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(-4.00) * Expected Volume: exp(0.00) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|******************* ******* ***** ****** * **** | +1.0e-10
Z=-inf(0.00%) | Like=-4300.42..-62.33 [-4300.4196..-307.0706] | it/evals=0/301 eff=0.0000% N=300
Z=-548.4(0.00%) | Like=-541.96..-62.33 [-4300.4196..-307.0706] | it/evals=22/324 eff=91.6667% N=300
Z=-542.0(0.00%) | Like=-535.58..-62.33 [-4300.4196..-307.0706] | it/evals=30/332 eff=93.7500% N=300
Z=-517.9(0.00%) | Like=-511.63..-62.33 [-4300.4196..-307.0706] | it/evals=51/355 eff=92.7273% N=300
Z=-501.7(0.00%) | Like=-495.74..-62.33 [-4300.4196..-307.0706] | it/evals=60/367 eff=89.5522% N=300
Mono-modal Volume: ~exp(-4.04) * Expected Volume: exp(-0.22) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|*************************** ***** ******** *** | +1.0e-10
Z=-491.4(0.00%) | Like=-484.90..-62.33 [-4300.4196..-307.0706] | it/evals=67/374 eff=90.5405% N=300
Z=-464.1(0.00%) | Like=-454.36..-62.16 [-4300.4196..-307.0706] | it/evals=88/397 eff=90.7216% N=300
Z=-457.0(0.00%) | Like=-450.76..-62.16 [-4300.4196..-307.0706] | it/evals=90/401 eff=89.1089% N=300
Z=-431.7(0.00%) | Like=-426.08..-62.16 [-4300.4196..-307.0706] | it/evals=109/425 eff=87.2000% N=300
Z=-416.0(0.00%) | Like=-409.37..-62.16 [-4300.4196..-307.0706] | it/evals=120/436 eff=88.2353% N=300
Mono-modal Volume: ~exp(-4.80) * Expected Volume: exp(-0.45) Quality: ok
index : +1.0| ***********************************************| +5.0
amplitude: +1.0e-12|********************************* ************ *| +1.0e-10
Z=-397.1(0.00%) | Like=-390.50..-62.16 [-4300.4196..-307.0706] | it/evals=134/453 eff=87.5817% N=300
Z=-377.3(0.00%) | Like=-368.51..-62.16 [-4300.4196..-307.0706] | it/evals=150/471 eff=87.7193% N=300
Z=-353.5(0.00%) | Like=-346.76..-62.16 [-4300.4196..-307.0706] | it/evals=168/495 eff=86.1538% N=300
Z=-343.8(0.00%) | Like=-336.09..-62.16 [-4300.4196..-307.0706] | it/evals=180/509 eff=86.1244% N=300
Z=-325.6(0.00%) | Like=-318.98..-60.52 [-4300.4196..-307.0706] | it/evals=200/532 eff=86.2069% N=300
Mono-modal Volume: ~exp(-4.80) Expected Volume: exp(-0.67) Quality: ok
index : +1.0| *********************************************| +5.0
amplitude: +1.0e-12| ********************************************* *| +1.0e-10
Z=-313.6(0.00%) | Like=-307.10..-60.52 [-4300.4196..-307.0706] | it/evals=210/547 eff=85.0202% N=300
Z=-302.0(0.00%) | Like=-294.39..-60.52 [-306.9740..-174.4427] | it/evals=228/571 eff=84.1328% N=300
Z=-290.0(0.00%) | Like=-283.57..-60.52 [-306.9740..-174.4427] | it/evals=240/586 eff=83.9161% N=300
Z=-274.0(0.00%) | Like=-266.81..-60.52 [-306.9740..-174.4427] | it/evals=256/609 eff=82.8479% N=300
Mono-modal Volume: ~exp(-4.95) * Expected Volume: exp(-0.89) Quality: ok
index : +1.0| *******************************************| +5.0
amplitude: +1.0e-12| ************************************ ******* *| +1.0e-10
Z=-266.0(0.00%) | Like=-259.59..-60.52 [-306.9740..-174.4427] | it/evals=268/626 eff=82.2086% N=300
Z=-265.0(0.00%) | Like=-256.41..-60.52 [-306.9740..-174.4427] | it/evals=270/628 eff=82.3171% N=300
Z=-253.3(0.00%) | Like=-245.32..-60.41 [-306.9740..-174.4427] | it/evals=289/651 eff=82.3362% N=300
Z=-244.9(0.00%) | Like=-238.26..-60.41 [-306.9740..-174.4427] | it/evals=300/664 eff=82.4176% N=300
Z=-226.9(0.00%) | Like=-219.65..-60.41 [-306.9740..-174.4427] | it/evals=318/687 eff=82.1705% N=300
Z=-220.6(0.00%) | Like=-213.20..-60.41 [-306.9740..-174.4427] | it/evals=330/704 eff=81.6832% N=300
Mono-modal Volume: ~exp(-5.53) * Expected Volume: exp(-1.12) Quality: ok
index : +1.0| ******************************************| +5.0
amplitude: +1.0e-12| ****************************************** *| +1.0e-10
Z=-214.8(0.00%) | Like=-208.55..-60.41 [-306.9740..-174.4427] | it/evals=335/709 eff=81.9071% N=300
Z=-204.2(0.00%) | Like=-196.11..-60.41 [-306.9740..-174.4427] | it/evals=357/732 eff=82.6389% N=300
Z=-200.4(0.00%) | Like=-193.52..-60.41 [-306.9740..-174.4427] | it/evals=360/737 eff=82.3799% N=300
Z=-193.8(0.00%) | Like=-187.69..-60.41 [-306.9740..-174.4427] | it/evals=378/760 eff=82.1739% N=300
Z=-189.3(0.00%) | Like=-183.55..-60.41 [-306.9740..-174.4427] | it/evals=390/778 eff=81.5900% N=300
Mono-modal Volume: ~exp(-5.53) Expected Volume: exp(-1.34) Quality: ok
index : +1.0| ************************************ | +5.0
amplitude: +1.0e-12| ***************************************** *| +1.0e-10
Z=-181.4(0.00%) | Like=-174.95..-60.41 [-306.9740..-174.4427] | it/evals=407/799 eff=81.5631% N=300
Z=-177.1(0.00%) | Like=-171.65..-60.41 [-174.2941..-118.0348] | it/evals=420/818 eff=81.0811% N=300
Z=-172.8(0.00%) | Like=-166.53..-60.14 [-174.2941..-118.0348] | it/evals=435/841 eff=80.4067% N=300
Z=-168.2(0.00%) | Like=-162.12..-60.14 [-174.2941..-118.0348] | it/evals=450/858 eff=80.6452% N=300
Z=-163.6(0.00%) | Like=-157.19..-60.14 [-174.2941..-118.0348] | it/evals=468/881 eff=80.5508% N=300
Mono-modal Volume: ~exp(-5.53) Expected Volume: exp(-1.56) Quality: ok
index : +1.0| ******************************* * | +5.0
amplitude: +1.0e-12| **************************************** | +1.0e-10
Z=-159.0(0.00%) | Like=-152.36..-60.14 [-174.2941..-118.0348] | it/evals=480/899 eff=80.1336% N=300
Z=-154.5(0.00%) | Like=-148.40..-60.14 [-174.2941..-118.0348] | it/evals=493/922 eff=79.2605% N=300
Z=-151.5(0.00%) | Like=-145.13..-60.14 [-174.2941..-118.0348] | it/evals=506/945 eff=78.4496% N=300
Z=-150.3(0.00%) | Like=-143.94..-60.14 [-174.2941..-118.0348] | it/evals=510/952 eff=78.2209% N=300
Z=-146.7(0.00%) | Like=-140.72..-59.48 [-174.2941..-118.0348] | it/evals=525/976 eff=77.6627% N=300
Mono-modal Volume: ~exp(-6.04) * Expected Volume: exp(-1.79) Quality: ok
index : +1.0| **************************** +4.1 | +5.0
amplitude: +1.0e-12| ******************************* **** * | +1.0e-10
Z=-144.9(0.00%) | Like=-139.30..-59.07 [-174.2941..-118.0348] | it/evals=536/1000 eff=76.5714% N=300
Z=-143.9(0.00%) | Like=-137.56..-59.07 [-174.2941..-118.0348] | it/evals=540/1006 eff=76.4873% N=300
Z=-138.8(0.00%) | Like=-132.61..-59.07 [-174.2941..-118.0348] | it/evals=559/1029 eff=76.6804% N=300
Z=-136.2(0.00%) | Like=-130.15..-59.07 [-174.2941..-118.0348] | it/evals=570/1043 eff=76.7160% N=300
Z=-131.9(0.00%) | Like=-125.26..-59.07 [-174.2941..-118.0348] | it/evals=587/1066 eff=76.6319% N=300
Z=-128.2(0.00%) | Like=-121.37..-59.07 [-174.2941..-118.0348] | it/evals=600/1089 eff=76.0456% N=300
Mono-modal Volume: ~exp(-6.31) * Expected Volume: exp(-2.01) Quality: ok
index : +1.0| ************************ +3.8 | +5.0
amplitude: +1.0e-12| ****************************** ** | +1.0e-10
Z=-127.4(0.00%) | Like=-121.12..-59.07 [-174.2941..-118.0348] | it/evals=603/1093 eff=76.0404% N=300
Z=-123.6(0.00%) | Like=-117.40..-59.07 [-118.0303..-87.0513] | it/evals=618/1116 eff=75.7353% N=300
Z=-121.7(0.00%) | Like=-116.04..-59.07 [-118.0303..-87.0513] | it/evals=630/1137 eff=75.2688% N=300
Z=-119.3(0.00%) | Like=-113.39..-59.07 [-118.0303..-87.0513] | it/evals=647/1161 eff=75.1452% N=300
Z=-117.1(0.00%) | Like=-110.17..-59.07 [-118.0303..-87.0513] | it/evals=660/1183 eff=74.7452% N=300
Mono-modal Volume: ~exp(-6.31) Expected Volume: exp(-2.23) Quality: ok
index : +1.0| +2.0 ********************** +3.7 | +5.0
amplitude: +1.0e-12| ****************************** | +1.0e-10
Z=-113.4(0.00%) | Like=-106.79..-59.07 [-118.0303..-87.0513] | it/evals=674/1204 eff=74.5575% N=300
Z=-110.3(0.00%) | Like=-104.02..-59.07 [-118.0303..-87.0513] | it/evals=690/1224 eff=74.6753% N=300
Z=-108.0(0.00%) | Like=-101.80..-59.07 [-118.0303..-87.0513] | it/evals=705/1249 eff=74.2887% N=300
Z=-105.6(0.00%) | Like=-98.93..-59.07 [-118.0303..-87.0513] | it/evals=720/1270 eff=74.2268% N=300
Mono-modal Volume: ~exp(-6.31) Expected Volume: exp(-2.46) Quality: ok
index : +1.0| +2.1 ******************* +3.6 | +5.0
amplitude: +1.0e-12| *************************** +7.5e-11 | +1.0e-10
Z=-103.4(0.00%) | Like=-97.51..-59.07 [-118.0303..-87.0513] | it/evals=737/1293 eff=74.2195% N=300
Z=-102.2(0.00%) | Like=-96.53..-59.07 [-118.0303..-87.0513] | it/evals=750/1315 eff=73.8916% N=300
Z=-101.3(0.00%) | Like=-95.25..-58.75 [-118.0303..-87.0513] | it/evals=761/1339 eff=73.2435% N=300
Z=-99.2(0.00%) | Like=-92.86..-58.75 [-118.0303..-87.0513] | it/evals=780/1362 eff=73.4463% N=300
Z=-97.5(0.00%) | Like=-91.43..-58.75 [-118.0303..-87.0513] | it/evals=795/1385 eff=73.2719% N=300
Mono-modal Volume: ~exp(-7.07) * Expected Volume: exp(-2.68) Quality: ok
index : +1.0| +2.1 ***************** +3.4 | +5.0
amplitude: +1.0e-12| +2.4e-11 ************************ +7.2e-11 | +1.0e-10
Z=-96.7(0.00%) | Like=-90.56..-58.75 [-118.0303..-87.0513] | it/evals=804/1396 eff=73.3577% N=300
Z=-96.1(0.00%) | Like=-90.33..-58.75 [-118.0303..-87.0513] | it/evals=810/1402 eff=73.5027% N=300
Z=-94.7(0.00%) | Like=-88.73..-58.75 [-118.0303..-87.0513] | it/evals=826/1426 eff=73.3570% N=300
Z=-93.5(0.00%) | Like=-87.40..-58.75 [-118.0303..-87.0513] | it/evals=840/1449 eff=73.1070% N=300
Z=-92.0(0.00%) | Like=-85.89..-58.75 [-87.0222..-72.4457] | it/evals=858/1472 eff=73.2082% N=300
Z=-91.1(0.00%) | Like=-85.04..-58.75 [-87.0222..-72.4457] | it/evals=870/1487 eff=73.2940% N=300
Mono-modal Volume: ~exp(-7.07) Expected Volume: exp(-2.90) Quality: ok
index : +1.0| +2.1 *************** +3.3 | +5.0
amplitude: +1.0e-12| +2.6e-11 ********************** +6.9e-11 | +1.0e-10
Z=-89.9(0.00%) | Like=-83.54..-58.75 [-87.0222..-72.4457] | it/evals=882/1508 eff=73.0132% N=300
Z=-88.8(0.00%) | Like=-82.40..-58.75 [-87.0222..-72.4457] | it/evals=895/1534 eff=72.5284% N=300
Z=-88.3(0.00%) | Like=-81.85..-58.75 [-87.0222..-72.4457] | it/evals=900/1540 eff=72.5806% N=300
Z=-87.0(0.00%) | Like=-80.91..-58.75 [-87.0222..-72.4457] | it/evals=916/1563 eff=72.5257% N=300
Z=-86.1(0.00%) | Like=-80.06..-58.75 [-87.0222..-72.4457] | it/evals=930/1582 eff=72.5429% N=300
Mono-modal Volume: ~exp(-7.07) Expected Volume: exp(-3.13) Quality: ok
index : +1.0| +2.2 ************** +3.3 | +5.0
amplitude: +1.0e-12| +2.8e-11 ******************** +6.5e-11 | +1.0e-10
Z=-85.2(0.00%) | Like=-79.08..-58.75 [-87.0222..-72.4457] | it/evals=944/1603 eff=72.4482% N=300
Z=-84.3(0.00%) | Like=-78.03..-58.75 [-87.0222..-72.4457] | it/evals=960/1624 eff=72.5076% N=300
Z=-83.2(0.00%) | Like=-77.13..-58.75 [-87.0222..-72.4457] | it/evals=977/1647 eff=72.5316% N=300
Z=-82.5(0.00%) | Like=-76.46..-58.75 [-87.0222..-72.4457] | it/evals=990/1668 eff=72.3684% N=300
Mono-modal Volume: ~exp(-7.37) * Expected Volume: exp(-3.35) Quality: ok
index : +1.0| +2.2 ************* +3.2 | +5.0
amplitude: +1.0e-12| +2.9e-11 ****************** +6.4e-11 | +1.0e-10
Z=-81.8(0.00%) | Like=-75.84..-58.75 [-87.0222..-72.4457] | it/evals=1005/1685 eff=72.5632% N=300
Z=-80.9(0.00%) | Like=-74.56..-58.75 [-87.0222..-72.4457] | it/evals=1020/1704 eff=72.6496% N=300
Z=-79.5(0.00%) | Like=-73.01..-58.75 [-87.0222..-72.4457] | it/evals=1041/1728 eff=72.8992% N=300
Z=-79.0(0.00%) | Like=-72.73..-58.75 [-87.0222..-72.4457] | it/evals=1050/1744 eff=72.7147% N=300
Z=-78.2(0.00%) | Like=-72.12..-58.75 [-72.4265..-66.1775] | it/evals=1066/1768 eff=72.6158% N=300
Mono-modal Volume: ~exp(-7.89) * Expected Volume: exp(-3.57) 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=-78.0(0.00%) | Like=-71.67..-58.75 [-72.4265..-66.1775] | it/evals=1072/1779 eff=72.4814% N=300
Z=-77.6(0.00%) | Like=-71.34..-58.75 [-72.4265..-66.1775] | it/evals=1080/1788 eff=72.5806% N=300
Z=-76.7(0.00%) | Like=-70.67..-58.75 [-72.4265..-66.1775] | it/evals=1101/1811 eff=72.8657% N=300
Z=-76.4(0.00%) | Like=-70.50..-58.75 [-72.4265..-66.1775] | it/evals=1110/1821 eff=72.9783% N=300
Z=-75.9(0.00%) | Like=-69.99..-58.75 [-72.4265..-66.1775] | it/evals=1128/1844 eff=73.0570% N=300
Mono-modal Volume: ~exp(-8.02) * Expected Volume: exp(-3.80) Quality: ok
index : +1.0| +2.3 *********** +3.1 | +5.0
amplitude: +1.0e-12| +3.2e-11 ************** +6.0e-11 | +1.0e-10
Z=-75.6(0.00%) | Like=-69.54..-58.75 [-72.4265..-66.1775] | it/evals=1139/1861 eff=72.9660% N=300
Z=-75.6(0.00%) | Like=-69.49..-58.75 [-72.4265..-66.1775] | it/evals=1140/1862 eff=72.9834% N=300
Z=-75.0(0.00%) | Like=-68.96..-58.75 [-72.4265..-66.1775] | it/evals=1159/1885 eff=73.1230% N=300
Z=-74.7(0.01%) | Like=-68.69..-58.75 [-72.4265..-66.1775] | it/evals=1170/1900 eff=73.1250% N=300
Z=-74.4(0.01%) | Like=-68.38..-58.75 [-72.4265..-66.1775] | it/evals=1183/1924 eff=72.8448% N=300
Z=-73.9(0.01%) | Like=-67.80..-58.75 [-72.4265..-66.1775] | it/evals=1200/1945 eff=72.9483% N=300
Mono-modal Volume: ~exp(-8.13) * Expected Volume: exp(-4.02) Quality: ok
index : +1.0| +2.3 ********* +3.0 | +5.0
amplitude: +1.0e-12| +3.3e-11 ************* +5.8e-11 | +1.0e-10
Z=-73.8(0.01%) | Like=-67.55..-58.75 [-72.4265..-66.1775] | it/evals=1206/1952 eff=73.0024% N=300
Z=-73.2(0.02%) | Like=-67.11..-58.75 [-72.4265..-66.1775] | it/evals=1226/1975 eff=73.1940% N=300
Z=-73.1(0.03%) | Like=-66.99..-58.75 [-72.4265..-66.1775] | it/evals=1230/1979 eff=73.2579% N=300
Z=-72.6(0.04%) | Like=-66.53..-58.75 [-72.4265..-66.1775] | it/evals=1251/2002 eff=73.5018% N=300
Z=-72.4(0.05%) | Like=-66.38..-58.75 [-72.4265..-66.1775] | it/evals=1260/2012 eff=73.5981% N=300
Mono-modal Volume: ~exp(-8.36) * Expected Volume: exp(-4.24) Quality: ok
index : +1.0| +2.4 ********* +3.0 | +5.0
amplitude: +1.0e-12| +3.4e-11 ************* +5.7e-11 | +1.0e-10
Z=-72.2(0.06%) | Like=-66.16..-58.75 [-66.1634..-65.0841] | it/evals=1273/2028 eff=73.6690% N=300
Z=-71.8(0.09%) | Like=-65.80..-58.75 [-66.1634..-65.0841] | it/evals=1290/2048 eff=73.7986% N=300
Z=-71.5(0.13%) | Like=-65.22..-58.75 [-66.1634..-65.0841] | it/evals=1309/2070 eff=73.9548% N=300
Z=-71.2(0.17%) | Like=-65.01..-58.75 [-65.0798..-64.9481] | it/evals=1320/2085 eff=73.9496% N=300
Z=-70.9(0.24%) | Like=-64.64..-58.75 [-64.6744..-64.6382] | it/evals=1335/2108 eff=73.8385% N=300
Mono-modal Volume: ~exp(-8.66) * Expected Volume: exp(-4.47) 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=-70.8(0.26%) | Like=-64.53..-58.75 [-64.5619..-64.5279] | it/evals=1340/2118 eff=73.7074% N=300
Z=-70.6(0.33%) | Like=-64.27..-58.75 [-64.2737..-64.2503] | it/evals=1350/2129 eff=73.8108% N=300
Z=-70.3(0.48%) | Like=-63.91..-58.75 [-63.9466..-63.9073] | it/evals=1367/2152 eff=73.8121% N=300
Z=-70.0(0.62%) | Like=-63.67..-58.75 [-63.6706..-63.6648]*| it/evals=1380/2169 eff=73.8363% N=300
Z=-69.7(0.86%) | Like=-63.35..-58.75 [-63.3721..-63.3462] | it/evals=1398/2191 eff=73.9291% N=300
Mono-modal Volume: ~exp(-8.66) Expected Volume: exp(-4.69) Quality: ok
index : +1.0| +2.4 ******* +2.9 | +5.0
amplitude: +1.0e-12| +3.6e-11 ********** +5.4e-11 | +1.0e-10
Z=-69.5(1.07%) | Like=-63.19..-58.75 [-63.1928..-63.1874]*| it/evals=1410/2210 eff=73.8220% N=300
Z=-69.2(1.34%) | Like=-63.00..-58.75 [-62.9951..-62.9804] | it/evals=1427/2233 eff=73.8231% N=300
Z=-69.0(1.58%) | Like=-62.80..-58.75 [-62.8034..-62.8011]*| it/evals=1440/2253 eff=73.7327% N=300
Z=-68.8(1.94%) | Like=-62.58..-58.75 [-62.6073..-62.5795] | it/evals=1455/2276 eff=73.6336% N=300
Z=-68.6(2.38%) | Like=-62.41..-58.75 [-62.4155..-62.3984] | it/evals=1470/2296 eff=73.6473% N=300
Mono-modal Volume: ~exp(-8.83) * Expected Volume: exp(-4.91) Quality: ok
index : +1.0| +2.5 ****** +2.9 | +5.0
amplitude: +1.0e-12| +3.7e-11 ********* +5.3e-11 | +1.0e-10
Z=-68.6(2.50%) | Like=-62.36..-58.75 [-62.3563..-62.3537]*| it/evals=1474/2301 eff=73.6632% N=300
Z=-68.4(3.13%) | Like=-62.11..-58.75 [-62.1108..-62.1090]*| it/evals=1493/2326 eff=73.6920% N=300
Z=-68.3(3.38%) | Like=-62.07..-58.75 [-62.0651..-62.0647]*| it/evals=1500/2338 eff=73.6016% N=300
Z=-68.1(4.07%) | Like=-61.90..-58.75 [-61.8954..-61.8868]*| it/evals=1516/2361 eff=73.5565% N=300
Z=-68.0(4.75%) | Like=-61.75..-58.75 [-61.7482..-61.7476]*| it/evals=1530/2380 eff=73.5577% N=300
Mono-modal Volume: ~exp(-9.42) * Expected Volume: exp(-5.14) 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(5.24%) | Like=-61.63..-58.75 [-61.6317..-61.6217] | it/evals=1541/2393 eff=73.6264% N=300
Z=-67.7(6.23%) | Like=-61.43..-58.75 [-61.4306..-61.4275]*| it/evals=1560/2414 eff=73.7938% N=300
Z=-67.5(7.18%) | Like=-61.31..-58.75 [-61.3144..-61.2959] | it/evals=1575/2437 eff=73.7015% N=300
Z=-67.4(8.30%) | Like=-61.14..-58.75 [-61.1370..-61.1359]*| it/evals=1590/2456 eff=73.7477% N=300
Z=-67.3(9.48%) | Like=-61.02..-58.75 [-61.0207..-61.0094] | it/evals=1605/2480 eff=73.6239% N=300
Mono-modal Volume: ~exp(-9.59) * Expected Volume: exp(-5.36) Quality: ok
index : +1.0| +2.5 ****** +2.9 | +5.0
amplitude: +1.0e-12| +3.8e-11 ******* +5.1e-11 | +1.0e-10
Z=-67.2(9.73%) | Like=-61.00..-58.75 [-61.0007..-60.9947]*| it/evals=1608/2484 eff=73.6264% N=300
Z=-67.2(10.60%) | Like=-60.93..-58.75 [-60.9339..-60.9298]*| it/evals=1620/2499 eff=73.6698% N=300
Z=-67.0(11.99%) | Like=-60.82..-58.75 [-60.8204..-60.8186]*| it/evals=1636/2522 eff=73.6274% N=300
Z=-66.9(13.34%) | Like=-60.76..-58.75 [-60.7608..-60.7456] | it/evals=1650/2542 eff=73.5950% N=300
Z=-66.8(15.08%) | Like=-60.59..-58.75 [-60.5914..-60.5861]*| it/evals=1669/2569 eff=73.5566% N=300
Mono-modal Volume: ~exp(-9.84) * Expected Volume: exp(-5.58) Quality: ok
index : +1.0| +2.5 **** +2.8 | +5.0
amplitude: +1.0e-12| +3.9e-11 ****** +5.0e-11 | +1.0e-10
Z=-66.8(15.67%) | Like=-60.56..-58.75 [-60.5794..-60.5645] | it/evals=1675/2577 eff=73.5617% N=300
Z=-66.7(16.18%) | Like=-60.51..-58.75 [-60.5107..-60.5066]*| it/evals=1680/2582 eff=73.6196% N=300
Z=-66.6(17.99%) | Like=-60.41..-58.75 [-60.4235..-60.4127] | it/evals=1696/2605 eff=73.5792% N=300
Z=-66.6(19.66%) | Like=-60.33..-58.75 [-60.3295..-60.3293]*| it/evals=1710/2622 eff=73.6434% N=300
Z=-66.5(21.65%) | Like=-60.24..-58.75 [-60.2368..-60.2343]*| it/evals=1728/2645 eff=73.6887% N=300
Z=-66.4(23.00%) | Like=-60.18..-58.75 [-60.1787..-60.1623] | it/evals=1740/2665 eff=73.5729% N=300
Mono-modal Volume: ~exp(-9.96) * 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=-66.4(23.21%) | Like=-60.16..-58.75 [-60.1620..-60.1615]*| it/evals=1742/2667 eff=73.5953% N=300
Z=-66.3(25.00%) | Like=-60.10..-58.75 [-60.1026..-60.1016]*| it/evals=1758/2691 eff=73.5257% N=300
Z=-66.3(26.40%) | Like=-60.07..-58.75 [-60.0693..-60.0576] | it/evals=1770/2708 eff=73.5050% N=300
Z=-66.2(28.42%) | Like=-59.99..-58.75 [-59.9864..-59.9805]*| it/evals=1786/2731 eff=73.4677% N=300
Z=-66.1(30.24%) | Like=-59.93..-58.75 [-59.9267..-59.9247]*| it/evals=1800/2754 eff=73.3496% N=300
Mono-modal Volume: ~exp(-10.55) * Expected Volume: exp(-6.03) Quality: ok
index : +1.0| +2.5 **** +2.8 | +5.0
amplitude: +1.0e-12| +4.0e-11 ****** +4.9e-11 | +1.0e-10
Z=-66.1(31.37%) | Like=-59.86..-58.75 [-59.8627..-59.8600]*| it/evals=1809/2766 eff=73.3577% N=300
Z=-66.0(33.59%) | Like=-59.79..-58.75 [-59.7918..-59.7891]*| it/evals=1827/2788 eff=73.4325% N=300
Z=-66.0(33.97%) | Like=-59.79..-58.75 [-59.7858..-59.7834]*| it/evals=1830/2791 eff=73.4645% N=300
Z=-65.9(36.09%) | Like=-59.71..-58.75 [-59.7063..-59.7058]*| it/evals=1847/2815 eff=73.4394% N=300
Z=-65.9(37.81%) | Like=-59.68..-58.75 [-59.6761..-59.6671]*| it/evals=1860/2834 eff=73.4017% N=300
Mono-modal Volume: ~exp(-10.72) * Expected Volume: exp(-6.25) 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=-65.9(39.76%) | Like=-59.63..-58.75 [-59.6309..-59.6267]*| it/evals=1876/2854 eff=73.4534% N=300
Z=-65.8(41.39%) | Like=-59.59..-58.75 [-59.5916..-59.5902]*| it/evals=1890/2874 eff=73.4266% N=300
Z=-65.8(43.80%) | Like=-59.55..-58.75 [-59.5490..-59.5425]*| it/evals=1909/2898 eff=73.4796% N=300
Z=-65.7(45.10%) | Like=-59.51..-58.75 [-59.5073..-59.4983]*| it/evals=1920/2911 eff=73.5350% N=300
Z=-65.7(47.47%) | Like=-59.46..-58.75 [-59.4629..-59.4611]*| it/evals=1938/2935 eff=73.5484% N=300
Mono-modal Volume: ~exp(-11.10) * Expected Volume: exp(-6.48) Quality: ok
index : +1.0| +2.6 **** +2.8 | +5.0
amplitude: +1.0e-12| +4.1e-11 **** +4.8e-11 | +1.0e-10
Z=-65.7(48.07%) | Like=-59.46..-58.75 [-59.4556..-59.4535]*| it/evals=1943/2941 eff=73.5706% N=300
Z=-65.6(48.89%) | Like=-59.44..-58.75 [-59.4447..-59.4422]*| it/evals=1950/2950 eff=73.5849% N=300
Z=-65.6(50.66%) | Like=-59.40..-58.75 [-59.4022..-59.4020]*| it/evals=1966/2972 eff=73.5778% N=300
Z=-65.6(52.24%) | Like=-59.38..-58.75 [-59.3788..-59.3786]*| it/evals=1980/2993 eff=73.5240% N=300
Z=-65.5(54.23%) | Like=-59.35..-58.75 [-59.3468..-59.3447]*| it/evals=1998/3015 eff=73.5912% N=300
Mono-modal Volume: ~exp(-11.10) 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=-65.5(55.55%) | Like=-59.33..-58.75 [-59.3296..-59.3295]*| it/evals=2010/3037 eff=73.4381% N=300
Z=-65.5(57.38%) | Like=-59.31..-58.75 [-59.3105..-59.3075]*| it/evals=2026/3060 eff=73.4058% N=300
Z=-65.5(58.83%) | Like=-59.29..-58.75 [-59.2902..-59.2889]*| it/evals=2040/3077 eff=73.4606% N=300
Z=-65.4(60.55%) | Like=-59.26..-58.75 [-59.2608..-59.2575]*| it/evals=2056/3101 eff=73.4024% N=300
Z=-65.4(61.90%) | Like=-59.24..-58.75 [-59.2397..-59.2396]*| it/evals=2070/3119 eff=73.4303% N=300
Mono-modal Volume: ~exp(-11.32) * Expected Volume: exp(-6.92) Quality: ok
index : +1.0| +2.6 *** +2.8 | +5.0
amplitude: +1.0e-12| +4.2e-11 **** +4.7e-11 | +1.0e-10
Z=-65.4(62.55%) | Like=-59.24..-58.75 [-59.2366..-59.2340]*| it/evals=2077/3129 eff=73.4182% N=300
Z=-65.4(64.22%) | Like=-59.21..-58.75 [-59.2127..-59.2123]*| it/evals=2094/3152 eff=73.4222% N=300
Z=-65.4(64.72%) | Like=-59.20..-58.75 [-59.2029..-59.2014]*| it/evals=2100/3159 eff=73.4523% N=300
Z=-65.3(66.47%) | Like=-59.18..-58.75 [-59.1820..-59.1817]*| it/evals=2119/3182 eff=73.5253% N=300
Z=-65.3(67.39%) | Like=-59.17..-58.75 [-59.1669..-59.1663]*| it/evals=2130/3200 eff=73.4483% N=300
Mono-modal Volume: ~exp(-11.38) * 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=-65.3(68.54%) | Like=-59.14..-58.75 [-59.1402..-59.1362]*| it/evals=2144/3221 eff=73.3995% N=300
Z=-65.3(69.90%) | Like=-59.12..-58.75 [-59.1182..-59.1169]*| it/evals=2160/3240 eff=73.4694% N=300
[ultranest] Explored until L=-6e+01
[ultranest] Likelihood function evaluations: 3241
[ultranest] logZ = -64.92 +- 0.09785
[ultranest] Effective samples strategy satisfied (ESS = 983.0, need >400)
[ultranest] Posterior uncertainty strategy is satisfied (KL: 0.46+-0.07 nat, need <0.50 nat)
[ultranest] Evidency uncertainty strategy is satisfied (dlogz=0.28, need <0.5)
[ultranest] logZ error budget: single: 0.13 bs:0.10 tail:0.26 total:0.28 required:<0.50
[ultranest] done iterating.
logZ = -64.921 +- 0.334
single instance: logZ = -64.921 +- 0.132
bootstrapped : logZ = -64.923 +- 0.207
tail : logZ = +- 0.262
insert order U test : converged: True correlation: inf iterations
index : 2.336 │ ▁ ▁▁▁▁▁▁▁▂▂▃▃▄▅▆▇▇▆▇▅▄▄▄▃▂▂▁▁▁▁▁▁▁▁▁ │2.982 2.672 +- 0.083
amplitude : 0.0000000000332│ ▁ ▁▁▁▁▁▂▂▃▄▆▆▆▇▇▇▆▆▆▅▄▃▂▂▁▁▁▁▁▁▁ ▁ │0.0000000000586 0.0000000000443 +- 0.0000000000030
Understanding the outputs#
In the Jupyter notebook, you should be able to see an interactive visualisation of how the parameter space shrinks which starts from the (min,max) shrinks down towards the optimal parameters.
The output above is filled with interesting information. Here we provide a short description of the most relevant information provided above. For more detailed information see the UltraNest docs.
During the sampling
Z=-68.8(0.53%) | Like=-63.96..-58.75 [-63.9570..-63.9539]*| it/evals=640/1068 eff=73.7327% N=300
Some important information here is:
Progress (0.53%): the completed fraction of the integral. This is not a time progress bar. Stays at zero for a good fraction of the run.
Efficiency (eff value) of the sampling: this indicates out of the proposed new points, how many were accepted. If your efficiency is too small (<<1%), maybe you should revise your priors (e.g use a LogUniform prior for the normalisation).
Final outputs
The final lines indicate that all three “convergence” strategies are satisfied (samples, posterior uncertainty, and evidence uncertainty).
logZ = -65.104 +- 0.292
The main goal of the Nested sampling algorithm is to estimate Z (the Bayesian evidence) which is given above together with an uncertainty. In a similar way to deltaLogLike and deltaAIC, deltaLogZ values can be used for model comparison. For more information see : on the use of the evidence for model comparison. An interesting comparison of the efficiency and false discovery rate of model selection with deltaLogLike and deltaLogZ is given in Appendix C of Buchner et al., 2014.
Results stored on disk
if log_dir is set to a name where the results will be stored, then
a directory is created containing many useful results and plots.
A description of these outputs is given in the Ultranest
docs.
Results#
Within a Bayesian analysis, the concept of best-fit has to be viewed differently from what is done in a gradient descent fit.
The output of the Bayesian analysis is the posterior distribution and there is no “best-fit” output. One has to define, based on the posteriors, what we want to consider as “best-fit” and several options are possible:
the mean of the distribution
the median
the lowest likelihood value
By default the DatasetModels will be updated with the mean of
the posterior distributions.
print(result_joint.models)
DatasetModels
Component 0: SkyModel
Name : crab
Datasets names : None
Spectral model type : PowerLawSpectralModel
Spatial model type :
Temporal model type :
Parameters:
index : 2.672 +/- 0.08
amplitude : 4.43e-11 +/- 3.0e-12 1 / (TeV s cm2)
reference (frozen): 1.000 TeV
The Sampler class returns a very rich dictionary.
The most “standard” information about the posterior distributions can
be found in :
print(result_joint.sampler_results["posterior"])
{'mean': [2.672106112724858, 4.427672156505491e-11], 'stdev': [0.08338132802725552, 2.9696520185358577e-12], 'median': [2.6657757708326453, 4.416698138357718e-11], 'errlo': [2.5951722429374215, 4.1242970444550054e-11], 'errup': [2.7591728307199346, 4.735413466118044e-11], 'information_gain_bits': [2.7035057519858374, 3.0978182768854174]}
Besides mean, errors, etc, an interesting value is the
information gain which estimates how much the posterior
distribution has shrunk with respect to the prior (i.e. how much
we’ve learned). A value < 1 means that the parameter is poorly
constrained within the prior range (we haven’t learned much with respect to our prior assumption).
For a physical interpretation of the information gain see this
example.
The SamplerResult dictionary contains also other interesting
information :
print(result_joint.sampler_results.keys())
dict_keys(['niter', 'logz', 'logzerr', 'logz_bs', 'logz_single', 'logzerr_tail', 'logzerr_bs', 'ess', 'H', 'Herr', 'posterior', 'weighted_samples', 'samples', 'maximum_likelihood', 'ncall', 'paramnames', 'logzerr_single', 'insertion_order_MWW_test'])
Of particular interest, the samples used in the process to approximate the posterior distribution can be accessed via :
for i, n in enumerate(model.parameters.free_parameters.names):
s = result_joint.samples[:, i]
fig, ax = plt.subplots()
ax.hist(s, bins=30)
ax.axvline(np.mean(s), ls="--", color="red")
ax.set_xlabel(n)
plt.show()
While the above plots are interesting, the real strength of the Bayesian analysis is to visualise all parameters correlations which is usually done using “corner plots”. Ultranest corner plot function is a wrapper around the corner package. See the above link for optional keywords. Other packages exist for corner plots, like chainconsumer which is discussed later in this tutorial.
from ultranest.plot import cornerplot
cornerplot(
result_joint.sampler_results,
plot_datapoints=True,
plot_density=True,
bins=20,
title_fmt=".2e",
smooth=False,
)
plt.show()
Spectral model error band from samples#
To compute the spectral error band (“butterfly plots”), we will directly use the samples of the posterior distribution. This is more robust as compared to the traditional method of using the covariance matrix of the parameters which implicitly assumes Gaussian errors while for the posterior distribution there is no shape assumed. This difference can become significant when the parameter errors are non-Gaussian. For this we will need to convert the list of samples back to the spectral model parameters with the relevant units (e.g. normalisation units).
def get_samples_from_posterior(spectral_model, results):
"""
Create a list of spectral parameters with correct units
from the unitless parameters returned by the sampler.
"""
n_samples = results.samples.shape[0]
samples = []
for p in spectral_model.parameters:
try:
idx = spectral_model.parameters.free_unique_parameters.index(p)
samples.append(results.samples[:, idx] * p.unit)
except ValueError:
samples.append(np.ones(n_samples) * p.quantity)
return samples
samples = get_samples_from_posterior(datasets.models[0].spectral_model, result_joint)
Next we can provide these samples to the plot_error
method.
Individual run analysis#
Now we’ll analyse several Crab runs individually so that we can compare them.
result_0 = sampler.run(datasets[0])
result_1 = sampler.run(datasets[1])
result_2 = sampler.run(datasets[2])
[ultranest] Sampling 300 live points from prior ...
Mono-modal Volume: ~exp(-3.87) * Expected Volume: exp(0.00) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|************************* ******** ********* ***| +1.0e-10
Z=-inf(0.00%) | Like=-827.54..-20.76 [-827.5407..-108.6713] | it/evals=0/301 eff=0.0000% N=300
Z=-174.4(0.00%) | Like=-169.54..-20.76 [-827.5407..-108.6713] | it/evals=30/333 eff=90.9091% N=300
Z=-161.1(0.00%) | Like=-155.36..-20.76 [-827.5407..-108.6713] | it/evals=60/366 eff=90.9091% N=300
Mono-modal Volume: ~exp(-4.17) * Expected Volume: exp(-0.22) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|********************* *** ******* ********** ***| +1.0e-10
Z=-157.2(0.00%) | Like=-152.35..-20.76 [-827.5407..-108.6713] | it/evals=67/374 eff=90.5405% N=300
Z=-148.8(0.00%) | Like=-143.15..-20.76 [-827.5407..-108.6713] | it/evals=90/400 eff=90.0000% N=300
Z=-138.3(0.00%) | Like=-133.68..-20.76 [-827.5407..-108.6713] | it/evals=120/435 eff=88.8889% N=300
Mono-modal Volume: ~exp(-4.30) * Expected Volume: exp(-0.45) Quality: ok
index : +1.0| ***********************************************| +5.0
amplitude: +1.0e-12|******************************************** ***| +1.0e-10
Z=-134.1(0.00%) | Like=-129.01..-20.76 [-827.5407..-108.6713] | it/evals=134/451 eff=88.7417% N=300
Z=-127.6(0.00%) | Like=-122.39..-20.76 [-827.5407..-108.6713] | it/evals=150/470 eff=88.2353% N=300
Z=-118.4(0.00%) | Like=-113.12..-20.76 [-827.5407..-108.6713] | it/evals=180/510 eff=85.7143% N=300
Mono-modal Volume: ~exp(-4.80) * Expected Volume: exp(-0.67) Quality: ok
index : +1.0| *********************************************| +5.0
amplitude: +1.0e-12| ****************************************** * | +1.0e-10
Z=-114.6(0.00%) | Like=-109.85..-20.76 [-827.5407..-108.6713] | it/evals=201/538 eff=84.4538% N=300
Z=-112.8(0.00%) | Like=-108.19..-20.76 [-108.6497..-67.6343] | it/evals=210/548 eff=84.6774% N=300
Z=-105.2(0.00%) | Like=-99.09..-20.76 [-108.6497..-67.6343] | it/evals=240/584 eff=84.5070% N=300
Mono-modal Volume: ~exp(-4.80) Expected Volume: exp(-0.89) Quality: ok
index : +1.0| ********************************************| +5.0
amplitude: +1.0e-12| ******************************* * ***** * *| +1.0e-10
Z=-97.2(0.00%) | Like=-91.82..-20.76 [-108.6497..-67.6343] | it/evals=268/629 eff=81.4590% N=300
Z=-96.7(0.00%) | Like=-91.48..-20.76 [-108.6497..-67.6343] | it/evals=270/632 eff=81.3253% N=300
Z=-91.1(0.00%) | Like=-86.12..-20.76 [-108.6497..-67.6343] | it/evals=300/667 eff=81.7439% N=300
Z=-84.6(0.00%) | Like=-79.19..-20.58 [-108.6497..-67.6343] | it/evals=330/705 eff=81.4815% N=300
Mono-modal Volume: ~exp(-4.80) Expected Volume: exp(-1.12) Quality: ok
index : +1.0| ******************************************| +5.0
amplitude: +1.0e-12| *********************** ********* ** ** | +1.0e-10
Z=-79.7(0.00%) | Like=-75.04..-20.58 [-108.6497..-67.6343] | it/evals=357/745 eff=80.2247% N=300
Z=-79.4(0.00%) | Like=-74.70..-20.58 [-108.6497..-67.6343] | it/evals=360/749 eff=80.1782% N=300
Z=-76.7(0.00%) | Like=-71.69..-20.58 [-108.6497..-67.6343] | it/evals=384/791 eff=78.2077% N=300
Z=-75.9(0.00%) | Like=-70.56..-20.58 [-108.6497..-67.6343] | it/evals=390/804 eff=77.3810% N=300
Mono-modal Volume: ~exp(-5.71) * Expected Volume: exp(-1.34) Quality: ok
index : +1.0| ************************************** | +5.0
amplitude: +1.0e-12| *********************************** ** | +1.0e-10
Z=-74.1(0.00%) | Like=-68.94..-20.58 [-108.6497..-67.6343] | it/evals=402/817 eff=77.7563% N=300
Z=-71.9(0.00%) | Like=-67.16..-20.58 [-67.6216..-49.3596] | it/evals=420/840 eff=77.7778% N=300
Z=-69.1(0.00%) | Like=-63.97..-20.58 [-67.6216..-49.3596] | it/evals=448/883 eff=76.8439% N=300
Z=-68.8(0.00%) | Like=-63.62..-20.58 [-67.6216..-49.3596] | it/evals=450/887 eff=76.6610% N=300
Mono-modal Volume: ~exp(-5.78) * Expected Volume: exp(-1.56) Quality: ok
index : +1.0| ********************************** | +5.0
amplitude: +1.0e-12| ********************************** +7.7e-11 | +1.0e-10
Z=-67.1(0.00%) | Like=-62.22..-20.58 [-67.6216..-49.3596] | it/evals=469/918 eff=75.8900% N=300
Z=-65.9(0.00%) | Like=-60.68..-20.58 [-67.6216..-49.3596] | it/evals=480/932 eff=75.9494% N=300
Z=-62.6(0.00%) | Like=-57.66..-20.56 [-67.6216..-49.3596] | it/evals=510/968 eff=76.3473% N=300
Mono-modal Volume: ~exp(-5.97) * Expected Volume: exp(-1.79) Quality: ok
index : +1.0| ***************************** +4.1 | +5.0
amplitude: +1.0e-12| ******************************* * +7.6e-11 | +1.0e-10
Z=-60.8(0.00%) | Like=-55.91..-20.56 [-67.6216..-49.3596] | it/evals=536/1001 eff=76.4622% N=300
Z=-60.4(0.00%) | Like=-55.15..-20.56 [-67.6216..-49.3596] | it/evals=540/1006 eff=76.4873% N=300
Z=-57.6(0.00%) | Like=-52.50..-20.56 [-67.6216..-49.3596] | it/evals=570/1044 eff=76.6129% N=300
Z=-55.6(0.00%) | Like=-50.48..-20.56 [-67.6216..-49.3596] | it/evals=598/1086 eff=76.0814% N=300
Z=-55.5(0.00%) | Like=-50.28..-20.56 [-67.6216..-49.3596] | it/evals=600/1090 eff=75.9494% N=300
Mono-modal Volume: ~exp(-5.97) Expected Volume: exp(-2.01) Quality: ok
index : +1.0| ************************* +3.8 | +5.0
amplitude: +1.0e-12| ***************************** +6.9e-11 | +1.0e-10
Z=-53.4(0.00%) | Like=-47.86..-20.56 [-49.3255..-36.6247] | it/evals=628/1128 eff=75.8454% N=300
Z=-53.2(0.00%) | Like=-47.35..-20.56 [-49.3255..-36.6247] | it/evals=630/1130 eff=75.9036% N=300
Z=-50.5(0.00%) | Like=-45.53..-20.56 [-49.3255..-36.6247] | it/evals=659/1177 eff=75.1425% N=300
Z=-50.5(0.00%) | Like=-45.42..-20.56 [-49.3255..-36.6247] | it/evals=660/1179 eff=75.0853% N=300
Mono-modal Volume: ~exp(-6.17) * Expected Volume: exp(-2.23) Quality: ok
index : +1.0| *********************** +3.7 | +5.0
amplitude: +1.0e-12| ************************** +6.6e-11 | +1.0e-10
Z=-49.9(0.00%) | Like=-44.81..-20.56 [-49.3255..-36.6247] | it/evals=670/1192 eff=75.1121% N=300
Z=-48.8(0.00%) | Like=-43.59..-20.56 [-49.3255..-36.6247] | it/evals=690/1219 eff=75.0816% N=300
Z=-47.1(0.00%) | Like=-41.89..-20.56 [-49.3255..-36.6247] | it/evals=718/1261 eff=74.7138% N=300
Z=-47.0(0.00%) | Like=-41.78..-20.56 [-49.3255..-36.6247] | it/evals=720/1263 eff=74.7664% N=300
Mono-modal Volume: ~exp(-6.17) 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=-45.3(0.00%) | Like=-39.93..-20.56 [-49.3255..-36.6247] | it/evals=750/1298 eff=75.1503% N=300
Z=-43.5(0.00%) | Like=-38.05..-20.56 [-49.3255..-36.6247] | it/evals=780/1339 eff=75.0722% N=300
Mono-modal Volume: ~exp(-6.73) * Expected Volume: exp(-2.68) Quality: ok
index : +1.0| +2.0 ******************* +3.4 | +5.0
amplitude: +1.0e-12| ********************** +6.0e-11 | +1.0e-10
Z=-42.1(0.00%) | Like=-36.52..-20.56 [-36.5936..-28.9339] | it/evals=804/1376 eff=74.7212% N=300
Z=-41.8(0.00%) | Like=-36.25..-20.56 [-36.5936..-28.9339] | it/evals=810/1384 eff=74.7232% N=300
Z=-39.9(0.00%) | Like=-34.52..-20.56 [-36.5936..-28.9339] | it/evals=840/1422 eff=74.8663% N=300
Z=-38.7(0.00%) | Like=-33.36..-20.56 [-36.5936..-28.9339] | it/evals=870/1462 eff=74.8709% N=300
Mono-modal Volume: ~exp(-6.73) Expected Volume: exp(-2.90) Quality: ok
index : +1.0| +2.0 **************** +3.3 | +5.0
amplitude: +1.0e-12| ******************* +5.6e-11 | +1.0e-10
Z=-37.8(0.00%) | Like=-32.52..-20.56 [-36.5936..-28.9339] | it/evals=894/1500 eff=74.5000% N=300
Z=-37.6(0.00%) | Like=-32.29..-20.56 [-36.5936..-28.9339] | it/evals=900/1508 eff=74.5033% N=300
Z=-36.8(0.00%) | Like=-31.71..-20.56 [-36.5936..-28.9339] | it/evals=925/1550 eff=74.0000% N=300
Z=-36.7(0.00%) | Like=-31.53..-20.52 [-36.5936..-28.9339] | it/evals=930/1557 eff=73.9857% N=300
Mono-modal Volume: ~exp(-7.15) * Expected Volume: exp(-3.13) Quality: ok
index : +1.0| +2.0 *************** +3.2 | +5.0
amplitude: +1.0e-12| ****************** +5.4e-11 | +1.0e-10
Z=-36.5(0.00%) | Like=-31.30..-20.52 [-36.5936..-28.9339] | it/evals=938/1569 eff=73.9165% N=300
Z=-35.9(0.00%) | Like=-30.59..-20.52 [-36.5936..-28.9339] | it/evals=960/1595 eff=74.1313% N=300
Z=-35.0(0.01%) | Like=-29.42..-20.49 [-36.5936..-28.9339] | it/evals=990/1631 eff=74.3802% N=300
Mono-modal Volume: ~exp(-7.33) * Expected Volume: exp(-3.35) Quality: ok
index : +1.0| +2.1 ************* +3.1 | +5.0
amplitude: +1.0e-12| **************** +5.1e-11 | +1.0e-10
Z=-34.5(0.02%) | Like=-28.95..-20.49 [-36.5936..-28.9339] | it/evals=1005/1652 eff=74.3343% N=300
Z=-34.1(0.03%) | Like=-28.58..-20.48 [-28.9303..-26.9950] | it/evals=1020/1668 eff=74.5614% N=300
Z=-33.3(0.06%) | Like=-27.63..-20.48 [-28.9303..-26.9950] | it/evals=1048/1710 eff=74.3262% N=300
Z=-33.3(0.06%) | Like=-27.57..-20.48 [-28.9303..-26.9950] | it/evals=1050/1712 eff=74.3626% N=300
Mono-modal Volume: ~exp(-7.78) * Expected Volume: exp(-3.57) Quality: ok
index : +1.0| +2.1 ************ +3.1 | +5.0
amplitude: +1.0e-12| ************** +4.9e-11 | +1.0e-10
Z=-32.7(0.11%) | Like=-27.18..-20.48 [-28.9303..-26.9950] | it/evals=1072/1737 eff=74.5999% N=300
Z=-32.5(0.13%) | Like=-27.07..-20.48 [-28.9303..-26.9950] | it/evals=1080/1746 eff=74.6888% N=300
Z=-31.9(0.24%) | Like=-26.41..-20.48 [-26.4149..-26.4148]*| it/evals=1110/1783 eff=74.8483% N=300
Z=-31.4(0.41%) | Like=-25.97..-20.48 [-25.9663..-25.9546] | it/evals=1138/1824 eff=74.6719% N=300
Mono-modal Volume: ~exp(-7.84) * 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=-31.4(0.42%) | Like=-25.95..-20.48 [-25.9663..-25.9546] | it/evals=1139/1825 eff=74.6885% N=300
Z=-31.4(0.43%) | Like=-25.89..-20.48 [-25.8903..-25.8872]*| it/evals=1140/1827 eff=74.6562% N=300
Z=-30.9(0.68%) | Like=-25.52..-20.48 [-25.5451..-25.5235] | it/evals=1170/1866 eff=74.7126% N=300
Z=-30.5(1.08%) | Like=-24.98..-20.48 [-25.0051..-24.9835] | it/evals=1198/1907 eff=74.5488% N=300
Z=-30.5(1.11%) | Like=-24.93..-20.48 [-24.9698..-24.9263] | it/evals=1200/1910 eff=74.5342% N=300
Mono-modal Volume: ~exp(-8.30) * Expected Volume: exp(-4.02) Quality: ok
index : +1.0| +2.2 ********** +3.0 | +5.0
amplitude: +1.0e-12| +2.4e-11 *********** +4.6e-11 | +1.0e-10
Z=-30.4(1.21%) | Like=-24.81..-20.48 [-24.8127..-24.8034]*| it/evals=1206/1919 eff=74.4904% N=300
Z=-30.0(1.73%) | Like=-24.51..-20.48 [-24.5103..-24.4727] | it/evals=1230/1948 eff=74.6359% N=300
Z=-29.6(2.44%) | Like=-24.13..-20.47 [-24.1585..-24.1327] | it/evals=1260/1988 eff=74.6445% N=300
Mono-modal Volume: ~exp(-8.30) Expected Volume: exp(-4.24) Quality: ok
index : +1.0| +2.2 ********* +2.9 | +5.0
amplitude: +1.0e-12| +2.5e-11 *********** +4.5e-11 | +1.0e-10
Z=-29.3(3.28%) | Like=-23.85..-20.47 [-23.8628..-23.8457] | it/evals=1286/2026 eff=74.5075% N=300
Z=-29.3(3.46%) | Like=-23.83..-20.47 [-23.8271..-23.8158] | it/evals=1290/2033 eff=74.4374% N=300
Z=-29.0(4.70%) | Like=-23.45..-20.47 [-23.4660..-23.4485] | it/evals=1320/2071 eff=74.5342% N=300
Mono-modal Volume: ~exp(-8.45) * Expected Volume: exp(-4.47) Quality: ok
index : +1.0| +2.3 ******** +2.9 | +5.0
amplitude: +1.0e-12| +2.6e-11 ********* +4.3e-11 | +1.0e-10
Z=-28.8(5.81%) | Like=-23.22..-20.47 [-23.2212..-23.2000] | it/evals=1340/2099 eff=74.4858% N=300
Z=-28.7(6.38%) | Like=-23.13..-20.47 [-23.1302..-23.1299]*| it/evals=1350/2112 eff=74.5033% N=300
Z=-28.4(8.30%) | Like=-22.88..-20.47 [-22.8834..-22.8698] | it/evals=1379/2153 eff=74.4199% N=300
Z=-28.4(8.38%) | Like=-22.87..-20.47 [-22.8834..-22.8698] | it/evals=1380/2156 eff=74.3534% N=300
Z=-28.2(10.39%) | Like=-22.65..-20.47 [-22.6721..-22.6520] | it/evals=1404/2198 eff=73.9726% N=300
Mono-modal Volume: ~exp(-9.02) * Expected Volume: exp(-4.69) Quality: ok
index : +1.0| +2.3 ******** +2.8 | +5.0
amplitude: +1.0e-12| +2.7e-11 ********* +4.2e-11 | +1.0e-10
Z=-28.2(10.63%) | Like=-22.63..-20.47 [-22.6384..-22.6256] | it/evals=1407/2202 eff=73.9748% N=300
Z=-28.2(10.83%) | Like=-22.60..-20.47 [-22.6197..-22.6027] | it/evals=1410/2205 eff=74.0157% N=300
Z=-27.9(13.64%) | Like=-22.39..-20.47 [-22.3972..-22.3869] | it/evals=1440/2238 eff=74.3034% N=300
Z=-27.8(16.32%) | Like=-22.23..-20.47 [-22.2312..-22.2215]*| it/evals=1466/2280 eff=74.0404% N=300
Z=-27.7(16.67%) | Like=-22.21..-20.47 [-22.2071..-22.2046]*| it/evals=1470/2285 eff=74.0554% N=300
Mono-modal Volume: ~exp(-9.24) * Expected Volume: exp(-4.91) Quality: ok
index : +1.0| +2.3 ****** +2.8 | +5.0
amplitude: +1.0e-12| +2.7e-11 ******** +4.1e-11 | +1.0e-10
Z=-27.7(17.11%) | Like=-22.20..-20.47 [-22.2020..-22.1969]*| it/evals=1474/2291 eff=74.0331% N=300
Z=-27.6(19.95%) | Like=-22.09..-20.47 [-22.0863..-22.0747] | it/evals=1500/2319 eff=74.2942% N=300
Z=-27.4(23.01%) | Like=-21.89..-20.47 [-21.8859..-21.8846]*| it/evals=1530/2359 eff=74.3079% N=300
Mono-modal Volume: ~exp(-9.51) * Expected Volume: exp(-5.14) 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(24.36%) | Like=-21.82..-20.47 [-21.8153..-21.8147]*| it/evals=1541/2373 eff=74.3367% N=300
Z=-27.3(26.79%) | Like=-21.75..-20.47 [-21.7467..-21.7392]*| it/evals=1560/2395 eff=74.4630% N=300
Z=-27.1(30.59%) | Like=-21.66..-20.47 [-21.6564..-21.6559]*| it/evals=1590/2435 eff=74.4731% N=300
Mono-modal Volume: ~exp(-9.65) * Expected Volume: exp(-5.36) 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(32.80%) | Like=-21.59..-20.47 [-21.5908..-21.5895]*| it/evals=1608/2461 eff=74.4100% N=300
Z=-27.0(34.29%) | Like=-21.55..-20.47 [-21.5478..-21.5397]*| it/evals=1620/2475 eff=74.4828% N=300
Z=-26.9(37.23%) | Like=-21.44..-20.47 [-21.4439..-21.4375]*| it/evals=1646/2516 eff=74.2780% N=300
Z=-26.9(37.77%) | Like=-21.42..-20.47 [-21.4217..-21.4158]*| it/evals=1650/2521 eff=74.2909% N=300
Mono-modal Volume: ~exp(-9.65) Expected Volume: exp(-5.58) Quality: ok
index : +1.0| +2.4 ***** +2.7 | +5.0
amplitude: +1.0e-12| +2.9e-11 ****** +3.9e-11 | +1.0e-10
Z=-26.8(41.02%) | Like=-21.37..-20.47 [-21.3703..-21.3656]*| it/evals=1676/2561 eff=74.1265% N=300
Z=-26.8(41.49%) | Like=-21.36..-20.47 [-21.3579..-21.3568]*| it/evals=1680/2569 eff=74.0414% N=300
Z=-26.8(44.38%) | Like=-21.30..-20.47 [-21.3020..-21.3007]*| it/evals=1705/2610 eff=73.8095% N=300
Z=-26.7(44.96%) | Like=-21.27..-20.46 [-21.2725..-21.2719]*| it/evals=1710/2617 eff=73.8023% N=300
Z=-26.7(48.21%) | Like=-21.19..-20.46 [-21.1857..-21.1836]*| it/evals=1738/2658 eff=73.7065% N=300
Z=-26.7(48.44%) | Like=-21.18..-20.46 [-21.1817..-21.1815]*| it/evals=1740/2661 eff=73.6976% N=300
Mono-modal Volume: ~exp(-9.83) * Expected Volume: exp(-5.81) Quality: ok
index : +1.0| +2.4 **** +2.7 | +5.0
amplitude: +1.0e-12| +3.0e-11 ****** +3.9e-11 | +1.0e-10
Z=-26.7(48.66%) | Like=-21.18..-20.46 [-21.1780..-21.1766]*| it/evals=1742/2664 eff=73.6887% N=300
Z=-26.6(51.95%) | Like=-21.10..-20.46 [-21.0953..-21.0945]*| it/evals=1770/2708 eff=73.5050% N=300
Z=-26.5(55.17%) | Like=-21.04..-20.46 [-21.0428..-21.0412]*| it/evals=1798/2749 eff=73.4177% N=300
Z=-26.5(55.41%) | Like=-21.04..-20.46 [-21.0408..-21.0390]*| it/evals=1800/2753 eff=73.3795% N=300
Mono-modal Volume: ~exp(-10.14) * Expected Volume: exp(-6.03) Quality: ok
index : +1.0| +2.4 **** +2.7 | +5.0
amplitude: +1.0e-12| +3.0e-11 ***** +3.8e-11 | +1.0e-10
Z=-26.5(56.31%) | Like=-21.03..-20.46 [-21.0303..-21.0295]*| it/evals=1809/2764 eff=73.4172% N=300
Z=-26.5(58.51%) | Like=-21.00..-20.46 [-21.0008..-21.0007]*| it/evals=1830/2790 eff=73.4940% N=300
Z=-26.4(61.56%) | Like=-20.97..-20.46 [-20.9653..-20.9634]*| it/evals=1860/2829 eff=73.5469% N=300
Mono-modal Volume: ~exp(-10.26) * Expected Volume: exp(-6.25) Quality: ok
index : +1.0| +2.4 **** +2.7 | +5.0
amplitude: +1.0e-12| +3.1e-11 **** +3.8e-11 | +1.0e-10
Z=-26.4(63.05%) | Like=-20.94..-20.46 [-20.9390..-20.9376]*| it/evals=1876/2850 eff=73.5686% N=300
Z=-26.4(64.35%) | Like=-20.92..-20.46 [-20.9184..-20.9159]*| it/evals=1890/2865 eff=73.6842% N=300
Z=-26.3(66.87%) | Like=-20.87..-20.46 [-20.8687..-20.8678]*| it/evals=1918/2905 eff=73.6276% N=300
Z=-26.3(67.05%) | Like=-20.87..-20.46 [-20.8660..-20.8660]*| it/evals=1920/2908 eff=73.6196% N=300
Mono-modal Volume: ~exp(-10.54) * Expected Volume: exp(-6.48) Quality: ok
index : +1.0| +2.5 **** +2.7 | +5.0
amplitude: +1.0e-12| +3.1e-11 **** +3.8e-11 | +1.0e-10
Z=-26.3(69.05%) | Like=-20.84..-20.46 [-20.8425..-20.8362]*| it/evals=1943/2945 eff=73.4594% N=300
Z=-26.3(69.65%) | Like=-20.82..-20.46 [-20.8241..-20.8215]*| it/evals=1950/2956 eff=73.4187% N=300
[ultranest] Explored until L=-2e+01
[ultranest] Likelihood function evaluations: 2962
[ultranest] logZ = -25.98 +- 0.09559
[ultranest] Effective samples strategy satisfied (ESS = 990.8, need >400)
[ultranest] Posterior uncertainty strategy is satisfied (KL: 0.45+-0.09 nat, need <0.50 nat)
[ultranest] Evidency uncertainty strategy is satisfied (dlogz=0.28, need <0.5)
[ultranest] logZ error budget: single: 0.12 bs:0.10 tail:0.26 total:0.28 required:<0.50
[ultranest] done iterating.
logZ = -25.940 +- 0.321
single instance: logZ = -25.940 +- 0.122
bootstrapped : logZ = -25.979 +- 0.186
tail : logZ = +- 0.262
insert order U test : converged: True correlation: inf iterations
index : 2.15 │ ▁ ▁▁▁▂▂▃▃▃▄▆▆▇▇▇▆▅▅▅▃▃▂▁▁▁▁▁▁▁ ▁▁▁ ▁ │3.18 2.57 +- 0.13
amplitude : 0.0000000000209│ ▁▁▁▁▁▁▁▂▂▂▄▄▆▅▅▇▇▆▇▆▇▆▄▅▄▃▃▂▂▁▁▁▁▁ ▁▁ │0.0000000000481 0.0000000000341 +- 0.0000000000038
[ultranest] Sampling 300 live points from prior ...
Mono-modal Volume: ~exp(-4.08) * Expected Volume: exp(0.00) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|********************************* ****** * ** **| +1.0e-10
Z=-inf(0.00%) | Like=-1048.64..-19.34 [-1048.6378..-127.2170] | it/evals=0/301 eff=0.0000% N=300
Z=-215.6(0.00%) | Like=-209.90..-19.34 [-1048.6378..-127.2170] | it/evals=30/336 eff=83.3333% N=300
Z=-199.5(0.00%) | Like=-193.35..-19.34 [-1048.6378..-127.2170] | it/evals=60/374 eff=81.0811% N=300
Mono-modal Volume: ~exp(-4.23) * Expected Volume: exp(-0.22) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|********************************* ****** * *****| +1.0e-10
Z=-195.8(0.00%) | Like=-190.36..-19.34 [-1048.6378..-127.2170] | it/evals=67/381 eff=82.7160% N=300
Z=-183.9(0.00%) | Like=-177.77..-19.34 [-1048.6378..-127.2170] | it/evals=90/408 eff=83.3333% N=300
Z=-170.9(0.00%) | Like=-165.26..-19.34 [-1048.6378..-127.2170] | it/evals=120/442 eff=84.5070% N=300
Mono-modal Volume: ~exp(-4.58) * Expected Volume: exp(-0.45) Quality: ok
index : +1.0| ***********************************************| +5.0
amplitude: +1.0e-12|********************************* ******** *****| +1.0e-10
Z=-166.1(0.00%) | Like=-160.57..-19.34 [-1048.6378..-127.2170] | it/evals=134/461 eff=83.2298% N=300
Z=-157.0(0.00%) | Like=-150.95..-19.34 [-1048.6378..-127.2170] | it/evals=150/482 eff=82.4176% N=300
Z=-143.6(0.00%) | Like=-138.28..-19.34 [-1048.6378..-127.2170] | it/evals=180/523 eff=80.7175% N=300
Mono-modal Volume: ~exp(-4.58) Expected Volume: exp(-0.67) Quality: ok
index : +1.0| *********************************************| +5.0
amplitude: +1.0e-12| **************************************** *****| +1.0e-10
Z=-126.4(0.00%) | Like=-121.49..-19.34 [-126.8097..-57.6895] | it/evals=210/557 eff=81.7121% N=300
Z=-113.9(0.00%) | Like=-107.32..-19.34 [-126.8097..-57.6895] | it/evals=240/595 eff=81.3559% N=300
Mono-modal Volume: ~exp(-4.71) * Expected Volume: exp(-0.89) Quality: ok
index : +1.0| *******************************************| +5.0
amplitude: +1.0e-12| ************************************** *****| +1.0e-10
Z=-100.2(0.00%) | Like=-94.51..-19.24 [-126.8097..-57.6895] | it/evals=268/636 eff=79.7619% N=300
Z=-99.5(0.00%) | Like=-93.94..-19.24 [-126.8097..-57.6895] | it/evals=270/638 eff=79.8817% N=300
Z=-88.5(0.00%) | Like=-82.20..-19.24 [-126.8097..-57.6895] | it/evals=300/673 eff=80.4290% N=300
Z=-78.0(0.00%) | Like=-72.67..-19.24 [-126.8097..-57.6895] | it/evals=330/714 eff=79.7101% N=300
Mono-modal Volume: ~exp(-5.19) * Expected Volume: exp(-1.12) Quality: ok
index : +1.0| *****************************************| +5.0
amplitude: +1.0e-12| ******************************************| +1.0e-10
Z=-76.6(0.00%) | Like=-70.60..-19.24 [-126.8097..-57.6895] | it/evals=335/721 eff=79.5724% N=300
Z=-69.8(0.00%) | Like=-64.09..-19.24 [-126.8097..-57.6895] | it/evals=360/757 eff=78.7746% N=300
Z=-64.7(0.00%) | Like=-59.61..-19.24 [-126.8097..-57.6895] | it/evals=387/799 eff=77.5551% N=300
Z=-64.3(0.00%) | Like=-59.35..-19.24 [-126.8097..-57.6895] | it/evals=390/802 eff=77.6892% N=300
Mono-modal Volume: ~exp(-5.19) Expected Volume: exp(-1.34) Quality: ok
index : +1.0| ***************************************| +5.0
amplitude: +1.0e-12| *****************************************| +1.0e-10
Z=-60.5(0.00%) | Like=-55.30..-19.24 [-57.4208..-38.5658] | it/evals=420/838 eff=78.0669% N=300
Z=-57.1(0.00%) | Like=-52.12..-19.24 [-57.4208..-38.5658] | it/evals=450/879 eff=77.7202% N=300
Mono-modal Volume: ~exp(-5.67) * Expected Volume: exp(-1.56) Quality: ok
index : +1.0| ********************************** | +5.0
amplitude: +1.0e-12| ********************************** ****| +1.0e-10
Z=-54.7(0.00%) | Like=-49.73..-19.24 [-57.4208..-38.5658] | it/evals=469/905 eff=77.5207% N=300
Z=-53.7(0.00%) | Like=-48.92..-19.24 [-57.4208..-38.5658] | it/evals=480/922 eff=77.1704% N=300
Z=-50.9(0.00%) | Like=-45.88..-19.24 [-57.4208..-38.5658] | it/evals=510/962 eff=77.0393% N=300
Mono-modal Volume: ~exp(-6.09) * Expected Volume: exp(-1.79) Quality: ok
index : +1.0| +2.0 ****************************** | +5.0
amplitude: +1.0e-12| ********************************* ****| +1.0e-10
Z=-48.6(0.00%) | Like=-43.51..-19.24 [-57.4208..-38.5658] | it/evals=536/998 eff=76.7908% N=300
Z=-48.2(0.00%) | Like=-43.19..-19.24 [-57.4208..-38.5658] | it/evals=540/1005 eff=76.5957% N=300
Z=-46.0(0.00%) | Like=-41.02..-19.24 [-57.4208..-38.5658] | it/evals=568/1047 eff=76.0375% N=300
Z=-45.8(0.00%) | Like=-40.94..-19.24 [-57.4208..-38.5658] | it/evals=570/1049 eff=76.1015% N=300
Z=-44.1(0.00%) | Like=-39.17..-19.24 [-57.4208..-38.5658] | it/evals=600/1091 eff=75.8534% N=300
Mono-modal Volume: ~exp(-6.09) Expected Volume: exp(-2.01) Quality: ok
index : +1.0| +2.0 *************************** | +5.0
amplitude: +1.0e-12| +2.6e-11 ******************************* ****| +1.0e-10
Z=-42.8(0.00%) | Like=-38.21..-19.22 [-38.5520..-29.3159] | it/evals=626/1129 eff=75.5127% N=300
Z=-42.6(0.00%) | Like=-37.63..-19.22 [-38.5520..-29.3159] | it/evals=630/1134 eff=75.5396% N=300
Z=-40.9(0.00%) | Like=-36.08..-19.22 [-38.5520..-29.3159] | it/evals=660/1176 eff=75.3425% N=300
Mono-modal Volume: ~exp(-6.57) * Expected Volume: exp(-2.23) Quality: ok
index : +1.0| +2.1 ************************ +4.0 | +5.0
amplitude: +1.0e-12| +2.9e-11 ********************************* | +1.0e-10
Z=-40.5(0.00%) | Like=-35.46..-19.22 [-38.5520..-29.3159] | it/evals=670/1190 eff=75.2809% N=300
Z=-39.6(0.00%) | Like=-34.93..-19.22 [-38.5520..-29.3159] | it/evals=690/1216 eff=75.3275% N=300
Z=-38.4(0.00%) | Like=-33.45..-19.22 [-38.5520..-29.3159] | it/evals=720/1253 eff=75.5509% N=300
Mono-modal Volume: ~exp(-6.57) 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=-37.4(0.00%) | Like=-32.58..-19.22 [-38.5520..-29.3159] | it/evals=745/1291 eff=75.1766% N=300
Z=-37.3(0.00%) | Like=-32.40..-19.22 [-38.5520..-29.3159] | it/evals=750/1296 eff=75.3012% N=300
Z=-36.3(0.00%) | Like=-31.44..-19.22 [-38.5520..-29.3159] | it/evals=779/1338 eff=75.0482% N=300
Z=-36.3(0.00%) | Like=-31.28..-19.22 [-38.5520..-29.3159] | it/evals=780/1339 eff=75.0722% N=300
Mono-modal Volume: ~exp(-6.63) * Expected Volume: exp(-2.68) Quality: ok
index : +1.0| +2.2 ******************* +3.7 | +5.0
amplitude: +1.0e-12| +3.1e-11 **************************** | +1.0e-10
Z=-35.4(0.00%) | Like=-30.45..-19.22 [-38.5520..-29.3159] | it/evals=804/1372 eff=75.0000% N=300
Z=-35.2(0.00%) | Like=-30.29..-19.22 [-38.5520..-29.3159] | it/evals=810/1381 eff=74.9306% N=300
Z=-34.5(0.00%) | Like=-29.63..-19.22 [-38.5520..-29.3159] | it/evals=835/1423 eff=74.3544% N=300
Z=-34.4(0.01%) | Like=-29.50..-19.22 [-38.5520..-29.3159] | it/evals=840/1429 eff=74.4021% N=300
Z=-33.6(0.01%) | Like=-28.74..-19.22 [-29.3063..-26.1955] | it/evals=870/1472 eff=74.2321% N=300
Mono-modal Volume: ~exp(-7.03) * Expected Volume: exp(-2.90) Quality: ok
index : +1.0| +2.2 ****************** +3.7 | +5.0
amplitude: +1.0e-12| +3.3e-11 ************************* | +1.0e-10
Z=-33.6(0.01%) | Like=-28.71..-19.22 [-29.3063..-26.1955] | it/evals=871/1473 eff=74.2540% N=300
Z=-33.0(0.02%) | Like=-28.02..-19.17 [-29.3063..-26.1955] | it/evals=900/1510 eff=74.3802% N=300
Z=-32.3(0.04%) | Like=-27.44..-19.17 [-29.3063..-26.1955] | it/evals=928/1552 eff=74.1214% N=300
Z=-32.3(0.04%) | Like=-27.38..-19.17 [-29.3063..-26.1955] | it/evals=930/1557 eff=73.9857% N=300
Mono-modal Volume: ~exp(-7.35) * Expected Volume: exp(-3.13) Quality: ok
index : +1.0| +2.3 **************** +3.6 | +5.0
amplitude: +1.0e-12| +3.5e-11 ************************ | +1.0e-10
Z=-32.1(0.05%) | Like=-27.28..-19.17 [-29.3063..-26.1955] | it/evals=938/1568 eff=73.9748% N=300
Z=-31.7(0.08%) | Like=-26.76..-19.17 [-29.3063..-26.1955] | it/evals=960/1599 eff=73.9030% N=300
Z=-31.2(0.13%) | Like=-26.14..-19.17 [-26.1598..-25.7482] | it/evals=987/1641 eff=73.6018% N=300
Z=-31.1(0.14%) | Like=-26.08..-19.17 [-26.1598..-25.7482] | it/evals=990/1645 eff=73.6059% N=300
Mono-modal Volume: ~exp(-7.41) * Expected Volume: exp(-3.35) Quality: ok
index : +1.0| +2.3 *************** +3.5 | +5.0
amplitude: +1.0e-12| +3.6e-11 ********************* +7.8e-11| +1.0e-10
Z=-30.9(0.18%) | Like=-25.78..-19.17 [-26.1598..-25.7482] | it/evals=1005/1661 eff=73.8428% N=300
Z=-30.6(0.23%) | Like=-25.50..-19.17 [-25.5026..-25.5022]*| it/evals=1020/1678 eff=74.0203% N=300
Z=-30.1(0.38%) | Like=-24.96..-19.17 [-24.9648..-24.9486] | it/evals=1048/1719 eff=73.8548% N=300
Z=-30.0(0.39%) | Like=-24.93..-19.17 [-24.9338..-24.9330]*| it/evals=1050/1721 eff=73.8916% N=300
Mono-modal Volume: ~exp(-7.61) * Expected Volume: exp(-3.57) Quality: ok
index : +1.0| +2.4 ************* +3.4 | +5.0
amplitude: +1.0e-12| +3.8e-11 ******************** +7.5e-11 | +1.0e-10
Z=-29.7(0.55%) | Like=-24.50..-19.17 [-24.4989..-24.4833] | it/evals=1072/1758 eff=73.5254% N=300
Z=-29.6(0.63%) | Like=-24.30..-19.17 [-24.3035..-24.2526] | it/evals=1080/1768 eff=73.5695% N=300
Z=-29.0(1.05%) | Like=-23.62..-19.17 [-23.6199..-23.6152]*| it/evals=1110/1807 eff=73.6563% N=300
Mono-modal Volume: ~exp(-8.06) * Expected Volume: exp(-3.80) Quality: ok
index : +1.0| +2.4 ************* +3.3 | +5.0
amplitude: +1.0e-12| +4.0e-11 ***************** +7.3e-11 | +1.0e-10
Z=-28.5(1.67%) | Like=-23.29..-19.17 [-23.2870..-23.2850]*| it/evals=1139/1850 eff=73.4839% N=300
Z=-28.5(1.69%) | Like=-23.28..-19.17 [-23.2850..-23.2736] | it/evals=1140/1853 eff=73.4063% N=300
Z=-28.1(2.49%) | Like=-22.92..-19.17 [-22.9371..-22.9210] | it/evals=1170/1890 eff=73.5849% N=300
Z=-27.8(3.54%) | Like=-22.56..-19.17 [-22.5592..-22.5542]*| it/evals=1199/1931 eff=73.5132% N=300
Z=-27.8(3.56%) | Like=-22.55..-19.17 [-22.5542..-22.5312] | it/evals=1200/1932 eff=73.5294% N=300
Mono-modal Volume: ~exp(-8.18) * Expected Volume: exp(-4.02) Quality: ok
index : +1.0| +2.4 *********** +3.3 | +5.0
amplitude: +1.0e-12| +4.2e-11 **************** +7.2e-11 | +1.0e-10
Z=-27.7(3.83%) | Like=-22.48..-19.17 [-22.4763..-22.4750]*| it/evals=1206/1939 eff=73.5815% N=300
Z=-27.5(4.88%) | Like=-22.15..-19.17 [-22.1508..-22.1405] | it/evals=1230/1971 eff=73.6086% N=300
Z=-27.2(6.34%) | Like=-21.96..-19.17 [-21.9561..-21.9488]*| it/evals=1259/2012 eff=73.5397% N=300
Z=-27.2(6.40%) | Like=-21.95..-19.17 [-21.9488..-21.9375] | it/evals=1260/2013 eff=73.5552% N=300
Mono-modal Volume: ~exp(-8.48) * Expected Volume: exp(-4.24) Quality: ok
index : +1.0| +2.5 ********** +3.2 | +5.0
amplitude: +1.0e-12| +4.3e-11 ************** +6.9e-11 | +1.0e-10
Z=-27.0(7.26%) | Like=-21.83..-19.17 [-21.8293..-21.8272]*| it/evals=1273/2034 eff=73.4141% N=300
Z=-26.9(8.25%) | Like=-21.69..-19.17 [-21.7104..-21.6919] | it/evals=1290/2057 eff=73.4206% N=300
Z=-26.7(10.37%) | Like=-21.50..-19.17 [-21.5039..-21.4998]*| it/evals=1320/2097 eff=73.4558% N=300
Mono-modal Volume: ~exp(-8.62) * Expected Volume: exp(-4.47) Quality: ok
index : +1.0| +2.5 ********** +3.2 | +5.0
amplitude: +1.0e-12| +4.4e-11 ************* +6.8e-11 | +1.0e-10
Z=-26.5(11.73%) | Like=-21.34..-19.17 [-21.3419..-21.3312] | it/evals=1340/2122 eff=73.5456% N=300
Z=-26.5(12.55%) | Like=-21.24..-19.17 [-21.2444..-21.2244] | it/evals=1350/2134 eff=73.6096% N=300
Z=-26.3(15.16%) | Like=-21.06..-19.17 [-21.0559..-21.0527]*| it/evals=1379/2175 eff=73.5467% N=300
Z=-26.3(15.26%) | Like=-21.05..-19.17 [-21.0527..-21.0446]*| it/evals=1380/2176 eff=73.5608% N=300
Z=-26.1(17.86%) | Like=-20.83..-19.17 [-20.8252..-20.8243]*| it/evals=1405/2218 eff=73.2534% N=300
Mono-modal Volume: ~exp(-8.62) Expected Volume: exp(-4.69) Quality: ok
index : +1.0| +2.5 ******** +3.1 | +5.0
amplitude: +1.0e-12| +4.5e-11 *********** +6.6e-11 | +1.0e-10
Z=-26.1(18.45%) | Like=-20.80..-19.17 [-20.8040..-20.8014]*| it/evals=1410/2226 eff=73.2087% N=300
Z=-25.9(21.27%) | Like=-20.69..-19.17 [-20.6855..-20.6835]*| it/evals=1435/2267 eff=72.9537% N=300
Z=-25.9(21.83%) | Like=-20.67..-19.17 [-20.6745..-20.6708]*| it/evals=1440/2276 eff=72.8745% N=300
Z=-25.8(25.26%) | Like=-20.55..-19.17 [-20.5492..-20.5454]*| it/evals=1468/2317 eff=72.7814% N=300
Z=-25.8(25.43%) | Like=-20.53..-19.17 [-20.5454..-20.5338] | it/evals=1470/2319 eff=72.8083% N=300
Mono-modal Volume: ~exp(-8.72) * Expected Volume: exp(-4.91) Quality: ok
index : +1.0| +2.6 ******** +3.1 | +5.0
amplitude: +1.0e-12| +4.6e-11 *********** +6.5e-11 | +1.0e-10
Z=-25.7(25.89%) | Like=-20.52..-19.17 [-20.5193..-20.5159]*| it/evals=1474/2325 eff=72.7901% N=300
Z=-25.6(29.06%) | Like=-20.43..-19.17 [-20.4310..-20.4288]*| it/evals=1500/2362 eff=72.7449% N=300
Z=-25.5(32.57%) | Like=-20.34..-19.17 [-20.3364..-20.3307]*| it/evals=1530/2399 eff=72.8919% N=300
Mono-modal Volume: ~exp(-9.33) * Expected Volume: exp(-5.14) Quality: ok
index : +1.0| +2.6 ******* +3.1 | +5.0
amplitude: +1.0e-12| +4.7e-11 ********* +6.4e-11 | +1.0e-10
Z=-25.5(33.83%) | Like=-20.29..-19.17 [-20.2942..-20.2932]*| it/evals=1541/2419 eff=72.7230% N=300
Z=-25.4(36.16%) | Like=-20.23..-19.17 [-20.2278..-20.2240]*| it/evals=1560/2441 eff=72.8631% N=300
Z=-25.3(39.75%) | Like=-20.14..-19.17 [-20.1422..-20.1415]*| it/evals=1590/2479 eff=72.9693% 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=-25.3(42.45%) | Like=-20.06..-19.17 [-20.0609..-20.0567]*| it/evals=1615/2515 eff=72.9120% N=300
Z=-25.2(43.04%) | Like=-20.04..-19.17 [-20.0418..-20.0402]*| it/evals=1620/2520 eff=72.9730% N=300
Z=-25.2(46.29%) | Like=-19.94..-19.17 [-19.9401..-19.9382]*| it/evals=1648/2560 eff=72.9204% N=300
Z=-25.2(46.53%) | Like=-19.94..-19.17 [-19.9362..-19.9349]*| it/evals=1650/2562 eff=72.9443% N=300
Mono-modal Volume: ~exp(-9.58) * Expected Volume: exp(-5.58) Quality: ok
index : +1.0| +2.6 ****** +3.0 | +5.0
amplitude: +1.0e-12| +4.8e-11 ******** +6.2e-11 | +1.0e-10
Z=-25.1(49.52%) | Like=-19.86..-19.17 [-19.8633..-19.8605]*| it/evals=1675/2606 eff=72.6366% N=300
Z=-25.1(50.10%) | Like=-19.85..-19.17 [-19.8523..-19.8488]*| it/evals=1680/2612 eff=72.6644% N=300
Z=-25.0(53.02%) | Like=-19.81..-19.17 [-19.8071..-19.8042]*| it/evals=1705/2655 eff=72.3992% N=300
Z=-25.0(53.59%) | Like=-19.80..-19.17 [-19.8013..-19.8006]*| it/evals=1710/2662 eff=72.3963% N=300
Z=-25.0(56.69%) | Like=-19.74..-19.17 [-19.7373..-19.7332]*| it/evals=1739/2703 eff=72.3679% N=300
Z=-25.0(56.81%) | Like=-19.73..-19.17 [-19.7332..-19.7326]*| it/evals=1740/2704 eff=72.3794% N=300
Mono-modal Volume: ~exp(-9.81) * Expected Volume: exp(-5.81) Quality: ok
index : +1.0| +2.6 ***** +3.0 | +5.0
amplitude: +1.0e-12| +4.9e-11 ******* +6.2e-11 | +1.0e-10
Z=-25.0(57.00%) | Like=-19.73..-19.17 [-19.7319..-19.7298]*| it/evals=1742/2706 eff=72.4023% N=300
Z=-24.9(59.95%) | Like=-19.69..-19.16 [-19.6861..-19.6856]*| it/evals=1770/2740 eff=72.5410% N=300
Z=-24.9(63.04%) | Like=-19.63..-19.16 [-19.6332..-19.6312]*| it/evals=1800/2779 eff=72.6099% N=300
Mono-modal Volume: ~exp(-9.81) Expected Volume: exp(-6.03) Quality: ok
index : +1.0| +2.6 ***** +3.0 | +5.0
amplitude: +1.0e-12| +5.0e-11 ******* +6.1e-11 | +1.0e-10
Z=-24.8(65.17%) | Like=-19.58..-19.16 [-19.5802..-19.5800]*| it/evals=1823/2818 eff=72.3987% N=300
Z=-24.8(65.80%) | Like=-19.57..-19.16 [-19.5716..-19.5704]*| it/evals=1830/2830 eff=72.3320% N=300
Z=-24.8(68.03%) | Like=-19.53..-19.16 [-19.5322..-19.5315]*| it/evals=1854/2872 eff=72.0840% N=300
Z=-24.8(68.57%) | Like=-19.53..-19.16 [-19.5253..-19.5246]*| it/evals=1860/2882 eff=72.0372% N=300
Mono-modal Volume: ~exp(-10.38) * Expected Volume: exp(-6.25) Quality: ok
index : +1.0| +2.7 **** +2.9 | +5.0
amplitude: +1.0e-12| +5.0e-11 ****** +6.0e-11 | +1.0e-10
Z=-24.8(69.93%) | Like=-19.51..-19.16 [-19.5136..-19.5129]*| it/evals=1876/2911 eff=71.8499% N=300
[ultranest] Explored until L=-2e+01
[ultranest] Likelihood function evaluations: 2911
[ultranest] logZ = -24.4 +- 0.07927
[ultranest] Effective samples strategy satisfied (ESS = 1000.7, need >400)
[ultranest] Posterior uncertainty strategy is satisfied (KL: 0.45+-0.07 nat, need <0.50 nat)
[ultranest] Evidency uncertainty strategy is satisfied (dlogz=0.27, 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 = -24.399 +- 0.327
single instance: logZ = -24.399 +- 0.119
bootstrapped : logZ = -24.399 +- 0.195
tail : logZ = +- 0.262
insert order U test : converged: True correlation: inf iterations
index : 2.26 │ ▁▁▁▁▁▁▁▂▃▃▅▆▆▆▇▇▇▇▆▆▄▃▃▃▃▁▁▁▁▁▁▁▁▁ ▁ │3.57 2.83 +- 0.17
amplitude : 0.0000000000308│ ▁ ▁▁▁▁▁▂▃▃▄▄▅▆▇▇▅▆▆▅▅▃▄▂▂▂▁▁▁▁▁▁▁▁ │0.0000000000775 0.0000000000548 +- 0.0000000000060
[ultranest] Sampling 300 live points from prior ...
Mono-modal Volume: ~exp(-3.87) * Expected Volume: exp(0.00) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|*********************** ************** **** *** | +1.0e-10
Z=-inf(0.00%) | Like=-1488.77..-13.39 [-1488.7692..-93.5836] | it/evals=0/301 eff=0.0000% N=300
Z=-150.2(0.00%) | Like=-145.06..-13.39 [-1488.7692..-93.5836] | it/evals=30/336 eff=83.3333% N=300
Z=-140.7(0.00%) | Like=-136.46..-13.39 [-1488.7692..-93.5836] | it/evals=60/369 eff=86.9565% N=300
Mono-modal Volume: ~exp(-4.19) * Expected Volume: exp(-0.22) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|*********************** ******* ****** **** * * | +1.0e-10
Z=-139.6(0.00%) | Like=-135.74..-13.39 [-1488.7692..-93.5836] | it/evals=67/376 eff=88.1579% N=300
Z=-133.3(0.00%) | Like=-128.25..-13.39 [-1488.7692..-93.5836] | it/evals=90/406 eff=84.9057% N=300
Z=-123.2(0.00%) | Like=-118.28..-13.39 [-1488.7692..-93.5836] | it/evals=120/443 eff=83.9161% N=300
Mono-modal Volume: ~exp(-4.56) * Expected Volume: exp(-0.45) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|******************************* ************* * | +1.0e-10
Z=-117.9(0.00%) | Like=-112.52..-13.39 [-1488.7692..-93.5836] | it/evals=134/462 eff=82.7160% N=300
Z=-113.6(0.00%) | Like=-108.50..-13.35 [-1488.7692..-93.5836] | it/evals=150/480 eff=83.3333% N=300
Z=-103.9(0.00%) | Like=-98.77..-13.35 [-1488.7692..-93.5836] | it/evals=180/515 eff=83.7209% N=300
Mono-modal Volume: ~exp(-4.64) * Expected Volume: exp(-0.67) Quality: ok
index : +1.0| **********************************************| +5.0
amplitude: +1.0e-12|*********************************************** | +1.0e-10
Z=-100.4(0.00%) | Like=-95.78..-13.35 [-1488.7692..-93.5836] | it/evals=201/541 eff=83.4025% N=300
Z=-98.5(0.00%) | Like=-93.30..-13.35 [-93.3092..-48.6880] | it/evals=210/553 eff=83.0040% N=300
Z=-88.9(0.00%) | Like=-84.20..-13.35 [-93.3092..-48.6880] | it/evals=240/590 eff=82.7586% N=300
Mono-modal Volume: ~exp(-5.09) * Expected Volume: exp(-0.89) Quality: ok
index : +1.0| ********************************************| +5.0
amplitude: +1.0e-12| ********************************************* | +1.0e-10
Z=-81.6(0.00%) | Like=-76.64..-13.35 [-93.3092..-48.6880] | it/evals=268/632 eff=80.7229% N=300
Z=-81.2(0.00%) | Like=-76.11..-13.35 [-93.3092..-48.6880] | it/evals=270/635 eff=80.5970% N=300
Z=-73.6(0.00%) | Like=-68.21..-13.35 [-93.3092..-48.6880] | it/evals=300/676 eff=79.7872% N=300
Z=-67.8(0.00%) | Like=-62.78..-13.35 [-93.3092..-48.6880] | it/evals=330/716 eff=79.3269% N=300
Mono-modal Volume: ~exp(-5.09) Expected Volume: exp(-1.12) Quality: ok
index : +1.0| ******************************************| +5.0
amplitude: +1.0e-12| *********************************************| +1.0e-10
Z=-61.6(0.00%) | Like=-56.10..-13.35 [-93.3092..-48.6880] | it/evals=360/752 eff=79.6460% N=300
Z=-57.0(0.00%) | Like=-51.94..-13.35 [-93.3092..-48.6880] | it/evals=390/787 eff=80.0821% N=300
Mono-modal Volume: ~exp(-5.43) * Expected Volume: exp(-1.34) Quality: ok
index : +1.0| ************************************ ****| +5.0
amplitude: +1.0e-12| ********************************************| +1.0e-10
Z=-55.6(0.00%) | Like=-50.43..-13.35 [-93.3092..-48.6880] | it/evals=402/803 eff=79.9205% N=300
Z=-53.2(0.00%) | Like=-48.21..-13.35 [-48.6253..-31.7799] | it/evals=420/826 eff=79.8479% N=300
Z=-49.6(0.00%) | Like=-44.62..-13.35 [-48.6253..-31.7799] | it/evals=450/861 eff=80.2139% N=300
Mono-modal Volume: ~exp(-5.46) * Expected Volume: exp(-1.56) Quality: ok
index : +1.0| *********************************** * | +5.0
amplitude: +1.0e-12| ******************************************| +1.0e-10
Z=-47.8(0.00%) | Like=-43.12..-13.35 [-48.6253..-31.7799] | it/evals=469/896 eff=78.6913% N=300
Z=-47.0(0.00%) | Like=-41.92..-13.35 [-48.6253..-31.7799] | it/evals=480/911 eff=78.5597% N=300
Z=-44.6(0.00%) | Like=-39.68..-13.35 [-48.6253..-31.7799] | it/evals=510/946 eff=78.9474% N=300
Mono-modal Volume: ~exp(-6.01) * Expected Volume: exp(-1.79) Quality: ok
index : +1.0| ********************************* | +5.0
amplitude: +1.0e-12| *****************************************| +1.0e-10
Z=-42.7(0.00%) | Like=-37.87..-13.35 [-48.6253..-31.7799] | it/evals=536/983 eff=78.4773% N=300
Z=-42.4(0.00%) | Like=-37.39..-13.35 [-48.6253..-31.7799] | it/evals=540/988 eff=78.4884% N=300
Z=-39.9(0.00%) | Like=-34.97..-13.35 [-48.6253..-31.7799] | it/evals=570/1028 eff=78.2967% N=300
Z=-38.0(0.00%) | Like=-32.82..-13.35 [-48.6253..-31.7799] | it/evals=600/1067 eff=78.2269% N=300
Mono-modal Volume: ~exp(-6.39) * Expected Volume: exp(-2.01) Quality: ok
index : +1.0| **************************** +4.2 | +5.0
amplitude: +1.0e-12| ***************************************| +1.0e-10
Z=-37.8(0.00%) | Like=-32.72..-13.35 [-48.6253..-31.7799] | it/evals=603/1070 eff=78.3117% N=300
Z=-36.1(0.00%) | Like=-31.23..-13.35 [-31.7743..-22.2117] | it/evals=630/1102 eff=78.5536% N=300
Z=-34.4(0.00%) | Like=-29.30..-13.32 [-31.7743..-22.2117] | it/evals=660/1136 eff=78.9474% N=300
Mono-modal Volume: ~exp(-6.49) * Expected Volume: exp(-2.23) Quality: ok
index : +1.0| +2.0 ************************* +4.0 | +5.0
amplitude: +1.0e-12| ***************************************| +1.0e-10
Z=-33.9(0.00%) | Like=-28.97..-13.32 [-31.7743..-22.2117] | it/evals=670/1152 eff=78.6385% N=300
Z=-33.0(0.00%) | Like=-28.28..-13.32 [-31.7743..-22.2117] | it/evals=690/1176 eff=78.7671% N=300
Z=-32.0(0.00%) | Like=-27.19..-13.32 [-31.7743..-22.2117] | it/evals=720/1217 eff=78.5169% N=300
Mono-modal Volume: ~exp(-6.49) Expected Volume: exp(-2.46) Quality: ok
index : +1.0| +2.0 ********************** +3.8 | +5.0
amplitude: +1.0e-12| **************************************| +1.0e-10
Z=-30.8(0.00%) | Like=-25.69..-13.32 [-31.7743..-22.2117] | it/evals=750/1254 eff=78.6164% N=300
Z=-29.6(0.00%) | Like=-24.44..-13.32 [-31.7743..-22.2117] | it/evals=780/1292 eff=78.6290% N=300
Mono-modal Volume: ~exp(-6.56) * Expected Volume: exp(-2.68) Quality: ok
index : +1.0| +2.1 ********************* +3.7 | +5.0
amplitude: +1.0e-12| +2.6e-11 *********************************** | +1.0e-10
Z=-28.7(0.00%) | Like=-23.62..-13.32 [-31.7743..-22.2117] | it/evals=804/1323 eff=78.5924% N=300
Z=-28.5(0.00%) | Like=-23.35..-13.32 [-31.7743..-22.2117] | it/evals=810/1335 eff=78.2609% N=300
Z=-27.3(0.01%) | Like=-21.95..-13.32 [-22.1599..-19.9653] | it/evals=837/1375 eff=77.8605% N=300
Z=-27.2(0.01%) | Like=-21.90..-13.32 [-22.1599..-19.9653] | it/evals=840/1378 eff=77.9221% N=300
Z=-26.2(0.04%) | Like=-21.11..-13.32 [-22.1599..-19.9653] | it/evals=869/1421 eff=77.5201% N=300
Z=-26.2(0.04%) | Like=-21.08..-13.32 [-22.1599..-19.9653] | it/evals=870/1424 eff=77.4021% N=300
Mono-modal Volume: ~exp(-6.79) * Expected Volume: exp(-2.90) Quality: ok
index : +1.0| +2.2 ****************** +3.5 | +5.0
amplitude: +1.0e-12| +3.0e-11 ****************************** | +1.0e-10
Z=-26.1(0.04%) | Like=-21.06..-13.32 [-22.1599..-19.9653] | it/evals=871/1426 eff=77.3535% N=300
Z=-25.5(0.08%) | Like=-20.66..-13.32 [-22.1599..-19.9653] | it/evals=900/1460 eff=77.5862% N=300
Z=-24.9(0.13%) | Like=-19.99..-13.32 [-22.1599..-19.9653] | it/evals=927/1501 eff=77.1857% N=300
Z=-24.9(0.14%) | Like=-19.92..-13.32 [-19.9314..-19.7098] | it/evals=930/1504 eff=77.2425% N=300
Mono-modal Volume: ~exp(-7.19) * Expected Volume: exp(-3.13) Quality: ok
index : +1.0| +2.2 *************** +3.4 | +5.0
amplitude: +1.0e-12| +3.0e-11 **************************** | +1.0e-10
Z=-24.7(0.17%) | Like=-19.78..-13.32 [-19.9314..-19.7098] | it/evals=938/1515 eff=77.2016% N=300
Z=-24.3(0.25%) | Like=-19.28..-13.32 [-19.2829..-19.2774]*| it/evals=960/1547 eff=76.9848% N=300
Z=-23.7(0.43%) | Like=-18.71..-13.31 [-18.7114..-18.7107]*| it/evals=990/1584 eff=77.1028% N=300
Mono-modal Volume: ~exp(-7.65) * Expected Volume: exp(-3.35) Quality: ok
index : +1.0| +2.3 ************** +3.3 | +5.0
amplitude: +1.0e-12| +3.2e-11 ************************ | +1.0e-10
Z=-23.4(0.56%) | Like=-18.40..-13.31 [-18.3970..-18.3917]*| it/evals=1005/1602 eff=77.1889% N=300
Z=-23.2(0.73%) | Like=-18.07..-13.31 [-18.0674..-18.0594]*| it/evals=1020/1619 eff=77.3313% N=300
Z=-22.7(1.18%) | Like=-17.70..-13.31 [-17.7219..-17.7014] | it/evals=1047/1660 eff=76.9853% N=300
Z=-22.7(1.25%) | Like=-17.63..-13.31 [-17.6275..-17.5688] | it/evals=1050/1664 eff=76.9795% N=300
Mono-modal Volume: ~exp(-7.78) * Expected Volume: exp(-3.57) Quality: ok
index : +1.0| +2.3 ************ +3.2 | +5.0
amplitude: +1.0e-12| +3.4e-11 ********************* +7.7e-11 | +1.0e-10
Z=-22.3(1.74%) | Like=-17.30..-13.31 [-17.3033..-17.2786] | it/evals=1072/1697 eff=76.7359% N=300
Z=-22.2(1.95%) | Like=-17.22..-13.31 [-17.2284..-17.2151] | it/evals=1080/1710 eff=76.5957% N=300
Z=-21.9(2.73%) | Like=-16.88..-13.31 [-16.8765..-16.8654] | it/evals=1107/1750 eff=76.3448% N=300
Z=-21.8(2.82%) | Like=-16.85..-13.31 [-16.8647..-16.8516] | it/evals=1110/1756 eff=76.2363% N=300
Z=-21.5(3.81%) | Like=-16.55..-13.31 [-16.5671..-16.5544] | it/evals=1135/1796 eff=75.8690% N=300
Mono-modal Volume: ~exp(-7.78) Expected Volume: exp(-3.80) Quality: ok
index : +1.0| +2.3 *********** +3.2 | +5.0
amplitude: +1.0e-12| +3.6e-11 ******************** +7.5e-11 | +1.0e-10
Z=-21.5(4.07%) | Like=-16.51..-13.31 [-16.5209..-16.5056] | it/evals=1140/1803 eff=75.8483% N=300
Z=-21.2(5.36%) | Like=-16.24..-13.31 [-16.2679..-16.2387] | it/evals=1166/1845 eff=75.4693% N=300
Z=-21.2(5.56%) | Like=-16.22..-13.31 [-16.2187..-16.1944] | it/evals=1170/1852 eff=75.3866% N=300
Z=-21.0(7.00%) | Like=-16.01..-13.31 [-16.0381..-16.0069] | it/evals=1195/1894 eff=74.9686% N=300
Z=-20.9(7.31%) | Like=-15.96..-13.31 [-15.9585..-15.9374] | it/evals=1200/1899 eff=75.0469% N=300
Mono-modal Volume: ~exp(-8.06) * Expected Volume: exp(-4.02) Quality: ok
index : +1.0| +2.4 ********** +3.1 | +5.0
amplitude: +1.0e-12| +3.7e-11 ****************** +7.2e-11 | +1.0e-10
Z=-20.9(7.69%) | Like=-15.88..-13.31 [-15.8834..-15.8756]*| it/evals=1206/1908 eff=75.0000% N=300
Z=-20.7(9.41%) | Like=-15.70..-13.31 [-15.6984..-15.6903]*| it/evals=1230/1943 eff=74.8631% N=300
Z=-20.4(11.94%) | Like=-15.50..-13.31 [-15.4997..-15.4930]*| it/evals=1260/1979 eff=75.0447% N=300
Mono-modal Volume: ~exp(-8.58) * Expected Volume: exp(-4.24) 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=-20.3(13.00%) | Like=-15.37..-13.31 [-15.3667..-15.3643]*| it/evals=1273/1998 eff=74.9706% N=300
Z=-20.2(14.66%) | Like=-15.25..-13.31 [-15.2516..-15.2246] | it/evals=1290/2022 eff=74.9129% N=300
Z=-20.1(17.33%) | Like=-15.11..-13.31 [-15.1119..-15.1115]*| it/evals=1317/2062 eff=74.7446% N=300
Z=-20.0(17.63%) | Like=-15.11..-13.31 [-15.1070..-15.1013]*| it/evals=1320/2068 eff=74.6606% N=300
Mono-modal Volume: ~exp(-8.58) Expected Volume: exp(-4.47) Quality: ok
index : +1.0| +2.4 ******** +3.1 | +5.0
amplitude: +1.0e-12| +4.0e-11 *************** +6.8e-11 | +1.0e-10
Z=-19.9(20.28%) | Like=-14.97..-13.31 [-14.9679..-14.9632]*| it/evals=1349/2104 eff=74.7783% N=300
Z=-19.9(20.40%) | Like=-14.96..-13.31 [-14.9632..-14.9561]*| it/evals=1350/2105 eff=74.7922% N=300
Z=-19.7(23.77%) | Like=-14.81..-13.31 [-14.8115..-14.8082]*| it/evals=1380/2143 eff=74.8779% N=300
Mono-modal Volume: ~exp(-8.67) * Expected Volume: exp(-4.69) 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=-19.6(26.75%) | Like=-14.68..-13.31 [-14.6812..-14.6759]*| it/evals=1407/2180 eff=74.8404% N=300
Z=-19.6(27.07%) | Like=-14.67..-13.31 [-14.6701..-14.6648]*| it/evals=1410/2184 eff=74.8408% N=300
Z=-19.5(30.34%) | Like=-14.55..-13.31 [-14.5473..-14.5417]*| it/evals=1438/2224 eff=74.7401% N=300
Z=-19.5(30.54%) | Like=-14.52..-13.31 [-14.5417..-14.5223] | it/evals=1440/2227 eff=74.7276% N=300
Z=-19.4(34.08%) | Like=-14.41..-13.31 [-14.4117..-14.4100]*| it/evals=1469/2268 eff=74.6443% N=300
Z=-19.4(34.22%) | Like=-14.41..-13.31 [-14.4100..-14.4009]*| it/evals=1470/2269 eff=74.6572% N=300
Mono-modal Volume: ~exp(-8.67) Expected Volume: exp(-4.91) Quality: ok
index : +1.0| +2.5 ******* +3.0 | +5.0
amplitude: +1.0e-12| +4.2e-11 ************ +6.5e-11 | +1.0e-10
Z=-19.3(36.60%) | Like=-14.36..-13.31 [-14.3595..-14.3587]*| it/evals=1490/2308 eff=74.2032% N=300
Z=-19.3(37.77%) | Like=-14.32..-13.31 [-14.3217..-14.3132]*| it/evals=1500/2323 eff=74.1473% N=300
Z=-19.2(41.11%) | Like=-14.24..-13.31 [-14.2394..-14.2391]*| it/evals=1529/2364 eff=74.0795% N=300
Z=-19.2(41.25%) | Like=-14.24..-13.31 [-14.2391..-14.2306]*| it/evals=1530/2366 eff=74.0561% N=300
Mono-modal Volume: ~exp(-8.86) * Expected Volume: exp(-5.14) 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=-19.2(42.61%) | Like=-14.20..-13.31 [-14.1975..-14.1951]*| it/evals=1541/2380 eff=74.0865% N=300
Z=-19.1(44.93%) | Like=-14.13..-13.31 [-14.1340..-14.1327]*| it/evals=1560/2402 eff=74.2150% N=300
Z=-19.0(47.95%) | Like=-14.05..-13.31 [-14.0507..-14.0505]*| it/evals=1586/2446 eff=73.9049% N=300
Z=-19.0(48.44%) | Like=-14.05..-13.31 [-14.0465..-14.0464]*| it/evals=1590/2450 eff=73.9535% N=300
Mono-modal Volume: ~exp(-8.87) * Expected Volume: exp(-5.36) Quality: ok
index : +1.0| +2.5 ****** +2.9 | +5.0
amplitude: +1.0e-12| +4.4e-11 ********* +6.2e-11 | +1.0e-10
Z=-19.0(50.37%) | Like=-14.01..-13.31 [-14.0115..-14.0090]*| it/evals=1608/2476 eff=73.8971% N=300
Z=-19.0(51.72%) | Like=-13.98..-13.31 [-13.9800..-13.9794]*| it/evals=1620/2493 eff=73.8714% N=300
Z=-18.9(55.19%) | Like=-13.90..-13.31 [-13.8998..-13.8985]*| it/evals=1650/2532 eff=73.9247% N=300
Mono-modal Volume: ~exp(-9.32) * Expected Volume: exp(-5.58) Quality: ok
index : +1.0| +2.6 ***** +2.9 | +5.0
amplitude: +1.0e-12| +4.5e-11 ********* +6.1e-11 | +1.0e-10
Z=-18.8(57.79%) | Like=-13.86..-13.31 [-13.8626..-13.8590]*| it/evals=1675/2567 eff=73.8862% N=300
Z=-18.8(58.32%) | Like=-13.85..-13.31 [-13.8510..-13.8508]*| it/evals=1680/2572 eff=73.9437% N=300
Z=-18.8(61.24%) | Like=-13.81..-13.31 [-13.8068..-13.8066]*| it/evals=1709/2613 eff=73.8867% N=300
Z=-18.8(61.34%) | Like=-13.81..-13.31 [-13.8066..-13.8034]*| it/evals=1710/2615 eff=73.8661% N=300
Z=-18.7(63.84%) | Like=-13.78..-13.31 [-13.7755..-13.7716]*| it/evals=1736/2657 eff=73.6529% N=300
Z=-18.7(64.18%) | Like=-13.77..-13.31 [-13.7676..-13.7675]*| it/evals=1740/2663 eff=73.6352% N=300
Mono-modal Volume: ~exp(-9.55) * Expected Volume: exp(-5.81) Quality: ok
index : +1.0| +2.6 ***** +2.9 | +5.0
amplitude: +1.0e-12| +4.6e-11 ******** +6.0e-11 | +1.0e-10
Z=-18.7(64.35%) | Like=-13.77..-13.31 [-13.7673..-13.7657]*| it/evals=1742/2666 eff=73.6264% N=300
Z=-18.7(66.89%) | Like=-13.71..-13.31 [-13.7148..-13.7143]*| it/evals=1770/2704 eff=73.6273% N=300
Z=-18.7(69.26%) | Like=-13.68..-13.30 [-13.6804..-13.6780]*| it/evals=1798/2746 eff=73.5078% N=300
Z=-18.7(69.42%) | Like=-13.67..-13.30 [-13.6735..-13.6730]*| it/evals=1800/2750 eff=73.4694% N=300
[ultranest] Explored until L=-1e+01
[ultranest] Likelihood function evaluations: 2756
[ultranest] logZ = -18.32 +- 0.1052
[ultranest] Effective samples strategy satisfied (ESS = 1008.2, need >400)
[ultranest] Posterior uncertainty strategy is satisfied (KL: 0.46+-0.08 nat, need <0.50 nat)
[ultranest] Evidency uncertainty strategy is satisfied (dlogz=0.28, need <0.5)
[ultranest] logZ error budget: single: 0.11 bs:0.11 tail:0.26 total:0.28 required:<0.50
[ultranest] done iterating.
logZ = -18.294 +- 0.332
single instance: logZ = -18.294 +- 0.115
bootstrapped : logZ = -18.318 +- 0.204
tail : logZ = +- 0.262
insert order U test : converged: True correlation: inf iterations
index : 2.20 │ ▁▁▁▁▁▁▂▂▃▅▄▅▅▆▇▅▅▅▅▄▃▃▃▃▂▁▁▁▁▁▁ ▁ ▁ ▁ │3.51 2.75 +- 0.17
amplitude : 0.0000000000291│ ▁▁▁▁▁▂▂▂▃▅▅▅▆▆▇▇▇▆▆▅▃▄▃▂▂▂▂▁▁▁▁▁ ▁▁▁▁ │0.0000000000856 0.0000000000532 +- 0.0000000000079
Comparing the posterior distribution of all runs#
For a comparison of different posterior distributions, we can use the package chainconsumer. As this is not a Gammapy dependency, you’ll need to install it. More info here : https://samreay.github.io/ChainConsumer/
# Uncomment this if you have installed `chainconsumer`.
# from chainconsumer import Chain, ChainConfig, ChainConsumer, PlotConfig, Truth, make_sample
# from pandas import DataFrame
# c = ChainConsumer()
# def create_chain(result, name, color="k"):
# return Chain(
# samples=DataFrame(result, columns=["index", "amplitude"]),
# name=name,
# color=color,
# smooth=7,
# shade=False,
# linewidth=1.0,
# cmap="magma",
# show_contour_labels=True,
# kde= True
# )
# c.add_chain(create_chain(result_joint.samples, "joint"))
# c.add_chain(create_chain(result_0.samples, "run0", "g"))
# c.add_chain(create_chain(result_1.samples, "run1", "b"))
# c.add_chain(create_chain(result_2.samples, "run2", "y"))
# fig = c.plotter.plot()
# plt.show()
Corner plot comparison#
Corner plot comparing the three Crab runs.#
We can see the joint analysis allows to better constrain the parameters than the individual runs (more observation time is of course better). One can note as well that one of the run has a notably different amplitude (possibly due to calibrations or/and atmospheric issues).
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 43.569 seconds)
