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(-4.12) * Expected Volume: exp(0.00) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|************************** **************** ** | +1.0e-10
Z=-inf(0.00%) | Like=-3177.80..-60.72 [-3177.7975..-318.1089] | it/evals=0/301 eff=0.0000% N=300
Z=-550.5(0.00%) | Like=-544.21..-60.72 [-3177.7975..-318.1089] | it/evals=21/322 eff=95.4545% N=300
Z=-533.9(0.00%) | Like=-525.89..-60.72 [-3177.7975..-318.1089] | it/evals=30/332 eff=93.7500% N=300
Z=-508.5(0.00%) | Like=-503.03..-60.72 [-3177.7975..-318.1089] | it/evals=49/353 eff=92.4528% N=300
Z=-493.2(0.00%) | Like=-486.04..-60.72 [-3177.7975..-318.1089] | it/evals=60/368 eff=88.2353% N=300
Mono-modal Volume: ~exp(-4.12) Expected Volume: exp(-0.22) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|************************** ************* ** * **| +1.0e-10
Z=-470.6(0.00%) | Like=-464.32..-60.72 [-3177.7975..-318.1089] | it/evals=78/388 eff=88.6364% N=300
Z=-459.9(0.00%) | Like=-453.09..-60.72 [-3177.7975..-318.1089] | it/evals=90/404 eff=86.5385% N=300
Z=-442.6(0.00%) | Like=-434.42..-60.72 [-3177.7975..-318.1089] | it/evals=106/426 eff=84.1270% N=300
Z=-430.8(0.00%) | Like=-422.69..-60.72 [-3177.7975..-318.1089] | it/evals=120/441 eff=85.1064% N=300
Mono-modal Volume: ~exp(-4.31) * Expected Volume: exp(-0.45) Quality: ok
index : +1.0| ***********************************************| +5.0
amplitude: +1.0e-12|**************************************** ** ****| +1.0e-10
Z=-414.8(0.00%) | Like=-409.25..-60.72 [-3177.7975..-318.1089] | it/evals=134/456 eff=85.8974% N=300
Z=-400.2(0.00%) | Like=-394.38..-59.18 [-3177.7975..-318.1089] | it/evals=150/476 eff=85.2273% N=300
Z=-371.3(0.00%) | Like=-364.73..-59.18 [-3177.7975..-318.1089] | it/evals=171/499 eff=85.9296% N=300
Z=-362.4(0.00%) | Like=-355.15..-59.18 [-3177.7975..-318.1089] | it/evals=180/509 eff=86.1244% N=300
Z=-343.0(0.00%) | Like=-335.18..-59.18 [-3177.7975..-318.1089] | it/evals=196/531 eff=84.8485% N=300
Mono-modal Volume: ~exp(-4.47) * Expected Volume: exp(-0.67) Quality: ok
index : +1.0| *********************************************| +5.0
amplitude: +1.0e-12| *************************************** ** ****| +1.0e-10
Z=-336.7(0.00%) | Like=-329.95..-59.18 [-3177.7975..-318.1089] | it/evals=201/538 eff=84.4538% N=300
Z=-324.9(0.00%) | Like=-317.21..-59.18 [-317.2129..-178.4013] | it/evals=210/548 eff=84.6774% N=300
Z=-300.0(0.00%) | Like=-292.25..-59.18 [-317.2129..-178.4013] | it/evals=228/570 eff=84.4444% N=300
Z=-290.7(0.00%) | Like=-283.96..-59.18 [-317.2129..-178.4013] | it/evals=240/583 eff=84.8057% N=300
Z=-269.9(0.00%) | Like=-263.92..-59.18 [-317.2129..-178.4013] | it/evals=259/605 eff=84.9180% N=300
Mono-modal Volume: ~exp(-4.47) Expected Volume: exp(-0.89) Quality: ok
index : +1.0| *******************************************| +5.0
amplitude: +1.0e-12| ************************************** ** ****| +1.0e-10
Z=-263.0(0.00%) | Like=-257.19..-59.18 [-317.2129..-178.4013] | it/evals=270/618 eff=84.9057% N=300
Z=-254.4(0.00%) | Like=-247.64..-59.18 [-317.2129..-178.4013] | it/evals=285/640 eff=83.8235% N=300
Z=-240.9(0.00%) | Like=-234.29..-59.18 [-317.2129..-178.4013] | it/evals=300/660 eff=83.3333% N=300
Z=-232.7(0.00%) | Like=-226.88..-59.18 [-317.2129..-178.4013] | it/evals=313/682 eff=81.9372% N=300
Z=-225.1(0.00%) | Like=-218.98..-59.18 [-317.2129..-178.4013] | it/evals=328/708 eff=80.3922% N=300
Z=-224.3(0.00%) | Like=-218.41..-59.18 [-317.2129..-178.4013] | it/evals=330/714 eff=79.7101% 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=-222.0(0.00%) | Like=-216.01..-59.18 [-317.2129..-178.4013] | it/evals=335/720 eff=79.7619% N=300
Z=-211.3(0.00%) | Like=-204.84..-59.18 [-317.2129..-178.4013] | it/evals=354/744 eff=79.7297% N=300
Z=-208.1(0.00%) | Like=-200.91..-59.18 [-317.2129..-178.4013] | it/evals=360/751 eff=79.8226% N=300
Z=-200.0(0.00%) | Like=-193.91..-59.18 [-317.2129..-178.4013] | it/evals=373/774 eff=78.6920% N=300
Z=-195.5(0.00%) | Like=-189.78..-59.18 [-317.2129..-178.4013] | it/evals=385/795 eff=77.7778% N=300
Z=-193.8(0.00%) | Like=-187.83..-58.99 [-317.2129..-178.4013] | it/evals=390/804 eff=77.3810% 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=-189.1(0.00%) | Like=-183.34..-58.99 [-317.2129..-178.4013] | it/evals=402/827 eff=76.2808% N=300
Z=-185.7(0.00%) | Like=-179.54..-58.99 [-317.2129..-178.4013] | it/evals=415/849 eff=75.5920% N=300
Z=-184.4(0.00%) | Like=-178.13..-58.99 [-178.1313..-124.8799] | it/evals=420/856 eff=75.5396% N=300
Z=-179.3(0.00%) | Like=-173.75..-58.99 [-178.1313..-124.8799] | it/evals=438/878 eff=75.7785% N=300
Z=-175.7(0.00%) | Like=-169.44..-58.99 [-178.1313..-124.8799] | it/evals=450/895 eff=75.6303% N=300
Z=-172.2(0.00%) | Like=-166.63..-58.99 [-178.1313..-124.8799] | it/evals=465/917 eff=75.3647% N=300
Mono-modal Volume: ~exp(-5.76) * Expected Volume: exp(-1.56) Quality: ok
index : +1.0| ********************************** | +5.0
amplitude: +1.0e-12| ************************************** ** | +1.0e-10
Z=-171.6(0.00%) | Like=-165.83..-58.99 [-178.1313..-124.8799] | it/evals=469/922 eff=75.4019% N=300
Z=-166.7(0.00%) | Like=-160.12..-58.99 [-178.1313..-124.8799] | it/evals=480/939 eff=75.1174% N=300
Z=-161.5(0.00%) | Like=-154.62..-58.78 [-178.1313..-124.8799] | it/evals=492/961 eff=74.4327% N=300
Z=-157.7(0.00%) | Like=-151.71..-58.78 [-178.1313..-124.8799] | it/evals=508/984 eff=74.2690% N=300
Z=-157.3(0.00%) | Like=-150.24..-58.78 [-178.1313..-124.8799] | it/evals=510/987 eff=74.2358% N=300
Z=-153.0(0.00%) | Like=-146.93..-58.78 [-178.1313..-124.8799] | it/evals=527/1009 eff=74.3300% N=300
Mono-modal Volume: ~exp(-5.76) Expected Volume: exp(-1.79) Quality: ok
index : +1.0| ****************************** | +5.0
amplitude: +1.0e-12| ************************************* * | +1.0e-10
Z=-150.1(0.00%) | Like=-144.01..-58.78 [-178.1313..-124.8799] | it/evals=540/1028 eff=74.1758% N=300
Z=-146.5(0.00%) | Like=-140.15..-58.78 [-178.1313..-124.8799] | it/evals=554/1051 eff=73.7683% N=300
Z=-142.9(0.00%) | Like=-136.36..-58.78 [-178.1313..-124.8799] | it/evals=570/1073 eff=73.7387% N=300
Z=-140.6(0.00%) | Like=-134.14..-58.78 [-178.1313..-124.8799] | it/evals=579/1097 eff=72.6474% N=300
Z=-137.4(0.00%) | Like=-130.80..-58.78 [-178.1313..-124.8799] | it/evals=592/1120 eff=72.1951% N=300
Z=-135.3(0.00%) | Like=-128.87..-58.78 [-178.1313..-124.8799] | it/evals=600/1132 eff=72.1154% N=300
Mono-modal Volume: ~exp(-6.12) * Expected Volume: exp(-2.01) Quality: ok
index : +1.0| *************************** +4.0 | +5.0
amplitude: +1.0e-12| *************************** ******* | +1.0e-10
Z=-134.7(0.00%) | Like=-128.32..-58.78 [-178.1313..-124.8799] | it/evals=603/1138 eff=71.9570% N=300
Z=-131.9(0.00%) | Like=-125.90..-58.78 [-178.1313..-124.8799] | it/evals=618/1160 eff=71.8605% N=300
Z=-130.1(0.00%) | Like=-123.47..-58.78 [-124.8393..-93.7998] | it/evals=630/1179 eff=71.6724% N=300
Z=-126.3(0.00%) | Like=-120.25..-58.78 [-124.8393..-93.7998] | it/evals=647/1201 eff=71.8091% N=300
Z=-123.6(0.00%) | Like=-117.20..-58.78 [-124.8393..-93.7998] | it/evals=660/1221 eff=71.6612% N=300
Mono-modal Volume: ~exp(-6.27) * Expected Volume: exp(-2.23) Quality: ok
index : +1.0| +2.0 *********************** +3.8 | +5.0
amplitude: +1.0e-12| ******************************** | +1.0e-10
Z=-121.8(0.00%) | Like=-115.35..-58.78 [-124.8393..-93.7998] | it/evals=670/1237 eff=71.5048% N=300
Z=-118.6(0.00%) | Like=-112.34..-58.78 [-124.8393..-93.7998] | it/evals=687/1259 eff=71.6371% N=300
Z=-118.1(0.00%) | Like=-111.59..-58.78 [-124.8393..-93.7998] | it/evals=690/1264 eff=71.5768% N=300
Z=-115.7(0.00%) | Like=-109.42..-58.78 [-124.8393..-93.7998] | it/evals=706/1286 eff=71.6024% N=300
Z=-113.9(0.00%) | Like=-107.57..-58.78 [-124.8393..-93.7998] | it/evals=719/1309 eff=71.2587% N=300
Z=-113.8(0.00%) | Like=-107.54..-58.78 [-124.8393..-93.7998] | it/evals=720/1310 eff=71.2871% N=300
Mono-modal Volume: ~exp(-6.58) * Expected Volume: exp(-2.46) Quality: ok
index : +1.0| +2.0 ********************* +3.7 | +5.0
amplitude: +1.0e-12| *************************** +7.7e-11 | +1.0e-10
Z=-110.9(0.00%) | Like=-104.07..-58.78 [-124.8393..-93.7998] | it/evals=737/1331 eff=71.4840% N=300
Z=-108.6(0.00%) | Like=-102.43..-58.78 [-124.8393..-93.7998] | it/evals=750/1350 eff=71.4286% N=300
Z=-106.2(0.00%) | Like=-100.03..-58.78 [-124.8393..-93.7998] | it/evals=767/1372 eff=71.5485% N=300
Z=-104.9(0.00%) | Like=-98.80..-58.78 [-124.8393..-93.7998] | it/evals=780/1391 eff=71.4940% N=300
Z=-102.8(0.00%) | Like=-96.61..-58.78 [-124.8393..-93.7998] | it/evals=799/1414 eff=71.7235% N=300
Mono-modal Volume: ~exp(-6.63) * Expected Volume: exp(-2.68) Quality: ok
index : +1.0| +2.1 ****************** +3.5 | +5.0
amplitude: +1.0e-12| ************************** +7.4e-11 | +1.0e-10
Z=-102.2(0.00%) | Like=-95.59..-58.78 [-124.8393..-93.7998] | it/evals=804/1423 eff=71.5939% N=300
Z=-101.4(0.00%) | Like=-95.33..-58.78 [-124.8393..-93.7998] | it/evals=810/1431 eff=71.6180% N=300
Z=-99.8(0.00%) | Like=-93.62..-58.78 [-93.6354..-76.3638] | it/evals=830/1452 eff=72.0486% N=300
Z=-98.8(0.00%) | Like=-92.20..-58.78 [-93.6354..-76.3638] | it/evals=840/1465 eff=72.1030% N=300
Z=-96.4(0.00%) | Like=-90.16..-58.78 [-93.6354..-76.3638] | it/evals=858/1487 eff=72.2831% N=300
Z=-95.4(0.00%) | Like=-89.35..-58.78 [-93.6354..-76.3638] | it/evals=870/1507 eff=72.0795% N=300
Mono-modal Volume: ~exp(-6.75) * Expected Volume: exp(-2.90) Quality: ok
index : +1.0| +2.1 ***************** +3.4 | +5.0
amplitude: +1.0e-12| +2.5e-11 ************************ +7.1e-11 | +1.0e-10
Z=-95.3(0.00%) | Like=-89.34..-58.78 [-93.6354..-76.3638] | it/evals=871/1508 eff=72.1026% N=300
Z=-93.8(0.00%) | Like=-87.71..-58.78 [-93.6354..-76.3638] | it/evals=889/1530 eff=72.2764% N=300
Z=-92.9(0.00%) | Like=-86.76..-58.78 [-93.6354..-76.3638] | it/evals=900/1542 eff=72.4638% N=300
Z=-91.5(0.00%) | Like=-85.26..-58.78 [-93.6354..-76.3638] | it/evals=917/1565 eff=72.4901% N=300
Z=-90.9(0.00%) | Like=-84.91..-58.78 [-93.6354..-76.3638] | it/evals=926/1588 eff=71.8944% N=300
Z=-90.6(0.00%) | Like=-84.54..-58.78 [-93.6354..-76.3638] | it/evals=930/1600 eff=71.5385% N=300
Mono-modal Volume: ~exp(-7.10) * Expected Volume: exp(-3.13) Quality: ok
index : +1.0| +2.2 **************** +3.3 | +5.0
amplitude: +1.0e-12| +2.6e-11 ********************* +6.7e-11 | +1.0e-10
Z=-90.0(0.00%) | Like=-83.82..-58.78 [-93.6354..-76.3638] | it/evals=938/1612 eff=71.4939% N=300
Z=-88.7(0.00%) | Like=-82.39..-58.78 [-93.6354..-76.3638] | it/evals=955/1633 eff=71.6429% N=300
Z=-88.4(0.00%) | Like=-81.95..-58.78 [-93.6354..-76.3638] | it/evals=960/1638 eff=71.7489% N=300
Z=-87.1(0.00%) | Like=-80.70..-58.78 [-93.6354..-76.3638] | it/evals=979/1662 eff=71.8796% N=300
Z=-86.4(0.00%) | Like=-80.18..-58.78 [-93.6354..-76.3638] | it/evals=990/1679 eff=71.7912% 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| +2.8e-11 ******************** +6.6e-11 | +1.0e-10
Z=-85.4(0.00%) | Like=-79.09..-58.78 [-93.6354..-76.3638] | it/evals=1005/1702 eff=71.6833% N=300
Z=-84.7(0.00%) | Like=-78.40..-58.78 [-93.6354..-76.3638] | it/evals=1017/1724 eff=71.4185% N=300
Z=-84.5(0.00%) | Like=-78.28..-58.78 [-93.6354..-76.3638] | it/evals=1020/1728 eff=71.4286% N=300
Z=-83.6(0.00%) | Like=-77.51..-58.78 [-93.6354..-76.3638] | it/evals=1037/1749 eff=71.5666% N=300
Z=-82.9(0.00%) | Like=-76.46..-58.78 [-93.6354..-76.3638] | it/evals=1050/1768 eff=71.5259% N=300
Z=-82.4(0.00%) | Like=-76.15..-58.78 [-76.3366..-67.4186] | it/evals=1058/1790 eff=71.0067% N=300
Z=-82.0(0.00%) | Like=-75.54..-58.78 [-76.3366..-67.4186] | it/evals=1066/1813 eff=70.4560% N=300
Mono-modal Volume: ~exp(-7.68) * Expected Volume: exp(-3.57) Quality: ok
index : +1.0| +2.2 ************* +3.2 | +5.0
amplitude: +1.0e-12| +2.9e-11 ***************** +6.3e-11 | +1.0e-10
Z=-81.7(0.00%) | Like=-75.27..-58.78 [-76.3366..-67.4186] | it/evals=1072/1825 eff=70.2951% N=300
Z=-81.3(0.00%) | Like=-74.82..-58.78 [-76.3366..-67.4186] | it/evals=1080/1834 eff=70.4042% N=300
Z=-80.3(0.00%) | Like=-73.77..-58.78 [-76.3366..-67.4186] | it/evals=1097/1856 eff=70.5013% N=300
Z=-79.6(0.00%) | Like=-73.29..-58.78 [-76.3366..-67.4186] | it/evals=1110/1877 eff=70.3868% N=300
Z=-78.8(0.00%) | Like=-72.24..-58.78 [-76.3366..-67.4186] | it/evals=1126/1899 eff=70.4190% N=300
Z=-78.2(0.00%) | Like=-71.68..-58.78 [-76.3366..-67.4186] | it/evals=1138/1922 eff=70.1603% N=300
Mono-modal Volume: ~exp(-7.92) * 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=-78.1(0.00%) | Like=-71.67..-58.78 [-76.3366..-67.4186] | it/evals=1139/1923 eff=70.1787% N=300
Z=-78.1(0.00%) | Like=-71.66..-58.78 [-76.3366..-67.4186] | it/evals=1140/1924 eff=70.1970% N=300
Z=-77.3(0.00%) | Like=-70.86..-58.78 [-76.3366..-67.4186] | it/evals=1157/1946 eff=70.2916% N=300
Z=-76.8(0.00%) | Like=-70.50..-58.78 [-76.3366..-67.4186] | it/evals=1170/1962 eff=70.3971% N=300
Z=-76.2(0.00%) | Like=-69.87..-58.78 [-76.3366..-67.4186] | it/evals=1186/1984 eff=70.4276% N=300
Z=-75.7(0.00%) | Like=-69.29..-58.78 [-76.3366..-67.4186] | it/evals=1200/2006 eff=70.3400% N=300
Mono-modal Volume: ~exp(-8.06) * 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=-75.5(0.00%) | Like=-69.07..-58.78 [-76.3366..-67.4186] | it/evals=1206/2017 eff=70.2388% N=300
Z=-74.8(0.01%) | Like=-68.54..-58.78 [-76.3366..-67.4186] | it/evals=1226/2039 eff=70.5003% N=300
Z=-74.7(0.01%) | Like=-68.41..-58.78 [-76.3366..-67.4186] | it/evals=1230/2044 eff=70.5275% N=300
Z=-74.3(0.01%) | Like=-67.91..-58.78 [-76.3366..-67.4186] | it/evals=1246/2066 eff=70.5549% N=300
Z=-73.9(0.02%) | Like=-67.59..-58.78 [-76.3366..-67.4186] | it/evals=1260/2083 eff=70.6674% N=300
Mono-modal Volume: ~exp(-8.50) * 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=-73.5(0.02%) | Like=-67.32..-58.78 [-67.3921..-65.3439] | it/evals=1273/2097 eff=70.8403% N=300
Z=-73.1(0.03%) | Like=-66.74..-58.78 [-67.3921..-65.3439] | it/evals=1290/2118 eff=70.9571% N=300
Z=-72.7(0.05%) | Like=-66.18..-58.78 [-67.3921..-65.3439] | it/evals=1307/2140 eff=71.0326% N=300
Z=-72.3(0.07%) | Like=-65.93..-58.78 [-67.3921..-65.3439] | it/evals=1320/2160 eff=70.9677% N=300
Z=-71.9(0.10%) | Like=-65.59..-58.78 [-67.3921..-65.3439] | it/evals=1337/2182 eff=71.0414% N=300
Mono-modal Volume: ~exp(-8.70) * Expected Volume: exp(-4.47) Quality: ok
index : +1.0| +2.4 ******** +3.0 | +5.0
amplitude: +1.0e-12| +3.4e-11 *********** +5.5e-11 | +1.0e-10
Z=-71.9(0.11%) | Like=-65.57..-58.78 [-67.3921..-65.3439] | it/evals=1340/2186 eff=71.0498% N=300
Z=-71.7(0.13%) | Like=-65.38..-58.78 [-67.3921..-65.3439] | it/evals=1350/2198 eff=71.1275% N=300
Z=-71.3(0.19%) | Like=-65.10..-58.78 [-65.3088..-65.0533] | it/evals=1367/2220 eff=71.1979% N=300
Z=-71.1(0.24%) | Like=-64.74..-58.78 [-64.7425..-64.7269] | it/evals=1380/2236 eff=71.2810% N=300
Z=-70.8(0.31%) | Like=-64.56..-58.78 [-64.5588..-64.5585]*| it/evals=1394/2259 eff=71.1588% N=300
Mono-modal Volume: ~exp(-8.70) Expected Volume: exp(-4.69) Quality: ok
index : +1.0| +2.4 ******** +3.0 | +5.0
amplitude: +1.0e-12| +3.5e-11 *********** +5.5e-11 | +1.0e-10
Z=-70.6(0.39%) | Like=-64.32..-58.78 [-64.3684..-64.3202] | it/evals=1408/2279 eff=71.1470% N=300
Z=-70.6(0.40%) | Like=-64.27..-58.78 [-64.3118..-64.2729] | it/evals=1410/2282 eff=71.1403% N=300
Z=-70.3(0.56%) | Like=-63.83..-58.78 [-63.8297..-63.8154] | it/evals=1428/2303 eff=71.2931% N=300
Z=-70.1(0.71%) | Like=-63.66..-58.78 [-63.6626..-63.6616]*| it/evals=1440/2321 eff=71.2519% N=300
Z=-69.8(0.89%) | Like=-63.39..-58.78 [-63.4114..-63.3896] | it/evals=1457/2346 eff=71.2121% N=300
Z=-69.6(1.11%) | Like=-63.10..-58.78 [-63.1148..-63.0996] | it/evals=1470/2360 eff=71.3592% N=300
Mono-modal Volume: ~exp(-8.70) Expected Volume: exp(-4.91) Quality: ok
index : +1.0| +2.4 ******* +2.9 | +5.0
amplitude: +1.0e-12| +3.6e-11 ********* +5.4e-11 | +1.0e-10
Z=-69.4(1.35%) | Like=-63.03..-58.78 [-63.0338..-63.0297]*| it/evals=1482/2380 eff=71.2500% N=300
Z=-69.2(1.74%) | Like=-62.71..-58.78 [-62.7269..-62.7087] | it/evals=1499/2404 eff=71.2452% N=300
Z=-69.2(1.76%) | Like=-62.70..-58.78 [-62.7017..-62.6919]*| it/evals=1500/2406 eff=71.2251% N=300
Z=-69.0(2.03%) | Like=-62.62..-58.78 [-62.6189..-62.5920] | it/evals=1511/2428 eff=71.0056% N=300
Z=-68.8(2.59%) | Like=-62.40..-58.78 [-62.3981..-62.3855] | it/evals=1529/2452 eff=71.0502% N=300
Z=-68.8(2.61%) | Like=-62.39..-58.78 [-62.3981..-62.3855] | it/evals=1530/2453 eff=71.0636% N=300
Mono-modal Volume: ~exp(-8.83) * 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=-68.6(2.94%) | Like=-62.21..-58.78 [-62.2760..-62.2125] | it/evals=1541/2475 eff=70.8506% N=300
Z=-68.4(3.59%) | Like=-61.91..-58.78 [-61.9262..-61.9143] | it/evals=1558/2497 eff=70.9149% N=300
Z=-68.4(3.70%) | Like=-61.89..-58.78 [-61.9014..-61.8895] | it/evals=1560/2499 eff=70.9413% N=300
Z=-68.2(4.29%) | Like=-61.72..-58.78 [-61.7486..-61.7211] | it/evals=1573/2521 eff=70.8240% N=300
Z=-68.2(4.63%) | Like=-61.63..-58.78 [-61.6317..-61.6306]*| it/evals=1579/2544 eff=70.3654% N=300
Z=-68.1(5.29%) | Like=-61.53..-58.78 [-61.5479..-61.5296] | it/evals=1590/2559 eff=70.3851% N=300
Z=-67.9(6.19%) | Like=-61.41..-58.78 [-61.4215..-61.4100] | it/evals=1605/2581 eff=70.3639% N=300
Mono-modal Volume: ~exp(-9.58) * Expected Volume: exp(-5.36) Quality: ok
index : +1.0| +2.5 ****** +2.9 | +5.0
amplitude: +1.0e-12| +3.8e-11 ******** +5.2e-11 | +1.0e-10
Z=-67.9(6.32%) | Like=-61.39..-58.78 [-61.3962..-61.3853] | it/evals=1608/2586 eff=70.3412% N=300
Z=-67.7(7.10%) | Like=-61.32..-58.78 [-61.3165..-61.3080]*| it/evals=1620/2602 eff=70.3736% N=300
Z=-67.6(8.30%) | Like=-61.21..-58.78 [-61.2068..-61.2064]*| it/evals=1638/2624 eff=70.4819% N=300
Z=-67.5(9.19%) | Like=-61.10..-58.78 [-61.1096..-61.0985] | it/evals=1650/2645 eff=70.3625% N=300
Z=-67.4(10.34%) | Like=-61.02..-58.78 [-61.0300..-61.0166] | it/evals=1666/2667 eff=70.3845% N=300
Mono-modal Volume: ~exp(-9.78) * Expected Volume: exp(-5.58) Quality: ok
index : +1.0| +2.5 ***** +2.8 | +5.0
amplitude: +1.0e-12| +3.8e-11 ******* +5.1e-11 | +1.0e-10
Z=-67.3(11.00%) | Like=-60.96..-58.78 [-60.9600..-60.9451] | it/evals=1675/2682 eff=70.3191% N=300
Z=-67.3(11.37%) | Like=-60.92..-58.78 [-60.9206..-60.9162]*| it/evals=1680/2687 eff=70.3812% N=300
Z=-67.2(12.27%) | Like=-60.84..-58.76 [-60.8438..-60.8296] | it/evals=1692/2710 eff=70.2075% N=300
Z=-67.1(13.92%) | Like=-60.70..-58.76 [-60.7151..-60.7027] | it/evals=1710/2730 eff=70.3704% N=300
Z=-67.0(15.07%) | Like=-60.61..-58.75 [-60.6114..-60.6112]*| it/evals=1723/2752 eff=70.2692% N=300
Z=-66.9(16.48%) | Like=-60.49..-58.75 [-60.4898..-60.4840]*| it/evals=1738/2774 eff=70.2506% N=300
Z=-66.9(16.67%) | Like=-60.48..-58.75 [-60.4840..-60.4620] | it/evals=1740/2776 eff=70.2746% N=300
Mono-modal Volume: ~exp(-10.03) * Expected Volume: exp(-5.81) Quality: ok
index : +1.0| +2.5 **** +2.8 | +5.0
amplitude: +1.0e-12| +3.9e-11 ****** +5.0e-11 | +1.0e-10
Z=-66.9(16.90%) | Like=-60.45..-58.75 [-60.4519..-60.4476]*| it/evals=1742/2779 eff=70.2703% N=300
Z=-66.8(18.77%) | Like=-60.37..-58.75 [-60.3659..-60.3652]*| it/evals=1760/2801 eff=70.3719% N=300
Z=-66.7(19.84%) | Like=-60.30..-58.75 [-60.2987..-60.2954]*| it/evals=1770/2814 eff=70.4057% N=300
Z=-66.6(21.77%) | Like=-60.22..-58.75 [-60.2210..-60.2208]*| it/evals=1786/2836 eff=70.4259% N=300
Z=-66.5(23.19%) | Like=-60.15..-58.75 [-60.1539..-60.1509]*| it/evals=1800/2857 eff=70.3950% N=300
Mono-modal Volume: ~exp(-10.25) * Expected Volume: exp(-6.03) Quality: ok
index : +1.0| +2.5 **** +2.8 | +5.0
amplitude: +1.0e-12| +4.0e-11 ****** +5.0e-11 | +1.0e-10
Z=-66.5(24.34%) | Like=-60.11..-58.75 [-60.1062..-60.1046]*| it/evals=1809/2869 eff=70.4165% N=300
Z=-66.4(26.62%) | Like=-60.02..-58.75 [-60.0298..-60.0167] | it/evals=1828/2890 eff=70.5792% N=300
Z=-66.4(26.90%) | Like=-60.01..-58.75 [-60.0119..-60.0110]*| it/evals=1830/2892 eff=70.6019% N=300
Z=-66.3(28.85%) | Like=-59.96..-58.75 [-59.9625..-59.9615]*| it/evals=1847/2914 eff=70.6580% N=300
Z=-66.3(30.42%) | Like=-59.90..-58.75 [-59.8992..-59.8946]*| it/evals=1860/2936 eff=70.5615% N=300
Mono-modal Volume: ~exp(-10.25) Expected Volume: exp(-6.25) Quality: ok
index : +1.0| +2.6 **** +2.8 | +5.0
amplitude: +1.0e-12| +4.0e-11 ****** +4.9e-11 | +1.0e-10
Z=-66.2(32.43%) | Like=-59.84..-58.75 [-59.8609..-59.8445] | it/evals=1876/2956 eff=70.6325% N=300
Z=-66.2(34.30%) | Like=-59.78..-58.75 [-59.7811..-59.7779]*| it/evals=1890/2974 eff=70.6806% N=300
Z=-66.1(36.61%) | Like=-59.72..-58.75 [-59.7248..-59.7167]*| it/evals=1907/2996 eff=70.7344% N=300
Z=-66.1(38.20%) | Like=-59.67..-58.75 [-59.6713..-59.6691]*| it/evals=1920/3015 eff=70.7182% N=300
Z=-66.0(39.86%) | Like=-59.64..-58.75 [-59.6433..-59.6409]*| it/evals=1933/3039 eff=70.5732% N=300
Mono-modal Volume: ~exp(-10.80) * Expected Volume: exp(-6.48) Quality: ok
index : +1.0| +2.6 **** +2.8 | +5.0
amplitude: +1.0e-12| +4.0e-11 **** +4.8e-11 | +1.0e-10
Z=-66.0(41.05%) | Like=-59.63..-58.75 [-59.6276..-59.6209]*| it/evals=1943/3054 eff=70.5519% N=300
Z=-66.0(41.97%) | Like=-59.60..-58.75 [-59.5999..-59.5980]*| it/evals=1950/3064 eff=70.5499% N=300
Z=-65.9(43.97%) | Like=-59.56..-58.75 [-59.5597..-59.5543]*| it/evals=1967/3085 eff=70.6284% N=300
Z=-65.9(45.59%) | Like=-59.53..-58.75 [-59.5253..-59.5247]*| it/evals=1980/3103 eff=70.6386% N=300
Z=-65.9(47.02%) | Like=-59.50..-58.75 [-59.5036..-59.4972]*| it/evals=1992/3125 eff=70.5133% N=300
Z=-65.8(48.86%) | Like=-59.47..-58.75 [-59.4688..-59.4687]*| it/evals=2008/3147 eff=70.5304% N=300
Mono-modal Volume: ~exp(-10.80) Expected Volume: exp(-6.70) Quality: ok
index : +1.0| +2.6 **** +2.8 | +5.0
amplitude: +1.0e-12| +4.1e-11 **** +4.8e-11 | +1.0e-10
Z=-65.8(49.06%) | Like=-59.47..-58.75 [-59.4659..-59.4582]*| it/evals=2010/3150 eff=70.5263% N=300
Z=-65.8(50.73%) | Like=-59.42..-58.75 [-59.4212..-59.4106] | it/evals=2024/3172 eff=70.4735% N=300
Z=-65.8(52.38%) | Like=-59.39..-58.75 [-59.3888..-59.3853]*| it/evals=2038/3194 eff=70.4216% N=300
Z=-65.7(52.61%) | Like=-59.39..-58.75 [-59.3851..-59.3827]*| it/evals=2040/3199 eff=70.3691% N=300
Z=-65.7(54.89%) | Like=-59.35..-58.75 [-59.3518..-59.3512]*| it/evals=2060/3221 eff=70.5238% N=300
Z=-65.7(55.97%) | Like=-59.32..-58.75 [-59.3230..-59.3207]*| it/evals=2070/3234 eff=70.5521% N=300
Mono-modal Volume: ~exp(-10.95) * Expected Volume: exp(-6.92) Quality: ok
index : +1.0| +2.6 **** +2.8 | +5.0
amplitude: +1.0e-12| +4.1e-11 **** +4.8e-11 | +1.0e-10
Z=-65.7(56.76%) | Like=-59.31..-58.75 [-59.3146..-59.3121]*| it/evals=2077/3242 eff=70.5982% N=300
Z=-65.6(58.67%) | Like=-59.29..-58.75 [-59.2890..-59.2865]*| it/evals=2095/3263 eff=70.7054% N=300
Z=-65.6(59.21%) | Like=-59.28..-58.75 [-59.2794..-59.2769]*| it/evals=2100/3268 eff=70.7547% N=300
Z=-65.6(60.57%) | Like=-59.25..-58.75 [-59.2517..-59.2468]*| it/evals=2114/3290 eff=70.7023% N=300
Z=-65.6(62.21%) | Like=-59.23..-58.75 [-59.2267..-59.2256]*| it/evals=2130/3310 eff=70.7641% N=300
Mono-modal Volume: ~exp(-11.26) * Expected Volume: exp(-7.15) Quality: ok
index : +1.0| +2.6 ** +2.7 | +5.0
amplitude: +1.0e-12| +4.2e-11 **** +4.7e-11 | +1.0e-10
Z=-65.6(63.55%) | Like=-59.21..-58.75 [-59.2097..-59.2095]*| it/evals=2144/3329 eff=70.7824% N=300
Z=-65.5(65.07%) | Like=-59.18..-58.75 [-59.1813..-59.1806]*| it/evals=2160/3351 eff=70.7965% N=300
Z=-65.5(66.35%) | Like=-59.16..-58.75 [-59.1591..-59.1581]*| it/evals=2175/3373 eff=70.7777% N=300
Z=-65.5(67.68%) | Like=-59.14..-58.75 [-59.1404..-59.1400]*| it/evals=2190/3392 eff=70.8279% N=300
Z=-65.5(68.95%) | Like=-59.12..-58.75 [-59.1192..-59.1191]*| it/evals=2205/3413 eff=70.8320% N=300
Mono-modal Volume: ~exp(-11.54) * Expected Volume: exp(-7.37) Quality: ok
index : +1.0| +2.6 ** +2.7 | +5.0
amplitude: +1.0e-12| +4.2e-11 **** +4.7e-11 | +1.0e-10
Z=-65.5(69.47%) | Like=-59.12..-58.75 [-59.1157..-59.1156]*| it/evals=2211/3420 eff=70.8654% N=300
[ultranest] Explored until L=-6e+01
[ultranest] Likelihood function evaluations: 3426
[ultranest] logZ = -65.1 +- 0.09134
[ultranest] Effective samples strategy satisfied (ESS = 990.6, 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: 0.13 bs:0.09 tail:0.26 total:0.28 required:<0.50
[ultranest] done iterating.
logZ = -65.104 +- 0.324
single instance: logZ = -65.104 +- 0.134
bootstrapped : logZ = -65.103 +- 0.192
tail : logZ = +- 0.262
insert order U test : converged: True correlation: inf iterations
index : 2.390 │ ▁▁▁▁▁▁▁▂▃▃▄▅▅▇▆▇▅▆▆▇▅▅▅▃▄▂▂▂▁▁▁▁▁▁▁ ▁ │2.993 2.670 +- 0.084
amplitude : 0.0000000000319│ ▁ ▁▁▁▁▁▁▂▁▂▂▄▄▅▆▆▆▆▇▇▇▅▄▄▄▃▂▃▂▁▁▁▁▁▁ │0.0000000000549 0.0000000000444 +- 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 : 2.670 +/- 0.08
amplitude : 4.44e-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': [2.6700622783022916, 4.444981152819273e-11], 'stdev': [0.0842810436295121, 3.1209813615420048e-12], 'median': [2.670463966366878, 4.4465710309548377e-11], 'errlo': [2.5844413808109037, 4.127836508466212e-11], 'errup': [2.753566646760844, 4.7600819055338806e-11], 'information_gain_bits': [2.6747286161981934, 3.116605132229592]}
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(-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=-1593.79..-20.71 [-1593.7938..-113.8419] | it/evals=0/301 eff=0.0000% N=300
Z=-177.2(0.00%) | Like=-172.45..-20.71 [-1593.7938..-113.8419] | it/evals=30/331 eff=96.7742% N=300
Z=-166.5(0.00%) | Like=-161.55..-20.71 [-1593.7938..-113.8419] | it/evals=60/370 eff=85.7143% N=300
Mono-modal Volume: ~exp(-4.34) * Expected Volume: exp(-0.22) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|************************** *********** *** * ***| +1.0e-10
Z=-164.2(0.00%) | Like=-159.03..-20.71 [-1593.7938..-113.8419] | it/evals=67/378 eff=85.8974% N=300
Z=-154.1(0.00%) | Like=-149.18..-20.71 [-1593.7938..-113.8419] | it/evals=90/405 eff=85.7143% N=300
Z=-143.5(0.00%) | Like=-138.62..-20.71 [-1593.7938..-113.8419] | it/evals=118/446 eff=80.8219% N=300
Z=-143.0(0.00%) | Like=-138.43..-20.71 [-1593.7938..-113.8419] | it/evals=120/448 eff=81.0811% N=300
Mono-modal Volume: ~exp(-4.34) Expected Volume: exp(-0.45) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|************************** *************** * ***| +1.0e-10
Z=-133.7(0.00%) | Like=-128.92..-20.71 [-1593.7938..-113.8419] | it/evals=150/483 eff=81.9672% N=300
Z=-124.8(0.00%) | Like=-119.88..-20.71 [-1593.7938..-113.8419] | it/evals=180/520 eff=81.8182% N=300
Mono-modal Volume: ~exp(-4.34) Expected Volume: exp(-0.67) Quality: ok
index : +1.0| *********************************************| +5.0
amplitude: +1.0e-12| ************************* ************ ** *****| +1.0e-10
Z=-118.3(0.00%) | Like=-113.26..-20.53 [-113.4242..-71.8835] | it/evals=209/556 eff=81.6406% N=300
Z=-118.0(0.00%) | Like=-113.17..-20.53 [-113.4242..-71.8835] | it/evals=210/557 eff=81.7121% N=300
Z=-110.2(0.00%) | Like=-104.92..-20.53 [-113.4242..-71.8835] | it/evals=238/597 eff=80.1347% N=300
Z=-109.7(0.00%) | Like=-104.73..-20.53 [-113.4242..-71.8835] | it/evals=240/599 eff=80.2676% N=300
Mono-modal Volume: ~exp(-4.41) * Expected Volume: exp(-0.89) Quality: ok
index : +1.0| *********************************************| +5.0
amplitude: +1.0e-12| ************************************** ** *****| +1.0e-10
Z=-100.7(0.00%) | Like=-95.36..-20.53 [-113.4242..-71.8835] | it/evals=268/637 eff=79.5252% N=300
Z=-100.1(0.00%) | Like=-94.34..-20.53 [-113.4242..-71.8835] | it/evals=270/639 eff=79.6460% N=300
Z=-93.5(0.00%) | Like=-88.81..-20.53 [-113.4242..-71.8835] | it/evals=299/679 eff=78.8918% N=300
Z=-93.4(0.00%) | Like=-88.69..-20.53 [-113.4242..-71.8835] | it/evals=300/680 eff=78.9474% N=300
Z=-88.1(0.00%) | Like=-83.11..-20.53 [-113.4242..-71.8835] | it/evals=330/718 eff=78.9474% N=300
Mono-modal Volume: ~exp(-5.20) * Expected Volume: exp(-1.12) Quality: ok
index : +1.0| *******************************************| +5.0
amplitude: +1.0e-12| *********************************** * * * | +1.0e-10
Z=-87.3(0.00%) | Like=-82.09..-20.53 [-113.4242..-71.8835] | it/evals=335/725 eff=78.8235% N=300
Z=-82.9(0.00%) | Like=-78.24..-20.53 [-113.4242..-71.8835] | it/evals=360/760 eff=78.2609% N=300
Z=-79.3(0.00%) | Like=-74.41..-20.53 [-113.4242..-71.8835] | it/evals=387/800 eff=77.4000% N=300
Z=-79.0(0.00%) | Like=-74.24..-20.53 [-113.4242..-71.8835] | it/evals=390/803 eff=77.5348% N=300
Mono-modal Volume: ~exp(-5.20) Expected Volume: exp(-1.34) Quality: ok
index : +1.0| ******************************************| +5.0
amplitude: +1.0e-12| ********************************** * | +1.0e-10
Z=-76.5(0.00%) | Like=-71.31..-20.53 [-71.8663..-50.3050] | it/evals=414/839 eff=76.8089% N=300
Z=-75.5(0.00%) | Like=-70.10..-20.53 [-71.8663..-50.3050] | it/evals=420/846 eff=76.9231% N=300
Z=-72.0(0.00%) | Like=-66.79..-20.53 [-71.8663..-50.3050] | it/evals=447/887 eff=76.1499% N=300
Z=-71.6(0.00%) | Like=-66.62..-20.53 [-71.8663..-50.3050] | it/evals=450/893 eff=75.8853% N=300
Mono-modal Volume: ~exp(-5.40) * Expected Volume: exp(-1.56) Quality: ok
index : +1.0| *********************************** | +5.0
amplitude: +1.0e-12| *********************************** +7.8e-11| +1.0e-10
Z=-69.7(0.00%) | Like=-64.22..-20.53 [-71.8663..-50.3050] | it/evals=469/925 eff=75.0400% N=300
Z=-68.5(0.00%) | Like=-63.57..-20.53 [-71.8663..-50.3050] | it/evals=480/941 eff=74.8830% N=300
Z=-66.5(0.00%) | Like=-61.31..-20.53 [-71.8663..-50.3050] | it/evals=507/982 eff=74.3402% N=300
Z=-66.1(0.00%) | Like=-60.95..-20.53 [-71.8663..-50.3050] | it/evals=510/987 eff=74.2358% N=300
Z=-64.3(0.00%) | Like=-59.59..-20.53 [-71.8663..-50.3050] | it/evals=532/1031 eff=72.7770% N=300
Mono-modal Volume: ~exp(-5.55) * Expected Volume: exp(-1.79) Quality: ok
index : +1.0| ******************************** | +5.0
amplitude: +1.0e-12| ********************************* +7.6e-11 | +1.0e-10
Z=-64.1(0.00%) | Like=-59.37..-20.53 [-71.8663..-50.3050] | it/evals=536/1038 eff=72.6287% N=300
Z=-63.9(0.00%) | Like=-59.15..-20.53 [-71.8663..-50.3050] | it/evals=540/1046 eff=72.3861% N=300
Z=-61.4(0.00%) | Like=-55.81..-20.53 [-71.8663..-50.3050] | it/evals=568/1087 eff=72.1728% N=300
Z=-61.1(0.00%) | Like=-55.55..-20.53 [-71.8663..-50.3050] | it/evals=570/1089 eff=72.2433% N=300
Z=-58.5(0.00%) | Like=-53.00..-20.53 [-71.8663..-50.3050] | it/evals=598/1129 eff=72.1351% N=300
Z=-58.3(0.00%) | Like=-52.87..-20.53 [-71.8663..-50.3050] | it/evals=600/1133 eff=72.0288% N=300
Mono-modal Volume: ~exp(-5.59) * Expected Volume: exp(-2.01) Quality: ok
index : +1.0| **************************** +4.0 | +5.0
amplitude: +1.0e-12| ****************************** +7.3e-11 | +1.0e-10
Z=-58.0(0.00%) | Like=-52.54..-20.53 [-71.8663..-50.3050] | it/evals=603/1139 eff=71.8713% N=300
Z=-55.2(0.00%) | Like=-49.86..-20.53 [-50.2177..-35.0589] | it/evals=627/1180 eff=71.2500% N=300
Z=-55.0(0.00%) | Like=-49.74..-20.53 [-50.2177..-35.0589] | it/evals=630/1184 eff=71.2670% N=300
Z=-52.2(0.00%) | Like=-46.29..-20.53 [-50.2177..-35.0589] | it/evals=658/1224 eff=71.2121% N=300
Z=-51.9(0.00%) | Like=-46.12..-20.53 [-50.2177..-35.0589] | it/evals=660/1226 eff=71.2743% N=300
Mono-modal Volume: ~exp(-6.19) * Expected Volume: exp(-2.23) Quality: ok
index : +1.0| ************************ +3.7 | +5.0
amplitude: +1.0e-12| ************************** +6.7e-11 | +1.0e-10
Z=-50.8(0.00%) | Like=-45.10..-20.53 [-50.2177..-35.0589] | it/evals=670/1246 eff=70.8245% N=300
Z=-49.2(0.00%) | Like=-43.84..-20.53 [-50.2177..-35.0589] | it/evals=690/1277 eff=70.6244% N=300
Z=-47.4(0.00%) | Like=-42.05..-20.53 [-50.2177..-35.0589] | it/evals=716/1317 eff=70.4031% N=300
Z=-47.1(0.00%) | Like=-41.92..-20.53 [-50.2177..-35.0589] | it/evals=720/1322 eff=70.4501% N=300
Mono-modal Volume: ~exp(-6.57) * Expected Volume: exp(-2.46) Quality: ok
index : +1.0| ********************** +3.6 | +5.0
amplitude: +1.0e-12| ************************* +6.4e-11 | +1.0e-10
Z=-46.2(0.00%) | Like=-41.05..-20.53 [-50.2177..-35.0589] | it/evals=737/1344 eff=70.5939% N=300
Z=-45.6(0.00%) | Like=-40.39..-20.53 [-50.2177..-35.0589] | it/evals=750/1362 eff=70.6215% N=300
Z=-43.9(0.00%) | Like=-38.51..-20.53 [-50.2177..-35.0589] | it/evals=779/1405 eff=70.4977% N=300
Z=-43.9(0.00%) | Like=-38.45..-20.53 [-50.2177..-35.0589] | it/evals=780/1406 eff=70.5244% N=300
Mono-modal Volume: ~exp(-6.57) Expected Volume: exp(-2.68) Quality: ok
index : +1.0| +1.9 ******************* +3.4 | +5.0
amplitude: +1.0e-12| ********************** +5.9e-11 | +1.0e-10
Z=-42.6(0.00%) | Like=-36.98..-20.53 [-50.2177..-35.0589] | it/evals=804/1447 eff=70.0959% N=300
Z=-42.3(0.00%) | Like=-36.86..-20.53 [-50.2177..-35.0589] | it/evals=810/1456 eff=70.0692% N=300
Z=-40.9(0.00%) | Like=-35.41..-20.53 [-50.2177..-35.0589] | it/evals=836/1496 eff=69.8997% N=300
Z=-40.7(0.00%) | Like=-35.02..-20.53 [-35.0216..-28.2204] | it/evals=840/1501 eff=69.9417% N=300
Z=-39.8(0.00%) | Like=-34.52..-20.53 [-35.0216..-28.2204] | it/evals=860/1540 eff=69.3548% N=300
Z=-39.4(0.00%) | Like=-33.97..-20.53 [-35.0216..-28.2204] | it/evals=870/1561 eff=68.9929% N=300
Mono-modal Volume: ~exp(-7.13) * Expected Volume: exp(-2.90) Quality: ok
index : +1.0| +2.0 ***************** +3.3 | +5.0
amplitude: +1.0e-12| ******************** +5.7e-11 | +1.0e-10
Z=-39.3(0.00%) | Like=-33.94..-20.53 [-35.0216..-28.2204] | it/evals=871/1563 eff=68.9628% N=300
Z=-38.4(0.00%) | Like=-33.11..-20.53 [-35.0216..-28.2204] | it/evals=897/1597 eff=69.1596% N=300
Z=-38.3(0.00%) | Like=-33.06..-20.53 [-35.0216..-28.2204] | it/evals=900/1601 eff=69.1776% N=300
Z=-37.4(0.00%) | Like=-32.01..-20.53 [-35.0216..-28.2204] | it/evals=930/1637 eff=69.5587% N=300
Mono-modal Volume: ~exp(-7.35) * Expected Volume: exp(-3.13) Quality: ok
index : +1.0| +2.0 **************** +3.3 | +5.0
amplitude: +1.0e-12| ****************** +5.4e-11 | +1.0e-10
Z=-37.1(0.00%) | Like=-31.83..-20.53 [-35.0216..-28.2204] | it/evals=938/1649 eff=69.5330% N=300
Z=-36.6(0.00%) | Like=-31.24..-20.53 [-35.0216..-28.2204] | it/evals=960/1680 eff=69.5652% N=300
Z=-35.7(0.01%) | Like=-30.12..-20.53 [-35.0216..-28.2204] | it/evals=990/1718 eff=69.8166% N=300
Mono-modal Volume: ~exp(-7.35) Expected Volume: exp(-3.35) Quality: ok
index : +1.0| +2.1 *************** +3.2 | +5.0
amplitude: +1.0e-12| **************** +5.2e-11 | +1.0e-10
Z=-34.9(0.01%) | Like=-29.47..-20.53 [-35.0216..-28.2204] | it/evals=1018/1752 eff=70.1102% N=300
Z=-34.9(0.01%) | Like=-29.45..-20.53 [-35.0216..-28.2204] | it/evals=1020/1754 eff=70.1513% N=300
Z=-34.1(0.03%) | Like=-28.52..-20.53 [-35.0216..-28.2204] | it/evals=1047/1795 eff=70.0334% N=300
Z=-34.0(0.03%) | Like=-28.35..-20.53 [-35.0216..-28.2204] | it/evals=1050/1800 eff=70.0000% N=300
Mono-modal Volume: ~exp(-7.44) * Expected Volume: exp(-3.57) Quality: ok
index : +1.0| +2.1 ************* +3.1 | +5.0
amplitude: +1.0e-12| **************** +5.1e-11 | +1.0e-10
Z=-33.5(0.06%) | Like=-28.02..-20.52 [-28.2028..-27.0394] | it/evals=1072/1833 eff=69.9282% N=300
Z=-33.3(0.07%) | Like=-27.77..-20.52 [-28.2028..-27.0394] | it/evals=1080/1844 eff=69.9482% N=300
Z=-32.7(0.14%) | Like=-27.04..-20.52 [-28.2028..-27.0394] | it/evals=1106/1882 eff=69.9115% N=300
Z=-32.6(0.15%) | Like=-26.96..-20.52 [-27.0339..-26.8857] | it/evals=1110/1887 eff=69.9433% N=300
Z=-32.0(0.27%) | Like=-26.39..-20.52 [-26.3949..-26.3911]*| it/evals=1135/1926 eff=69.8032% N=300
Mono-modal Volume: ~exp(-7.99) * Expected Volume: exp(-3.80) Quality: ok
index : +1.0| +2.2 *********** +3.0 | +5.0
amplitude: +1.0e-12| ************* +4.8e-11 | +1.0e-10
Z=-31.9(0.29%) | Like=-26.32..-20.52 [-26.3729..-26.3157] | it/evals=1139/1932 eff=69.7917% N=300
Z=-31.9(0.29%) | Like=-26.29..-20.52 [-26.2938..-26.2408] | it/evals=1140/1933 eff=69.8102% N=300
Z=-31.3(0.54%) | Like=-25.75..-20.52 [-25.7469..-25.7429]*| it/evals=1170/1971 eff=70.0180% N=300
Z=-30.9(0.82%) | Like=-25.37..-20.52 [-25.3833..-25.3710] | it/evals=1196/2011 eff=69.9006% N=300
Z=-30.8(0.87%) | Like=-25.34..-20.52 [-25.3434..-25.3181] | it/evals=1200/2016 eff=69.9301% N=300
Mono-modal Volume: ~exp(-8.04) * 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=-30.7(0.95%) | Like=-25.25..-20.52 [-25.2733..-25.2523] | it/evals=1206/2022 eff=70.0348% N=300
Z=-30.4(1.31%) | Like=-24.93..-20.52 [-24.9428..-24.9281] | it/evals=1230/2056 eff=70.0456% N=300
Z=-30.1(1.88%) | Like=-24.57..-20.52 [-24.5909..-24.5672] | it/evals=1258/2095 eff=70.0836% N=300
Z=-30.0(1.92%) | Like=-24.55..-20.52 [-24.5669..-24.5478] | it/evals=1260/2098 eff=70.0779% N=300
Mono-modal Volume: ~exp(-8.07) * Expected Volume: exp(-4.24) Quality: ok
index : +1.0| +2.2 ********** +2.9 | +5.0
amplitude: +1.0e-12| +2.5e-11 *********** +4.6e-11 | +1.0e-10
Z=-29.9(2.25%) | Like=-24.37..-20.52 [-24.3689..-24.3672]*| it/evals=1273/2117 eff=70.0605% N=300
Z=-29.7(2.78%) | Like=-24.12..-20.52 [-24.1223..-24.0607] | it/evals=1290/2142 eff=70.0326% N=300
Z=-29.4(3.89%) | Like=-23.74..-20.46 [-23.7431..-23.7396]*| it/evals=1319/2181 eff=70.1223% N=300
Z=-29.3(3.94%) | Like=-23.74..-20.46 [-23.7396..-23.7343]*| it/evals=1320/2186 eff=69.9894% N=300
Mono-modal Volume: ~exp(-8.07) Expected Volume: exp(-4.47) Quality: ok
index : +1.0| +2.3 ******** +2.9 | +5.0
amplitude: +1.0e-12| +2.6e-11 ********* +4.4e-11 | +1.0e-10
Z=-29.0(5.22%) | Like=-23.38..-20.46 [-23.3812..-23.3770]*| it/evals=1348/2222 eff=70.1353% N=300
Z=-29.0(5.34%) | Like=-23.37..-20.46 [-23.3738..-23.3725]*| it/evals=1350/2226 eff=70.0935% N=300
Z=-28.8(7.00%) | Like=-23.11..-20.46 [-23.1114..-23.1044]*| it/evals=1374/2264 eff=69.9593% N=300
Z=-28.7(7.42%) | Like=-23.07..-20.46 [-23.0729..-23.0669]*| it/evals=1380/2272 eff=69.9797% N=300
Z=-28.5(9.08%) | Like=-22.94..-20.46 [-22.9437..-22.9417]*| it/evals=1402/2311 eff=69.7166% N=300
Mono-modal Volume: ~exp(-8.62) * Expected Volume: exp(-4.69) Quality: ok
index : +1.0| +2.3 ******** +2.9 | +5.0
amplitude: +1.0e-12| +2.7e-11 ********* +4.3e-11 | +1.0e-10
Z=-28.5(9.44%) | Like=-22.92..-20.46 [-22.9169..-22.9055] | it/evals=1407/2320 eff=69.6535% N=300
Z=-28.5(9.64%) | Like=-22.90..-20.46 [-22.8984..-22.8927]*| it/evals=1410/2325 eff=69.6296% N=300
Z=-28.2(12.00%) | Like=-22.73..-20.46 [-22.7296..-22.7149] | it/evals=1440/2365 eff=69.7337% N=300
Z=-28.1(14.29%) | Like=-22.55..-20.46 [-22.5530..-22.5520]*| it/evals=1469/2411 eff=69.5879% N=300
Z=-28.0(14.39%) | Like=-22.55..-20.46 [-22.5520..-22.5503]*| it/evals=1470/2415 eff=69.5035% N=300
Mono-modal Volume: ~exp(-8.88) * Expected Volume: exp(-4.91) Quality: ok
index : +1.0| +2.3 ******* +2.8 | +5.0
amplitude: +1.0e-12| +2.7e-11 ********* +4.2e-11 | +1.0e-10
Z=-28.0(14.75%) | Like=-22.54..-20.46 [-22.5370..-22.5331]*| it/evals=1474/2420 eff=69.5283% N=300
Z=-27.9(17.11%) | Like=-22.41..-20.46 [-22.4132..-22.4084]*| it/evals=1500/2451 eff=69.7350% N=300
Z=-27.7(20.11%) | Like=-22.26..-20.46 [-22.2772..-22.2580] | it/evals=1530/2487 eff=69.9588% N=300
Mono-modal Volume: ~exp(-9.13) * Expected Volume: exp(-5.14) Quality: ok
index : +1.0| +2.3 ****** +2.8 | +5.0
amplitude: +1.0e-12| +2.8e-11 ******** +4.1e-11 | +1.0e-10
Z=-27.7(21.21%) | Like=-22.17..-20.46 [-22.1673..-22.1649]*| it/evals=1541/2501 eff=70.0136% N=300
Z=-27.6(23.19%) | Like=-22.05..-20.46 [-22.0724..-22.0549] | it/evals=1560/2525 eff=70.1124% N=300
Z=-27.5(26.53%) | Like=-21.90..-20.46 [-21.8951..-21.8892]*| it/evals=1588/2564 eff=70.1413% N=300
Z=-27.5(26.73%) | Like=-21.88..-20.46 [-21.8892..-21.8792] | it/evals=1590/2566 eff=70.1677% N=300
Mono-modal Volume: ~exp(-9.30) * Expected Volume: exp(-5.36) 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=-27.4(29.01%) | Like=-21.80..-20.46 [-21.8039..-21.8016]*| it/evals=1608/2592 eff=70.1571% N=300
Z=-27.3(30.32%) | Like=-21.76..-20.46 [-21.7638..-21.7616]*| it/evals=1620/2609 eff=70.1602% N=300
Z=-27.2(33.66%) | Like=-21.67..-20.46 [-21.6748..-21.6704]*| it/evals=1647/2649 eff=70.1149% N=300
Z=-27.2(33.97%) | Like=-21.66..-20.46 [-21.6636..-21.6570]*| it/evals=1650/2652 eff=70.1531% N=300
Mono-modal Volume: ~exp(-9.47) * Expected Volume: exp(-5.58) Quality: ok
index : +1.0| +2.4 ****** +2.8 | +5.0
amplitude: +1.0e-12| +2.9e-11 ****** +4.0e-11 | +1.0e-10
Z=-27.1(36.83%) | Like=-21.59..-20.46 [-21.5851..-21.5816]*| it/evals=1675/2688 eff=70.1424% N=300
Z=-27.1(37.41%) | Like=-21.57..-20.46 [-21.5735..-21.5627] | it/evals=1680/2695 eff=70.1461% N=300
Z=-27.0(40.99%) | Like=-21.49..-20.46 [-21.4852..-21.4818]*| it/evals=1710/2733 eff=70.2836% N=300
Z=-26.9(44.22%) | Like=-21.39..-20.46 [-21.3934..-21.3919]*| it/evals=1738/2772 eff=70.3074% N=300
Z=-26.9(44.43%) | Like=-21.39..-20.46 [-21.3857..-21.3854]*| it/evals=1740/2775 eff=70.3030% N=300
Mono-modal Volume: ~exp(-9.70) * 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=-26.9(44.69%) | Like=-21.38..-20.46 [-21.3808..-21.3738]*| it/evals=1742/2778 eff=70.2986% N=300
Z=-26.9(47.82%) | Like=-21.30..-20.46 [-21.2991..-21.2918]*| it/evals=1770/2814 eff=70.4057% N=300
Z=-26.8(51.12%) | Like=-21.23..-20.46 [-21.2299..-21.2290]*| it/evals=1798/2854 eff=70.3994% N=300
Z=-26.8(51.32%) | Like=-21.23..-20.46 [-21.2287..-21.2275]*| it/evals=1800/2856 eff=70.4225% N=300
Mono-modal Volume: ~exp(-9.98) * 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=-26.8(52.20%) | Like=-21.20..-20.46 [-21.2011..-21.1989]*| it/evals=1809/2868 eff=70.4439% N=300
Z=-26.7(54.53%) | Like=-21.15..-20.46 [-21.1463..-21.1455]*| it/evals=1830/2895 eff=70.5202% N=300
Z=-26.7(57.67%) | Like=-21.10..-20.46 [-21.0954..-21.0951]*| it/evals=1860/2934 eff=70.6150% N=300
Mono-modal Volume: ~exp(-9.98) 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=-26.6(60.40%) | Like=-21.05..-20.46 [-21.0465..-21.0427]*| it/evals=1887/2968 eff=70.7271% N=300
Z=-26.6(60.69%) | Like=-21.04..-20.46 [-21.0396..-21.0393]*| it/evals=1890/2971 eff=70.7600% N=300
Z=-26.6(63.14%) | Like=-21.01..-20.46 [-21.0065..-21.0063]*| it/evals=1916/3009 eff=70.7272% N=300
Z=-26.6(63.56%) | Like=-21.00..-20.46 [-21.0046..-21.0009]*| it/evals=1920/3015 eff=70.7182% N=300
Mono-modal Volume: ~exp(-10.33) * 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=-26.5(65.55%) | Like=-20.96..-20.46 [-20.9585..-20.9542]*| it/evals=1943/3050 eff=70.6545% N=300
Z=-26.5(66.20%) | Like=-20.94..-20.46 [-20.9424..-20.9414]*| it/evals=1950/3060 eff=70.6522% N=300
Z=-26.5(68.58%) | Like=-20.92..-20.46 [-20.9179..-20.9125]*| it/evals=1978/3100 eff=70.6429% N=300
Z=-26.5(68.74%) | Like=-20.91..-20.46 [-20.9112..-20.9110]*| it/evals=1980/3104 eff=70.6134% N=300
[ultranest] Explored until L=-2e+01
[ultranest] Likelihood function evaluations: 3123
[ultranest] logZ = -26.13 +- 0.08106
[ultranest] Effective samples strategy satisfied (ESS = 1019.1, need >400)
[ultranest] Posterior uncertainty strategy is satisfied (KL: 0.46+-0.10 nat, need <0.50 nat)
[ultranest] Evidency uncertainty strategy is satisfied (dlogz=0.27, need <0.5)
[ultranest] logZ error budget: single: 0.12 bs:0.08 tail:0.26 total:0.27 required:<0.50
[ultranest] done iterating.
logZ = -26.127 +- 0.302
single instance: logZ = -26.127 +- 0.123
bootstrapped : logZ = -26.130 +- 0.151
tail : logZ = +- 0.262
insert order U test : converged: True correlation: inf iterations
index : 2.09 │ ▁ ▁▁▁▁▁▂▂▄▄▃▅▄▇▇▇▇▆▇▇▆▅▃▃▂▂▂▁▁▁▁▁▁ ▁ │3.10 2.57 +- 0.13
amplitude : 0.0000000000201│ ▁ ▁▁▁▁▁▂▃▄▅▆▅▇▆▆▇▆▆▆▄▆▄▃▃▁▂▂▂▁▁▁▁▁ ▁ │0.0000000000497 0.0000000000341 +- 0.0000000000040
[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=-741.50..-19.72 [-741.4992..-122.1910] | it/evals=0/301 eff=0.0000% N=300
Z=-217.8(0.00%) | Like=-211.79..-19.72 [-741.4992..-122.1910] | it/evals=30/334 eff=88.2353% N=300
Z=-201.3(0.00%) | Like=-195.06..-19.72 [-741.4992..-122.1910] | it/evals=60/370 eff=85.7143% N=300
Mono-modal Volume: ~exp(-4.10) * Expected Volume: exp(-0.22) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|******************************** * ****** ** **| +1.0e-10
Z=-197.9(0.00%) | Like=-193.09..-19.72 [-741.4992..-122.1910] | it/evals=67/379 eff=84.8101% N=300
Z=-183.1(0.00%) | Like=-177.83..-19.72 [-741.4992..-122.1910] | it/evals=90/406 eff=84.9057% N=300
Z=-170.8(0.00%) | Like=-165.04..-19.72 [-741.4992..-122.1910] | it/evals=120/442 eff=84.5070% N=300
Mono-modal Volume: ~exp(-4.66) * Expected Volume: exp(-0.45) Quality: ok
index : +1.0| ***********************************************| +5.0
amplitude: +1.0e-12|******************************** * ****** ******| +1.0e-10
Z=-162.8(0.00%) | Like=-156.92..-19.72 [-741.4992..-122.1910] | it/evals=134/458 eff=84.8101% N=300
Z=-156.0(0.00%) | Like=-150.94..-19.72 [-741.4992..-122.1910] | it/evals=150/477 eff=84.7458% N=300
Z=-143.7(0.00%) | Like=-138.88..-19.72 [-741.4992..-122.1910] | it/evals=180/517 eff=82.9493% N=300
Mono-modal Volume: ~exp(-5.00) * Expected Volume: exp(-0.67) Quality: ok
index : +1.0| **********************************************| +5.0
amplitude: +1.0e-12| ******************************** ****** ******| +1.0e-10
Z=-136.3(0.00%) | Like=-131.13..-19.72 [-741.4992..-122.1910] | it/evals=201/546 eff=81.7073% N=300
Z=-131.7(0.00%) | Like=-126.03..-19.71 [-741.4992..-122.1910] | it/evals=210/558 eff=81.3953% N=300
Z=-119.2(0.00%) | Like=-111.56..-19.71 [-121.9295..-63.6349] | it/evals=240/592 eff=82.1918% N=300
Mono-modal Volume: ~exp(-5.00) Expected Volume: exp(-0.89) Quality: ok
index : +1.0| *******************************************| +5.0
amplitude: +1.0e-12| ************************************** ******| +1.0e-10
Z=-107.6(0.00%) | Like=-102.22..-19.71 [-121.9295..-63.6349] | it/evals=268/630 eff=81.2121% N=300
Z=-107.1(0.00%) | Like=-101.33..-19.71 [-121.9295..-63.6349] | it/evals=270/633 eff=81.0811% N=300
Z=-97.1(0.00%) | Like=-91.85..-19.71 [-121.9295..-63.6349] | it/evals=300/665 eff=82.1918% N=300
Z=-87.1(0.00%) | Like=-81.32..-19.71 [-121.9295..-63.6349] | it/evals=330/699 eff=82.7068% N=300
Mono-modal Volume: ~exp(-5.00) Expected Volume: exp(-1.12) Quality: ok
index : +1.0| ******************************************| +5.0
amplitude: +1.0e-12| *******************************************| +1.0e-10
Z=-79.1(0.00%) | Like=-73.93..-19.71 [-121.9295..-63.6349] | it/evals=358/736 eff=82.1101% N=300
Z=-78.7(0.00%) | Like=-73.33..-19.71 [-121.9295..-63.6349] | it/evals=360/739 eff=82.0046% N=300
Z=-71.9(0.00%) | Like=-66.24..-19.71 [-121.9295..-63.6349] | it/evals=390/775 eff=82.1053% N=300
Mono-modal Volume: ~exp(-5.38) * 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=-63.60..-19.42 [-63.6038..-42.4920] | it/evals=402/789 eff=82.2086% N=300
Z=-65.3(0.00%) | Like=-59.71..-19.24 [-63.6038..-42.4920] | it/evals=420/819 eff=80.9249% N=300
Z=-61.3(0.00%) | Like=-56.21..-19.24 [-63.6038..-42.4920] | it/evals=450/858 eff=80.6452% N=300
Mono-modal Volume: ~exp(-5.52) * Expected Volume: exp(-1.56) Quality: ok
index : +1.0| ************************************* | +5.0
amplitude: +1.0e-12| *********************************** ***| +1.0e-10
Z=-59.3(0.00%) | Like=-54.34..-19.24 [-63.6038..-42.4920] | it/evals=469/887 eff=79.8978% N=300
Z=-58.0(0.00%) | Like=-52.90..-19.24 [-63.6038..-42.4920] | it/evals=480/900 eff=80.0000% N=300
Z=-54.8(0.00%) | Like=-49.54..-19.24 [-63.6038..-42.4920] | it/evals=510/936 eff=80.1887% N=300
Z=-52.5(0.00%) | Like=-47.27..-19.24 [-63.6038..-42.4920] | it/evals=535/976 eff=79.1420% N=300
Mono-modal Volume: ~exp(-5.69) * Expected Volume: exp(-1.79) Quality: ok
index : +1.0| ********************************* | +5.0
amplitude: +1.0e-12| **************************************| +1.0e-10
Z=-52.4(0.00%) | Like=-47.27..-19.24 [-63.6038..-42.4920] | it/evals=536/977 eff=79.1728% N=300
Z=-52.1(0.00%) | Like=-46.80..-19.24 [-63.6038..-42.4920] | it/evals=540/982 eff=79.1789% N=300
Z=-49.7(0.00%) | Like=-44.59..-19.24 [-63.6038..-42.4920] | it/evals=570/1017 eff=79.4979% N=300
Z=-47.5(0.00%) | Like=-42.15..-19.24 [-42.4344..-30.9759] | it/evals=597/1057 eff=78.8639% N=300
Z=-47.2(0.00%) | Like=-41.59..-19.24 [-42.4344..-30.9759] | it/evals=600/1061 eff=78.8436% N=300
Mono-modal Volume: ~exp(-5.88) * Expected Volume: exp(-2.01) Quality: ok
index : +1.0| +2.0 **************************** | +5.0
amplitude: +1.0e-12| +2.4e-11 *************************************| +1.0e-10
Z=-46.9(0.00%) | Like=-41.42..-19.24 [-42.4344..-30.9759] | it/evals=603/1067 eff=78.6180% N=300
Z=-44.7(0.00%) | Like=-39.39..-19.24 [-42.4344..-30.9759] | it/evals=630/1106 eff=78.1638% N=300
Z=-42.9(0.00%) | Like=-37.92..-19.24 [-42.4344..-30.9759] | it/evals=660/1149 eff=77.7385% N=300
Mono-modal Volume: ~exp(-6.20) * 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=-42.3(0.00%) | Like=-37.15..-19.24 [-42.4344..-30.9759] | it/evals=670/1161 eff=77.8165% N=300
Z=-41.3(0.00%) | Like=-36.40..-19.24 [-42.4344..-30.9759] | it/evals=690/1192 eff=77.3543% N=300
Z=-39.8(0.00%) | Like=-34.39..-19.24 [-42.4344..-30.9759] | it/evals=716/1233 eff=76.7417% N=300
Z=-39.6(0.00%) | Like=-34.05..-19.24 [-42.4344..-30.9759] | it/evals=720/1238 eff=76.7591% N=300
Mono-modal Volume: ~exp(-6.49) * 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=-38.5(0.00%) | Like=-33.12..-19.24 [-42.4344..-30.9759] | it/evals=737/1263 eff=76.5317% N=300
Z=-37.8(0.00%) | Like=-32.80..-19.24 [-42.4344..-30.9759] | it/evals=750/1282 eff=76.3747% N=300
Z=-36.6(0.00%) | Like=-31.48..-19.24 [-42.4344..-30.9759] | it/evals=780/1320 eff=76.4706% N=300
Mono-modal Volume: ~exp(-6.99) * 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=-35.7(0.00%) | Like=-30.73..-19.24 [-30.9722..-25.9967] | it/evals=804/1351 eff=76.4986% N=300
Z=-35.5(0.00%) | Like=-30.54..-19.24 [-30.9722..-25.9967] | it/evals=810/1361 eff=76.3431% N=300
Z=-34.3(0.00%) | Like=-28.87..-19.24 [-30.9722..-25.9967] | it/evals=840/1399 eff=76.4331% N=300
Z=-33.2(0.01%) | Like=-27.87..-19.24 [-30.9722..-25.9967] | it/evals=868/1439 eff=76.2072% N=300
Z=-33.1(0.01%) | Like=-27.77..-19.24 [-30.9722..-25.9967] | it/evals=870/1445 eff=75.9825% N=300
Mono-modal Volume: ~exp(-6.99) Expected Volume: exp(-2.90) Quality: ok
index : +1.0| +2.3 ***************** +3.6 | +5.0
amplitude: +1.0e-12| +3.4e-11 ************************ | +1.0e-10
Z=-32.1(0.04%) | Like=-26.79..-19.24 [-30.9722..-25.9967] | it/evals=897/1483 eff=75.8242% N=300
Z=-32.0(0.04%) | Like=-26.74..-19.24 [-30.9722..-25.9967] | it/evals=900/1487 eff=75.8214% N=300
Z=-31.3(0.08%) | Like=-26.16..-19.24 [-30.9722..-25.9967] | it/evals=922/1527 eff=75.1426% N=300
Z=-31.1(0.10%) | Like=-25.97..-19.24 [-25.9838..-25.6221] | it/evals=930/1539 eff=75.0605% N=300
Mono-modal Volume: ~exp(-7.20) * Expected Volume: exp(-3.13) Quality: ok
index : +1.0| +2.3 *************** +3.5 | +5.0
amplitude: +1.0e-12| +3.7e-11 ********************* +7.8e-11| +1.0e-10
Z=-30.9(0.12%) | Like=-25.76..-19.24 [-25.9838..-25.6221] | it/evals=938/1549 eff=75.1001% N=300
Z=-30.4(0.21%) | Like=-25.26..-19.24 [-25.2704..-25.2559] | it/evals=960/1578 eff=75.1174% N=300
Z=-29.7(0.41%) | Like=-24.53..-19.22 [-24.5284..-24.4805] | it/evals=990/1612 eff=75.4573% N=300
Mono-modal Volume: ~exp(-7.64) * Expected Volume: exp(-3.35) Quality: ok
index : +1.0| +2.3 ************* +3.4 | +5.0
amplitude: +1.0e-12| +3.8e-11 ******************* +7.6e-11 | +1.0e-10
Z=-29.4(0.56%) | Like=-24.29..-19.22 [-24.2863..-24.2698] | it/evals=1005/1634 eff=75.3373% N=300
Z=-29.1(0.74%) | Like=-24.00..-19.22 [-24.0000..-23.9980]*| it/evals=1020/1654 eff=75.3323% N=300
Z=-28.7(1.14%) | Like=-23.65..-19.20 [-23.6526..-23.6412] | it/evals=1047/1692 eff=75.2155% N=300
Z=-28.6(1.19%) | Like=-23.62..-19.20 [-23.6190..-23.5909] | it/evals=1050/1695 eff=75.2688% N=300
Mono-modal Volume: ~exp(-7.89) * Expected Volume: exp(-3.57) Quality: ok
index : +1.0| +2.4 ************* +3.3 | +5.0
amplitude: +1.0e-12| +4.0e-11 ****************** +7.4e-11 | +1.0e-10
Z=-28.3(1.63%) | Like=-23.37..-19.20 [-23.3786..-23.3679] | it/evals=1072/1727 eff=75.1226% N=300
Z=-28.2(1.80%) | Like=-23.31..-19.19 [-23.3298..-23.3107] | it/evals=1080/1737 eff=75.1566% N=300
Z=-27.9(2.53%) | Like=-22.95..-19.19 [-22.9464..-22.9309] | it/evals=1110/1773 eff=75.3564% N=300
Z=-27.6(3.46%) | Like=-22.63..-19.19 [-22.6335..-22.6322]*| it/evals=1136/1812 eff=75.1323% N=300
Mono-modal Volume: ~exp(-7.89) Expected Volume: exp(-3.80) Quality: ok
index : +1.0| +2.4 *********** +3.3 | +5.0
amplitude: +1.0e-12| +4.1e-11 **************** +7.2e-11 | +1.0e-10
Z=-27.5(3.60%) | Like=-22.60..-19.19 [-22.6232..-22.6038] | it/evals=1140/1817 eff=75.1483% N=300
Z=-27.3(4.64%) | Like=-22.42..-19.19 [-22.4234..-22.4063] | it/evals=1166/1858 eff=74.8395% N=300
Z=-27.3(4.81%) | Like=-22.36..-19.19 [-22.3848..-22.3648] | it/evals=1170/1863 eff=74.8560% N=300
Z=-27.0(6.08%) | Like=-22.07..-19.18 [-22.0653..-22.0607]*| it/evals=1198/1903 eff=74.7349% N=300
Z=-27.0(6.20%) | Like=-22.06..-19.18 [-22.0575..-22.0323] | it/evals=1200/1907 eff=74.6733% N=300
Mono-modal Volume: ~exp(-8.33) * Expected Volume: exp(-4.02) Quality: ok
index : +1.0| +2.5 ********** +3.2 | +5.0
amplitude: +1.0e-12| +4.3e-11 ************** +7.0e-11 | +1.0e-10
Z=-26.9(6.54%) | Like=-22.00..-19.18 [-21.9987..-21.9984]*| it/evals=1206/1916 eff=74.6287% N=300
Z=-26.8(7.84%) | Like=-21.85..-19.17 [-21.8488..-21.8482]*| it/evals=1230/1948 eff=74.6359% N=300
Z=-26.6(9.53%) | Like=-21.61..-19.17 [-21.6110..-21.6053]*| it/evals=1256/1988 eff=74.4076% N=300
Z=-26.5(9.83%) | Like=-21.59..-19.17 [-21.5870..-21.5616] | it/evals=1260/1993 eff=74.4241% N=300
Mono-modal Volume: ~exp(-8.42) * Expected Volume: exp(-4.24) Quality: ok
index : +1.0| +2.5 ********** +3.2 | +5.0
amplitude: +1.0e-12| +4.4e-11 ************* +6.8e-11 | +1.0e-10
Z=-26.4(10.70%) | Like=-21.47..-19.17 [-21.4703..-21.4694]*| it/evals=1273/2013 eff=74.3141% N=300
Z=-26.3(11.94%) | Like=-21.35..-19.17 [-21.3506..-21.3407]*| it/evals=1290/2038 eff=74.2232% N=300
Z=-26.2(14.14%) | Like=-21.14..-19.17 [-21.1389..-21.1338]*| it/evals=1317/2078 eff=74.0720% N=300
Z=-26.1(14.40%) | Like=-21.11..-19.17 [-21.1139..-21.1011] | it/evals=1320/2081 eff=74.1157% N=300
Mono-modal Volume: ~exp(-8.63) * Expected Volume: exp(-4.47) Quality: ok
index : +1.0| +2.5 ******** +3.1 | +5.0
amplitude: +1.0e-12| +4.5e-11 *********** +6.7e-11 | +1.0e-10
Z=-26.0(16.55%) | Like=-20.97..-19.17 [-20.9694..-20.9597]*| it/evals=1340/2105 eff=74.2382% N=300
Z=-26.0(17.56%) | Like=-20.91..-19.17 [-20.9076..-20.9048]*| it/evals=1350/2116 eff=74.3392% N=300
Z=-25.8(20.84%) | Like=-20.75..-19.17 [-20.7481..-20.7446]*| it/evals=1380/2153 eff=74.4738% N=300
Z=-25.7(23.74%) | Like=-20.61..-19.16 [-20.6092..-20.5987] | it/evals=1406/2193 eff=74.2736% N=300
Mono-modal Volume: ~exp(-9.06) * Expected Volume: exp(-4.69) Quality: ok
index : +1.0| +2.5 ******** +3.1 | +5.0
amplitude: +1.0e-12| +4.6e-11 *********** +6.5e-11 | +1.0e-10
Z=-25.7(23.87%) | Like=-20.60..-19.16 [-20.6092..-20.5987] | it/evals=1407/2194 eff=74.2872% N=300
Z=-25.6(24.25%) | Like=-20.59..-19.16 [-20.5906..-20.5822]*| it/evals=1410/2197 eff=74.3279% N=300
Z=-25.5(27.85%) | Like=-20.44..-19.16 [-20.4377..-20.4180] | it/evals=1440/2236 eff=74.3802% N=300
Z=-25.4(30.85%) | Like=-20.34..-19.16 [-20.3418..-20.3411]*| it/evals=1464/2276 eff=74.0891% N=300
Z=-25.4(31.66%) | Like=-20.33..-19.16 [-20.3251..-20.3149] | it/evals=1470/2287 eff=73.9809% N=300
Mono-modal Volume: ~exp(-9.21) * Expected Volume: exp(-4.91) 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(32.11%) | Like=-20.29..-19.16 [-20.2948..-20.2929]*| it/evals=1474/2293 eff=73.9589% N=300
Z=-25.3(35.35%) | Like=-20.22..-19.16 [-20.2224..-20.2209]*| it/evals=1500/2328 eff=73.9645% N=300
Z=-25.2(39.04%) | Like=-20.16..-19.16 [-20.1564..-20.1563]*| it/evals=1530/2367 eff=74.0203% N=300
Mono-modal Volume: ~exp(-9.33) * Expected Volume: exp(-5.14) Quality: ok
index : +1.0| +2.6 ****** +3.0 | +5.0
amplitude: +1.0e-12| +4.7e-11 ********* +6.3e-11 | +1.0e-10
Z=-25.1(40.47%) | Like=-20.13..-19.16 [-20.1313..-20.1247]*| it/evals=1541/2381 eff=74.0509% N=300
Z=-25.1(42.65%) | Like=-20.06..-19.16 [-20.0638..-20.0610]*| it/evals=1560/2406 eff=74.0741% N=300
Z=-25.0(46.23%) | Like=-20.00..-19.16 [-20.0019..-19.9999]*| it/evals=1590/2444 eff=74.1604% N=300
Mono-modal Volume: ~exp(-9.33) 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=-25.0(48.72%) | Like=-19.95..-19.16 [-19.9547..-19.9537]*| it/evals=1611/2480 eff=73.8991% N=300
Z=-24.9(49.70%) | Like=-19.94..-19.16 [-19.9362..-19.9361]*| it/evals=1620/2494 eff=73.8377% N=300
Z=-24.9(52.32%) | Like=-19.87..-19.16 [-19.8708..-19.8678]*| it/evals=1645/2533 eff=73.6677% N=300
Z=-24.9(52.89%) | Like=-19.86..-19.16 [-19.8580..-19.8573]*| it/evals=1650/2539 eff=73.6936% N=300
Z=-24.8(55.45%) | Like=-19.79..-19.16 [-19.7946..-19.7927]*| it/evals=1674/2579 eff=73.4533% 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| +4.9e-11 ******* +6.2e-11 | +1.0e-10
Z=-24.8(55.54%) | Like=-19.79..-19.16 [-19.7927..-19.7926]*| it/evals=1675/2580 eff=73.4649% N=300
Z=-24.8(56.03%) | Like=-19.78..-19.16 [-19.7776..-19.7756]*| it/evals=1680/2586 eff=73.4908% N=300
Z=-24.8(59.06%) | Like=-19.72..-19.16 [-19.7220..-19.7209]*| it/evals=1710/2623 eff=73.6117% N=300
Z=-24.7(62.06%) | Like=-19.66..-19.16 [-19.6602..-19.6590]*| it/evals=1740/2662 eff=73.6664% N=300
Mono-modal Volume: ~exp(-10.17) * Expected Volume: exp(-5.81) 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=-24.7(62.28%) | Like=-19.66..-19.16 [-19.6585..-19.6563]*| it/evals=1742/2664 eff=73.6887% N=300
Z=-24.7(64.94%) | Like=-19.60..-19.16 [-19.6039..-19.6031]*| it/evals=1770/2698 eff=73.8115% N=300
Z=-24.6(67.68%) | Like=-19.57..-19.16 [-19.5667..-19.5650]*| it/evals=1800/2736 eff=73.8916% N=300
Mono-modal Volume: ~exp(-10.22) * Expected Volume: exp(-6.03) Quality: ok
index : +1.0| +2.7 ***** +3.0 | +5.0
amplitude: +1.0e-12| +5.0e-11 ****** +6.0e-11 | +1.0e-10
Z=-24.6(68.46%) | Like=-19.55..-19.16 [-19.5525..-19.5521]*| it/evals=1809/2746 eff=73.9575% N=300
[ultranest] Explored until L=-2e+01
[ultranest] Likelihood function evaluations: 2769
[ultranest] logZ = -24.25 +- 0.08577
[ultranest] Effective samples strategy satisfied (ESS = 1009.1, 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 = -24.233 +- 0.307
single instance: logZ = -24.233 +- 0.116
bootstrapped : logZ = -24.247 +- 0.161
tail : logZ = +- 0.262
insert order U test : converged: True correlation: inf iterations
index : 2.29 │ ▁▁▁▁▁▁▁▂▃▃▃▆▅▇▆▇▇▆▇▆▆▄▃▄▃▂▂▁▁▁▁▁▁▁ ▁▁ │3.48 2.83 +- 0.16
amplitude : 0.0000000000360│ ▁▁▁▁▁▁▂▂▃▅▅▅▇▅▆▅▇▇▅▅▆▄▃▃▂▂▁▁▁▁▁▁▁▁ ▁▁ │0.0000000000801 0.0000000000551 +- 0.0000000000061
[ultranest] Sampling 300 live points from prior ...
Mono-modal Volume: ~exp(-4.10) * Expected Volume: exp(0.00) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|***************** *************** **** *********| +1.0e-10
Z=-inf(0.00%) | Like=-967.05..-13.37 [-967.0459..-100.9819] | it/evals=0/301 eff=0.0000% N=300
Z=-150.4(0.00%) | Like=-146.08..-13.37 [-967.0459..-100.9819] | it/evals=30/331 eff=96.7742% N=300
Z=-140.1(0.00%) | Like=-135.65..-13.37 [-967.0459..-100.9819] | it/evals=60/364 eff=93.7500% N=300
Mono-modal Volume: ~exp(-4.10) Expected Volume: exp(-0.22) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|***************** *************** **************| +1.0e-10
Z=-132.5(0.00%) | Like=-127.77..-13.37 [-967.0459..-100.9819] | it/evals=90/401 eff=89.1089% N=300
Z=-122.7(0.00%) | Like=-117.71..-13.37 [-967.0459..-100.9819] | it/evals=120/439 eff=86.3309% N=300
Mono-modal Volume: ~exp(-4.20) * Expected Volume: exp(-0.45) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|****************************** *****************| +1.0e-10
Z=-119.7(0.00%) | Like=-115.36..-13.37 [-967.0459..-100.9819] | it/evals=134/455 eff=86.4516% N=300
Z=-115.1(0.00%) | Like=-110.04..-13.37 [-967.0459..-100.9819] | it/evals=150/473 eff=86.7052% N=300
Z=-107.7(0.00%) | Like=-102.72..-13.37 [-967.0459..-100.9819] | it/evals=180/507 eff=86.9565% N=300
Mono-modal Volume: ~exp(-4.49) * Expected Volume: exp(-0.67) Quality: ok
index : +1.0| ***********************************************| +5.0
amplitude: +1.0e-12|************************************************| +1.0e-10
Z=-103.5(0.00%) | Like=-98.45..-13.37 [-100.9704..-55.2360] | it/evals=201/537 eff=84.8101% N=300
Z=-101.0(0.00%) | Like=-95.81..-13.37 [-100.9704..-55.2360] | it/evals=210/547 eff=85.0202% N=300
Z=-93.5(0.00%) | Like=-87.80..-13.37 [-100.9704..-55.2360] | it/evals=240/584 eff=84.5070% N=300
Mono-modal Volume: ~exp(-4.49) Expected Volume: exp(-0.89) Quality: ok
index : +1.0| *********************************************| +5.0
amplitude: +1.0e-12| ***********************************************| +1.0e-10
Z=-86.8(0.00%) | Like=-81.98..-13.37 [-100.9704..-55.2360] | it/evals=268/620 eff=83.7500% N=300
Z=-86.5(0.00%) | Like=-81.81..-13.37 [-100.9704..-55.2360] | it/evals=270/623 eff=83.5913% N=300
Z=-78.6(0.00%) | Like=-72.77..-13.37 [-100.9704..-55.2360] | it/evals=300/661 eff=83.1025% N=300
Z=-71.7(0.00%) | Like=-66.66..-13.37 [-100.9704..-55.2360] | it/evals=329/701 eff=82.0449% N=300
Z=-71.5(0.00%) | Like=-66.52..-13.37 [-100.9704..-55.2360] | it/evals=330/702 eff=82.0896% N=300
Mono-modal Volume: ~exp(-4.49) Expected Volume: exp(-1.12) Quality: ok
index : +1.0| *******************************************| +5.0
amplitude: +1.0e-12| **********************************************| +1.0e-10
Z=-66.2(0.00%) | Like=-60.75..-13.37 [-100.9704..-55.2360] | it/evals=358/738 eff=81.7352% N=300
Z=-65.8(0.00%) | Like=-60.55..-13.37 [-100.9704..-55.2360] | it/evals=360/740 eff=81.8182% N=300
Z=-62.0(0.00%) | Like=-56.81..-13.37 [-100.9704..-55.2360] | it/evals=384/781 eff=79.8337% N=300
Z=-61.0(0.00%) | Like=-55.61..-13.37 [-100.9704..-55.2360] | it/evals=390/788 eff=79.9180% N=300
Mono-modal Volume: ~exp(-5.40) * Expected Volume: exp(-1.34) Quality: ok
index : +1.0| *****************************************| +5.0
amplitude: +1.0e-12| ********************************************| +1.0e-10
Z=-59.1(0.00%) | Like=-53.65..-13.37 [-55.1218..-35.7195] | it/evals=402/812 eff=78.5156% N=300
Z=-55.7(0.00%) | Like=-49.90..-13.37 [-55.1218..-35.7195] | it/evals=420/835 eff=78.5047% N=300
Z=-51.2(0.00%) | Like=-46.18..-13.37 [-55.1218..-35.7195] | it/evals=450/873 eff=78.5340% N=300
Mono-modal Volume: ~exp(-5.57) * Expected Volume: exp(-1.56) Quality: ok
index : +1.0| ************************************** | +5.0
amplitude: +1.0e-12| *******************************************| +1.0e-10
Z=-49.1(0.00%) | Like=-43.92..-13.37 [-55.1218..-35.7195] | it/evals=469/898 eff=78.4281% N=300
Z=-48.1(0.00%) | Like=-43.48..-13.37 [-55.1218..-35.7195] | it/evals=480/911 eff=78.5597% N=300
Z=-46.2(0.00%) | Like=-41.52..-13.37 [-55.1218..-35.7195] | it/evals=507/951 eff=77.8802% N=300
Z=-46.0(0.00%) | Like=-41.01..-13.37 [-55.1218..-35.7195] | it/evals=510/956 eff=77.7439% N=300
Mono-modal Volume: ~exp(-5.57) Expected Volume: exp(-1.79) Quality: ok
index : +1.0| ********************************* | +5.0
amplitude: +1.0e-12| ******************************************| +1.0e-10
Z=-43.6(0.00%) | Like=-38.88..-13.37 [-55.1218..-35.7195] | it/evals=539/992 eff=77.8902% N=300
Z=-43.5(0.00%) | Like=-38.83..-13.37 [-55.1218..-35.7195] | it/evals=540/993 eff=77.9221% N=300
Z=-41.9(0.00%) | Like=-37.01..-13.34 [-55.1218..-35.7195] | it/evals=567/1034 eff=77.2480% N=300
Z=-41.7(0.00%) | Like=-36.72..-13.34 [-55.1218..-35.7195] | it/evals=570/1042 eff=76.8194% N=300
Z=-40.1(0.00%) | Like=-35.22..-13.34 [-35.5284..-24.4026] | it/evals=599/1082 eff=76.5985% N=300
Z=-40.1(0.00%) | Like=-35.22..-13.34 [-35.5284..-24.4026] | it/evals=600/1083 eff=76.6284% N=300
Mono-modal Volume: ~exp(-5.91) * Expected Volume: exp(-2.01) Quality: ok
index : +1.0| ****************************** | +5.0
amplitude: +1.0e-12| *****************************************| +1.0e-10
Z=-39.9(0.00%) | Like=-35.17..-13.34 [-35.5284..-24.4026] | it/evals=603/1090 eff=76.3291% N=300
Z=-37.9(0.00%) | Like=-32.30..-13.34 [-35.5284..-24.4026] | it/evals=630/1126 eff=76.2712% N=300
Z=-36.0(0.00%) | Like=-30.96..-13.34 [-35.5284..-24.4026] | it/evals=657/1166 eff=75.8661% N=300
Z=-35.9(0.00%) | Like=-30.78..-13.34 [-35.5284..-24.4026] | it/evals=660/1170 eff=75.8621% N=300
Mono-modal Volume: ~exp(-6.20) * Expected Volume: exp(-2.23) Quality: ok
index : +1.0| +2.0 ************************** +4.1 | +5.0
amplitude: +1.0e-12| ***************************************| +1.0e-10
Z=-35.3(0.00%) | Like=-30.30..-13.34 [-35.5284..-24.4026] | it/evals=670/1185 eff=75.7062% N=300
Z=-34.2(0.00%) | Like=-28.95..-13.34 [-35.5284..-24.4026] | it/evals=690/1209 eff=75.9076% N=300
Z=-32.5(0.00%) | Like=-27.37..-13.34 [-35.5284..-24.4026] | it/evals=720/1247 eff=76.0296% N=300
Mono-modal Volume: ~exp(-6.34) * Expected Volume: exp(-2.46) Quality: ok
index : +1.0| +2.0 *********************** +3.8 | +5.0
amplitude: +1.0e-12| **************************************| +1.0e-10
Z=-31.7(0.00%) | Like=-26.59..-13.34 [-35.5284..-24.4026] | it/evals=737/1269 eff=76.0578% N=300
Z=-31.1(0.00%) | Like=-26.09..-13.34 [-35.5284..-24.4026] | it/evals=750/1283 eff=76.2970% N=300
Z=-30.2(0.00%) | Like=-25.26..-13.34 [-35.5284..-24.4026] | it/evals=774/1324 eff=75.5859% N=300
Z=-30.0(0.00%) | Like=-25.03..-13.34 [-35.5284..-24.4026] | it/evals=780/1330 eff=75.7282% N=300
Mono-modal Volume: ~exp(-6.68) * Expected Volume: exp(-2.68) Quality: ok
index : +1.0| +2.1 ******************** +3.7 | +5.0
amplitude: +1.0e-12| +2.5e-11 ********************************** * | +1.0e-10
Z=-29.2(0.00%) | Like=-24.29..-13.34 [-24.3895..-20.2241] | it/evals=804/1362 eff=75.7062% N=300
Z=-29.0(0.00%) | Like=-24.09..-13.34 [-24.3895..-20.2241] | it/evals=810/1368 eff=75.8427% N=300
Z=-28.2(0.01%) | Like=-23.42..-13.34 [-24.3895..-20.2241] | it/evals=840/1402 eff=76.2250% N=300
Z=-27.4(0.01%) | Like=-22.24..-13.34 [-24.3895..-20.2241] | it/evals=870/1437 eff=76.5172% N=300
Mono-modal Volume: ~exp(-6.68) Expected Volume: exp(-2.90) Quality: ok
index : +1.0| +2.1 ****************** +3.6 | +5.0
amplitude: +1.0e-12| +2.7e-11 ********************************* | +1.0e-10
Z=-26.7(0.03%) | Like=-21.63..-13.34 [-24.3895..-20.2241] | it/evals=893/1476 eff=75.9354% N=300
Z=-26.5(0.03%) | Like=-21.42..-13.34 [-24.3895..-20.2241] | it/evals=900/1485 eff=75.9494% N=300
Z=-25.8(0.06%) | Like=-20.69..-13.34 [-24.3895..-20.2241] | it/evals=929/1524 eff=75.8987% N=300
Z=-25.8(0.06%) | Like=-20.67..-13.34 [-24.3895..-20.2241] | it/evals=930/1525 eff=75.9184% N=300
Mono-modal Volume: ~exp(-6.92) * Expected Volume: exp(-3.13) Quality: ok
index : +1.0| +2.2 **************** +3.5 | +5.0
amplitude: +1.0e-12| +2.9e-11 ****************************** | +1.0e-10
Z=-25.6(0.08%) | Like=-20.34..-13.34 [-24.3895..-20.2241] | it/evals=938/1537 eff=75.8286% N=300
Z=-25.0(0.14%) | Like=-19.89..-13.34 [-20.2082..-19.7510] | it/evals=960/1563 eff=76.0095% N=300
Z=-24.5(0.24%) | Like=-19.48..-13.34 [-19.4752..-19.4701]*| it/evals=988/1603 eff=75.8250% N=300
Z=-24.4(0.24%) | Like=-19.47..-13.34 [-19.4682..-19.4607]*| it/evals=990/1607 eff=75.7460% N=300
Mono-modal Volume: ~exp(-7.31) * Expected Volume: exp(-3.35) Quality: ok
index : +1.0| +2.2 *************** +3.4 | +5.0
amplitude: +1.0e-12| +3.1e-11 *************************** | +1.0e-10
Z=-24.2(0.32%) | Like=-19.21..-13.34 [-19.2273..-19.2133] | it/evals=1005/1626 eff=75.7919% N=300
Z=-23.9(0.42%) | Like=-18.88..-13.34 [-18.8957..-18.8794] | it/evals=1020/1647 eff=75.7238% N=300
Z=-23.5(0.62%) | Like=-18.54..-13.34 [-18.5362..-18.4856] | it/evals=1043/1685 eff=75.3069% N=300
Z=-23.4(0.70%) | Like=-18.42..-13.34 [-18.4158..-18.4048] | it/evals=1050/1697 eff=75.1611% N=300
Mono-modal Volume: ~exp(-7.85) * Expected Volume: exp(-3.57) Quality: ok
index : +1.0| +2.3 ************* +3.3 | +5.0
amplitude: +1.0e-12| +3.2e-11 *********************** +7.9e-11| +1.0e-10
Z=-23.1(0.97%) | Like=-18.08..-13.34 [-18.1098..-18.0840] | it/evals=1072/1734 eff=74.7559% N=300
Z=-23.0(1.09%) | Like=-17.93..-13.34 [-17.9266..-17.9145] | it/evals=1080/1744 eff=74.7922% N=300
Z=-22.6(1.68%) | Like=-17.36..-13.34 [-17.4242..-17.3596] | it/evals=1110/1782 eff=74.8988% N=300
Z=-22.2(2.39%) | Like=-17.13..-13.34 [-17.1260..-17.1147] | it/evals=1135/1821 eff=74.6220% N=300
Mono-modal Volume: ~exp(-7.85) Expected Volume: exp(-3.80) Quality: ok
index : +1.0| +2.3 ************ +3.2 | +5.0
amplitude: +1.0e-12| +3.5e-11 ********************* +7.7e-11 | +1.0e-10
Z=-22.2(2.53%) | Like=-17.04..-13.34 [-17.0438..-17.0384]*| it/evals=1140/1826 eff=74.7051% N=300
Z=-21.8(3.60%) | Like=-16.72..-13.34 [-16.7514..-16.7185] | it/evals=1170/1865 eff=74.7604% N=300
Z=-21.6(4.49%) | Like=-16.48..-13.34 [-16.5006..-16.4790] | it/evals=1192/1905 eff=74.2679% N=300
Z=-21.5(4.88%) | Like=-16.42..-13.34 [-16.4155..-16.3946] | it/evals=1200/1920 eff=74.0741% N=300
Mono-modal Volume: ~exp(-8.01) * Expected Volume: exp(-4.02) Quality: ok
index : +1.0| +2.4 *********** +3.2 | +5.0
amplitude: +1.0e-12| +3.7e-11 ******************* +7.4e-11 | +1.0e-10
Z=-21.5(5.21%) | Like=-16.37..-13.34 [-16.3812..-16.3652] | it/evals=1206/1927 eff=74.1242% N=300
Z=-21.2(6.54%) | Like=-16.12..-13.34 [-16.1189..-16.1186]*| it/evals=1230/1959 eff=74.1410% N=300
Z=-21.0(8.22%) | Like=-15.95..-13.34 [-15.9497..-15.9488]*| it/evals=1258/1998 eff=74.0872% N=300
Z=-21.0(8.38%) | Like=-15.94..-13.34 [-15.9432..-15.9422]*| it/evals=1260/2000 eff=74.1176% N=300
Mono-modal Volume: ~exp(-8.17) * Expected Volume: exp(-4.24) Quality: ok
index : +1.0| +2.4 ********** +3.1 | +5.0
amplitude: +1.0e-12| +3.8e-11 ****************** +7.2e-11 | +1.0e-10
Z=-20.9(9.30%) | Like=-15.84..-13.34 [-15.8433..-15.8359]*| it/evals=1273/2021 eff=73.9686% N=300
Z=-20.8(10.72%) | Like=-15.77..-13.34 [-15.7726..-15.7678]*| it/evals=1290/2044 eff=73.9679% N=300
Z=-20.6(12.74%) | Like=-15.53..-13.34 [-15.5294..-15.5229]*| it/evals=1320/2082 eff=74.0741% N=300
Mono-modal Volume: ~exp(-8.17) Expected Volume: exp(-4.47) Quality: ok
index : +1.0| +2.4 ********** +3.1 | +5.0
amplitude: +1.0e-12| +3.9e-11 **************** +7.0e-11 | +1.0e-10
Z=-20.4(14.69%) | Like=-15.40..-13.34 [-15.4013..-15.3757] | it/evals=1346/2118 eff=74.0374% N=300
Z=-20.4(15.13%) | Like=-15.36..-13.34 [-15.3756..-15.3646] | it/evals=1350/2129 eff=73.8108% N=300
Z=-20.2(17.90%) | Like=-15.19..-13.34 [-15.1942..-15.1928]*| it/evals=1377/2168 eff=73.7152% N=300
Z=-20.2(18.12%) | Like=-15.18..-13.34 [-15.1799..-15.1768]*| it/evals=1380/2172 eff=73.7179% N=300
Z=-20.1(20.53%) | Like=-15.05..-13.34 [-15.0499..-15.0444]*| it/evals=1404/2212 eff=73.4310% N=300
Mono-modal Volume: ~exp(-8.63) * Expected Volume: exp(-4.69) 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=-20.1(20.86%) | Like=-15.04..-13.34 [-15.0418..-15.0412]*| it/evals=1407/2215 eff=73.4726% N=300
Z=-20.1(21.16%) | Like=-15.03..-13.34 [-15.0406..-15.0306] | it/evals=1410/2221 eff=73.3993% N=300
Z=-19.9(24.18%) | Like=-14.88..-13.34 [-14.8821..-14.8821]*| it/evals=1440/2260 eff=73.4694% N=300
Z=-19.8(27.53%) | Like=-14.77..-13.34 [-14.7702..-14.7702]*| it/evals=1469/2299 eff=73.4867% N=300
Z=-19.8(27.60%) | Like=-14.77..-13.34 [-14.7702..-14.7647]*| it/evals=1470/2300 eff=73.5000% N=300
Mono-modal Volume: ~exp(-8.79) * Expected Volume: exp(-4.91) Quality: ok
index : +1.0| +2.5 ******** +3.0 | +5.0
amplitude: +1.0e-12| +4.1e-11 ************* +6.7e-11 | +1.0e-10
Z=-19.8(28.00%) | Like=-14.75..-13.34 [-14.7620..-14.7499] | it/evals=1474/2305 eff=73.5162% N=300
Z=-19.7(30.72%) | Like=-14.65..-13.34 [-14.6466..-14.6412]*| it/evals=1499/2344 eff=73.3366% N=300
Z=-19.7(30.80%) | Like=-14.64..-13.34 [-14.6412..-14.6374]*| it/evals=1500/2346 eff=73.3138% N=300
Z=-19.6(34.31%) | Like=-14.48..-13.31 [-14.4798..-14.4797]*| it/evals=1530/2381 eff=73.5223% N=300
Mono-modal Volume: ~exp(-9.03) * Expected Volume: exp(-5.14) 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=-19.6(35.58%) | Like=-14.44..-13.31 [-14.4352..-14.4335]*| it/evals=1541/2396 eff=73.5210% N=300
Z=-19.5(37.97%) | Like=-14.38..-13.31 [-14.3772..-14.3769]*| it/evals=1560/2418 eff=73.6544% N=300
Z=-19.4(40.98%) | Like=-14.29..-13.31 [-14.2915..-14.2803] | it/evals=1587/2457 eff=73.5744% N=300
Z=-19.4(41.38%) | Like=-14.27..-13.31 [-14.2731..-14.2705]*| it/evals=1590/2462 eff=73.5430% N=300
Mono-modal Volume: ~exp(-9.03) Expected Volume: exp(-5.36) Quality: ok
index : +1.0| +2.5 ****** +3.0 | +5.0
amplitude: +1.0e-12| +4.3e-11 *********** +6.4e-11 | +1.0e-10
Z=-19.3(44.17%) | Like=-14.21..-13.31 [-14.2110..-14.2091]*| it/evals=1615/2497 eff=73.5093% N=300
Z=-19.3(44.75%) | Like=-14.20..-13.31 [-14.1995..-14.1972]*| it/evals=1620/2507 eff=73.4028% N=300
Z=-19.3(47.91%) | Like=-14.12..-13.31 [-14.1228..-14.1221]*| it/evals=1647/2545 eff=73.3630% N=300
Z=-19.3(48.26%) | Like=-14.12..-13.31 [-14.1198..-14.1166]*| it/evals=1650/2551 eff=73.3008% N=300
Z=-19.2(50.48%) | Like=-14.07..-13.31 [-14.0702..-14.0681]*| it/evals=1670/2587 eff=73.0214% N=300
Mono-modal Volume: ~exp(-9.40) * Expected Volume: exp(-5.58) Quality: ok
index : +1.0| +2.5 ****** +2.9 | +5.0
amplitude: +1.0e-12| +4.4e-11 *********** +6.3e-11 | +1.0e-10
Z=-19.2(51.07%) | Like=-14.06..-13.31 [-14.0609..-14.0591]*| it/evals=1675/2596 eff=72.9530% N=300
Z=-19.2(51.65%) | Like=-14.05..-13.31 [-14.0495..-14.0485]*| it/evals=1680/2601 eff=73.0117% N=300
Z=-19.1(54.78%) | Like=-13.99..-13.31 [-13.9884..-13.9851]*| it/evals=1710/2635 eff=73.2334% N=300
Z=-19.1(57.77%) | Like=-13.91..-13.31 [-13.9141..-13.9101]*| it/evals=1738/2676 eff=73.1481% N=300
Z=-19.1(57.96%) | Like=-13.91..-13.31 [-13.9100..-13.9089]*| it/evals=1740/2679 eff=73.1400% N=300
Mono-modal Volume: ~exp(-9.72) * Expected Volume: exp(-5.81) 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=-19.1(58.17%) | Like=-13.91..-13.31 [-13.9064..-13.9021]*| it/evals=1742/2682 eff=73.1318% N=300
Z=-19.0(60.93%) | Like=-13.85..-13.31 [-13.8549..-13.8527]*| it/evals=1770/2716 eff=73.2616% N=300
Z=-19.0(63.85%) | Like=-13.79..-13.30 [-13.7914..-13.7908]*| it/evals=1800/2754 eff=73.3496% N=300
Mono-modal Volume: ~exp(-9.86) * Expected Volume: exp(-6.03) Quality: ok
index : +1.0| +2.6 ***** +2.9 | +5.0
amplitude: +1.0e-12| +4.6e-11 ********* +6.1e-11 | +1.0e-10
Z=-19.0(64.71%) | Like=-13.78..-13.30 [-13.7796..-13.7780]*| it/evals=1809/2766 eff=73.3577% N=300
Z=-18.9(66.68%) | Like=-13.74..-13.30 [-13.7442..-13.7392]*| it/evals=1830/2792 eff=73.4350% N=300
Z=-18.9(68.99%) | Like=-13.71..-13.30 [-13.7109..-13.7104]*| it/evals=1857/2831 eff=73.3702% N=300
Z=-18.9(69.24%) | Like=-13.71..-13.30 [-13.7059..-13.7055]*| it/evals=1860/2835 eff=73.3728% N=300
[ultranest] Explored until L=-1e+01
[ultranest] Likelihood function evaluations: 2846
[ultranest] logZ = -18.52 +- 0.08709
[ultranest] Effective samples strategy satisfied (ESS = 1032.7, need >400)
[ultranest] Posterior uncertainty strategy is satisfied (KL: 0.47+-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 = -18.534 +- 0.300
single instance: logZ = -18.534 +- 0.117
bootstrapped : logZ = -18.519 +- 0.146
tail : logZ = +- 0.262
insert order U test : converged: True correlation: inf iterations
index : 2.05 │ ▁ ▁▁▁▁▁▂▂▃▄▅▆▆▆▇▇▇▆▇▅▅▃▃▂▂▂▂▁▁▁▁▁▁ ▁ │3.49 2.75 +- 0.18
amplitude : 0.0000000000236│ ▁ ▁▁▁▁▁▁▂▂▃▄▅▇▇▇▇▇▇▇▄▆▅▄▃▃▂▂▂▁▂▁▁▁▁ ▁ │0.0000000000838 0.0000000000531 +- 0.0000000000083
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 46.674 seconds)
