..-58.75 [-60.6957..-60.6862]*| it/evals=1680/2631 eff=72.0721% N=300
Z=-67.0(14.99%) | Like=-60.59..-58.75 [-60.5924..-60.5898]*| it/evals=1697/2654 eff=72.0901% N=300
Z=-66.9(16.25%) | Like=-60.55..-58.75 [-60.5474..-60.5412]*| it/evals=1710/2671 eff=72.1215% N=300
Z=-66.8(17.96%) | Like=-60.48..-58.75 [-60.4795..-60.4655] | it/evals=1728/2693 eff=72.2106% N=300
Z=-66.7(19.24%) | Like=-60.41..-58.75 [-60.4143..-60.4117]*| it/evals=1740/2709 eff=72.2291% N=300
Mono-modal Volume: ~exp(-9.88) Expected Volume: exp(-5.81) Quality: ok
index : +1.0| +2.5 **** +2.8 | +5.0
amplitude: +1.0e-12| +3.9e-11 ****** +5.0e-11 | +1.0e-10
Z=-66.6(20.73%) | Like=-60.34..-58.75 [-60.3449..-60.3403]*| it/evals=1754/2729 eff=72.2108% N=300
Z=-66.5(22.55%) | Like=-60.25..-58.75 [-60.2509..-60.2503]*| it/evals=1770/2747 eff=72.3335% N=300
Z=-66.5(24.56%) | Like=-60.19..-58.75 [-60.1935..-60.1740] | it/evals=1788/2768 eff=72.4473% N=300
Z=-66.4(25.88%) | Like=-60.13..-58.75 [-60.1443..-60.1288] | it/evals=1800/2788 eff=72.3473% N=300
Mono-modal Volume: ~exp(-10.15) * Expected Volume: exp(-6.03) Quality: ok
index : +1.0| +2.5 **** +2.8 | +5.0
amplitude: +1.0e-12| +4.0e-11 ****** +4.9e-11 | +1.0e-10
Z=-66.4(26.95%) | Like=-60.09..-58.75 [-60.0900..-60.0703] | it/evals=1809/2799 eff=72.3890% N=300
Z=-66.3(28.92%) | Like=-60.02..-58.75 [-60.0196..-60.0178]*| it/evals=1826/2820 eff=72.4603% N=300
Z=-66.3(29.35%) | Like=-59.98..-58.75 [-59.9844..-59.9840]*| it/evals=1830/2826 eff=72.4466% N=300
Z=-66.2(31.32%) | Like=-59.91..-58.75 [-59.9068..-59.9061]*| it/evals=1847/2850 eff=72.4314% N=300
Z=-66.2(32.95%) | Like=-59.84..-58.75 [-59.8382..-59.8365]*| it/evals=1860/2869 eff=72.4017% N=300
Mono-modal Volume: ~exp(-10.19) * 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=-66.1(34.91%) | Like=-59.79..-58.75 [-59.7851..-59.7820]*| it/evals=1876/2890 eff=72.4324% N=300
Z=-66.1(36.67%) | Like=-59.73..-58.75 [-59.7348..-59.7316]*| it/evals=1890/2906 eff=72.5249% N=300
Z=-66.0(38.58%) | Like=-59.69..-58.75 [-59.6880..-59.6845]*| it/evals=1906/2928 eff=72.5266% N=300
Z=-66.0(40.38%) | Like=-59.64..-58.75 [-59.6449..-59.6402]*| it/evals=1920/2949 eff=72.4802% N=300
Z=-65.9(41.98%) | Like=-59.60..-58.75 [-59.5973..-59.5940]*| it/evals=1933/2971 eff=72.3699% N=300
Mono-modal Volume: ~exp(-10.90) * Expected Volume: exp(-6.48) Quality: ok
index : +1.0| +2.6 **** +2.8 | +5.0
amplitude: +1.0e-12| +4.1e-11 **** +4.8e-11 | +1.0e-10
Z=-65.9(43.23%) | Like=-59.57..-58.75 [-59.5675..-59.5662]*| it/evals=1943/2982 eff=72.4459% N=300
Z=-65.9(43.92%) | Like=-59.54..-58.75 [-59.5426..-59.5390]*| it/evals=1950/2990 eff=72.4907% N=300
Z=-65.8(46.01%) | Like=-59.50..-58.75 [-59.4961..-59.4915]*| it/evals=1967/3013 eff=72.5028% N=300
Z=-65.8(47.54%) | Like=-59.46..-58.75 [-59.4583..-59.4573]*| it/evals=1980/3033 eff=72.4479% N=300
Z=-65.7(49.46%) | Like=-59.44..-58.75 [-59.4363..-59.4353]*| it/evals=1996/3055 eff=72.4501% N=300
Mono-modal Volume: ~exp(-10.96) * Expected Volume: exp(-6.70) Quality: ok
index : +1.0| +2.6 **** +2.8 | +5.0
amplitude: +1.0e-12| +4.1e-11 **** +4.8e-11 | +1.0e-10
Z=-65.7(51.04%) | Like=-59.42..-58.75 [-59.4214..-59.4211]*| it/evals=2010/3076 eff=72.4063% N=300
Z=-65.7(53.53%) | Like=-59.37..-58.75 [-59.3711..-59.3698]*| it/evals=2031/3097 eff=72.6135% N=300
Z=-65.7(54.39%) | Like=-59.36..-58.75 [-59.3582..-59.3573]*| it/evals=2040/3108 eff=72.6496% N=300
Z=-65.6(56.44%) | Like=-59.33..-58.75 [-59.3330..-59.3314]*| it/evals=2058/3130 eff=72.7208% N=300
Z=-65.6(57.69%) | Like=-59.31..-58.75 [-59.3109..-59.3106]*| it/evals=2070/3146 eff=72.7337% N=300
Mono-modal Volume: ~exp(-11.24) * Expected Volume: exp(-6.92) Quality: ok
index : +1.0| +2.6 **** +2.8 | +5.0
amplitude: +1.0e-12| +4.1e-11 **** +4.7e-11 | +1.0e-10
Z=-65.6(58.43%) | Like=-59.30..-58.75 [-59.2964..-59.2910]*| it/evals=2077/3154 eff=72.7751% N=300
Z=-65.6(60.33%) | Like=-59.26..-58.75 [-59.2599..-59.2529]*| it/evals=2096/3175 eff=72.9043% N=300
Z=-65.5(60.76%) | Like=-59.25..-58.75 [-59.2505..-59.2497]*| it/evals=2100/3180 eff=72.9167% N=300
Z=-65.5(62.27%) | Like=-59.22..-58.75 [-59.2228..-59.2199]*| it/evals=2116/3202 eff=72.9152% N=300
Z=-65.5(63.73%) | Like=-59.19..-58.75 [-59.1947..-59.1941]*| it/evals=2130/3220 eff=72.9452% N=300
Mono-modal Volume: ~exp(-11.24) Expected Volume: exp(-7.15) Quality: ok
index : +1.0| +2.6 ** +2.7 | +5.0
amplitude: +1.0e-12| +4.2e-11 **** +4.7e-11 | +1.0e-10
Z=-65.5(65.03%) | Like=-59.18..-58.75 [-59.1771..-59.1767]*| it/evals=2144/3242 eff=72.8756% N=300
Z=-65.5(66.26%) | Like=-59.16..-58.75 [-59.1571..-59.1563]*| it/evals=2157/3264 eff=72.7733% N=300
Z=-65.5(66.54%) | Like=-59.15..-58.75 [-59.1495..-59.1493]*| it/evals=2160/3269 eff=72.7518% N=300
Z=-65.4(67.86%) | Like=-59.14..-58.75 [-59.1357..-59.1329]*| it/evals=2175/3291 eff=72.7182% N=300
Z=-65.4(69.04%) | Like=-59.12..-58.75 [-59.1214..-59.1202]*| it/evals=2189/3313 eff=72.6518% N=300
Z=-65.4(69.11%) | Like=-59.12..-58.75 [-59.1202..-59.1201]*| it/evals=2190/3315 eff=72.6368% N=300
[ultranest] Explored until L=-6e+01
[ultranest] Likelihood function evaluations: 3329
[ultranest] logZ = -65.06 +- 0.08653
[ultranest] Effective samples strategy satisfied (ESS = 991.8, need >400)
[ultranest] Posterior uncertainty strategy is satisfied (KL: 0.45+-0.07 nat, need <0.50 nat)
[ultranest] Evidency uncertainty strategy is satisfied (dlogz=0.28, need <0.5)
[ultranest] logZ error budget: single: 0.13 bs:0.09 tail:0.26 total:0.28 required:<0.50
[ultranest] done iterating.
logZ = -65.048 +- 0.301
single instance: logZ = -65.048 +- 0.133
bootstrapped : logZ = -65.065 +- 0.148
tail : logZ = +- 0.262
insert order U test : converged: True correlation: inf iterations
index : 2.370 │ ▁▁▁▁▁▁▁▁▃▂▂▄▄▅▅▇▆▇▇▆▆▇▆▅▄▃▂▂▂▁▁▁▁▁▁▁▁ │2.961 2.669 +- 0.084
amplitude : 0.0000000000335│ ▁ ▁▁▁▁▁▂▂▂▃▄▄▆▄▇▇▆▆▆▅▄▄▄▂▂▂▁▁▁▁▁▁▁▁ ▁ │0.0000000000569 0.0000000000445 +- 0.0000000000031
Understanding the outputs#
In the Jupyter notebook, you should be able to see an interactive visualisation of how the parameter space shrinks which starts from the (min,max) shrinks down towards the optimal parameters.
The output above is filled with interesting information. Here we provide a short description of the most relevant information provided above. For more detailed information see the UltraNest docs.
During the sampling
Z=-68.8(0.53%) | Like=-63.96..-58.75 [-63.9570..-63.9539]*| it/evals=640/1068 eff=73.7327% N=300
Some important information here is:
Progress (0.53%): the completed fraction of the integral. This is not a time progress bar. Stays at zero for a good fraction of the run.
Efficiency (eff value) of the sampling: this indicates out of the proposed new points, how many were accepted. If your efficiency is too small (<<1%), maybe you should revise your priors (e.g use a LogUniform prior for the normalisation).
Final outputs
The final lines indicate that all three “convergence” strategies are satisfied (samples, posterior uncertainty, and evidence uncertainty).
logZ = -65.104 +- 0.292
The main goal of the Nested sampling algorithm is to estimate Z (the Bayesian evidence) which is given above together with an uncertainty. In a similar way to deltaLogLike and deltaAIC, deltaLogZ values can be used for model comparison. For more information see : on the use of the evidence for model comparison. An interesting comparison of the efficiency and false discovery rate of model selection with deltaLogLike and deltaLogZ is given in Appendix C of Buchner et al., 2014.
Results stored on disk
if log_dir is set to a name where the results will be stored, then
a directory is created containing many useful results and plots.
A description of these outputs is given in the Ultranest
docs.
Results#
Within a Bayesian analysis, the concept of best-fit has to be viewed differently from what is done in a gradient descent fit.
The output of the Bayesian analysis is the posterior distribution and there is no “best-fit” output. One has to define, based on the posteriors, what we want to consider as “best-fit” and several options are possible:
the mean of the distribution
the median
the lowest likelihood value
By default the DatasetModels will be updated with the mean of
the posterior distributions.
print(result_joint.models)
DatasetModels
Component 0: SkyModel
Name : crab
Datasets names : None
Spectral model type : PowerLawSpectralModel
Spatial model type :
Temporal model type :
Parameters:
index : 2.669 +/- 0.08
amplitude : 4.45e-11 +/- 3.1e-12 1 / (TeV s cm2)
reference (frozen): 1.000 TeV
The Sampler class returns a very rich dictionary.
The most “standard” information about the posterior distributions can
be found in :
print(result_joint.sampler_results["posterior"])
{'mean': [2.669497685883489, 4.4491714066825525e-11], 'stdev': [0.08382636082838323, 3.0630664475317227e-12], 'median': [2.668545538147532, 4.438322378124178e-11], 'errlo': [2.585912959377391, 4.143727762278298e-11], 'errup': [2.751055245574489, 4.7644488968299695e-11], 'information_gain_bits': [2.685305164989858, 3.1111136335100977]}
Besides mean, errors, etc, an interesting value is the
information gain which estimates how much the posterior
distribution has shrunk with respect to the prior (i.e. how much
we’ve learned). A value < 1 means that the parameter is poorly
constrained within the prior range (we haven’t learned much with respect to our prior assumption).
For a physical example see this
example.
The SamplerResult dictionary contains also other interesting
information :
print(result_joint.sampler_results.keys())
dict_keys(['niter', 'logz', 'logzerr', 'logz_bs', 'logz_single', 'logzerr_tail', 'logzerr_bs', 'ess', 'H', 'Herr', 'posterior', 'weighted_samples', 'samples', 'maximum_likelihood', 'ncall', 'paramnames', 'logzerr_single', 'insertion_order_MWW_test'])
Of particular interest, the samples used in the process to approximate the posterior distribution can be accessed via :
for i, n in enumerate(model.parameters.free_parameters.names):
s = result_joint.samples[:, i]
fig, ax = plt.subplots()
ax.hist(s, bins=30)
ax.axvline(np.mean(s), ls="--", color="red")
ax.set_xlabel(n)
plt.show()
While the above plots are interesting, the real strength of the Bayesian analysis is to visualise all parameters correlations which is usually done using “corner plots”. Ultranest corner plot function is a wrapper around the corner package. See the above link for optional keywords. Other packages exist for corner plots, like chainconsumer which is discussed later in this tutorial.
from ultranest.plot import cornerplot
cornerplot(
result_joint.sampler_results,
plot_datapoints=True,
plot_density=True,
bins=20,
title_fmt=".2e",
smooth=False,
)
plt.show()
Spectral model error band from samples#
To compute the spectral error band (“butterfly plots”), we will directly use the samples of the posterior distribution. This is more robust as compared to the traditional method of using the covariance matrix of the parameters which implicitly assumes Gaussian errors while for the posterior distribution there is no shape assumed. This difference can become significant when the parameter errors are non-Gaussian. For this we will need to convert the list of samples back to the spectral model parameters with the relevant units (e.g. normalisation units).
def get_samples_from_posterior(spectral_model, results):
"""
Create a list of spectral parameters with correct units
from the unitless parameters returned by the sampler.
"""
n_samples = results.samples.shape[0]
samples = []
for p in spectral_model.parameters:
try:
idx = spectral_model.parameters.free_unique_parameters.index(p)
samples.append(results.samples[:, idx] * p.unit)
except ValueError:
samples.append(np.ones(n_samples) * p.quantity)
return samples
samples = get_samples_from_posterior(datasets.models[0].spectral_model, result_joint)
Next we can provide these samples to the plot_error
method.
Individual run analysis#
Now we’ll analyse several Crab runs individually so that we can compare them.
result_0 = sampler.run(datasets[0])
result_1 = sampler.run(datasets[1])
result_2 = sampler.run(datasets[2])
[ultranest] Sampling 300 live points from prior ...
Mono-modal Volume: ~exp(-4.04) * Expected Volume: exp(0.00) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|********************* ******************** *****| +1.0e-10
Z=-inf(0.00%) | Like=-1356.70..-21.20 [-1356.6957..-111.4298] | it/evals=0/301 eff=0.0000% N=300
Z=-178.0(0.00%) | Like=-172.90..-21.20 [-1356.6957..-111.4298] | it/evals=30/335 eff=85.7143% N=300
Z=-165.8(0.00%) | Like=-161.12..-21.20 [-1356.6957..-111.4298] | it/evals=60/372 eff=83.3333% N=300
Mono-modal Volume: ~exp(-4.09) * Expected Volume: exp(-0.22) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|****************************** ** ******* *****| +1.0e-10
Z=-163.2(0.00%) | Like=-157.45..-21.20 [-1356.6957..-111.4298] | it/evals=67/379 eff=84.8101% N=300
Z=-153.1(0.00%) | Like=-148.04..-21.20 [-1356.6957..-111.4298] | it/evals=90/411 eff=81.0811% N=300
Z=-141.8(0.00%) | Like=-136.62..-21.20 [-1356.6957..-111.4298] | it/evals=120/447 eff=81.6327% N=300
Mono-modal Volume: ~exp(-4.55) * Expected Volume: exp(-0.45) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|********************************* ****** * **** | +1.0e-10
Z=-136.2(0.00%) | Like=-131.12..-21.20 [-1356.6957..-111.4298] | it/evals=134/463 eff=82.2086% N=300
Z=-130.4(0.00%) | Like=-125.36..-21.20 [-1356.6957..-111.4298] | it/evals=150/482 eff=82.4176% N=300
Z=-121.7(0.00%) | Like=-116.65..-20.94 [-1356.6957..-111.4298] | it/evals=180/517 eff=82.9493% N=300
Mono-modal Volume: ~exp(-4.78) * Expected Volume: exp(-0.67) Quality: ok
index : +1.0| *********************************************| +5.0
amplitude: +1.0e-12| ********************************** **** * *****| +1.0e-10
Z=-115.5(0.00%) | Like=-110.15..-20.94 [-111.4110..-65.8189] | it/evals=201/542 eff=83.0579% N=300
Z=-113.1(0.00%) | Like=-107.53..-20.94 [-111.4110..-65.8189] | it/evals=210/551 eff=83.6653% N=300
Z=-104.5(0.00%) | Like=-99.52..-20.94 [-111.4110..-65.8189] | it/evals=240/589 eff=83.0450% N=300
Mono-modal Volume: ~exp(-4.78) Expected Volume: exp(-0.89) Quality: ok
index : +1.0| ********************************************| +5.0
amplitude: +1.0e-12| ********************************* ***** ** | +1.0e-10
Z=-98.1(0.00%) | Like=-93.10..-20.94 [-111.4110..-65.8189] | it/evals=268/627 eff=81.9572% N=300
Z=-97.7(0.00%) | Like=-92.68..-20.94 [-111.4110..-65.8189] | it/evals=270/629 eff=82.0669% N=300
Z=-88.2(0.00%) | Like=-83.20..-20.94 [-111.4110..-65.8189] | it/evals=299/668 eff=81.2500% N=300
Z=-88.1(0.00%) | Like=-83.16..-20.94 [-111.4110..-65.8189] | it/evals=300/669 eff=81.3008% N=300
Z=-82.1(0.00%) | Like=-76.67..-20.94 [-111.4110..-65.8189] | it/evals=330/708 eff=80.8824% N=300
Mono-modal Volume: ~exp(-5.38) * Expected Volume: exp(-1.12) Quality: ok
index : +1.0| *******************************************| +5.0
amplitude: +1.0e-12| ********************************* ***** | +1.0e-10
Z=-81.2(0.00%) | Like=-75.97..-20.94 [-111.4110..-65.8189] | it/evals=335/714 eff=80.9179% N=300
Z=-77.0(0.00%) | Like=-72.20..-20.94 [-111.4110..-65.8189] | it/evals=360/746 eff=80.7175% N=300
Z=-73.6(0.00%) | Like=-68.35..-20.94 [-111.4110..-65.8189] | it/evals=386/787 eff=79.2608% N=300
Z=-73.0(0.00%) | Like=-67.97..-20.94 [-111.4110..-65.8189] | it/evals=390/793 eff=79.1075% N=300
Mono-modal Volume: ~exp(-5.57) * Expected Volume: exp(-1.34) Quality: ok
index : +1.0| *********************************** * | +5.0
amplitude: +1.0e-12| ******************************** * +7.8e-11| +1.0e-10
Z=-71.4(0.00%) | Like=-66.24..-20.69 [-111.4110..-65.8189] | it/evals=402/814 eff=78.2101% N=300
Z=-69.4(0.00%) | Like=-64.53..-20.69 [-65.7901..-45.7510] | it/evals=420/839 eff=77.9221% N=300
Z=-66.4(0.00%) | Like=-61.46..-20.69 [-65.7901..-45.7510] | it/evals=445/880 eff=76.7241% N=300
Z=-65.9(0.00%) | Like=-61.17..-20.69 [-65.7901..-45.7510] | it/evals=450/890 eff=76.2712% N=300
Mono-modal Volume: ~exp(-5.57) Expected Volume: exp(-1.56) Quality: ok
index : +1.0| ******************************** | +5.0
amplitude: +1.0e-12| ******************************* +7.2e-11 | +1.0e-10
Z=-63.6(0.00%) | Like=-58.41..-20.69 [-65.7901..-45.7510] | it/evals=477/927 eff=76.0766% N=300
Z=-63.2(0.00%) | Like=-57.67..-20.69 [-65.7901..-45.7510] | it/evals=480/932 eff=75.9494% N=300
Z=-59.9(0.00%) | Like=-54.70..-20.69 [-65.7901..-45.7510] | it/evals=504/973 eff=74.8886% N=300
Z=-59.3(0.00%) | Like=-54.06..-20.69 [-65.7901..-45.7510] | it/evals=510/981 eff=74.8899% N=300
Z=-57.1(0.00%) | Like=-51.72..-20.69 [-65.7901..-45.7510] | it/evals=532/1023 eff=73.5823% N=300
Mono-modal Volume: ~exp(-5.80) * Expected Volume: exp(-1.79) Quality: ok
index : +1.0| *************************** +3.9 | +5.0
amplitude: +1.0e-12| *************************** * +7.1e-11 | +1.0e-10
Z=-56.6(0.00%) | Like=-51.05..-20.69 [-65.7901..-45.7510] | it/evals=536/1028 eff=73.6264% N=300
Z=-56.1(0.00%) | Like=-50.71..-20.69 [-65.7901..-45.7510] | it/evals=540/1032 eff=73.7705% N=300
Z=-54.2(0.00%) | Like=-49.17..-20.69 [-65.7901..-45.7510] | it/evals=566/1073 eff=73.2212% N=300
Z=-53.9(0.00%) | Like=-49.05..-20.69 [-65.7901..-45.7510] | it/evals=570/1079 eff=73.1707% N=300
Z=-52.1(0.00%) | Like=-47.06..-20.67 [-65.7901..-45.7510] | it/evals=596/1119 eff=72.7717% N=300
Z=-51.8(0.00%) | Like=-46.95..-20.67 [-65.7901..-45.7510] | it/evals=600/1123 eff=72.9040% N=300
Mono-modal Volume: ~exp(-6.16) * Expected Volume: exp(-2.01) Quality: ok
index : +1.0| ************************* +3.8 | +5.0
amplitude: +1.0e-12| ************************** +6.6e-11 | +1.0e-10
Z=-51.7(0.00%) | Like=-46.75..-20.67 [-65.7901..-45.7510] | it/evals=603/1129 eff=72.7382% N=300
Z=-50.0(0.00%) | Like=-44.86..-20.67 [-45.7385..-33.2318] | it/evals=627/1169 eff=72.1519% N=300
Z=-49.7(0.00%) | Like=-44.39..-20.67 [-45.7385..-33.2318] | it/evals=630/1172 eff=72.2477% N=300
Z=-47.5(0.00%) | Like=-42.29..-20.61 [-45.7385..-33.2318] | it/evals=660/1206 eff=72.8477% N=300
Mono-modal Volume: ~exp(-6.64) * Expected Volume: exp(-2.23) Quality: ok
index : +1.0| ********************* +3.6 | +5.0
amplitude: +1.0e-12| *********************** * +6.3e-11 | +1.0e-10
Z=-46.8(0.00%) | Like=-41.51..-20.61 [-45.7385..-33.2318] | it/evals=670/1220 eff=72.8261% N=300
Z=-45.7(0.00%) | Like=-40.57..-20.61 [-45.7385..-33.2318] | it/evals=690/1245 eff=73.0159% N=300
Z=-44.1(0.00%) | Like=-38.84..-20.61 [-45.7385..-33.2318] | it/evals=720/1283 eff=73.2452% N=300
Mono-modal Volume: ~exp(-6.88) * Expected Volume: exp(-2.46) Quality: ok
index : +1.0| +1.9 ******************* +3.4 | +5.0
amplitude: +1.0e-12| ********************** +5.9e-11 | +1.0e-10
Z=-43.1(0.00%) | Like=-37.67..-20.61 [-45.7385..-33.2318] | it/evals=737/1304 eff=73.4064% N=300
Z=-42.2(0.00%) | Like=-36.76..-20.61 [-45.7385..-33.2318] | it/evals=750/1323 eff=73.3138% N=300
Z=-40.8(0.00%) | Like=-35.62..-20.61 [-45.7385..-33.2318] | it/evals=780/1360 eff=73.5849% N=300
Mono-modal Volume: ~exp(-6.88) Expected Volume: exp(-2.68) Quality: ok
index : +1.0| +2.0 ****************** +3.4 | +5.0
amplitude: +1.0e-12| ******************** +5.8e-11 | +1.0e-10
Z=-39.6(0.00%) | Like=-34.42..-20.61 [-45.7385..-33.2318] | it/evals=804/1401 eff=73.0245% N=300
Z=-39.4(0.00%) | Like=-34.07..-20.61 [-45.7385..-33.2318] | it/evals=810/1411 eff=72.9073% N=300
Z=-38.2(0.00%) | Like=-32.73..-20.61 [-33.2092..-27.6234] | it/evals=838/1452 eff=72.7431% N=300
Z=-38.1(0.00%) | Like=-32.58..-20.61 [-33.2092..-27.6234] | it/evals=840/1455 eff=72.7273% N=300
Z=-37.1(0.00%) | Like=-31.63..-20.61 [-33.2092..-27.6234] | it/evals=865/1495 eff=72.3849% N=300
Z=-36.9(0.00%) | Like=-31.51..-20.61 [-33.2092..-27.6234] | it/evals=870/1505 eff=72.1992% N=300
Mono-modal Volume: ~exp(-6.88) Expected Volume: exp(-2.90) Quality: ok
index : +1.0| +2.0 *************** +3.2 | +5.0
amplitude: +1.0e-12| ******************* +5.5e-11 | +1.0e-10
Z=-36.2(0.00%) | Like=-30.99..-20.61 [-33.2092..-27.6234] | it/evals=891/1544 eff=71.6238% N=300
Z=-35.9(0.00%) | Like=-30.69..-20.61 [-33.2092..-27.6234] | it/evals=900/1555 eff=71.7131% N=300
Z=-35.1(0.01%) | Like=-29.62..-20.58 [-33.2092..-27.6234] | it/evals=925/1595 eff=71.4286% N=300
Z=-34.9(0.01%) | Like=-29.51..-20.58 [-33.2092..-27.6234] | it/evals=930/1602 eff=71.4286% N=300
Mono-modal Volume: ~exp(-7.47) * Expected Volume: exp(-3.13) Quality: ok
index : +1.0| +2.1 ************* +3.1 | +5.0
amplitude: +1.0e-12| **************** +5.2e-11 | +1.0e-10
Z=-34.7(0.01%) | Like=-29.27..-20.58 [-33.2092..-27.6234] | it/evals=938/1616 eff=71.2766% N=300
Z=-33.9(0.03%) | Like=-28.46..-20.51 [-33.2092..-27.6234] | it/evals=960/1651 eff=71.0585% N=300
Z=-33.1(0.07%) | Like=-27.79..-20.51 [-33.2092..-27.6234] | it/evals=990/1691 eff=71.1718% N=300
Mono-modal Volume: ~exp(-7.47) Expected Volume: exp(-3.35) Quality: ok
index : +1.0| +2.1 ************* +3.1 | +5.0
amplitude: +1.0e-12| ************** +5.0e-11 | +1.0e-10
Z=-32.5(0.13%) | Like=-27.17..-20.51 [-27.6114..-26.9194] | it/evals=1017/1728 eff=71.2185% N=300
Z=-32.4(0.14%) | Like=-27.05..-20.51 [-27.6114..-26.9194] | it/evals=1020/1733 eff=71.1793% N=300
Z=-31.8(0.24%) | Like=-26.48..-20.51 [-26.4798..-26.4720]*| it/evals=1047/1774 eff=71.0312% N=300
Z=-31.8(0.26%) | Like=-26.33..-20.51 [-26.4674..-26.3347] | it/evals=1050/1778 eff=71.0419% N=300
Mono-modal Volume: ~exp(-7.75) * Expected Volume: exp(-3.57) Quality: ok
index : +1.0| +2.2 *********** +3.0 | +5.0
amplitude: +1.0e-12| ************* +4.8e-11 | +1.0e-10
Z=-31.3(0.41%) | Like=-25.93..-20.51 [-25.9716..-25.9338] | it/evals=1072/1819 eff=70.5727% N=300
Z=-31.1(0.49%) | Like=-25.78..-20.51 [-25.8126..-25.7833] | it/evals=1080/1828 eff=70.6806% N=300
Z=-30.6(0.79%) | Like=-25.38..-20.51 [-25.4017..-25.3774] | it/evals=1108/1867 eff=70.7084% N=300
Z=-30.6(0.82%) | Like=-25.33..-20.51 [-25.3709..-25.3343] | it/evals=1110/1870 eff=70.7006% N=300
Z=-30.2(1.25%) | Like=-25.01..-20.51 [-25.0341..-25.0087] | it/evals=1137/1910 eff=70.6211% N=300
Mono-modal Volume: ~exp(-8.03) * Expected Volume: exp(-3.80) Quality: ok
index : +1.0| +2.2 ********** +3.0 | +5.0
amplitude: +1.0e-12| +2.4e-11 ************ +4.7e-11 | +1.0e-10
Z=-30.2(1.28%) | Like=-24.98..-20.51 [-25.0048..-24.9819] | it/evals=1139/1912 eff=70.6576% N=300
Z=-30.2(1.30%) | Like=-24.97..-20.51 [-24.9696..-24.9388] | it/evals=1140/1913 eff=70.6758% N=300
Z=-29.8(1.86%) | Like=-24.62..-20.51 [-24.6174..-24.6053] | it/evals=1166/1953 eff=70.5384% N=300
Z=-29.8(1.94%) | Like=-24.59..-20.51 [-24.5896..-24.5839]*| it/evals=1170/1957 eff=70.6095% N=300
Z=-29.4(2.85%) | Like=-24.12..-20.51 [-24.1205..-24.0957] | it/evals=1199/1996 eff=70.6958% N=300
Z=-29.4(2.89%) | Like=-24.10..-20.51 [-24.1205..-24.0957] | it/evals=1200/1997 eff=70.7130% N=300
Mono-modal Volume: ~exp(-8.03) Expected Volume: exp(-4.02) Quality: ok
index : +1.0| +2.3 ********* +2.9 | +5.0
amplitude: +1.0e-12| +2.5e-11 *********** +4.5e-11 | +1.0e-10
Z=-29.1(3.75%) | Like=-23.86..-20.51 [-23.8831..-23.8635] | it/evals=1225/2034 eff=70.6459% N=300
Z=-29.1(3.96%) | Like=-23.82..-20.51 [-23.8373..-23.8152] | it/evals=1230/2040 eff=70.6897% N=300
Z=-28.8(4.98%) | Like=-23.53..-20.51 [-23.5288..-23.5198]*| it/evals=1256/2079 eff=70.6015% N=300
Z=-28.8(5.13%) | Like=-23.46..-20.51 [-23.4630..-23.4589]*| it/evals=1260/2083 eff=70.6674% N=300
Mono-modal Volume: ~exp(-8.42) * Expected Volume: exp(-4.24) Quality: ok
index : +1.0| +2.3 ******** +2.9 | +5.0
amplitude: +1.0e-12| +2.6e-11 ********** +4.4e-11 | +1.0e-10
Z=-28.6(5.73%) | Like=-23.39..-20.47 [-23.3861..-23.3719] | it/evals=1273/2105 eff=70.5263% N=300
Z=-28.5(6.77%) | Like=-23.17..-20.47 [-23.1749..-23.1695]*| it/evals=1290/2125 eff=70.6849% N=300
Z=-28.3(8.58%) | Like=-23.01..-20.47 [-23.0260..-23.0091] | it/evals=1318/2164 eff=70.7082% N=300
Z=-28.2(8.72%) | Like=-22.94..-20.47 [-22.9609..-22.9431] | it/evals=1320/2166 eff=70.7395% N=300
Mono-modal Volume: ~exp(-8.63) * Expected Volume: exp(-4.47) Quality: ok
index : +1.0| +2.3 ******** +2.8 | +5.0
amplitude: +1.0e-12| +2.7e-11 ******** +4.2e-11 | +1.0e-10
Z=-28.1(10.23%) | Like=-22.82..-20.47 [-22.8152..-22.8011] | it/evals=1340/2191 eff=70.8620% N=300
Z=-28.0(11.02%) | Like=-22.71..-20.47 [-22.7097..-22.7029]*| it/evals=1350/2202 eff=70.9779% N=300
Z=-27.8(13.58%) | Like=-22.46..-20.47 [-22.4602..-22.4577]*| it/evals=1380/2240 eff=71.1340% N=300
Mono-modal Volume: ~exp(-8.80) * Expected Volume: exp(-4.69) Quality: ok
index : +1.0| +2.3 ****** +2.8 | +5.0
amplitude: +1.0e-12| +2.7e-11 ******** +4.2e-11 | +1.0e-10
Z=-27.6(16.01%) | Like=-22.26..-20.47 [-22.2891..-22.2623] | it/evals=1407/2274 eff=71.2766% N=300
Z=-27.6(16.27%) | Like=-22.25..-20.47 [-22.2473..-22.2437]*| it/evals=1410/2278 eff=71.2841% N=300
Z=-27.4(19.64%) | Like=-22.05..-20.47 [-22.0519..-22.0471]*| it/evals=1440/2317 eff=71.3932% N=300
Z=-27.3(22.77%) | Like=-21.94..-20.47 [-21.9429..-21.9417]*| it/evals=1468/2356 eff=71.4008% N=300
Z=-27.3(23.01%) | Like=-21.94..-20.47 [-21.9373..-21.9337]*| it/evals=1470/2358 eff=71.4286% N=300
Mono-modal Volume: ~exp(-9.39) * Expected Volume: exp(-4.91) Quality: ok
index : +1.0| +2.4 ****** +2.8 | +5.0
amplitude: +1.0e-12| +2.8e-11 ******* +4.1e-11 | +1.0e-10
Z=-27.2(23.49%) | Like=-21.92..-20.47 [-21.9191..-21.9038] | it/evals=1474/2362 eff=71.4840% N=300
Z=-27.1(26.44%) | Like=-21.77..-20.47 [-21.7653..-21.7603]*| it/evals=1500/2390 eff=71.7703% N=300
Z=-27.0(30.21%) | Like=-21.64..-20.47 [-21.6366..-21.6310]*| it/evals=1530/2427 eff=71.9323% N=300
Mono-modal Volume: ~exp(-9.39) Expected Volume: exp(-5.14) Quality: ok
index : +1.0| +2.4 ****** +2.8 | +5.0
amplitude: +1.0e-12| +2.9e-11 ****** +4.0e-11 | +1.0e-10
Z=-26.9(33.16%) | Like=-21.55..-20.47 [-21.5521..-21.5512]*| it/evals=1556/2463 eff=71.9371% N=300
Z=-26.9(33.67%) | Like=-21.54..-20.47 [-21.5438..-21.5433]*| it/evals=1560/2470 eff=71.8894% N=300
Z=-26.8(37.22%) | Like=-21.46..-20.46 [-21.4554..-21.4484]*| it/evals=1588/2509 eff=71.8877% N=300
Z=-26.7(37.43%) | Like=-21.45..-20.46 [-21.4460..-21.4450]*| it/evals=1590/2511 eff=71.9132% N=300
Mono-modal Volume: ~exp(-9.54) * Expected Volume: exp(-5.36) Quality: ok
index : +1.0| +2.4 ***** +2.7 | +5.0
amplitude: +1.0e-12| +2.9e-11 ****** +3.9e-11 | +1.0e-10
Z=-26.7(39.52%) | Like=-21.39..-20.46 [-21.3938..-21.3905]*| it/evals=1608/2540 eff=71.7857% N=300
Z=-26.7(41.00%) | Like=-21.36..-20.46 [-21.3610..-21.3595]*| it/evals=1620/2555 eff=71.8404% N=300
Z=-26.6(44.66%) | Like=-21.26..-20.46 [-21.2615..-21.2600]*| it/evals=1650/2591 eff=72.0210% N=300
Mono-modal Volume: ~exp(-9.95) * Expected Volume: exp(-5.58) Quality: ok
index : +1.0| +2.4 ***** +2.7 | +5.0
amplitude: +1.0e-12| +3.0e-11 ****** +3.9e-11 | +1.0e-10
Z=-26.5(47.62%) | Like=-21.21..-20.46 [-21.2126..-21.2073]*| it/evals=1675/2627 eff=71.9811% N=300
Z=-26.5(48.22%) | Like=-21.20..-20.46 [-21.2035..-21.2025]*| it/evals=1680/2633 eff=72.0103% N=300
Z=-26.4(51.63%) | Like=-21.13..-20.46 [-21.1298..-21.1274]*| it/evals=1709/2672 eff=72.0489% N=300
Z=-26.4(51.73%) | Like=-21.13..-20.46 [-21.1274..-21.1244]*| it/evals=1710/2673 eff=72.0607% N=300
Z=-26.4(54.71%) | Like=-21.05..-20.46 [-21.0533..-21.0531]*| it/evals=1737/2712 eff=72.0149% N=300
Z=-26.4(55.05%) | Like=-21.05..-20.46 [-21.0506..-21.0454]*| it/evals=1740/2720 eff=71.9008% N=300
Mono-modal Volume: ~exp(-10.22) * Expected Volume: exp(-5.81) 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=-26.4(55.23%) | Like=-21.04..-20.46 [-21.0418..-21.0415]*| it/evals=1742/2722 eff=71.9240% N=300
Z=-26.3(58.29%) | Like=-20.99..-20.46 [-20.9909..-20.9874]*| it/evals=1770/2757 eff=72.0391% N=300
Z=-26.2(61.17%) | Like=-20.96..-20.46 [-20.9573..-20.9558]*| it/evals=1798/2796 eff=72.0353% N=300
Z=-26.2(61.38%) | Like=-20.95..-20.46 [-20.9543..-20.9543]*| it/evals=1800/2798 eff=72.0576% N=300
Mono-modal Volume: ~exp(-10.22) Expected Volume: exp(-6.03) 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=-26.2(63.95%) | Like=-20.91..-20.46 [-20.9140..-20.9134]*| it/evals=1827/2835 eff=72.0710% N=300
Z=-26.2(64.21%) | Like=-20.91..-20.46 [-20.9080..-20.9024]*| it/evals=1830/2839 eff=72.0756% N=300
Z=-26.2(66.70%) | Like=-20.87..-20.46 [-20.8711..-20.8710]*| it/evals=1857/2878 eff=72.0326% N=300
Z=-26.2(66.97%) | Like=-20.87..-20.46 [-20.8664..-20.8657]*| it/evals=1860/2881 eff=72.0651% N=300
Mono-modal Volume: ~exp(-10.59) * Expected Volume: exp(-6.25) Quality: ok
index : +1.0| +2.5 **** +2.7 | +5.0
amplitude: +1.0e-12| +3.1e-11 **** +3.7e-11 | +1.0e-10
Z=-26.1(68.38%) | Like=-20.84..-20.46 [-20.8417..-20.8385]*| it/evals=1876/2899 eff=72.1816% N=300
Z=-26.1(69.57%) | Like=-20.83..-20.46 [-20.8263..-20.8257]*| it/evals=1890/2914 eff=72.3030% N=300
[ultranest] Explored until L=-2e+01
[ultranest] Likelihood function evaluations: 2920
[ultranest] logZ = -25.76 +- 0.1079
[ultranest] Effective samples strategy satisfied (ESS = 989.6, need >400)
[ultranest] Posterior uncertainty strategy is satisfied (KL: 0.46+-0.06 nat, need <0.50 nat)
[ultranest] Evidency uncertainty strategy is satisfied (dlogz=0.28, need <0.5)
[ultranest] logZ error budget: single: 0.12 bs:0.11 tail:0.26 total:0.28 required:<0.50
[ultranest] done iterating.
logZ = -25.754 +- 0.320
single instance: logZ = -25.754 +- 0.120
bootstrapped : logZ = -25.758 +- 0.184
tail : logZ = +- 0.262
insert order U test : converged: True correlation: inf iterations
index : 2.08 │ ▁ ▁▁▁▁▁▁▁▂▃▄▅▅▇▅▇▆▇▆▅▅▄▄▂▂▂▁▁▁▁▁ ▁▁▁▁ │3.12 2.58 +- 0.13
amplitude : 0.0000000000211│ ▁ ▁▁▁▁▁▁▂▂▄▅▅▅▇▅▇▆▆▅▅▅▅▄▃▂▂▂▂▁▁▁▁▁▁▁▁ │0.0000000000487 0.0000000000341 +- 0.0000000000039
[ultranest] Sampling 300 live points from prior ...
Mono-modal Volume: ~exp(-3.97) * Expected Volume: exp(0.00) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|****************** ***** * * ********* * **** *| +1.0e-10
Z=-inf(0.00%) | Like=-1065.64..-19.48 [-1065.6386..-125.5591] | it/evals=0/301 eff=0.0000% N=300
Z=-217.7(0.00%) | Like=-212.85..-19.48 [-1065.6386..-125.5591] | it/evals=30/333 eff=90.9091% N=300
Z=-199.4(0.00%) | Like=-191.97..-19.48 [-1065.6386..-125.5591] | it/evals=60/371 eff=84.5070% N=300
Mono-modal Volume: ~exp(-4.71) * Expected Volume: exp(-0.22) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|****************** *** *** * ***** ** ** *** *| +1.0e-10
Z=-195.0(0.00%) | Like=-189.21..-19.48 [-1065.6386..-125.5591] | it/evals=67/379 eff=84.8101% N=300
Z=-181.5(0.00%) | Like=-176.37..-19.48 [-1065.6386..-125.5591] | it/evals=90/410 eff=81.8182% N=300
Z=-169.9(0.00%) | Like=-165.28..-19.48 [-1065.6386..-125.5591] | it/evals=120/446 eff=82.1918% N=300
Mono-modal Volume: ~exp(-4.71) Expected Volume: exp(-0.45) Quality: ok
index : +1.0|************************************** *********| +5.0
amplitude: +1.0e-12|****************** ********** ***** ** ** *** *| +1.0e-10
Z=-158.4(0.00%) | Like=-153.07..-19.48 [-1065.6386..-125.5591] | it/evals=149/482 eff=81.8681% N=300
Z=-158.0(0.00%) | Like=-153.05..-19.48 [-1065.6386..-125.5591] | it/evals=150/483 eff=81.9672% N=300
Z=-143.7(0.00%) | Like=-137.48..-19.48 [-1065.6386..-125.5591] | it/evals=180/522 eff=81.0811% N=300
Mono-modal Volume: ~exp(-4.71) Expected Volume: exp(-0.67) Quality: ok
index : +1.0| *********************************************| +5.0
amplitude: +1.0e-12| ********************************* ************| +1.0e-10
Z=-130.3(0.00%) | Like=-123.76..-19.48 [-123.9869..-63.7122] | it/evals=206/558 eff=79.8450% N=300
Z=-128.4(0.00%) | Like=-122.92..-19.48 [-123.9869..-63.7122] | it/evals=210/563 eff=79.8479% N=300
Z=-116.6(0.00%) | Like=-111.00..-19.48 [-123.9869..-63.7122] | it/evals=240/601 eff=79.7342% N=300
Mono-modal Volume: ~exp(-4.71) Expected Volume: exp(-0.89) Quality: ok
index : +1.0| *******************************************| +5.0
amplitude: +1.0e-12| ********************************************| +1.0e-10
Z=-105.0(0.00%) | Like=-99.59..-19.48 [-123.9869..-63.7122] | it/evals=269/637 eff=79.8220% N=300
Z=-104.8(0.00%) | Like=-97.34..-19.48 [-123.9869..-63.7122] | it/evals=270/638 eff=79.8817% N=300
Z=-92.5(0.00%) | Like=-86.70..-19.48 [-123.9869..-63.7122] | it/evals=300/677 eff=79.5756% N=300
Z=-82.8(0.00%) | Like=-77.00..-19.48 [-123.9869..-63.7122] | it/evals=330/715 eff=79.5181% N=300
Mono-modal Volume: ~exp(-4.75) * Expected Volume: exp(-1.12) Quality: ok
index : +1.0| *****************************************| +5.0
amplitude: +1.0e-12| *******************************************| +1.0e-10
Z=-81.6(0.00%) | Like=-76.12..-19.48 [-123.9869..-63.7122] | it/evals=335/722 eff=79.3839% N=300
Z=-75.6(0.00%) | Like=-70.25..-19.48 [-123.9869..-63.7122] | it/evals=360/753 eff=79.4702% N=300
Z=-70.4(0.00%) | Like=-65.46..-19.48 [-123.9869..-63.7122] | it/evals=389/793 eff=78.9047% N=300
Z=-70.3(0.00%) | Like=-65.31..-19.48 [-123.9869..-63.7122] | it/evals=390/795 eff=78.7879% N=300
Mono-modal Volume: ~exp(-5.26) * Expected Volume: exp(-1.34) Quality: ok
index : +1.0| ****************************************| +5.0
amplitude: +1.0e-12| *****************************************| +1.0e-10
Z=-69.0(0.00%) | Like=-64.09..-19.48 [-123.9869..-63.7122] | it/evals=402/809 eff=78.9784% N=300
Z=-66.8(0.00%) | Like=-61.68..-19.48 [-63.6854..-43.6647] | it/evals=420/830 eff=79.2453% N=300
Z=-63.1(0.00%) | Like=-58.14..-19.48 [-63.6854..-43.6647] | it/evals=450/869 eff=79.0861% N=300
Mono-modal Volume: ~exp(-5.44) * Expected Volume: exp(-1.56) Quality: ok
index : +1.0| ***************************************| +5.0
amplitude: +1.0e-12| ****************************************| +1.0e-10
Z=-61.1(0.00%) | Like=-56.03..-19.48 [-63.6854..-43.6647] | it/evals=469/893 eff=79.0894% N=300
Z=-59.8(0.00%) | Like=-54.45..-19.48 [-63.6854..-43.6647] | it/evals=480/905 eff=79.3388% N=300
Z=-56.6(0.00%) | Like=-51.54..-19.48 [-63.6854..-43.6647] | it/evals=507/946 eff=78.4830% N=300
Z=-56.3(0.00%) | Like=-51.44..-19.48 [-63.6854..-43.6647] | it/evals=510/949 eff=78.5824% 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=-54.3(0.00%) | Like=-49.37..-19.48 [-63.6854..-43.6647] | it/evals=536/988 eff=77.9070% N=300
Z=-54.0(0.00%) | Like=-49.05..-19.48 [-63.6854..-43.6647] | it/evals=540/993 eff=77.9221% N=300
Z=-51.6(0.00%) | Like=-46.34..-19.48 [-63.6854..-43.6647] | it/evals=568/1033 eff=77.4898% N=300
Z=-51.5(0.00%) | Like=-46.15..-19.48 [-63.6854..-43.6647] | it/evals=570/1035 eff=77.5510% N=300
Z=-49.3(0.00%) | Like=-44.34..-19.48 [-63.6854..-43.6647] | it/evals=600/1073 eff=77.6197% 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| **************************************| +1.0e-10
Z=-47.3(0.00%) | Like=-41.88..-19.48 [-43.5857..-31.4918] | it/evals=629/1109 eff=77.7503% N=300
Z=-47.2(0.00%) | Like=-41.81..-19.48 [-43.5857..-31.4918] | it/evals=630/1110 eff=77.7778% N=300
Z=-44.6(0.00%) | Like=-38.99..-19.48 [-43.5857..-31.4918] | it/evals=659/1150 eff=77.5294% N=300
Z=-44.5(0.00%) | Like=-38.98..-19.48 [-43.5857..-31.4918] | it/evals=660/1151 eff=77.5558% N=300
Mono-modal Volume: ~exp(-6.19) * Expected Volume: exp(-2.23) Quality: ok
index : +1.0| +2.0 ************************** +4.1 | +5.0
amplitude: +1.0e-12| +2.7e-11 *********************************** | +1.0e-10
Z=-43.7(0.00%) | Like=-38.22..-19.48 [-43.5857..-31.4918] | it/evals=670/1168 eff=77.1889% N=300
Z=-42.2(0.00%) | Like=-36.91..-19.23 [-43.5857..-31.4918] | it/evals=690/1195 eff=77.0950% N=300
Z=-40.6(0.00%) | Like=-35.37..-19.23 [-43.5857..-31.4918] | it/evals=719/1237 eff=76.7343% N=300
Z=-40.6(0.00%) | Like=-35.31..-19.23 [-43.5857..-31.4918] | it/evals=720/1238 eff=76.7591% N=300
Mono-modal Volume: ~exp(-6.42) * Expected Volume: exp(-2.46) Quality: ok
index : +1.0| +2.1 *********************** +4.0 | +5.0
amplitude: +1.0e-12| +3.0e-11 ******************************** | +1.0e-10
Z=-39.7(0.00%) | Like=-34.66..-19.18 [-43.5857..-31.4918] | it/evals=737/1264 eff=76.4523% N=300
Z=-39.2(0.00%) | Like=-34.21..-19.18 [-43.5857..-31.4918] | it/evals=750/1280 eff=76.5306% N=300
Z=-38.1(0.00%) | Like=-32.89..-19.18 [-43.5857..-31.4918] | it/evals=777/1320 eff=76.1765% N=300
Z=-38.0(0.00%) | Like=-32.78..-19.18 [-43.5857..-31.4918] | it/evals=780/1330 eff=75.7282% N=300
Mono-modal Volume: ~exp(-6.82) * Expected Volume: exp(-2.68) Quality: ok
index : +1.0| +2.2 ******************** +3.8 | +5.0
amplitude: +1.0e-12| +3.0e-11 ***************************** | +1.0e-10
Z=-37.0(0.00%) | Like=-31.88..-19.18 [-43.5857..-31.4918] | it/evals=804/1362 eff=75.7062% N=300
Z=-36.8(0.00%) | Like=-31.63..-19.18 [-43.5857..-31.4918] | it/evals=810/1371 eff=75.6303% N=300
Z=-35.7(0.00%) | Like=-30.35..-19.18 [-31.4558..-26.2349] | it/evals=840/1409 eff=75.7439% N=300
Z=-34.7(0.00%) | Like=-29.47..-19.18 [-31.4558..-26.2349] | it/evals=868/1450 eff=75.4783% N=300
Z=-34.6(0.00%) | Like=-29.45..-19.18 [-31.4558..-26.2349] | it/evals=870/1452 eff=75.5208% N=300
Mono-modal Volume: ~exp(-6.82) Expected Volume: exp(-2.90) Quality: ok
index : +1.0| +2.2 ******************* +3.7 | +5.0
amplitude: +1.0e-12| +3.3e-11 ************************** | +1.0e-10
Z=-33.7(0.01%) | Like=-28.45..-19.18 [-31.4558..-26.2349] | it/evals=900/1486 eff=75.8853% N=300
Z=-32.7(0.03%) | Like=-27.52..-19.18 [-31.4558..-26.2349] | it/evals=930/1524 eff=75.9804% N=300
Mono-modal Volume: ~exp(-6.83) * Expected Volume: exp(-3.13) Quality: ok
index : +1.0| +2.3 **************** +3.5 | +5.0
amplitude: +1.0e-12| +3.5e-11 *********************** | +1.0e-10
Z=-32.5(0.03%) | Like=-27.28..-19.18 [-31.4558..-26.2349] | it/evals=938/1534 eff=76.0130% N=300
Z=-31.9(0.06%) | Like=-26.61..-19.18 [-31.4558..-26.2349] | it/evals=960/1571 eff=75.5311% N=300
Z=-31.2(0.12%) | Like=-26.07..-19.18 [-26.2181..-25.6820] | it/evals=990/1609 eff=75.6303% N=300
Mono-modal Volume: ~exp(-7.61) * 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=-30.9(0.15%) | Like=-25.77..-19.18 [-26.2181..-25.6820] | it/evals=1005/1629 eff=75.6208% N=300
Z=-30.6(0.21%) | Like=-25.40..-19.18 [-25.4035..-25.3875] | it/evals=1020/1645 eff=75.8364% N=300
Z=-30.0(0.36%) | Like=-24.74..-19.18 [-24.7856..-24.7377] | it/evals=1050/1685 eff=75.8123% N=300
Mono-modal Volume: ~exp(-7.75) * Expected Volume: exp(-3.57) Quality: ok
index : +1.0| +2.4 ************* +3.4 | +5.0
amplitude: +1.0e-12| +3.8e-11 ******************** +7.6e-11 | +1.0e-10
Z=-29.6(0.54%) | Like=-24.41..-19.18 [-24.4123..-24.3814] | it/evals=1072/1721 eff=75.4398% N=300
Z=-29.5(0.61%) | Like=-24.27..-19.18 [-24.2748..-24.2739]*| it/evals=1080/1730 eff=75.5245% N=300
Z=-29.0(1.01%) | Like=-23.61..-19.18 [-23.6094..-23.6045]*| it/evals=1110/1767 eff=75.6646% N=300
Mono-modal Volume: ~exp(-7.75) 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=-28.5(1.54%) | Like=-23.34..-19.17 [-23.3512..-23.3409] | it/evals=1139/1805 eff=75.6811% N=300
Z=-28.5(1.56%) | Like=-23.34..-19.17 [-23.3366..-23.3344]*| it/evals=1140/1806 eff=75.6972% N=300
Z=-28.1(2.28%) | Like=-22.92..-19.17 [-22.9199..-22.9197]*| it/evals=1169/1845 eff=75.6634% N=300
Z=-28.1(2.32%) | Like=-22.92..-19.17 [-22.9197..-22.8981] | it/evals=1170/1847 eff=75.6303% N=300
Z=-27.8(3.31%) | Like=-22.60..-19.17 [-22.6003..-22.5910]*| it/evals=1199/1886 eff=75.5990% N=300
Z=-27.8(3.35%) | Like=-22.59..-19.17 [-22.5910..-22.5672] | it/evals=1200/1889 eff=75.5192% N=300
Mono-modal Volume: ~exp(-7.75) Expected Volume: exp(-4.02) Quality: ok
index : +1.0| +2.4 *********** +3.3 | +5.0
amplitude: +1.0e-12| +4.1e-11 **************** +7.1e-11 | +1.0e-10
Z=-27.5(4.24%) | Like=-22.29..-19.17 [-22.2882..-22.2827]*| it/evals=1223/1928 eff=75.1229% N=300
Z=-27.5(4.57%) | Like=-22.22..-19.17 [-22.2222..-22.2159]*| it/evals=1230/1936 eff=75.1834% N=300
Z=-27.2(6.01%) | Like=-21.89..-19.17 [-21.8934..-21.8921]*| it/evals=1258/1976 eff=75.0597% N=300
Z=-27.2(6.12%) | Like=-21.88..-19.17 [-21.8921..-21.8802] | it/evals=1260/1979 eff=75.0447% N=300
Mono-modal Volume: ~exp(-8.54) * 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(7.01%) | Like=-21.78..-19.17 [-21.7772..-21.7688]*| it/evals=1273/1993 eff=75.1920% N=300
Z=-26.9(8.14%) | Like=-21.58..-19.17 [-21.5772..-21.5616] | it/evals=1290/2015 eff=75.2187% N=300
Z=-26.6(10.47%) | Like=-21.30..-19.17 [-21.2968..-21.2946]*| it/evals=1320/2052 eff=75.3425% N=300
Mono-modal Volume: ~exp(-8.54) Expected Volume: exp(-4.47) Quality: ok
index : +1.0| +2.5 ******** +3.2 | +5.0
amplitude: +1.0e-12| +4.4e-11 ************* +6.7e-11 | +1.0e-10
Z=-26.4(12.68%) | Like=-21.14..-19.17 [-21.1393..-21.1306]*| it/evals=1347/2091 eff=75.2094% N=300
Z=-26.4(13.01%) | Like=-21.12..-19.17 [-21.1187..-21.1070] | it/evals=1350/2095 eff=75.2089% N=300
Z=-26.2(15.72%) | Like=-20.95..-19.17 [-20.9477..-20.9456]*| it/evals=1379/2134 eff=75.1908% N=300
Z=-26.2(15.84%) | Like=-20.95..-19.17 [-20.9456..-20.9450]*| it/evals=1380/2137 eff=75.1225% N=300
Z=-26.0(18.58%) | Like=-20.80..-19.17 [-20.8045..-20.8030]*| it/evals=1405/2178 eff=74.8136% N=300
Mono-modal Volume: ~exp(-8.55) * Expected Volume: exp(-4.69) Quality: ok
index : +1.0| +2.5 ******** +3.1 | +5.0
amplitude: +1.0e-12| +4.6e-11 *********** +6.6e-11 | +1.0e-10
Z=-26.0(18.75%) | Like=-20.80..-19.17 [-20.7983..-20.7951]*| it/evals=1407/2180 eff=74.8404% N=300
Z=-26.0(19.13%) | Like=-20.78..-19.17 [-20.7820..-20.7600] | it/evals=1410/2183 eff=74.8805% N=300
Z=-25.9(22.41%) | Like=-20.61..-19.17 [-20.6079..-20.6027]*| it/evals=1438/2222 eff=74.8179% N=300
Z=-25.8(22.62%) | Like=-20.59..-19.17 [-20.5950..-20.5907]*| it/evals=1440/2226 eff=74.7664% N=300
Z=-25.7(26.39%) | Like=-20.47..-19.17 [-20.4673..-20.4644]*| it/evals=1470/2266 eff=74.7711% N=300
Mono-modal Volume: ~exp(-9.19) * Expected Volume: exp(-4.91) 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=-25.7(26.86%) | Like=-20.46..-19.17 [-20.4580..-20.4576]*| it/evals=1474/2271 eff=74.7844% N=300
Z=-25.6(29.95%) | Like=-20.37..-19.17 [-20.3652..-20.3638]*| it/evals=1500/2302 eff=74.9251% N=300
Z=-25.5(33.51%) | Like=-20.26..-19.17 [-20.2578..-20.2542]*| it/evals=1530/2341 eff=74.9633% N=300
Mono-modal Volume: ~exp(-9.19) Expected Volume: exp(-5.14) Quality: ok
index : +1.0| +2.6 ****** +3.1 | +5.0
amplitude: +1.0e-12| +4.7e-11 ********* +6.4e-11 | +1.0e-10
Z=-25.4(36.58%) | Like=-20.15..-19.17 [-20.1491..-20.1412]*| it/evals=1555/2377 eff=74.8676% N=300
Z=-25.3(37.07%) | Like=-20.12..-19.17 [-20.1240..-20.1229]*| it/evals=1560/2384 eff=74.8560% N=300
Z=-25.2(40.66%) | Like=-20.06..-19.17 [-20.0586..-20.0496]*| it/evals=1590/2423 eff=74.8940% N=300
Mono-modal Volume: ~exp(-9.27) * Expected Volume: exp(-5.36) Quality: ok
index : +1.0| +2.6 ****** +3.0 | +5.0
amplitude: +1.0e-12| +4.8e-11 ********* +6.3e-11 | +1.0e-10
Z=-25.2(42.78%) | Like=-20.00..-19.17 [-20.0023..-20.0001]*| it/evals=1608/2453 eff=74.6865% N=300
Z=-25.2(44.29%) | Like=-19.97..-19.17 [-19.9735..-19.9587] | it/evals=1620/2475 eff=74.4828% N=300
Z=-25.1(47.69%) | Like=-19.90..-19.17 [-19.9044..-19.9013]*| it/evals=1648/2514 eff=74.4354% N=300
Z=-25.1(47.92%) | Like=-19.90..-19.17 [-19.9000..-19.8851] | it/evals=1650/2516 eff=74.4585% N=300
Mono-modal Volume: ~exp(-9.59) * Expected Volume: exp(-5.58) Quality: ok
index : +1.0| +2.6 ***** +3.0 | +5.0
amplitude: +1.0e-12| +4.9e-11 ******* +6.2e-11 | +1.0e-10
Z=-25.0(50.83%) | Like=-19.83..-19.16 [-19.8326..-19.8323]*| it/evals=1675/2551 eff=74.4114% N=300
Z=-25.0(51.41%) | Like=-19.83..-19.16 [-19.8271..-19.8255]*| it/evals=1680/2556 eff=74.4681% N=300
Z=-25.0(54.88%) | Like=-19.76..-19.16 [-19.7638..-19.7631]*| it/evals=1710/2595 eff=74.5098% N=300
Z=-24.9(57.72%) | Like=-19.72..-19.16 [-19.7235..-19.7207]*| it/evals=1736/2635 eff=74.3469% N=300
Z=-24.9(58.17%) | Like=-19.71..-19.16 [-19.7089..-19.7085]*| it/evals=1740/2642 eff=74.2955% N=300
Mono-modal Volume: ~exp(-9.59) Expected Volume: exp(-5.81) Quality: ok
index : +1.0| +2.6 ***** +3.0 | +5.0
amplitude: +1.0e-12| +4.9e-11 ******* +6.1e-11 | +1.0e-10
Z=-24.9(60.63%) | Like=-19.68..-19.16 [-19.6757..-19.6732]*| it/evals=1764/2678 eff=74.1800% N=300
Z=-24.8(61.28%) | Like=-19.67..-19.16 [-19.6689..-19.6676]*| it/evals=1770/2688 eff=74.1206% N=300
Z=-24.8(63.61%) | Like=-19.63..-19.16 [-19.6334..-19.6319]*| it/evals=1794/2728 eff=73.8880% N=300
Z=-24.8(64.18%) | Like=-19.62..-19.16 [-19.6202..-19.6192]*| it/evals=1800/2736 eff=73.8916% N=300
Mono-modal Volume: ~exp(-10.11) * Expected Volume: exp(-6.03) Quality: ok
index : +1.0| +2.7 ***** +3.0 | +5.0
amplitude: +1.0e-12| +5.0e-11 ******* +6.1e-11 | +1.0e-10
Z=-24.8(65.03%) | Like=-19.61..-19.16 [-19.6119..-19.6111]*| it/evals=1809/2748 eff=73.8971% N=300
Z=-24.8(66.93%) | Like=-19.57..-19.16 [-19.5742..-19.5725]*| it/evals=1830/2771 eff=74.0591% N=300
Z=-24.7(69.35%) | Like=-19.53..-19.16 [-19.5343..-19.5339]*| it/evals=1858/2811 eff=73.9944% N=300
Z=-24.7(69.53%) | Like=-19.53..-19.16 [-19.5323..-19.5309]*| it/evals=1860/2813 eff=74.0151% N=300
[ultranest] Explored until L=-2e+01
[ultranest] Likelihood function evaluations: 2818
[ultranest] logZ = -24.37 +- 0.09285
[ultranest] Effective samples strategy satisfied (ESS = 986.0, need >400)
[ultranest] Posterior uncertainty strategy is satisfied (KL: 0.46+-0.07 nat, need <0.50 nat)
[ultranest] Evidency uncertainty strategy is satisfied (dlogz=0.28, need <0.5)
[ultranest] logZ error budget: single: 0.12 bs:0.09 tail:0.26 total:0.28 required:<0.50
[ultranest] done iterating.
logZ = -24.356 +- 0.340
single instance: logZ = -24.356 +- 0.119
bootstrapped : logZ = -24.367 +- 0.217
tail : logZ = +- 0.262
insert order U test : converged: True correlation: inf iterations
index : 2.30 │ ▁▁▁▁▁▁▂▃▃▅▅▅▆▇▇▇▇▇▇▅▅▆▆▃▄▃▂▂▂▁▁▁▁▁▁▁▁ │3.45 2.83 +- 0.17
amplitude : 0.0000000000360│ ▁▁▁▁▁▁▂▃▄▅▆▇▇▇▇▇▇▆▆▅▄▃▃▃▂▁▁▁▁▁▁ ▁ │0.0000000000822 0.0000000000549 +- 0.0000000000058
[ultranest] Sampling 300 live points from prior ...
Mono-modal Volume: ~exp(-3.66) * Expected Volume: exp(0.00) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|*********************************** ****** **| +1.0e-10
Z=-inf(0.00%) | Like=-534.49..-13.35 [-534.4857..-89.3211] | it/evals=0/301 eff=0.0000% N=300
Z=-151.0(0.00%) | Like=-146.32..-13.35 [-534.4857..-89.3211] | it/evals=30/331 eff=96.7742% N=300
Z=-140.2(0.00%) | Like=-135.64..-13.35 [-534.4857..-89.3211] | it/evals=60/368 eff=88.2353% N=300
Mono-modal Volume: ~exp(-4.43) * Expected Volume: exp(-0.22) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|*********************************** *** ** **| +1.0e-10
Z=-137.7(0.00%) | Like=-132.87..-13.35 [-534.4857..-89.3211] | it/evals=67/375 eff=89.3333% N=300
Z=-131.0(0.00%) | Like=-126.06..-13.35 [-534.4857..-89.3211] | it/evals=90/403 eff=87.3786% N=300
Z=-122.5(0.00%) | Like=-116.43..-13.35 [-534.4857..-89.3211] | it/evals=120/437 eff=87.5912% N=300
Mono-modal Volume: ~exp(-4.44) * Expected Volume: exp(-0.45) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|************************************* *** ** **| +1.0e-10
Z=-117.1(0.00%) | Like=-112.11..-13.35 [-534.4857..-89.3211] | it/evals=134/453 eff=87.5817% N=300
Z=-113.3(0.00%) | Like=-108.17..-13.35 [-534.4857..-89.3211] | it/evals=150/473 eff=86.7052% N=300
Z=-104.1(0.00%) | Like=-99.13..-13.35 [-534.4857..-89.3211] | it/evals=180/508 eff=86.5385% N=300
Mono-modal Volume: ~exp(-4.44) Expected Volume: exp(-0.67) Quality: ok
index : +1.0| **********************************************| +5.0
amplitude: +1.0e-12| ************************************* *** ** **| +1.0e-10
Z=-95.9(0.00%) | Like=-90.57..-13.35 [-534.4857..-89.3211] | it/evals=210/543 eff=86.4198% N=300
Z=-87.6(0.00%) | Like=-82.19..-13.35 [-89.2046..-48.6708] | it/evals=238/583 eff=84.0989% N=300
Z=-87.1(0.00%) | Like=-82.15..-13.35 [-89.2046..-48.6708] | it/evals=240/586 eff=83.9161% N=300
Mono-modal Volume: ~exp(-5.10) * Expected Volume: exp(-0.89) Quality: ok
index : +1.0| ********************************************| +5.0
amplitude: +1.0e-12| ************************************ *** *****| +1.0e-10
Z=-80.6(0.00%) | Like=-75.30..-13.35 [-89.2046..-48.6708] | it/evals=268/624 eff=82.7160% N=300
Z=-80.2(0.00%) | Like=-75.24..-13.35 [-89.2046..-48.6708] | it/evals=270/626 eff=82.8221% N=300
Z=-74.5(0.00%) | Like=-69.35..-13.35 [-89.2046..-48.6708] | it/evals=298/666 eff=81.4208% N=300
Z=-74.1(0.00%) | Like=-69.03..-13.35 [-89.2046..-48.6708] | it/evals=300/671 eff=80.8625% N=300
Z=-69.5(0.00%) | Like=-64.81..-13.35 [-89.2046..-48.6708] | it/evals=330/708 eff=80.8824% N=300
Mono-modal Volume: ~exp(-5.42) * Expected Volume: exp(-1.12) Quality: ok
index : +1.0| *******************************************| +5.0
amplitude: +1.0e-12| *************************************** *****| +1.0e-10
Z=-68.8(0.00%) | Like=-63.88..-13.35 [-89.2046..-48.6708] | it/evals=335/717 eff=80.3357% N=300
Z=-63.5(0.00%) | Like=-58.37..-13.35 [-89.2046..-48.6708] | it/evals=360/750 eff=80.0000% N=300
Z=-58.0(0.00%) | Like=-52.91..-13.35 [-89.2046..-48.6708] | it/evals=390/786 eff=80.2469% 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=-54.3(0.00%) | Like=-49.23..-13.35 [-89.2046..-48.6708] | it/evals=418/822 eff=80.0766% N=300
Z=-54.0(0.00%) | Like=-48.85..-13.35 [-89.2046..-48.6708] | it/evals=420/824 eff=80.1527% N=300
Z=-51.0(0.00%) | Like=-46.44..-13.35 [-48.6405..-30.6663] | it/evals=450/863 eff=79.9290% N=300
Mono-modal Volume: ~exp(-5.55) * Expected Volume: exp(-1.56) Quality: ok
index : +1.0| ************************************** | +5.0
amplitude: +1.0e-12| ******************************************| +1.0e-10
Z=-49.4(0.00%) | Like=-44.27..-13.35 [-48.6405..-30.6663] | it/evals=469/883 eff=80.4460% N=300
Z=-48.2(0.00%) | Like=-43.11..-13.35 [-48.6405..-30.6663] | it/evals=480/896 eff=80.5369% N=300
Z=-45.3(0.00%) | Like=-39.91..-13.35 [-48.6405..-30.6663] | it/evals=508/936 eff=79.8742% N=300
Z=-45.0(0.00%) | Like=-39.26..-13.35 [-48.6405..-30.6663] | it/evals=510/939 eff=79.8122% N=300
Mono-modal Volume: ~exp(-5.85) * Expected Volume: exp(-1.79) Quality: ok
index : +1.0| ******************************* | +5.0
amplitude: +1.0e-12| *****************************************| +1.0e-10
Z=-42.3(0.00%) | Like=-37.04..-13.35 [-48.6405..-30.6663] | it/evals=536/971 eff=79.8808% N=300
Z=-41.9(0.00%) | Like=-36.85..-13.35 [-48.6405..-30.6663] | it/evals=540/979 eff=79.5287% N=300
Z=-39.7(0.00%) | Like=-34.48..-13.35 [-48.6405..-30.6663] | it/evals=569/1019 eff=79.1377% N=300
Z=-39.6(0.00%) | Like=-34.43..-13.35 [-48.6405..-30.6663] | it/evals=570/1022 eff=78.9474% N=300
Z=-37.5(0.00%) | Like=-32.33..-13.35 [-48.6405..-30.6663] | it/evals=600/1057 eff=79.2602% N=300
Mono-modal Volume: ~exp(-6.04) * Expected Volume: exp(-2.01) Quality: ok
index : +1.0| **************************** +4.2 | +5.0
amplitude: +1.0e-12| ****************************************| +1.0e-10
Z=-37.3(0.00%) | Like=-32.22..-13.35 [-48.6405..-30.6663] | it/evals=603/1060 eff=79.3421% N=300
Z=-35.5(0.00%) | Like=-30.28..-13.35 [-30.5529..-22.0571] | it/evals=630/1095 eff=79.2453% N=300
Z=-33.8(0.00%) | Like=-28.79..-13.35 [-30.5529..-22.0571] | it/evals=655/1136 eff=78.3493% N=300
Z=-33.5(0.00%) | Like=-28.36..-13.35 [-30.5529..-22.0571] | it/evals=660/1143 eff=78.2918% N=300
Mono-modal Volume: ~exp(-6.22) * Expected Volume: exp(-2.23) Quality: ok
index : +1.0| +2.0 ************************ +3.8 | +5.0
amplitude: +1.0e-12| **************************************| +1.0e-10
Z=-32.8(0.00%) | Like=-27.45..-13.35 [-30.5529..-22.0571] | it/evals=670/1153 eff=78.5463% N=300
Z=-31.5(0.00%) | Like=-26.37..-13.35 [-30.5529..-22.0571] | it/evals=690/1181 eff=78.3201% N=300
Z=-30.3(0.00%) | Like=-25.31..-13.35 [-30.5529..-22.0571] | it/evals=715/1221 eff=77.6330% N=300
Z=-30.1(0.00%) | Like=-25.22..-13.35 [-30.5529..-22.0571] | it/evals=720/1226 eff=77.7538% N=300
Mono-modal Volume: ~exp(-6.61) * Expected Volume: exp(-2.46) Quality: ok
index : +1.0| +2.1 ********************* +3.7 | +5.0
amplitude: +1.0e-12| +2.5e-11 *************************************| +1.0e-10
Z=-29.5(0.00%) | Like=-24.75..-13.31 [-30.5529..-22.0571] | it/evals=737/1250 eff=77.5789% N=300
Z=-29.1(0.00%) | Like=-24.52..-13.31 [-30.5529..-22.0571] | it/evals=750/1268 eff=77.4793% N=300
Z=-28.4(0.00%) | Like=-23.74..-13.31 [-30.5529..-22.0571] | it/evals=777/1308 eff=77.0833% N=300
Z=-28.3(0.00%) | Like=-23.67..-13.31 [-30.5529..-22.0571] | it/evals=780/1314 eff=76.9231% N=300
Mono-modal Volume: ~exp(-6.67) * Expected Volume: exp(-2.68) Quality: ok
index : +1.0| +2.1 ******************* +3.6 | +5.0
amplitude: +1.0e-12| +2.7e-11 ********************************* | +1.0e-10
Z=-27.7(0.01%) | Like=-22.98..-13.31 [-30.5529..-22.0571] | it/evals=804/1348 eff=76.7176% N=300
Z=-27.5(0.01%) | Like=-22.69..-13.31 [-30.5529..-22.0571] | it/evals=810/1355 eff=76.7773% N=300
Z=-26.7(0.02%) | Like=-21.82..-13.31 [-22.0564..-20.0033] | it/evals=840/1389 eff=77.1350% N=300
Z=-26.0(0.04%) | Like=-21.24..-13.31 [-22.0564..-20.0033] | it/evals=868/1429 eff=76.8822% N=300
Z=-26.0(0.04%) | Like=-21.14..-13.31 [-22.0564..-20.0033] | it/evals=870/1432 eff=76.8551% N=300
Mono-modal Volume: ~exp(-6.96) * 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=-26.0(0.05%) | Like=-21.08..-13.31 [-22.0564..-20.0033] | it/evals=871/1433 eff=76.8756% N=300
Z=-25.3(0.09%) | Like=-20.49..-13.31 [-22.0564..-20.0033] | it/evals=900/1472 eff=76.7918% N=300
Z=-24.8(0.16%) | Like=-19.94..-13.31 [-19.9889..-19.7376] | it/evals=930/1509 eff=76.9231% N=300
Mono-modal Volume: ~exp(-7.09) * 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(0.19%) | Like=-19.71..-13.31 [-19.7146..-19.6993] | it/evals=938/1522 eff=76.7594% N=300
Z=-24.2(0.27%) | Like=-19.30..-13.31 [-19.2987..-19.2891]*| it/evals=960/1550 eff=76.8000% N=300
Z=-23.7(0.46%) | Like=-18.82..-13.31 [-18.8383..-18.8234] | it/evals=990/1589 eff=76.8037% N=300
Mono-modal Volume: ~exp(-7.23) * Expected Volume: exp(-3.35) Quality: ok
index : +1.0| +2.3 ************** +3.4 | +5.0
amplitude: +1.0e-12| +3.2e-11 ************************** | +1.0e-10
Z=-23.4(0.59%) | Like=-18.49..-13.31 [-18.5420..-18.4888] | it/evals=1005/1610 eff=76.7176% N=300
Z=-23.2(0.75%) | Like=-18.19..-13.31 [-18.1923..-18.1887]*| it/evals=1020/1627 eff=76.8651% N=300
Z=-22.8(1.09%) | Like=-17.90..-13.31 [-17.9210..-17.8957] | it/evals=1044/1667 eff=76.3716% N=300
Z=-22.8(1.19%) | Like=-17.77..-13.31 [-17.7712..-17.7606] | it/evals=1050/1676 eff=76.3081% N=300
Mono-modal Volume: ~exp(-7.59) * Expected Volume: exp(-3.57) Quality: ok
index : +1.0| +2.3 ************* +3.3 | +5.0
amplitude: +1.0e-12| +3.4e-11 *********************** +7.8e-11| +1.0e-10
Z=-22.4(1.67%) | Like=-17.41..-13.31 [-17.4233..-17.4130] | it/evals=1072/1706 eff=76.2447% N=300
Z=-22.3(1.82%) | Like=-17.37..-13.31 [-17.3652..-17.3650]*| it/evals=1080/1716 eff=76.2712% N=300
Z=-21.9(2.73%) | Like=-16.92..-13.31 [-16.9328..-16.9154] | it/evals=1110/1753 eff=76.3937% N=300
Z=-21.6(3.73%) | Like=-16.64..-13.31 [-16.6604..-16.6403] | it/evals=1136/1792 eff=76.1394% N=300
Mono-modal Volume: ~exp(-7.59) Expected Volume: exp(-3.80) Quality: ok
index : +1.0| +2.3 *********** +3.2 | +5.0
amplitude: +1.0e-12| +3.6e-11 ******************** +7.5e-11 | +1.0e-10
Z=-21.6(3.88%) | Like=-16.60..-13.31 [-16.5968..-16.5912]*| it/evals=1140/1798 eff=76.1015% N=300
Z=-21.3(5.12%) | Like=-16.37..-13.31 [-16.3698..-16.3527] | it/evals=1166/1840 eff=75.7143% N=300
Z=-21.3(5.28%) | Like=-16.34..-13.31 [-16.3378..-16.3340]*| it/evals=1170/1844 eff=75.7772% N=300
Z=-21.0(6.79%) | Like=-16.03..-13.31 [-16.0310..-16.0275]*| it/evals=1199/1886 eff=75.5990% N=300
Z=-21.0(6.86%) | Like=-16.03..-13.31 [-16.0275..-16.0172] | it/evals=1200/1887 eff=75.6144% N=300
Mono-modal Volume: ~exp(-7.65) * Expected Volume: exp(-4.02) Quality: ok
index : +1.0| +2.4 ********** +3.2 | +5.0
amplitude: +1.0e-12| +3.7e-11 ****************** +7.1e-11 | +1.0e-10
Z=-21.0(7.25%) | Like=-15.97..-13.31 [-15.9715..-15.9532] | it/evals=1206/1897 eff=75.5166% N=300
Z=-20.8(8.97%) | Like=-15.78..-13.31 [-15.7758..-15.7663]*| it/evals=1230/1928 eff=75.5528% N=300
Z=-20.5(11.05%) | Like=-15.59..-13.31 [-15.5857..-15.5846]*| it/evals=1258/1967 eff=75.4649% N=300
Z=-20.5(11.22%) | Like=-15.56..-13.31 [-15.5846..-15.5615] | it/evals=1260/1970 eff=75.4491% N=300
Mono-modal Volume: ~exp(-8.10) * Expected Volume: exp(-4.24) Quality: ok
index : +1.0| +2.4 ********** +3.1 | +5.0
amplitude: +1.0e-12| +3.8e-11 **************** +7.1e-11 | +1.0e-10
Z=-20.4(12.42%) | Like=-15.49..-13.31 [-15.4933..-15.4785] | it/evals=1273/1988 eff=75.4147% N=300
Z=-20.3(13.79%) | Like=-15.38..-13.31 [-15.3767..-15.3722]*| it/evals=1290/2009 eff=75.4827% N=300
Z=-20.2(16.27%) | Like=-15.24..-13.31 [-15.2404..-15.2376]*| it/evals=1317/2049 eff=75.3002% N=300
Z=-20.1(16.60%) | Like=-15.22..-13.31 [-15.2164..-15.2114]*| it/evals=1320/2054 eff=75.2566% N=300
Mono-modal Volume: ~exp(-8.16) * Expected Volume: exp(-4.47) Quality: ok
index : +1.0| +2.4 ******** +3.1 | +5.0
amplitude: +1.0e-12| +4.0e-11 *************** +6.9e-11 | +1.0e-10
Z=-20.0(18.64%) | Like=-15.08..-13.31 [-15.0759..-15.0749]*| it/evals=1340/2087 eff=74.9860% N=300
Z=-20.0(19.65%) | Like=-15.03..-13.31 [-15.0324..-15.0255]*| it/evals=1350/2104 eff=74.8337% N=300
Z=-19.8(22.48%) | Like=-14.88..-13.31 [-14.8805..-14.8783]*| it/evals=1378/2143 eff=74.7694% N=300
Z=-19.8(22.70%) | Like=-14.88..-13.31 [-14.8781..-14.8729]*| it/evals=1380/2145 eff=74.7967% N=300
Mono-modal Volume: ~exp(-8.68) * Expected Volume: exp(-4.69) Quality: ok
index : +1.0| +2.5 ******** +3.0 | +5.0
amplitude: +1.0e-12| +4.1e-11 ************* +6.6e-11 | +1.0e-10
Z=-19.7(25.63%) | Like=-14.73..-13.31 [-14.7414..-14.7278] | it/evals=1407/2179 eff=74.8803% N=300
Z=-19.7(25.98%) | Like=-14.72..-13.31 [-14.7234..-14.7176]*| it/evals=1410/2183 eff=74.8805% N=300
Z=-19.6(29.27%) | Like=-14.63..-13.31 [-14.6285..-14.6135] | it/evals=1440/2219 eff=75.0391% N=300
Z=-19.5(32.00%) | Like=-14.51..-13.31 [-14.5146..-14.4977] | it/evals=1465/2259 eff=74.7831% N=300
Z=-19.5(32.58%) | Like=-14.48..-13.31 [-14.4757..-14.4730]*| it/evals=1470/2265 eff=74.8092% N=300
Mono-modal Volume: ~exp(-8.68) Expected Volume: exp(-4.91) Quality: ok
index : +1.0| +2.5 ******* +3.0 | +5.0
amplitude: +1.0e-12| +4.2e-11 ************* +6.5e-11 | +1.0e-10
Z=-19.4(35.26%) | Like=-14.40..-13.31 [-14.3997..-14.3900]*| it/evals=1493/2303 eff=74.5382% N=300
Z=-19.4(35.99%) | Like=-14.36..-13.31 [-14.3797..-14.3600] | it/evals=1500/2311 eff=74.5898% N=300
Z=-19.3(38.87%) | Like=-14.27..-13.31 [-14.2730..-14.2600] | it/evals=1524/2351 eff=74.3052% N=300
Z=-19.3(39.54%) | Like=-14.24..-13.31 [-14.2414..-14.2404]*| it/evals=1530/2360 eff=74.2718% N=300
Mono-modal Volume: ~exp(-9.19) * Expected Volume: exp(-5.14) Quality: ok
index : +1.0| +2.5 ****** +3.0 | +5.0
amplitude: +1.0e-12| +4.4e-11 *********** +6.4e-11 | +1.0e-10
Z=-19.2(40.86%) | Like=-14.20..-13.31 [-14.1959..-14.1958]*| it/evals=1541/2384 eff=73.9443% N=300
Z=-19.2(43.30%) | Like=-14.15..-13.31 [-14.1454..-14.1439]*| it/evals=1560/2406 eff=74.0741% N=300
Z=-19.1(46.63%) | Like=-14.08..-13.30 [-14.0825..-14.0820]*| it/evals=1589/2445 eff=74.0793% N=300
Z=-19.1(46.76%) | Like=-14.08..-13.30 [-14.0820..-14.0716] | it/evals=1590/2446 eff=74.0913% N=300
Mono-modal Volume: ~exp(-9.32) * Expected Volume: exp(-5.36) Quality: ok
index : +1.0| +2.5 ****** +2.9 | +5.0
amplitude: +1.0e-12| +4.4e-11 ********** +6.2e-11 | +1.0e-10
Z=-19.0(48.88%) | Like=-14.04..-13.30 [-14.0445..-14.0394]*| it/evals=1608/2466 eff=74.2382% N=300
Z=-19.0(50.24%) | Like=-14.00..-13.30 [-14.0039..-14.0013]*| it/evals=1620/2485 eff=74.1419% N=300
Z=-19.0(53.26%) | Like=-13.94..-13.30 [-13.9377..-13.9349]*| it/evals=1647/2524 eff=74.0558% N=300
Z=-19.0(53.62%) | Like=-13.93..-13.30 [-13.9329..-13.9329]*| it/evals=1650/2528 eff=74.0575% N=300
Mono-modal Volume: ~exp(-9.67) * Expected Volume: exp(-5.58) Quality: ok
index : +1.0| +2.6 ****** +2.9 | +5.0
amplitude: +1.0e-12| +4.5e-11 ********* +6.2e-11 | +1.0e-10
Z=-18.9(56.48%) | Like=-13.88..-13.30 [-13.8769..-13.8742]*| it/evals=1675/2564 eff=73.9841% N=300
Z=-18.9(57.00%) | Like=-13.87..-13.30 [-13.8693..-13.8691]*| it/evals=1680/2571 eff=73.9762% N=300
Z=-18.8(60.01%) | Like=-13.82..-13.30 [-13.8194..-13.8188]*| it/evals=1709/2610 eff=73.9827% N=300
Z=-18.8(60.12%) | Like=-13.82..-13.30 [-13.8188..-13.8175]*| it/evals=1710/2611 eff=73.9939% N=300
Z=-18.8(62.91%) | Like=-13.78..-13.30 [-13.7771..-13.7766]*| it/evals=1738/2650 eff=73.9574% N=300
Z=-18.8(63.11%) | Like=-13.78..-13.30 [-13.7766..-13.7730]*| it/evals=1740/2652 eff=73.9796% N=300
Mono-modal Volume: ~exp(-9.67) Expected Volume: exp(-5.81) Quality: ok
index : +1.0| +2.6 ***** +2.9 | +5.0
amplitude: +1.0e-12| +4.6e-11 ******** +6.0e-11 | +1.0e-10
Z=-18.8(65.65%) | Like=-13.73..-13.30 [-13.7315..-13.7309]*| it/evals=1767/2689 eff=73.9640% N=300
Z=-18.8(65.92%) | Like=-13.73..-13.30 [-13.7292..-13.7275]*| it/evals=1770/2693 eff=73.9657% N=300
Z=-18.7(68.53%) | Like=-13.69..-13.30 [-13.6924..-13.6910]*| it/evals=1800/2732 eff=74.0132% N=300
Mono-modal Volume: ~exp(-9.67) * Expected Volume: exp(-6.03) 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=-18.7(69.28%) | Like=-13.68..-13.30 [-13.6803..-13.6783]*| it/evals=1809/2746 eff=73.9575% N=300
[ultranest] Explored until L=-1e+01
[ultranest] Likelihood function evaluations: 2754
[ultranest] logZ = -18.34 +- 0.0888
[ultranest] Effective samples strategy satisfied (ESS = 1012.7, need >400)
[ultranest] Posterior uncertainty strategy is satisfied (KL: 0.46+-0.07 nat, need <0.50 nat)
[ultranest] Evidency uncertainty strategy is satisfied (dlogz=0.28, need <0.5)
[ultranest] logZ error budget: single: 0.12 bs:0.09 tail:0.26 total:0.28 required:<0.50
[ultranest] done iterating.
logZ = -18.333 +- 0.324
single instance: logZ = -18.333 +- 0.115
bootstrapped : logZ = -18.335 +- 0.190
tail : logZ = +- 0.262
insert order U test : converged: True correlation: inf iterations
index : 2.21 │ ▁▁▁▁▂▂▄▄▅▆▇▇▆▇▆▅▅▃▃▃▂▁▁▁▁▁▁▁▁▁▁ ▁ │3.76 2.75 +- 0.18
amplitude : 0.0000000000274│ ▁▁▁ ▁▁▁▃▃▃▄▄▆▅▇▅▇▆▅▅▄▃▃▂▂▁▁▁▁▁▁▁▁▁ ▁ │0.0000000000869 0.0000000000531 +- 0.0000000000079
Comparing the posterior distribution of all runs#
For a comparison of different posterior distributions, we can use the package chainconsumer. As this is not a Gammapy dependency, you’ll need to install it. More info here : https://samreay.github.io/ChainConsumer/
# Uncomment this if you have installed `chainconsumer`.
# from chainconsumer import Chain, ChainConfig, ChainConsumer, PlotConfig, Truth, make_sample
# from pandas import DataFrame
# c = ChainConsumer()
# def create_chain(result, name, color="k"):
# return Chain(
# samples=DataFrame(result, columns=["index", "amplitude"]),
# name=name,
# color=color,
# smooth=7,
# shade=False,
# linewidth=1.0,
# cmap="magma",
# show_contour_labels=True,
# kde= True
# )
# c.add_chain(create_chain(result_joint.samples, "joint"))
# c.add_chain(create_chain(result_0.samples, "run0", "g"))
# c.add_chain(create_chain(result_1.samples, "run1", "b"))
# c.add_chain(create_chain(result_2.samples, "run2", "y"))
# fig = c.plotter.plot()
# plt.show()
Corner plot comparison#
Corner plot comparing the three Crab runs.#
We can see the joint analysis allows to better constrain the parameters than the individual runs (more observation time is of course better). One can note as well that one of the run has a notably different amplitude (possibly due to calibrations or/and atmospheric issues).
Total running time of the script: (0 minutes 40.286 seconds)
