Note
Go to the end to download the full example code or to run this example in your browser via Binder.
Bayesian analysis with nested sampling#
A demonstration of a Bayesian analysis using the nested sampling technique.
Context#
1. Bayesian analysis#
Bayesian inference uses prior knowledge, in the form of a prior distribution, in order to estimate posterior probabilities which we traditionally visualise in the form of corner plots. These distributions contain more information than a maximum likelihood fit as they reveal not only the “best model” but provide a more accurate representation of errors and correlation between parameters. In particular, non-Gaussian degeneracies are complex to estimate with a maximum likelihood approach.
2. Limitations of the Markov Chain Monte Carlo approach#
A well-known approach to estimate this posterior distribution is the Markov Chain Monte Carlo (MCMC). This uses an ensemble of walkers to produce a chain of samples that after a convergence period will reach a stationary state. Once convergence is reached, the successive elements of the chain are samples of the target posterior distribution. However, the weakness of the MCMC approach lies in the “Once convergence” part. If the walkers are started far from the best likelihood region, the convergence time can be long or never reached if the walkers fall in a local minima. The choice of the initialisation point can become critical for complex models with a high number of dimensions and the ability of these walkers to escape a local minimum or to accurately describe a complex likelihood space is not guaranteed.
3. Nested sampling approach#
To overcome these issues, the nested sampling (NS) algorithm has gained traction in physics and astronomy. It is a Monte Carlo algorithm for computing an integral of the likelihood function over the prior model parameter space introduced in Skilling, 2004. The method performs this integral by evolving a collection of points through the parameter space (see recent reviews from Ashton et al., 2022, and Buchner, 2023). Without going into too many details, one important specificity of the NS method is that it starts from the entire parameter space and evolves a collection of live points to map all minima (including multiple modes if any), whereas Markov Chain Monte Carlo methods require an initialisation point and the walkers will explore the local likelihood. The ability of these walkers to escape a local minimum or to accurately describe a complex likelihood space is not guaranteed. This is a fundamental difference with MCMC or Minuit which will only ever probe the vicinity along their minimisation paths and do not have an overview of the global likelihood landscape. The analysis using the NS framework is more CPU time consuming than a standard classical fit, but it provides the full posterior distribution for all parameters, which is out of reach with traditional fitting techniques (N*(N-1)/2 contour plots to generate). In addition, it is more robust to the choice of initialisation, requires less human intervention and is therefore readily integrated in pipeline analysis. In Gammapy, we used the NS implementation of the UltraNest package (see here for more information), one of the leading package in Astronomy (already used in Cosmology and in X-rays). For a nice visualisation of the NS method see here : sampling visualisation. And for a tutorial of UltraNest applied to X-ray fitting with concrete examples and questions see : BXA Tutorial.
Note: please cite UltraNest if used for a paper
If you are using the “UltraNest” library for a paper, please follow its citation scheme: Cite UltraNest.
Proposed approach#
In this example, we will perform a Bayesian analysis with multiple 1D spectra of the Crab nebula data and investigate their posterior distributions.
Setup#
As usual, we’ll start with some setup …
import matplotlib.pyplot as plt
import numpy as np
import astropy.units as u
from gammapy.datasets import Datasets
from gammapy.datasets import SpectrumDatasetOnOff
from gammapy.modeling.models import (
SkyModel,
UniformPrior,
LogUniformPrior,
)
from gammapy.modeling.sampler import Sampler
Loading the spectral datasets#
Here we will load a few Crab 1D spectral data for which we will do a fit.
Model definition#
Now we want to define the spectral model that will be fitted to the data. The Crab spectra will be fitted here with a simple powerlaw for simplicity.
model = SkyModel.create(spectral_model="pl", name="crab")
Warning
Priors definition: Unlike a traditional fit where priors on the parameters are optional, here it is inherent to the Bayesian approach and are therefore mandatory.
In this case we will set (min,max) prior that will define the
boundaries in which the sampling will be performed.
Note that it is usually recommended to use a LogUniformPrior for
the parameters that have a large amplitude range like the
amplitude parameter.
A UniformPrior means that the samples will be drawn with uniform
probability between the (min,max) values in the linear or log space
in the case of a LogUniformPrior.
model.spectral_model.amplitude.prior = LogUniformPrior(min=1e-12, max=1e-10)
model.spectral_model.index.prior = UniformPrior(min=1, max=5)
datasets.models = [model]
print(datasets.models)
DatasetModels
Component 0: SkyModel
Name : crab
Datasets names : None
Spectral model type : PowerLawSpectralModel
Spatial model type :
Temporal model type :
Parameters:
index : 2.000 +/- 0.00
amplitude : 1.00e-12 +/- 0.0e+00 1 / (TeV s cm2)
reference (frozen): 1.000 TeV
Defining the sampler and options#
As for the Fit object, the Sampler object can receive
different backend (although just one is available for now).
The Sampler comes with “reasonable” default parameters, but you can
change them via the sampler_opts dictionary.
Here is a short description of the most relevant parameters that you
could change :
live_points: minimum number of live points throughout the run. More points allow to discover multiple peaks if existing, but is slower. To test the Prior boundaries and for debugging, a lower number (~100) can be used before a production run with more points (~400 or more).frac_remain: the cut-off condition for the integration, set by the maximum allowed fraction of posterior mass left in the live points vs the dead points. High values (e.g., 0.5) are faster and can be used if the posterior distribution is a relatively simple shape. A low value (1e-1, 1e-2) is optimal for finding peaks, but slower.log_dir: directory where the output files will be stored. If set to None, no files will be written. If set to a string, a directory will be created containing the ongoing status of the run and final results. For time consuming analysis, it is highly recommended to use that option to monitor the run and restart it in case of a crash (withresume=True).
Important note: unlike the MCMC method, you don’t need to define the number of steps for which the sampler will run. The algorithm will automatically stop once a convergence criteria has been reached.
sampler_opts = {
"live_points": 300,
"frac_remain": 0.3,
"log_dir": None,
}
sampler = Sampler(backend="ultranest", sampler_opts=sampler_opts)
Next we can run the sampler on a given dataset. No options are accepted in the run method.
[ultranest] Sampling 300 live points from prior ...
Mono-modal Volume: ~exp(-4.11) * Expected Volume: exp(0.00) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|************************* ******** ***********| +1.0e-10
Z=-inf(0.00%) | Like=-3781.70..-59.38 [-3781.6993..-321.0273] | it/evals=0/301 eff=0.0000% N=300
Z=-713.7(0.00%) | Like=-684.45..-59.38 [-3781.6993..-321.0273] | it/evals=18/320 eff=90.0000% N=300
Z=-545.2(0.00%) | Like=-539.18..-59.38 [-3781.6993..-321.0273] | it/evals=30/334 eff=88.2353% N=300
Z=-531.1(0.00%) | Like=-524.29..-59.38 [-3781.6993..-321.0273] | it/evals=47/353 eff=88.6792% N=300
Z=-512.7(0.00%) | Like=-506.80..-59.38 [-3781.6993..-321.0273] | it/evals=60/368 eff=88.2353% N=300
Mono-modal Volume: ~exp(-4.11) Expected Volume: exp(-0.22) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|************************* ********* ***********| +1.0e-10
Z=-486.3(0.00%) | Like=-480.81..-59.38 [-3781.6993..-321.0273] | it/evals=75/386 eff=87.2093% N=300
Z=-470.2(0.00%) | Like=-462.43..-59.38 [-3781.6993..-321.0273] | it/evals=90/403 eff=87.3786% N=300
Z=-450.3(0.00%) | Like=-443.32..-59.38 [-3781.6993..-321.0273] | it/evals=105/422 eff=86.0656% N=300
Z=-430.8(0.00%) | Like=-422.98..-59.38 [-3781.6993..-321.0273] | it/evals=120/440 eff=85.7143% N=300
Mono-modal Volume: ~exp(-4.11) Expected Volume: exp(-0.45) Quality: ok
index : +1.0| ***********************************************| +5.0
amplitude: +1.0e-12|************************************ ***********| +1.0e-10
Z=-415.0(0.00%) | Like=-408.67..-59.38 [-3781.6993..-321.0273] | it/evals=134/459 eff=84.2767% N=300
Z=-405.6(0.00%) | Like=-399.84..-59.38 [-3781.6993..-321.0273] | it/evals=147/480 eff=81.6667% N=300
Z=-403.7(0.00%) | Like=-397.43..-59.38 [-3781.6993..-321.0273] | it/evals=150/483 eff=81.9672% N=300
Z=-390.1(0.00%) | Like=-384.26..-59.38 [-3781.6993..-321.0273] | it/evals=164/502 eff=81.1881% N=300
Z=-375.5(0.00%) | Like=-367.08..-59.38 [-3781.6993..-321.0273] | it/evals=180/521 eff=81.4480% N=300
Z=-358.0(0.00%) | Like=-347.03..-59.38 [-3781.6993..-321.0273] | it/evals=194/541 eff=80.4979% N=300
Mono-modal Volume: ~exp(-4.70) * Expected Volume: exp(-0.67) Quality: ok
index : +1.0| **********************************************| +5.0
amplitude: +1.0e-12| *********************************** ***********| +1.0e-10
Z=-345.1(0.00%) | Like=-338.16..-59.38 [-3781.6993..-321.0273] | it/evals=201/551 eff=80.0797% N=300
Z=-330.7(0.00%) | Like=-324.01..-59.38 [-3781.6993..-321.0273] | it/evals=210/561 eff=80.4598% N=300
Z=-316.4(0.00%) | Like=-309.92..-59.38 [-320.6964..-188.5568] | it/evals=226/582 eff=80.1418% N=300
Z=-307.5(0.00%) | Like=-300.70..-59.38 [-320.6964..-188.5568] | it/evals=237/602 eff=78.4768% N=300
Z=-305.8(0.00%) | Like=-300.15..-59.38 [-320.6964..-188.5568] | it/evals=240/607 eff=78.1759% N=300
Z=-294.8(0.00%) | Like=-287.20..-59.38 [-320.6964..-188.5568] | it/evals=255/626 eff=78.2209% N=300
Mono-modal Volume: ~exp(-4.70) Expected Volume: exp(-0.89) Quality: ok
index : +1.0| ********************************************| +5.0
amplitude: +1.0e-12| ********************************* ******** **| +1.0e-10
Z=-281.4(0.00%) | Like=-275.23..-59.38 [-320.6964..-188.5568] | it/evals=270/642 eff=78.9474% N=300
Z=-265.5(0.00%) | Like=-259.33..-59.38 [-320.6964..-188.5568] | it/evals=288/661 eff=79.7784% N=300
Z=-254.9(0.00%) | Like=-248.96..-59.17 [-320.6964..-188.5568] | it/evals=300/675 eff=80.0000% N=300
Z=-247.3(0.00%) | Like=-237.23..-59.17 [-320.6964..-188.5568] | it/evals=315/695 eff=79.7468% N=300
Z=-235.5(0.00%) | Like=-228.96..-59.17 [-320.6964..-188.5568] | it/evals=330/714 eff=79.7101% N=300
Mono-modal Volume: ~exp(-5.04) * Expected Volume: exp(-1.12) Quality: ok
index : +1.0| ******************************************| +5.0
amplitude: +1.0e-12| ********************************* ******** **| +1.0e-10
Z=-233.3(0.00%) | Like=-227.08..-59.17 [-320.6964..-188.5568] | it/evals=335/720 eff=79.7619% N=300
Z=-226.0(0.00%) | Like=-217.83..-59.17 [-320.6964..-188.5568] | it/evals=349/739 eff=79.4989% N=300
Z=-219.2(0.00%) | Like=-212.18..-59.17 [-320.6964..-188.5568] | it/evals=360/753 eff=79.4702% N=300
Z=-210.5(0.00%) | Like=-204.49..-59.17 [-320.6964..-188.5568] | it/evals=376/772 eff=79.6610% N=300
Z=-203.6(0.00%) | Like=-196.98..-59.17 [-320.6964..-188.5568] | it/evals=390/789 eff=79.7546% N=300
Mono-modal Volume: ~exp(-5.23) * Expected Volume: exp(-1.34) Quality: ok
index : +1.0| *****************************************| +5.0
amplitude: +1.0e-12| ***************************************** **| +1.0e-10
Z=-199.1(0.00%) | Like=-192.31..-59.17 [-320.6964..-188.5568] | it/evals=402/804 eff=79.7619% N=300
Z=-191.3(0.00%) | Like=-184.95..-59.17 [-188.3337..-131.4907] | it/evals=416/823 eff=79.5411% N=300
Z=-189.2(0.00%) | Like=-182.09..-59.17 [-188.3337..-131.4907] | it/evals=420/827 eff=79.6964% N=300
Z=-183.3(0.00%) | Like=-177.28..-59.17 [-188.3337..-131.4907] | it/evals=435/846 eff=79.6703% N=300
Z=-179.7(0.00%) | Like=-173.76..-59.17 [-188.3337..-131.4907] | it/evals=448/865 eff=79.2920% N=300
Z=-179.2(0.00%) | Like=-173.28..-59.17 [-188.3337..-131.4907] | it/evals=450/873 eff=78.5340% N=300
Z=-175.5(0.00%) | Like=-170.01..-59.17 [-188.3337..-131.4907] | it/evals=466/892 eff=78.7162% N=300
Mono-modal Volume: ~exp(-5.50) * Expected Volume: exp(-1.56) Quality: ok
index : +1.0| *********************************** | +5.0
amplitude: +1.0e-12| **************************************** **| +1.0e-10
Z=-175.0(0.00%) | Like=-169.41..-59.17 [-188.3337..-131.4907] | it/evals=469/895 eff=78.8235% N=300
Z=-171.6(0.00%) | Like=-165.58..-59.17 [-188.3337..-131.4907] | it/evals=480/909 eff=78.8177% N=300
Z=-168.1(0.00%) | Like=-162.02..-59.17 [-188.3337..-131.4907] | it/evals=493/929 eff=78.3784% N=300
Z=-165.2(0.00%) | Like=-159.16..-59.17 [-188.3337..-131.4907] | it/evals=507/949 eff=78.1202% N=300
Z=-164.2(0.00%) | Like=-157.58..-59.17 [-188.3337..-131.4907] | it/evals=510/953 eff=78.1011% N=300
Z=-158.2(0.00%) | Like=-151.03..-59.17 [-188.3337..-131.4907] | it/evals=527/974 eff=78.1899% N=300
Mono-modal Volume: ~exp(-5.71) * Expected Volume: exp(-1.79) Quality: ok
index : +1.0| ******************************* | +5.0
amplitude: +1.0e-12| *************************************** * | +1.0e-10
Z=-155.3(0.00%) | Like=-149.23..-59.17 [-188.3337..-131.4907] | it/evals=536/987 eff=78.0204% N=300
Z=-154.3(0.00%) | Like=-148.29..-59.17 [-188.3337..-131.4907] | it/evals=540/994 eff=77.8098% N=300
Z=-150.8(0.00%) | Like=-144.08..-59.17 [-188.3337..-131.4907] | it/evals=555/1013 eff=77.8401% N=300
Z=-147.8(0.00%) | Like=-141.63..-59.17 [-188.3337..-131.4907] | it/evals=568/1033 eff=77.4898% N=300
Z=-147.4(0.00%) | Like=-141.33..-59.17 [-188.3337..-131.4907] | it/evals=570/1037 eff=77.3406% N=300
Z=-144.4(0.00%) | Like=-137.24..-59.17 [-188.3337..-131.4907] | it/evals=583/1057 eff=77.0145% N=300
Z=-140.2(0.00%) | Like=-133.99..-59.17 [-188.3337..-131.4907] | it/evals=599/1076 eff=77.1907% N=300
Z=-140.0(0.00%) | Like=-133.92..-59.17 [-188.3337..-131.4907] | it/evals=600/1078 eff=77.1208% N=300
Mono-modal Volume: ~exp(-5.71) Expected Volume: exp(-2.01) Quality: ok
index : +1.0| **************************** +4.1 | +5.0
amplitude: +1.0e-12| ************************************* | +1.0e-10
Z=-136.8(0.00%) | Like=-130.67..-59.17 [-131.3996..-96.3077] | it/evals=614/1095 eff=77.2327% N=300
Z=-133.7(0.00%) | Like=-126.99..-59.17 [-131.3996..-96.3077] | it/evals=627/1117 eff=76.7442% N=300
Z=-132.7(0.00%) | Like=-126.21..-59.17 [-131.3996..-96.3077] | it/evals=630/1122 eff=76.6423% N=300
Z=-130.2(0.00%) | Like=-123.90..-59.17 [-131.3996..-96.3077] | it/evals=642/1143 eff=76.1566% N=300
Z=-128.1(0.00%) | Like=-122.12..-59.17 [-131.3996..-96.3077] | it/evals=654/1163 eff=75.7822% N=300
Z=-127.1(0.00%) | Like=-120.87..-59.17 [-131.3996..-96.3077] | it/evals=660/1177 eff=75.2566% N=300
Mono-modal Volume: ~exp(-6.44) * Expected Volume: exp(-2.23) Quality: ok
index : +1.0| ************************* +3.9 | +5.0
amplitude: +1.0e-12| ********************************* | +1.0e-10
Z=-125.6(0.00%) | Like=-119.82..-59.17 [-131.3996..-96.3077] | it/evals=670/1196 eff=74.7768% N=300
Z=-123.6(0.00%) | Like=-117.34..-59.17 [-131.3996..-96.3077] | it/evals=684/1216 eff=74.6725% N=300
Z=-122.8(0.00%) | Like=-116.88..-59.17 [-131.3996..-96.3077] | it/evals=690/1223 eff=74.7562% N=300
Z=-121.3(0.00%) | Like=-114.81..-59.17 [-131.3996..-96.3077] | it/evals=704/1243 eff=74.6554% N=300
Z=-118.4(0.00%) | Like=-112.29..-59.17 [-131.3996..-96.3077] | it/evals=717/1263 eff=74.4548% N=300
Z=-118.1(0.00%) | Like=-112.15..-59.17 [-131.3996..-96.3077] | it/evals=720/1271 eff=74.1504% N=300
Mono-modal Volume: ~exp(-6.44) Expected Volume: exp(-2.46) Quality: ok
index : +1.0| +2.0 ********************** +3.7 | +5.0
amplitude: +1.0e-12| ***************************** * | +1.0e-10
Z=-116.0(0.00%) | Like=-110.26..-59.17 [-131.3996..-96.3077] | it/evals=737/1288 eff=74.5951% N=300
Z=-114.9(0.00%) | Like=-108.98..-59.17 [-131.3996..-96.3077] | it/evals=749/1308 eff=74.3056% N=300
Z=-114.8(0.00%) | Like=-108.96..-59.17 [-131.3996..-96.3077] | it/evals=750/1309 eff=74.3310% N=300
Z=-113.3(0.00%) | Like=-107.08..-59.17 [-131.3996..-96.3077] | it/evals=764/1328 eff=74.3191% N=300
Z=-111.6(0.00%) | Like=-105.67..-59.17 [-131.3996..-96.3077] | it/evals=778/1347 eff=74.3075% N=300
Z=-111.5(0.00%) | Like=-105.47..-59.17 [-131.3996..-96.3077] | it/evals=780/1349 eff=74.3565% N=300
Z=-110.5(0.00%) | Like=-104.24..-59.17 [-131.3996..-96.3077] | it/evals=789/1369 eff=73.8073% N=300
Z=-108.7(0.00%) | Like=-102.22..-59.17 [-131.3996..-96.3077] | it/evals=803/1388 eff=73.8051% N=300
Mono-modal Volume: ~exp(-6.44) Expected Volume: exp(-2.68) Quality: ok
index : +1.0| +2.0 ******************** +3.6 | +5.0
amplitude: +1.0e-12| *************************** +7.7e-11 | +1.0e-10
Z=-107.5(0.00%) | Like=-100.77..-59.17 [-131.3996..-96.3077] | it/evals=810/1399 eff=73.7034% N=300
Z=-105.7(0.00%) | Like=-99.33..-59.17 [-131.3996..-96.3077] | it/evals=824/1419 eff=73.6372% N=300
Z=-104.3(0.00%) | Like=-97.56..-59.17 [-131.3996..-96.3077] | it/evals=835/1439 eff=73.3099% N=300
Z=-103.3(0.00%) | Like=-96.43..-58.78 [-131.3996..-96.3077] | it/evals=840/1446 eff=73.2984% N=300
Z=-101.3(0.00%) | Like=-95.04..-58.78 [-96.1848..-78.3690] | it/evals=854/1465 eff=73.3047% N=300
Z=-100.1(0.00%) | Like=-94.05..-58.78 [-96.1848..-78.3690] | it/evals=867/1485 eff=73.1646% N=300
Z=-99.9(0.00%) | Like=-93.86..-58.78 [-96.1848..-78.3690] | it/evals=870/1488 eff=73.2323% N=300
Mono-modal Volume: ~exp(-6.86) * Expected Volume: exp(-2.90) Quality: ok
index : +1.0| +2.1 ******************* +3.5 | +5.0
amplitude: +1.0e-12| +2.4e-11 ************************* +7.4e-11 | +1.0e-10
Z=-99.8(0.00%) | Like=-93.81..-58.78 [-96.1848..-78.3690] | it/evals=871/1489 eff=73.2548% N=300
Z=-98.7(0.00%) | Like=-92.64..-58.78 [-96.1848..-78.3690] | it/evals=886/1508 eff=73.3444% N=300
Z=-97.6(0.00%) | Like=-91.19..-58.78 [-96.1848..-78.3690] | it/evals=900/1525 eff=73.4694% N=300
Z=-96.2(0.00%) | Like=-89.89..-58.78 [-96.1848..-78.3690] | it/evals=914/1544 eff=73.4727% N=300
Z=-94.9(0.00%) | Like=-88.06..-58.78 [-96.1848..-78.3690] | it/evals=926/1563 eff=73.3175% N=300
Z=-94.4(0.00%) | Like=-87.80..-58.78 [-96.1848..-78.3690] | it/evals=930/1568 eff=73.3438% N=300
Mono-modal Volume: ~exp(-6.89) * Expected Volume: exp(-3.13) Quality: ok
index : +1.0| +2.1 **************** +3.4 | +5.0
amplitude: +1.0e-12| +2.5e-11 ********************** +6.9e-11 | +1.0e-10
Z=-93.3(0.00%) | Like=-86.51..-58.78 [-96.1848..-78.3690] | it/evals=938/1581 eff=73.2240% N=300
Z=-91.6(0.00%) | Like=-85.09..-58.78 [-96.1848..-78.3690] | it/evals=955/1600 eff=73.4615% N=300
Z=-91.2(0.00%) | Like=-84.63..-58.78 [-96.1848..-78.3690] | it/evals=960/1606 eff=73.5069% N=300
Z=-89.8(0.00%) | Like=-83.36..-58.78 [-96.1848..-78.3690] | it/evals=976/1625 eff=73.6604% N=300
Z=-88.9(0.00%) | Like=-82.62..-58.78 [-96.1848..-78.3690] | it/evals=990/1643 eff=73.7156% N=300
Mono-modal Volume: ~exp(-7.15) * Expected Volume: exp(-3.35) Quality: ok
index : +1.0| +2.2 ************** +3.3 | +5.0
amplitude: +1.0e-12| +2.7e-11 ********************* +6.7e-11 | +1.0e-10
Z=-87.9(0.00%) | Like=-81.64..-58.78 [-96.1848..-78.3690] | it/evals=1005/1664 eff=73.6804% N=300
Z=-87.0(0.00%) | Like=-80.49..-58.78 [-96.1848..-78.3690] | it/evals=1020/1682 eff=73.8061% N=300
Z=-86.0(0.00%) | Like=-79.31..-58.78 [-96.1848..-78.3690] | it/evals=1035/1702 eff=73.8231% N=300
Z=-84.9(0.00%) | Like=-78.41..-58.78 [-96.1848..-78.3690] | it/evals=1050/1718 eff=74.0480% N=300
Z=-84.1(0.00%) | Like=-77.74..-58.78 [-78.3327..-68.6839] | it/evals=1063/1738 eff=73.9221% N=300
Mono-modal Volume: ~exp(-7.21) * Expected Volume: exp(-3.57) Quality: ok
index : +1.0| +2.2 ************* +3.2 | +5.0
amplitude: +1.0e-12| +2.9e-11 ******************* +6.5e-11 | +1.0e-10
Z=-83.7(0.00%) | Like=-77.23..-58.78 [-78.3327..-68.6839] | it/evals=1072/1754 eff=73.7276% N=300
Z=-83.2(0.00%) | Like=-76.58..-58.78 [-78.3327..-68.6839] | it/evals=1080/1763 eff=73.8209% N=300
Z=-82.4(0.00%) | Like=-75.98..-58.78 [-78.3327..-68.6839] | it/evals=1094/1782 eff=73.8192% N=300
Z=-81.7(0.00%) | Like=-75.39..-58.78 [-78.3327..-68.6839] | it/evals=1107/1802 eff=73.7017% N=300
Z=-81.6(0.00%) | Like=-75.05..-58.78 [-78.3327..-68.6839] | it/evals=1110/1805 eff=73.7542% N=300
Z=-80.8(0.00%) | Like=-74.49..-58.78 [-78.3327..-68.6839] | it/evals=1125/1824 eff=73.8189% N=300
Z=-80.3(0.00%) | Like=-73.92..-58.78 [-78.3327..-68.6839] | it/evals=1138/1844 eff=73.7047% N=300
Mono-modal Volume: ~exp(-7.58) * Expected Volume: exp(-3.80) Quality: ok
index : +1.0| +2.3 ************ +3.2 | +5.0
amplitude: +1.0e-12| +3.0e-11 ***************** +6.2e-11 | +1.0e-10
Z=-80.2(0.00%) | Like=-73.89..-58.78 [-78.3327..-68.6839] | it/evals=1139/1845 eff=73.7217% N=300
Z=-80.2(0.00%) | Like=-73.75..-58.78 [-78.3327..-68.6839] | it/evals=1140/1846 eff=73.7387% N=300
Z=-79.4(0.00%) | Like=-72.95..-58.78 [-78.3327..-68.6839] | it/evals=1156/1867 eff=73.7715% N=300
Z=-78.9(0.00%) | Like=-72.71..-58.78 [-78.3327..-68.6839] | it/evals=1169/1886 eff=73.7074% N=300
Z=-78.9(0.00%) | Like=-72.70..-58.78 [-78.3327..-68.6839] | it/evals=1170/1887 eff=73.7240% N=300
Z=-78.4(0.00%) | Like=-72.03..-58.76 [-78.3327..-68.6839] | it/evals=1184/1907 eff=73.6777% N=300
Z=-78.0(0.00%) | Like=-71.64..-58.76 [-78.3327..-68.6839] | it/evals=1196/1926 eff=73.5547% N=300
Z=-77.9(0.00%) | Like=-71.51..-58.76 [-78.3327..-68.6839] | it/evals=1200/1933 eff=73.4844% N=300
Mono-modal Volume: ~exp(-7.58) Expected Volume: exp(-4.02) Quality: ok
index : +1.0| +2.3 *********** +3.1 | +5.0
amplitude: +1.0e-12| +3.1e-11 **************** +6.1e-11 | +1.0e-10
Z=-77.4(0.00%) | Like=-71.17..-58.76 [-78.3327..-68.6839] | it/evals=1214/1952 eff=73.4867% N=300
Z=-76.9(0.00%) | Like=-70.45..-58.76 [-78.3327..-68.6839] | it/evals=1228/1971 eff=73.4889% N=300
Z=-76.9(0.00%) | Like=-70.37..-58.76 [-78.3327..-68.6839] | it/evals=1230/1975 eff=73.4328% N=300
Z=-76.3(0.00%) | Like=-69.54..-58.76 [-78.3327..-68.6839] | it/evals=1244/1994 eff=73.4357% N=300
Z=-75.7(0.00%) | Like=-68.86..-58.76 [-78.3327..-68.6839] | it/evals=1257/2013 eff=73.3800% N=300
Z=-75.5(0.00%) | Like=-68.77..-58.76 [-78.3327..-68.6839] | it/evals=1260/2016 eff=73.4266% N=300
Mono-modal Volume: ~exp(-8.10) * Expected Volume: exp(-4.24) 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=-75.0(0.01%) | Like=-68.45..-58.76 [-68.6732..-65.6084] | it/evals=1273/2031 eff=73.5413% N=300
Z=-74.5(0.01%) | Like=-67.97..-58.76 [-68.6732..-65.6084] | it/evals=1289/2050 eff=73.6571% N=300
Z=-74.5(0.01%) | Like=-67.90..-58.76 [-68.6732..-65.6084] | it/evals=1290/2051 eff=73.6722% N=300
Z=-74.1(0.02%) | Like=-67.65..-58.76 [-68.6732..-65.6084] | it/evals=1304/2070 eff=73.6723% N=300
Z=-73.7(0.02%) | Like=-67.38..-58.76 [-68.6732..-65.6084] | it/evals=1317/2090 eff=73.5754% N=300
Z=-73.7(0.02%) | Like=-67.32..-58.76 [-68.6732..-65.6084] | it/evals=1320/2093 eff=73.6196% N=300
Z=-73.3(0.03%) | Like=-66.78..-58.76 [-68.6732..-65.6084] | it/evals=1336/2113 eff=73.6900% N=300
Mono-modal Volume: ~exp(-8.10) Expected Volume: exp(-4.47) Quality: ok
index : +1.0| +2.4 ********* +3.0 | +5.0
amplitude: +1.0e-12| +3.4e-11 ************* +5.7e-11 | +1.0e-10
Z=-72.9(0.05%) | Like=-66.37..-58.76 [-68.6732..-65.6084] | it/evals=1350/2129 eff=73.8108% N=300
Z=-72.6(0.07%) | Like=-66.07..-58.76 [-68.6732..-65.6084] | it/evals=1363/2150 eff=73.6757% N=300
Z=-72.3(0.09%) | Like=-65.86..-58.76 [-68.6732..-65.6084] | it/evals=1376/2171 eff=73.5436% N=300
Z=-72.2(0.10%) | Like=-65.68..-58.76 [-68.6732..-65.6084] | it/evals=1380/2177 eff=73.5216% N=300
Z=-71.9(0.14%) | Like=-65.32..-58.76 [-65.5941..-65.1843] | it/evals=1396/2196 eff=73.6287% N=300
Z=-71.6(0.18%) | Like=-65.08..-58.76 [-65.0960..-65.0768] | it/evals=1406/2216 eff=73.3820% N=300
Mono-modal Volume: ~exp(-8.18) * Expected Volume: exp(-4.69) Quality: ok
index : +1.0| +2.4 ******** +3.0 | +5.0
amplitude: +1.0e-12| +3.5e-11 *********** +5.6e-11 | +1.0e-10
Z=-71.6(0.18%) | Like=-65.07..-58.76 [-65.0742..-65.0669]*| it/evals=1407/2219 eff=73.3194% N=300
Z=-71.6(0.20%) | Like=-65.00..-58.76 [-65.0609..-65.0000] | it/evals=1410/2223 eff=73.3229% N=300
Z=-71.3(0.27%) | Like=-64.61..-58.76 [-64.6109..-64.5867] | it/evals=1424/2243 eff=73.2887% N=300
Z=-71.0(0.36%) | Like=-64.37..-58.76 [-64.3703..-64.3166] | it/evals=1438/2262 eff=73.2926% N=300
Z=-70.9(0.37%) | Like=-64.28..-58.76 [-64.2820..-64.2666] | it/evals=1440/2265 eff=73.2824% N=300
Z=-70.7(0.44%) | Like=-64.14..-58.76 [-64.1352..-64.0841] | it/evals=1449/2285 eff=72.9975% N=300
Z=-70.6(0.53%) | Like=-63.92..-58.76 [-63.9507..-63.9228] | it/evals=1458/2305 eff=72.7182% N=300
Z=-70.3(0.65%) | Like=-63.66..-58.76 [-63.6612..-63.6520]*| it/evals=1470/2325 eff=72.5926% N=300
Mono-modal Volume: ~exp(-8.80) * Expected Volume: exp(-4.91) Quality: ok
index : +1.0| +2.4 ******* +2.9 | +5.0
amplitude: +1.0e-12| +3.6e-11 ********** +5.4e-11 | +1.0e-10
Z=-70.3(0.70%) | Like=-63.64..-58.76 [-63.6446..-63.5917] | it/evals=1474/2333 eff=72.5037% N=300
Z=-70.0(0.90%) | Like=-63.50..-58.76 [-63.4979..-63.4865] | it/evals=1491/2352 eff=72.6608% N=300
Z=-69.8(1.04%) | Like=-63.31..-58.76 [-63.3077..-63.2972] | it/evals=1500/2366 eff=72.6041% N=300
Z=-69.6(1.35%) | Like=-63.06..-58.76 [-63.0558..-63.0433] | it/evals=1517/2385 eff=72.7578% N=300
Z=-69.4(1.62%) | Like=-62.84..-58.76 [-62.8590..-62.8198] | it/evals=1530/2404 eff=72.7186% N=300
Z=-69.3(1.87%) | Like=-62.71..-58.76 [-62.7129..-62.7067]*| it/evals=1540/2424 eff=72.5047% N=300
Mono-modal Volume: ~exp(-8.80) Expected Volume: exp(-5.14) Quality: ok
index : +1.0| +2.4 ****** +2.9 | +5.0
amplitude: +1.0e-12| +3.7e-11 ********* +5.3e-11 | +1.0e-10
Z=-69.1(2.20%) | Like=-62.59..-58.76 [-62.6161..-62.5928] | it/evals=1553/2443 eff=72.4685% N=300
Z=-69.0(2.40%) | Like=-62.56..-58.76 [-62.5620..-62.5552]*| it/evals=1560/2453 eff=72.4570% N=300
Z=-68.9(2.73%) | Like=-62.45..-58.76 [-62.4712..-62.4490] | it/evals=1573/2472 eff=72.4217% N=300
Z=-68.8(3.01%) | Like=-62.38..-58.76 [-62.3812..-62.3708] | it/evals=1583/2491 eff=72.2501% N=300
Z=-68.7(3.20%) | Like=-62.27..-58.76 [-62.2926..-62.2698] | it/evals=1590/2499 eff=72.3056% N=300
Z=-68.6(3.62%) | Like=-62.15..-58.76 [-62.1466..-62.1456]*| it/evals=1601/2519 eff=72.1496% N=300
Mono-modal Volume: ~exp(-9.05) * Expected Volume: exp(-5.36) 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=-68.5(3.81%) | Like=-62.06..-58.76 [-62.0678..-62.0569] | it/evals=1608/2529 eff=72.1400% N=300
Z=-68.4(4.33%) | Like=-61.97..-58.76 [-61.9739..-61.9697]*| it/evals=1620/2546 eff=72.1282% N=300
Z=-68.3(4.90%) | Like=-61.72..-58.76 [-61.7427..-61.7158] | it/evals=1634/2565 eff=72.1413% N=300
Z=-68.1(5.43%) | Like=-61.61..-58.76 [-61.6077..-61.5954] | it/evals=1646/2585 eff=72.0350% N=300
Z=-68.1(5.61%) | Like=-61.58..-58.76 [-61.5841..-61.5769]*| it/evals=1650/2592 eff=71.9895% N=300
Z=-68.0(6.23%) | Like=-61.48..-58.76 [-61.4914..-61.4802] | it/evals=1661/2612 eff=71.8426% N=300
Mono-modal Volume: ~exp(-9.14) * Expected Volume: exp(-5.58) 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(7.13%) | Like=-61.32..-58.76 [-61.3228..-61.3211]*| it/evals=1675/2628 eff=71.9502% N=300
Z=-67.8(7.47%) | Like=-61.29..-58.76 [-61.2913..-61.2895]*| it/evals=1680/2636 eff=71.9178% N=300
Z=-67.7(8.54%) | Like=-61.13..-58.76 [-61.1550..-61.1294] | it/evals=1697/2655 eff=72.0594% N=300
Z=-67.6(9.54%) | Like=-60.99..-58.76 [-60.9949..-60.9927]*| it/evals=1710/2672 eff=72.0911% N=300
Z=-67.5(10.65%) | Like=-60.86..-58.75 [-60.8580..-60.8530]*| it/evals=1726/2691 eff=72.1874% N=300
Z=-67.4(11.57%) | Like=-60.71..-58.75 [-60.7138..-60.7101]*| it/evals=1737/2710 eff=72.0747% N=300
Z=-67.4(11.85%) | Like=-60.70..-58.75 [-60.7028..-60.6795] | it/evals=1740/2713 eff=72.1094% N=300
Mono-modal Volume: ~exp(-9.87) * Expected Volume: exp(-5.81) Quality: ok
index : +1.0| +2.5 ***** +2.8 | +5.0
amplitude: +1.0e-12| +3.9e-11 ****** +5.0e-11 | +1.0e-10
Z=-67.3(12.05%) | Like=-60.68..-58.75 [-60.6782..-60.6752]*| it/evals=1742/2715 eff=72.1325% N=300
Z=-67.2(13.58%) | Like=-60.59..-58.75 [-60.5913..-60.5891]*| it/evals=1757/2734 eff=72.1857% N=300
Z=-67.1(14.92%) | Like=-60.53..-58.75 [-60.5346..-60.5222] | it/evals=1770/2749 eff=72.2744% N=300
Z=-67.0(16.49%) | Like=-60.43..-58.75 [-60.4265..-60.4254]*| it/evals=1785/2768 eff=72.3258% N=300
Z=-67.0(17.91%) | Like=-60.35..-58.75 [-60.3493..-60.3380] | it/evals=1798/2790 eff=72.2088% N=300
Z=-66.9(18.11%) | Like=-60.34..-58.75 [-60.3354..-60.3265]*| it/evals=1800/2792 eff=72.2311% N=300
Mono-modal Volume: ~exp(-9.87) Expected Volume: exp(-6.03) Quality: ok
index : +1.0| +2.5 **** +2.8 | +5.0
amplitude: +1.0e-12| +3.9e-11 ****** +5.0e-11 | +1.0e-10
Z=-66.9(19.28%) | Like=-60.30..-58.75 [-60.3087..-60.2984] | it/evals=1811/2811 eff=72.1227% N=300
Z=-66.8(20.50%) | Like=-60.22..-58.75 [-60.2209..-60.2160]*| it/evals=1824/2830 eff=72.0949% N=300
Z=-66.8(21.21%) | Like=-60.20..-58.75 [-60.1977..-60.1954]*| it/evals=1830/2837 eff=72.1324% N=300
Z=-66.7(22.71%) | Like=-60.14..-58.75 [-60.1360..-60.1307]*| it/evals=1843/2856 eff=72.1049% N=300
Z=-66.6(24.27%) | Like=-60.06..-58.75 [-60.0566..-60.0445] | it/evals=1857/2875 eff=72.1165% N=300
Z=-66.6(24.64%) | Like=-60.04..-58.75 [-60.0388..-60.0367]*| it/evals=1860/2879 eff=72.1210% N=300
Z=-66.5(26.48%) | Like=-59.98..-58.75 [-59.9928..-59.9824] | it/evals=1875/2899 eff=72.1431% N=300
Mono-modal Volume: ~exp(-10.34) * Expected Volume: exp(-6.25) Quality: ok
index : +1.0| +2.5 **** +2.8 | +5.0
amplitude: +1.0e-12| +4.0e-11 ****** +4.9e-11 | +1.0e-10
Z=-66.5(26.63%) | Like=-59.98..-58.75 [-59.9817..-59.9799]*| it/evals=1876/2900 eff=72.1538% N=300
Z=-66.5(28.27%) | Like=-59.93..-58.75 [-59.9269..-59.9236]*| it/evals=1889/2919 eff=72.1268% N=300
Z=-66.5(28.39%) | Like=-59.92..-58.75 [-59.9236..-59.9233]*| it/evals=1890/2920 eff=72.1374% N=300
Z=-66.4(30.61%) | Like=-59.87..-58.75 [-59.8697..-59.8679]*| it/evals=1907/2939 eff=72.2622% N=300
Z=-66.4(32.33%) | Like=-59.81..-58.75 [-59.8090..-59.8066]*| it/evals=1920/2960 eff=72.1805% N=300
Z=-66.3(33.92%) | Like=-59.76..-58.75 [-59.7575..-59.7572]*| it/evals=1933/2979 eff=72.1538% N=300
Mono-modal Volume: ~exp(-10.34) Expected Volume: exp(-6.48) 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=-66.3(35.62%) | Like=-59.72..-58.75 [-59.7173..-59.7171]*| it/evals=1947/2998 eff=72.1646% N=300
Z=-66.2(36.05%) | Like=-59.71..-58.75 [-59.7146..-59.7127]*| it/evals=1950/3004 eff=72.1154% N=300
Z=-66.2(37.62%) | Like=-59.68..-58.75 [-59.6769..-59.6720]*| it/evals=1964/3023 eff=72.1263% N=300
Z=-66.2(39.36%) | Like=-59.62..-58.75 [-59.6361..-59.6185] | it/evals=1978/3042 eff=72.1371% N=300
Z=-66.2(39.60%) | Like=-59.62..-58.75 [-59.6180..-59.6106]*| it/evals=1980/3045 eff=72.1311% N=300
Z=-66.1(41.01%) | Like=-59.58..-58.75 [-59.5776..-59.5772]*| it/evals=1991/3065 eff=72.0072% N=300
Z=-66.1(42.62%) | Like=-59.55..-58.75 [-59.5506..-59.5505]*| it/evals=2004/3085 eff=71.9569% N=300
Mono-modal Volume: ~exp(-10.89) * Expected Volume: exp(-6.70) Quality: ok
index : +1.0| +2.6 **** +2.8 | +5.0
amplitude: +1.0e-12| +4.1e-11 **** +4.8e-11 | +1.0e-10
Z=-66.1(43.43%) | Like=-59.54..-58.75 [-59.5436..-59.5409]*| it/evals=2010/3093 eff=71.9656% N=300
Z=-66.0(45.78%) | Like=-59.48..-58.75 [-59.4788..-59.4754]*| it/evals=2028/3112 eff=72.1195% N=300
Z=-66.0(47.18%) | Like=-59.44..-58.75 [-59.4417..-59.4406]*| it/evals=2040/3129 eff=72.1103% N=300
Z=-65.9(48.92%) | Like=-59.41..-58.75 [-59.4115..-59.4112]*| it/evals=2055/3148 eff=72.1559% N=300
Z=-65.9(50.23%) | Like=-59.39..-58.75 [-59.3874..-59.3847]*| it/evals=2066/3168 eff=72.0363% N=300
Z=-65.9(50.68%) | Like=-59.38..-58.75 [-59.3757..-59.3720]*| it/evals=2070/3173 eff=72.0501% N=300
Mono-modal Volume: ~exp(-10.89) Expected Volume: exp(-6.92) Quality: ok
index : +1.0| +2.6 **** +2.8 | +5.0
amplitude: +1.0e-12| +4.1e-11 **** +4.8e-11 | +1.0e-10
Z=-65.9(52.03%) | Like=-59.34..-58.75 [-59.3407..-59.3393]*| it/evals=2082/3192 eff=71.9917% N=300
Z=-65.9(53.36%) | Like=-59.32..-58.75 [-59.3180..-59.3171]*| it/evals=2094/3211 eff=71.9340% N=300
Z=-65.8(54.05%) | Like=-59.31..-58.75 [-59.3084..-59.3081]*| it/evals=2100/3219 eff=71.9424% N=300
Z=-65.8(55.67%) | Like=-59.29..-58.75 [-59.2920..-59.2918]*| it/evals=2114/3238 eff=71.9537% N=300
Z=-65.8(56.76%) | Like=-59.28..-58.75 [-59.2786..-59.2779]*| it/evals=2124/3258 eff=71.8053% N=300
Z=-65.8(57.42%) | Like=-59.27..-58.75 [-59.2675..-59.2576]*| it/evals=2130/3270 eff=71.7172% N=300
Z=-65.8(58.66%) | Like=-59.24..-58.75 [-59.2400..-59.2380]*| it/evals=2141/3290 eff=71.6054% N=300
Mono-modal Volume: ~exp(-10.97) * Expected Volume: exp(-7.15) Quality: ok
index : +1.0| +2.6 ** +2.7 | +5.0
amplitude: +1.0e-12| +4.1e-11 **** +4.7e-11 | +1.0e-10
Z=-65.8(59.00%) | Like=-59.24..-58.75 [-59.2365..-59.2347]*| it/evals=2144/3293 eff=71.6338% N=300
Z=-65.7(60.64%) | Like=-59.21..-58.75 [-59.2121..-59.2106]*| it/evals=2160/3310 eff=71.7608% N=300
Z=-65.7(62.18%) | Like=-59.19..-58.75 [-59.1908..-59.1874]*| it/evals=2175/3329 eff=71.8059% N=300
Z=-65.7(63.57%) | Like=-59.17..-58.75 [-59.1676..-59.1674]*| it/evals=2189/3348 eff=71.8176% N=300
Z=-65.7(63.67%) | Like=-59.17..-58.75 [-59.1674..-59.1660]*| it/evals=2190/3349 eff=71.8268% N=300
Z=-65.7(64.91%) | Like=-59.15..-58.75 [-59.1512..-59.1504]*| it/evals=2203/3368 eff=71.8057% N=300
Mono-modal Volume: ~exp(-11.30) * Expected Volume: exp(-7.37) Quality: ok
index : +1.0| +2.6 ** +2.7 | +5.0
amplitude: +1.0e-12| +4.2e-11 **** +4.7e-11 | +1.0e-10
Z=-65.7(65.63%) | Like=-59.14..-58.75 [-59.1427..-59.1413]*| it/evals=2211/3380 eff=71.7857% N=300
Z=-65.6(66.46%) | Like=-59.12..-58.75 [-59.1249..-59.1245]*| it/evals=2220/3392 eff=71.7982% N=300
Z=-65.6(68.06%) | Like=-59.09..-58.75 [-59.0922..-59.0914]*| it/evals=2238/3411 eff=71.9383% N=300
Z=-65.6(69.14%) | Like=-59.08..-58.75 [-59.0781..-59.0768]*| it/evals=2250/3428 eff=71.9309% N=300
[ultranest] Explored until L=-6e+01
[ultranest] Likelihood function evaluations: 3445
[ultranest] logZ = -65.2 +- 0.0998
[ultranest] Effective samples strategy satisfied (ESS = 981.8, need >400)
[ultranest] Posterior uncertainty strategy is satisfied (KL: 0.46+-0.07 nat, need <0.50 nat)
[ultranest] Evidency uncertainty strategy is satisfied (dlogz=0.28, need <0.5)
[ultranest] logZ error budget: single: 0.14 bs:0.10 tail:0.26 total:0.28 required:<0.50
[ultranest] done iterating.
logZ = -65.231 +- 0.328
single instance: logZ = -65.231 +- 0.136
bootstrapped : logZ = -65.199 +- 0.198
tail : logZ = +- 0.262
insert order U test : converged: True correlation: inf iterations
index : 2.388 │ ▁▁▁▁▁▁▁▂▂▃▄▄▆▇▇▇▅▆▅▅▃▃▃▂▁▁▁▁▁▁▁▁ ▁ │3.073 2.675 +- 0.081
amplitude : 0.0000000000337│ ▁▁▁▁▁▁▁▁▂▃▃▃▅▄▆▇▇▇▆▇▆▅▆▅▄▄▃▁▁▁▁▁▁▁▁▁▁ │0.0000000000554 0.0000000000444 +- 0.0000000000030
Understanding the outputs#
In the Jupyter notebook, you should be able to see an interactive visualisation of how the parameter space shrinks which starts from the (min,max) shrinks down towards the optimal parameters.
The output above is filled with interesting information. Here we provide a short description of the most relevant information provided above. For more detailed information see the UltraNest docs.
During the sampling
Z=-68.8(0.53%) | Like=-63.96..-58.75 [-63.9570..-63.9539]*| it/evals=640/1068 eff=73.7327% N=300
Some important information here is:
Progress (0.53%): the completed fraction of the integral. This is not a time progress bar. Stays at zero for a good fraction of the run.
Efficiency (eff value) of the sampling: this indicates out of the proposed new points, how many were accepted. If your efficiency is too small (<<1%), maybe you should revise your priors (e.g use a LogUniform prior for the normalisation).
Final outputs
The final lines indicate that all three “convergence” strategies are satisfied (samples, posterior uncertainty, and evidence uncertainty).
logZ = -65.104 +- 0.292
The main goal of the Nested sampling algorithm is to estimate Z (the Bayesian evidence) which is given above together with an uncertainty. In a similar way to deltaLogLike and deltaAIC, deltaLogZ values can be used for model comparison. For more information see : on the use of the evidence for model comparison. An interesting comparison of the efficiency and false discovery rate of model selection with deltaLogLike and deltaLogZ is given in Appendix C of Buchner et al., 2014.
Results stored on disk
if log_dir is set to a name where the results will be stored, then
a directory is created containing many useful results and plots.
A description of these outputs is given in the Ultranest
docs.
Results#
Within a Bayesian analysis, the concept of best-fit has to be viewed differently from what is done in a gradient descent fit.
The output of the Bayesian analysis is the posterior distribution and there is no “best-fit” output. One has to define, based on the posteriors, what we want to consider as “best-fit” and several options are possible:
the mean of the distribution
the median
the lowest likelihood value
By default the DatasetModels will be updated with the mean of
the posterior distributions.
print(result_joint.models)
DatasetModels
Component 0: SkyModel
Name : crab
Datasets names : None
Spectral model type : PowerLawSpectralModel
Spatial model type :
Temporal model type :
Parameters:
index : 2.675 +/- 0.08
amplitude : 4.44e-11 +/- 3.0e-12 1 / (TeV s cm2)
reference (frozen): 1.000 TeV
The Sampler class returns a very rich dictionary.
The most “standard” information about the posterior distributions can
be found in :
print(result_joint.sampler_results["posterior"])
{'mean': [2.6754587662650917, 4.444527651755381e-11], 'stdev': [0.08073367798072967, 2.9941422276635043e-12], 'median': [2.67074254811291, 4.434248825331601e-11], 'errlo': [2.597201148830644, 4.143404161739705e-11], 'errup': [2.7556721996712534, 4.7537262359334296e-11], 'information_gain_bits': [2.728201650477127, 3.127037561682504]}
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(-3.88) * Expected Volume: exp(0.00) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|***************************** ** * ****** ** **| +1.0e-10
Z=-inf(0.00%) | Like=-1523.78..-20.88 [-1523.7849..-112.0425] | it/evals=0/301 eff=0.0000% N=300
Z=-177.6(0.00%) | Like=-172.65..-20.88 [-1523.7849..-112.0425] | it/evals=30/331 eff=96.7742% N=300
Z=-165.2(0.00%) | Like=-159.31..-20.88 [-1523.7849..-112.0425] | it/evals=60/365 eff=92.3077% N=300
Mono-modal Volume: ~exp(-4.15) * Expected Volume: exp(-0.22) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|************ **************** ** ******* * * | +1.0e-10
Z=-161.3(0.00%) | Like=-155.47..-20.88 [-1523.7849..-112.0425] | it/evals=67/373 eff=91.7808% N=300
Z=-154.1(0.00%) | Like=-148.55..-20.88 [-1523.7849..-112.0425] | it/evals=90/401 eff=89.1089% N=300
Z=-142.9(0.00%) | Like=-137.20..-20.88 [-1523.7849..-112.0425] | it/evals=120/436 eff=88.2353% N=300
Mono-modal Volume: ~exp(-4.41) * Expected Volume: exp(-0.45) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|***************************** *** ***** *** * * | +1.0e-10
Z=-137.9(0.00%) | Like=-133.08..-20.88 [-1523.7849..-112.0425] | it/evals=134/451 eff=88.7417% N=300
Z=-131.2(0.00%) | Like=-126.16..-20.88 [-1523.7849..-112.0425] | it/evals=150/473 eff=86.7052% N=300
Z=-120.0(0.00%) | Like=-115.40..-20.71 [-1523.7849..-112.0425] | it/evals=179/508 eff=86.0577% N=300
Z=-119.9(0.00%) | Like=-115.36..-20.71 [-1523.7849..-112.0425] | it/evals=180/509 eff=86.1244% N=300
Mono-modal Volume: ~exp(-4.41) Expected Volume: exp(-0.67) Quality: ok
index : +1.0| *********************************************| +5.0
amplitude: +1.0e-12| **************************************** * * * | +1.0e-10
Z=-115.7(0.00%) | Like=-109.99..-20.71 [-112.0390..-73.8858] | it/evals=201/543 eff=82.7160% N=300
Z=-113.1(0.00%) | Like=-107.93..-20.71 [-112.0390..-73.8858] | it/evals=210/553 eff=83.0040% N=300
Z=-106.1(0.00%) | Like=-100.73..-20.71 [-112.0390..-73.8858] | it/evals=237/588 eff=82.2917% N=300
Z=-105.4(0.00%) | Like=-100.22..-20.71 [-112.0390..-73.8858] | it/evals=240/591 eff=82.4742% N=300
Z=-100.7(0.00%) | Like=-96.06..-20.71 [-112.0390..-73.8858] | it/evals=265/626 eff=81.2883% N=300
Mono-modal Volume: ~exp(-5.03) * Expected Volume: exp(-0.89) Quality: ok
index : +1.0| ********************************************| +5.0
amplitude: +1.0e-12| ************************************** * * * * | +1.0e-10
Z=-100.3(0.00%) | Like=-95.65..-20.71 [-112.0390..-73.8858] | it/evals=268/630 eff=81.2121% N=300
Z=-100.0(0.00%) | Like=-95.43..-20.71 [-112.0390..-73.8858] | it/evals=270/632 eff=81.3253% N=300
Z=-94.9(0.00%) | Like=-89.95..-20.71 [-112.0390..-73.8858] | it/evals=299/667 eff=81.4714% N=300
Z=-94.8(0.00%) | Like=-89.58..-20.71 [-112.0390..-73.8858] | it/evals=300/668 eff=81.5217% N=300
Z=-89.6(0.00%) | Like=-84.44..-20.71 [-112.0390..-73.8858] | it/evals=326/703 eff=80.8933% N=300
Z=-89.0(0.00%) | Like=-83.78..-20.63 [-112.0390..-73.8858] | it/evals=330/707 eff=81.0811% N=300
Mono-modal Volume: ~exp(-5.28) * Expected Volume: exp(-1.12) Quality: ok
index : +1.0| *******************************************| +5.0
amplitude: +1.0e-12| ************************************* * * | +1.0e-10
Z=-88.1(0.00%) | Like=-83.10..-20.63 [-112.0390..-73.8858] | it/evals=335/717 eff=80.3357% N=300
Z=-84.2(0.00%) | Like=-78.72..-20.63 [-112.0390..-73.8858] | it/evals=360/754 eff=79.2952% N=300
Z=-79.9(0.00%) | Like=-75.09..-20.63 [-112.0390..-73.8858] | it/evals=386/790 eff=78.7755% N=300
Z=-79.5(0.00%) | Like=-74.92..-20.63 [-112.0390..-73.8858] | it/evals=390/795 eff=78.7879% 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=-78.3(0.00%) | Like=-73.37..-20.63 [-73.6868..-52.5511] | it/evals=402/811 eff=78.6693% N=300
Z=-76.2(0.00%) | Like=-71.06..-20.63 [-73.6868..-52.5511] | it/evals=420/839 eff=77.9221% N=300
Z=-72.9(0.00%) | Like=-68.24..-20.63 [-73.6868..-52.5511] | it/evals=449/874 eff=78.2230% N=300
Z=-72.8(0.00%) | Like=-68.20..-20.63 [-73.6868..-52.5511] | it/evals=450/875 eff=78.2609% N=300
Mono-modal Volume: ~exp(-5.40) Expected Volume: exp(-1.56) Quality: ok
index : +1.0| ************************************ | +5.0
amplitude: +1.0e-12| ******************************* *** | +1.0e-10
Z=-70.3(0.00%) | Like=-64.93..-20.63 [-73.6868..-52.5511] | it/evals=475/907 eff=78.2537% N=300
Z=-69.6(0.00%) | Like=-64.14..-20.63 [-73.6868..-52.5511] | it/evals=480/914 eff=78.1759% N=300
Z=-67.6(0.00%) | Like=-62.65..-20.63 [-73.6868..-52.5511] | it/evals=500/951 eff=76.8049% N=300
Z=-66.8(0.00%) | Like=-61.65..-20.63 [-73.6868..-52.5511] | it/evals=510/967 eff=76.4618% N=300
Z=-64.9(0.00%) | Like=-59.55..-20.63 [-73.6868..-52.5511] | it/evals=531/1005 eff=75.3191% N=300
Mono-modal Volume: ~exp(-5.40) Expected Volume: exp(-1.79) Quality: ok
index : +1.0| ******************************** | +5.0
amplitude: +1.0e-12| ****************************** ** +7.8e-11| +1.0e-10
Z=-63.9(0.00%) | Like=-58.81..-20.63 [-73.6868..-52.5511] | it/evals=540/1016 eff=75.4190% N=300
Z=-61.9(0.00%) | Like=-56.72..-20.63 [-73.6868..-52.5511] | it/evals=563/1053 eff=74.7676% N=300
Z=-61.2(0.00%) | Like=-55.63..-20.63 [-73.6868..-52.5511] | it/evals=570/1065 eff=74.5098% N=300
Z=-58.3(0.00%) | Like=-52.26..-20.63 [-52.2624..-37.9452] | it/evals=592/1102 eff=73.8155% N=300
Z=-57.2(0.00%) | Like=-51.89..-20.63 [-52.2624..-37.9452] | it/evals=600/1113 eff=73.8007% N=300
Mono-modal Volume: ~exp(-6.10) * Expected Volume: exp(-2.01) Quality: ok
index : +1.0| *************************** +3.9 | +5.0
amplitude: +1.0e-12| ****************************** +7.2e-11 | +1.0e-10
Z=-56.9(0.00%) | Like=-51.71..-20.63 [-52.2624..-37.9452] | it/evals=603/1118 eff=73.7164% N=300
Z=-54.7(0.00%) | Like=-49.25..-20.63 [-52.2624..-37.9452] | it/evals=630/1152 eff=73.9437% N=300
Z=-52.9(0.00%) | Like=-47.27..-20.63 [-52.2624..-37.9452] | it/evals=652/1188 eff=73.4234% N=300
Z=-52.1(0.00%) | Like=-46.40..-20.63 [-52.2624..-37.9452] | it/evals=660/1203 eff=73.0897% N=300
Mono-modal Volume: ~exp(-6.36) * 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=-51.3(0.00%) | Like=-45.68..-20.63 [-52.2624..-37.9452] | it/evals=670/1221 eff=72.7470% N=300
Z=-49.6(0.00%) | Like=-44.19..-20.63 [-52.2624..-37.9452] | it/evals=690/1243 eff=73.1707% N=300
Z=-48.0(0.00%) | Like=-42.44..-20.63 [-52.2624..-37.9452] | it/evals=714/1281 eff=72.7829% N=300
Z=-47.6(0.00%) | Like=-42.21..-20.63 [-52.2624..-37.9452] | it/evals=720/1289 eff=72.8008% N=300
Mono-modal Volume: ~exp(-6.50) * 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=-46.5(0.00%) | Like=-41.27..-20.63 [-52.2624..-37.9452] | it/evals=737/1308 eff=73.1151% N=300
Z=-45.9(0.00%) | Like=-40.85..-20.63 [-52.2624..-37.9452] | it/evals=750/1324 eff=73.2422% N=300
Z=-44.7(0.00%) | Like=-39.24..-20.63 [-52.2624..-37.9452] | it/evals=776/1360 eff=73.2075% N=300
Z=-44.5(0.00%) | Like=-38.90..-20.63 [-52.2624..-37.9452] | it/evals=780/1365 eff=73.2394% N=300
Mono-modal Volume: ~exp(-6.55) * Expected Volume: exp(-2.68) Quality: ok
index : +1.0| +1.9 ******************* +3.5 | +5.0
amplitude: +1.0e-12| ********************** +6.0e-11 | +1.0e-10
Z=-43.1(0.00%) | Like=-37.53..-20.54 [-37.9331..-29.5926] | it/evals=804/1394 eff=73.4918% N=300
Z=-42.8(0.00%) | Like=-37.29..-20.54 [-37.9331..-29.5926] | it/evals=810/1402 eff=73.5027% N=300
Z=-41.5(0.00%) | Like=-36.15..-20.54 [-37.9331..-29.5926] | it/evals=834/1438 eff=73.2865% N=300
Z=-41.3(0.00%) | Like=-35.90..-20.54 [-37.9331..-29.5926] | it/evals=840/1445 eff=73.3624% N=300
Z=-40.3(0.00%) | Like=-34.80..-20.54 [-37.9331..-29.5926] | it/evals=863/1481 eff=73.0737% N=300
Z=-40.0(0.00%) | Like=-34.49..-20.54 [-37.9331..-29.5926] | it/evals=870/1491 eff=73.0479% N=300
Mono-modal Volume: ~exp(-7.02) * 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=-39.9(0.00%) | Like=-34.40..-20.54 [-37.9331..-29.5926] | it/evals=871/1492 eff=73.0705% N=300
Z=-38.9(0.00%) | Like=-33.40..-20.54 [-37.9331..-29.5926] | it/evals=897/1527 eff=73.1051% N=300
Z=-38.8(0.00%) | Like=-33.29..-20.54 [-37.9331..-29.5926] | it/evals=900/1531 eff=73.1113% N=300
Z=-37.7(0.00%) | Like=-32.26..-20.54 [-37.9331..-29.5926] | it/evals=929/1566 eff=73.3807% N=300
Z=-37.7(0.00%) | Like=-32.24..-20.54 [-37.9331..-29.5926] | it/evals=930/1567 eff=73.4017% N=300
Mono-modal Volume: ~exp(-7.04) * Expected Volume: exp(-3.13) Quality: ok
index : +1.0| +2.0 **************** +3.3 | +5.0
amplitude: +1.0e-12| ******************* +5.5e-11 | +1.0e-10
Z=-37.4(0.00%) | Like=-32.01..-20.54 [-37.9331..-29.5926] | it/evals=938/1577 eff=73.4534% N=300
Z=-36.8(0.00%) | Like=-31.50..-20.54 [-37.9331..-29.5926] | it/evals=960/1607 eff=73.4507% N=300
Z=-36.0(0.01%) | Like=-30.42..-20.54 [-37.9331..-29.5926] | it/evals=985/1644 eff=73.2887% N=300
Z=-35.8(0.01%) | Like=-30.36..-20.54 [-37.9331..-29.5926] | it/evals=990/1650 eff=73.3333% N=300
Mono-modal Volume: ~exp(-7.08) * Expected Volume: exp(-3.35) Quality: ok
index : +1.0| +2.1 *************** +3.2 | +5.0
amplitude: +1.0e-12| ***************** +5.3e-11 | +1.0e-10
Z=-35.4(0.01%) | Like=-29.89..-20.54 [-37.9331..-29.5926] | it/evals=1005/1668 eff=73.4649% N=300
Z=-35.0(0.01%) | Like=-29.52..-20.54 [-29.5266..-27.2410] | it/evals=1020/1687 eff=73.5400% N=300
Z=-34.4(0.03%) | Like=-28.83..-20.54 [-29.5266..-27.2410] | it/evals=1045/1725 eff=73.3333% N=300
Z=-34.2(0.03%) | Like=-28.67..-20.54 [-29.5266..-27.2410] | it/evals=1050/1731 eff=73.3753% N=300
Z=-33.7(0.05%) | Like=-28.11..-20.54 [-29.5266..-27.2410] | it/evals=1071/1767 eff=73.0061% N=300
Mono-modal Volume: ~exp(-7.55) * 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=-33.7(0.06%) | Like=-28.06..-20.54 [-29.5266..-27.2410] | it/evals=1072/1768 eff=73.0245% N=300
Z=-33.5(0.07%) | Like=-27.86..-20.54 [-29.5266..-27.2410] | it/evals=1080/1777 eff=73.1212% N=300
Z=-32.8(0.12%) | Like=-27.23..-20.54 [-27.2362..-26.9610] | it/evals=1106/1812 eff=73.1481% N=300
Z=-32.8(0.14%) | Like=-27.15..-20.54 [-27.2362..-26.9610] | it/evals=1110/1818 eff=73.1225% N=300
Z=-32.2(0.22%) | Like=-26.62..-20.49 [-26.6459..-26.6243] | it/evals=1133/1855 eff=72.8617% N=300
Mono-modal Volume: ~exp(-8.13) * Expected Volume: exp(-3.80) Quality: ok
index : +1.0| +2.2 ************ +3.0 | +5.0
amplitude: +1.0e-12| ************** +4.9e-11 | +1.0e-10
Z=-32.1(0.24%) | Like=-26.45..-20.49 [-26.4489..-26.4408]*| it/evals=1139/1863 eff=72.8727% N=300
Z=-32.1(0.25%) | Like=-26.44..-20.49 [-26.4408..-26.4064] | it/evals=1140/1864 eff=72.8900% N=300
Z=-31.6(0.40%) | Like=-26.00..-20.49 [-25.9996..-25.9966]*| it/evals=1163/1900 eff=72.6875% N=300
Z=-31.5(0.45%) | Like=-25.91..-20.49 [-25.9133..-25.8984] | it/evals=1170/1909 eff=72.7160% N=300
Z=-31.1(0.66%) | Like=-25.67..-20.49 [-25.6655..-25.6492] | it/evals=1195/1944 eff=72.6886% N=300
Z=-31.0(0.71%) | Like=-25.60..-20.49 [-25.6036..-25.5706] | it/evals=1200/1949 eff=72.7714% N=300
Mono-modal Volume: ~exp(-8.13) 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(1.04%) | Like=-25.19..-20.49 [-25.2094..-25.1942] | it/evals=1226/1980 eff=72.9762% N=300
Z=-30.6(1.10%) | Like=-25.16..-20.49 [-25.1554..-25.1255] | it/evals=1230/1987 eff=72.9105% N=300
Z=-30.3(1.53%) | Like=-24.88..-20.49 [-24.8849..-24.8436] | it/evals=1255/2022 eff=72.8804% N=300
Z=-30.3(1.60%) | Like=-24.78..-20.49 [-24.8070..-24.7786] | it/evals=1260/2028 eff=72.9167% N=300
Mono-modal Volume: ~exp(-8.13) Expected Volume: exp(-4.24) 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=-30.0(2.07%) | Like=-24.49..-20.49 [-24.5036..-24.4861] | it/evals=1282/2060 eff=72.8409% N=300
Z=-29.9(2.24%) | Like=-24.34..-20.49 [-24.3593..-24.3427] | it/evals=1290/2070 eff=72.8814% N=300
Z=-29.7(2.64%) | Like=-24.18..-20.49 [-24.2022..-24.1801] | it/evals=1306/2106 eff=72.3145% N=300
Z=-29.6(3.08%) | Like=-24.05..-20.49 [-24.0535..-24.0489]*| it/evals=1320/2130 eff=72.1311% N=300
Mono-modal Volume: ~exp(-8.13) Expected Volume: exp(-4.47) Quality: ok
index : +1.0| +2.3 ******** +2.9 | +5.0
amplitude: +1.0e-12| +2.5e-11 ********** +4.4e-11 | +1.0e-10
Z=-29.4(3.83%) | Like=-23.85..-20.49 [-23.8463..-23.8300] | it/evals=1341/2162 eff=72.0193% N=300
Z=-29.3(4.19%) | Like=-23.71..-20.49 [-23.7278..-23.7067] | it/evals=1350/2177 eff=71.9233% N=300
Z=-29.1(5.04%) | Like=-23.46..-20.49 [-23.4710..-23.4595] | it/evals=1372/2212 eff=71.7573% N=300
Z=-29.0(5.50%) | Like=-23.34..-20.49 [-23.3553..-23.3410] | it/evals=1380/2228 eff=71.5768% N=300
Z=-28.8(6.67%) | Like=-23.13..-20.49 [-23.1309..-23.1135] | it/evals=1400/2264 eff=71.2831% N=300
Mono-modal Volume: ~exp(-8.91) * Expected Volume: exp(-4.69) 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=-28.7(7.06%) | Like=-23.05..-20.49 [-23.0497..-23.0438]*| it/evals=1407/2277 eff=71.1684% N=300
Z=-28.7(7.24%) | Like=-23.03..-20.49 [-23.0285..-23.0116] | it/evals=1410/2284 eff=71.0685% N=300
Z=-28.5(9.20%) | Like=-22.75..-20.47 [-22.7643..-22.7455] | it/evals=1439/2319 eff=71.2729% N=300
Z=-28.5(9.28%) | Like=-22.73..-20.47 [-22.7265..-22.7208]*| it/evals=1440/2321 eff=71.2519% N=300
Z=-28.2(11.66%) | Like=-22.56..-20.47 [-22.5635..-22.5539]*| it/evals=1467/2357 eff=71.3175% N=300
Z=-28.2(11.92%) | Like=-22.55..-20.47 [-22.5462..-22.5420]*| it/evals=1470/2360 eff=71.3592% N=300
Mono-modal Volume: ~exp(-8.92) * 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=-28.2(12.20%) | Like=-22.51..-20.47 [-22.5317..-22.5128] | it/evals=1474/2365 eff=71.3801% N=300
Z=-28.0(14.55%) | Like=-22.36..-20.47 [-22.3632..-22.3553]*| it/evals=1500/2393 eff=71.6675% N=300
Z=-27.9(16.91%) | Like=-22.22..-20.47 [-22.2172..-22.1945] | it/evals=1524/2428 eff=71.6165% N=300
Z=-27.8(17.53%) | Like=-22.16..-20.47 [-22.1610..-22.1467] | it/evals=1530/2434 eff=71.6963% N=300
Mono-modal Volume: ~exp(-8.92) 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(19.79%) | Like=-22.05..-20.47 [-22.0476..-22.0471]*| it/evals=1554/2465 eff=71.7783% N=300
Z=-27.7(20.45%) | Like=-22.02..-20.47 [-22.0160..-22.0155]*| it/evals=1560/2471 eff=71.8563% N=300
Z=-27.5(22.94%) | Like=-21.93..-20.47 [-21.9393..-21.9290] | it/evals=1585/2506 eff=71.8495% N=300
Z=-27.5(23.55%) | Like=-21.91..-20.47 [-21.9092..-21.9072]*| it/evals=1590/2512 eff=71.8807% N=300
Mono-modal Volume: ~exp(-9.15) * 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(25.43%) | Like=-21.79..-20.47 [-21.8140..-21.7870] | it/evals=1608/2538 eff=71.8499% N=300
Z=-27.4(26.89%) | Like=-21.72..-20.47 [-21.7206..-21.7056] | it/evals=1620/2558 eff=71.7449% N=300
Z=-27.3(29.87%) | Like=-21.59..-20.47 [-21.5871..-21.5849]*| it/evals=1645/2593 eff=71.7401% N=300
Z=-27.3(30.58%) | Like=-21.57..-20.47 [-21.5726..-21.5720]*| it/evals=1650/2600 eff=71.7391% N=300
Mono-modal Volume: ~exp(-9.26) * Expected Volume: exp(-5.58) Quality: ok
index : +1.0| +2.4 ***** +2.7 | +5.0
amplitude: +1.0e-12| +2.9e-11 ****** +4.0e-11 | +1.0e-10
Z=-27.2(33.63%) | Like=-21.48..-20.47 [-21.4850..-21.4748] | it/evals=1675/2630 eff=71.8884% N=300
Z=-27.2(34.27%) | Like=-21.46..-20.47 [-21.4591..-21.4566]*| it/evals=1680/2635 eff=71.9486% N=300
Z=-27.1(37.11%) | Like=-21.39..-20.47 [-21.3878..-21.3865]*| it/evals=1703/2670 eff=71.8565% N=300
Z=-27.0(38.04%) | Like=-21.37..-20.47 [-21.3732..-21.3710]*| it/evals=1710/2682 eff=71.7884% N=300
Z=-27.0(40.28%) | Like=-21.31..-20.47 [-21.3063..-21.3031]*| it/evals=1728/2719 eff=71.4345% N=300
Z=-27.0(41.72%) | Like=-21.28..-20.47 [-21.2831..-21.2721] | it/evals=1740/2733 eff=71.5166% N=300
Mono-modal Volume: ~exp(-9.58) * 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=-26.9(41.96%) | Like=-21.27..-20.47 [-21.2715..-21.2690]*| it/evals=1742/2736 eff=71.5107% N=300
Z=-26.9(45.38%) | Like=-21.18..-20.47 [-21.1823..-21.1783]*| it/evals=1770/2769 eff=71.6889% N=300
Z=-26.8(48.21%) | Like=-21.11..-20.47 [-21.1121..-21.1077]*| it/evals=1792/2807 eff=71.4799% N=300
Z=-26.8(49.19%) | Like=-21.10..-20.47 [-21.0954..-21.0924]*| it/evals=1800/2816 eff=71.5421% N=300
Mono-modal Volume: ~exp(-9.91) * 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=-26.8(50.34%) | Like=-21.08..-20.47 [-21.0794..-21.0791]*| it/evals=1809/2830 eff=71.5020% N=300
Z=-26.7(52.69%) | Like=-21.04..-20.46 [-21.0430..-21.0425]*| it/evals=1830/2860 eff=71.4844% N=300
Z=-26.7(55.67%) | Like=-20.98..-20.46 [-20.9826..-20.9806]*| it/evals=1856/2897 eff=71.4671% N=300
Z=-26.7(56.09%) | Like=-20.97..-20.46 [-20.9703..-20.9646]*| it/evals=1860/2906 eff=71.3738% N=300
Mono-modal Volume: ~exp(-10.24) * 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(57.86%) | Like=-20.94..-20.46 [-20.9408..-20.9407]*| it/evals=1876/2932 eff=71.2766% N=300
Z=-26.6(59.37%) | Like=-20.92..-20.46 [-20.9248..-20.9241]*| it/evals=1890/2951 eff=71.2939% N=300
Z=-26.6(62.30%) | Like=-20.88..-20.46 [-20.8825..-20.8817]*| it/evals=1918/2986 eff=71.4073% N=300
Z=-26.6(62.50%) | Like=-20.88..-20.46 [-20.8800..-20.8740]*| it/evals=1920/2988 eff=71.4286% N=300
Mono-modal Volume: ~exp(-10.38) * 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=-26.5(64.74%) | Like=-20.85..-20.46 [-20.8534..-20.8517]*| it/evals=1943/3018 eff=71.4864% N=300
Z=-26.5(65.40%) | Like=-20.84..-20.46 [-20.8448..-20.8437]*| it/evals=1950/3026 eff=71.5334% N=300
Z=-26.5(67.87%) | Like=-20.81..-20.46 [-20.8112..-20.8075]*| it/evals=1977/3061 eff=71.6045% N=300
Z=-26.5(68.14%) | Like=-20.81..-20.46 [-20.8064..-20.8031]*| it/evals=1980/3064 eff=71.6353% N=300
[ultranest] Explored until L=-2e+01
[ultranest] Likelihood function evaluations: 3092
[ultranest] logZ = -26.13 +- 0.1114
[ultranest] Effective samples strategy satisfied (ESS = 990.5, 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=0.28, need <0.5)
[ultranest] logZ error budget: single: 0.12 bs:0.11 tail:0.26 total:0.28 required:<0.50
[ultranest] done iterating.
logZ = -26.081 +- 0.315
single instance: logZ = -26.081 +- 0.125
bootstrapped : logZ = -26.132 +- 0.174
tail : logZ = +- 0.262
insert order U test : converged: True correlation: inf iterations
index : 2.16 │ ▁▁▁▁▁▁▂▂▂▃▄▅▆▅▆▆▇▇▅▃▅▄▃▂▂▂▁▁▁▁▁▁▁▁ ▁▁ │3.09 2.57 +- 0.13
amplitude : 0.0000000000210│ ▁ ▁▁▁▁▁▂▂▃▄▄▅▅▇▇▇▇▇▇▅▅▄▃▂▂▂▂▁▁▁▁▁▁▁▁▁ │0.0000000000492 0.0000000000340 +- 0.0000000000037
[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=-665.77..-21.05 [-665.7677..-122.3421] | it/evals=0/301 eff=0.0000% N=300
Z=-216.4(0.00%) | Like=-211.57..-21.05 [-665.7677..-122.3421] | it/evals=30/332 eff=93.7500% N=300
Z=-203.9(0.00%) | Like=-199.17..-21.05 [-665.7677..-122.3421] | it/evals=60/366 eff=90.9091% 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=-192.3(0.00%) | Like=-187.31..-21.05 [-665.7677..-122.3421] | it/evals=85/398 eff=86.7347% N=300
Z=-189.9(0.00%) | Like=-184.51..-21.05 [-665.7677..-122.3421] | it/evals=90/403 eff=87.3786% N=300
Z=-171.6(0.00%) | Like=-165.70..-21.05 [-665.7677..-122.3421] | it/evals=120/435 eff=88.8889% N=300
Mono-modal Volume: ~exp(-4.48) * Expected Volume: exp(-0.45) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|************************************ ******* **| +1.0e-10
Z=-164.5(0.00%) | Like=-159.14..-21.05 [-665.7677..-122.3421] | it/evals=134/452 eff=88.1579% N=300
Z=-154.7(0.00%) | Like=-149.44..-21.05 [-665.7677..-122.3421] | it/evals=150/473 eff=86.7052% N=300
Z=-142.0(0.00%) | Like=-136.68..-20.17 [-665.7677..-122.3421] | it/evals=180/508 eff=86.5385% N=300
Mono-modal Volume: ~exp(-4.48) Expected Volume: exp(-0.67) Quality: ok
index : +1.0| *********************************************| +5.0
amplitude: +1.0e-12| **********************************************| +1.0e-10
Z=-130.0(0.00%) | Like=-123.25..-20.17 [-665.7677..-122.3421] | it/evals=206/540 eff=85.8333% N=300
Z=-127.8(0.00%) | Like=-122.62..-20.17 [-665.7677..-122.3421] | it/evals=210/546 eff=85.3659% N=300
Z=-116.8(0.00%) | Like=-111.18..-20.17 [-122.2976..-61.9589] | it/evals=240/580 eff=85.7143% N=300
Mono-modal Volume: ~exp(-4.54) * Expected Volume: exp(-0.89) Quality: ok
index : +1.0| *******************************************| +5.0
amplitude: +1.0e-12| *********************************************| +1.0e-10
Z=-107.3(0.00%) | Like=-101.10..-19.45 [-122.2976..-61.9589] | it/evals=268/613 eff=85.6230% N=300
Z=-106.4(0.00%) | Like=-100.85..-19.45 [-122.2976..-61.9589] | it/evals=270/617 eff=85.1735% N=300
Z=-98.2(0.00%) | Like=-92.40..-19.45 [-122.2976..-61.9589] | it/evals=297/653 eff=84.1360% N=300
Z=-97.2(0.00%) | Like=-91.42..-19.45 [-122.2976..-61.9589] | it/evals=300/656 eff=84.2697% N=300
Z=-89.3(0.00%) | Like=-83.51..-19.45 [-122.2976..-61.9589] | it/evals=328/691 eff=83.8875% N=300
Z=-88.7(0.00%) | Like=-83.16..-19.45 [-122.2976..-61.9589] | it/evals=330/693 eff=83.9695% N=300
Mono-modal Volume: ~exp(-4.70) * Expected Volume: exp(-1.12) Quality: ok
index : +1.0| ******************************************| +5.0
amplitude: +1.0e-12| *******************************************| +1.0e-10
Z=-86.9(0.00%) | Like=-80.69..-19.45 [-122.2976..-61.9589] | it/evals=335/699 eff=83.9599% N=300
Z=-81.6(0.00%) | Like=-75.46..-19.45 [-122.2976..-61.9589] | it/evals=360/726 eff=84.5070% N=300
Z=-74.6(0.00%) | Like=-68.83..-19.35 [-122.2976..-61.9589] | it/evals=388/761 eff=84.1649% N=300
Z=-74.1(0.00%) | Like=-67.92..-19.35 [-122.2976..-61.9589] | it/evals=390/763 eff=84.2333% N=300
Mono-modal Volume: ~exp(-5.33) * Expected Volume: exp(-1.34) Quality: ok
index : +1.0| ****************************************| +5.0
amplitude: +1.0e-12| ******************************************| +1.0e-10
Z=-70.7(0.00%) | Like=-65.22..-19.35 [-122.2976..-61.9589] | it/evals=402/779 eff=83.9248% N=300
Z=-67.3(0.00%) | Like=-61.42..-19.35 [-61.8823..-41.3967] | it/evals=420/801 eff=83.8323% N=300
Z=-62.7(0.00%) | Like=-57.12..-19.35 [-61.8823..-41.3967] | it/evals=448/839 eff=83.1169% N=300
Z=-62.3(0.00%) | Like=-57.00..-19.35 [-61.8823..-41.3967] | it/evals=450/841 eff=83.1793% N=300
Mono-modal Volume: ~exp(-5.33) Expected Volume: exp(-1.56) Quality: ok
index : +1.0| **************************************| +5.0
amplitude: +1.0e-12| *************************************** | +1.0e-10
Z=-59.1(0.00%) | Like=-54.11..-19.35 [-61.8823..-41.3967] | it/evals=475/874 eff=82.7526% N=300
Z=-58.6(0.00%) | Like=-53.50..-19.35 [-61.8823..-41.3967] | it/evals=480/880 eff=82.7586% N=300
Z=-56.3(0.00%) | Like=-51.07..-19.35 [-61.8823..-41.3967] | it/evals=502/916 eff=81.4935% N=300
Z=-55.5(0.00%) | Like=-50.19..-19.35 [-61.8823..-41.3967] | it/evals=510/929 eff=81.0811% N=300
Z=-53.2(0.00%) | Like=-47.95..-19.34 [-61.8823..-41.3967] | it/evals=535/965 eff=80.4511% N=300
Mono-modal Volume: ~exp(-5.93) * Expected Volume: exp(-1.79) Quality: ok
index : +1.0| ********************************** | +5.0
amplitude: +1.0e-12| ************************************* | +1.0e-10
Z=-53.1(0.00%) | Like=-47.76..-19.34 [-61.8823..-41.3967] | it/evals=536/967 eff=80.3598% N=300
Z=-52.6(0.00%) | Like=-47.27..-19.34 [-61.8823..-41.3967] | it/evals=540/974 eff=80.1187% N=300
Z=-49.7(0.00%) | Like=-44.31..-19.24 [-61.8823..-41.3967] | it/evals=570/1012 eff=80.0562% N=300
Z=-48.0(0.00%) | Like=-43.07..-19.24 [-61.8823..-41.3967] | it/evals=595/1048 eff=79.5455% N=300
Z=-47.7(0.00%) | Like=-42.70..-19.24 [-61.8823..-41.3967] | it/evals=600/1054 eff=79.5756% N=300
Mono-modal Volume: ~exp(-5.93) Expected Volume: exp(-2.01) Quality: ok
index : +1.0| +1.9 ****************************** | +5.0
amplitude: +1.0e-12| +2.5e-11 ************************************ | +1.0e-10
Z=-46.4(0.00%) | Like=-41.42..-19.24 [-61.8823..-41.3967] | it/evals=619/1091 eff=78.2554% N=300
Z=-45.8(0.00%) | Like=-40.71..-19.24 [-41.3750..-31.3950] | it/evals=630/1107 eff=78.0669% N=300
Z=-44.0(0.00%) | Like=-38.98..-19.24 [-41.3750..-31.3950] | it/evals=656/1144 eff=77.7251% N=300
Z=-43.8(0.00%) | Like=-38.55..-19.24 [-41.3750..-31.3950] | it/evals=660/1148 eff=77.8302% N=300
Mono-modal Volume: ~exp(-6.17) * Expected Volume: exp(-2.23) Quality: ok
index : +1.0| +2.1 ************************** +4.1 | +5.0
amplitude: +1.0e-12| +2.7e-11 *********************************** | +1.0e-10
Z=-43.1(0.00%) | Like=-37.92..-19.24 [-41.3750..-31.3950] | it/evals=670/1162 eff=77.7262% N=300
Z=-42.0(0.00%) | Like=-36.97..-19.24 [-41.3750..-31.3950] | it/evals=690/1193 eff=77.2676% N=300
Z=-40.8(0.00%) | Like=-35.71..-19.24 [-41.3750..-31.3950] | it/evals=714/1229 eff=76.8568% N=300
Z=-40.5(0.00%) | Like=-35.24..-19.24 [-41.3750..-31.3950] | it/evals=720/1239 eff=76.6773% N=300
Mono-modal Volume: ~exp(-6.62) * Expected Volume: exp(-2.46) Quality: ok
index : +1.0| +2.1 *********************** +4.0 | +5.0
amplitude: +1.0e-12| +2.9e-11 ******************************* | +1.0e-10
Z=-39.6(0.00%) | Like=-34.35..-19.24 [-41.3750..-31.3950] | it/evals=737/1266 eff=76.2940% N=300
Z=-38.9(0.00%) | Like=-33.59..-19.24 [-41.3750..-31.3950] | it/evals=750/1289 eff=75.8342% N=300
Z=-37.7(0.00%) | Like=-32.47..-19.24 [-41.3750..-31.3950] | it/evals=775/1324 eff=75.6836% N=300
Z=-37.5(0.00%) | Like=-32.18..-19.24 [-41.3750..-31.3950] | it/evals=780/1336 eff=75.2896% N=300
Mono-modal Volume: ~exp(-6.62) Expected Volume: exp(-2.68) Quality: ok
index : +1.0| +2.1 ********************* +3.8 | +5.0
amplitude: +1.0e-12| +3.1e-11 *************************** * | +1.0e-10
Z=-36.5(0.00%) | Like=-31.40..-19.24 [-41.3750..-31.3950] | it/evals=804/1368 eff=75.2809% N=300
Z=-36.3(0.00%) | Like=-31.34..-19.19 [-31.3866..-26.2071] | it/evals=810/1380 eff=75.0000% N=300
Z=-35.5(0.00%) | Like=-30.56..-19.19 [-31.3866..-26.2071] | it/evals=837/1417 eff=74.9329% N=300
Z=-35.4(0.00%) | Like=-30.39..-19.19 [-31.3866..-26.2071] | it/evals=840/1420 eff=75.0000% N=300
Z=-34.7(0.00%) | Like=-29.49..-19.19 [-31.3866..-26.2071] | it/evals=863/1455 eff=74.7186% N=300
Z=-34.4(0.00%) | Like=-29.25..-19.19 [-31.3866..-26.2071] | it/evals=870/1470 eff=74.3590% N=300
Mono-modal Volume: ~exp(-6.94) * Expected Volume: exp(-2.90) Quality: ok
index : +1.0| +2.2 ****************** +3.6 | +5.0
amplitude: +1.0e-12| +3.3e-11 ************************** | +1.0e-10
Z=-34.4(0.00%) | Like=-29.21..-19.19 [-31.3866..-26.2071] | it/evals=871/1471 eff=74.3809% N=300
Z=-33.4(0.01%) | Like=-28.24..-19.19 [-31.3866..-26.2071] | it/evals=900/1506 eff=74.6269% N=300
Z=-32.6(0.03%) | Like=-27.30..-19.19 [-31.3866..-26.2071] | it/evals=927/1541 eff=74.6978% N=300
Z=-32.5(0.03%) | Like=-27.20..-19.19 [-31.3866..-26.2071] | it/evals=930/1545 eff=74.6988% N=300
Mono-modal Volume: ~exp(-7.17) * 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=-32.3(0.04%) | Like=-26.97..-19.19 [-31.3866..-26.2071] | it/evals=938/1558 eff=74.5628% N=300
Z=-31.7(0.07%) | Like=-26.53..-19.16 [-31.3866..-26.2071] | it/evals=960/1586 eff=74.6501% N=300
Z=-31.2(0.12%) | Like=-26.14..-19.16 [-26.1763..-25.6417] | it/evals=985/1624 eff=74.3958% N=300
Z=-31.1(0.13%) | Like=-26.02..-19.16 [-26.1763..-25.6417] | it/evals=990/1632 eff=74.3243% N=300
Mono-modal Volume: ~exp(-7.56) * Expected Volume: exp(-3.35) 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=-30.8(0.17%) | Like=-25.65..-19.16 [-26.1763..-25.6417] | it/evals=1005/1652 eff=74.3343% N=300
Z=-30.5(0.23%) | Like=-25.32..-19.16 [-25.3669..-25.3198] | it/evals=1020/1675 eff=74.1818% N=300
Z=-29.9(0.40%) | Like=-24.59..-19.16 [-24.6093..-24.5931] | it/evals=1049/1709 eff=74.4500% N=300
Z=-29.9(0.41%) | Like=-24.57..-19.16 [-24.5670..-24.5571]*| it/evals=1050/1710 eff=74.4681% N=300
Mono-modal Volume: ~exp(-7.56) Expected Volume: exp(-3.57) Quality: ok
index : +1.0| +2.4 ************* +3.4 | +5.0
amplitude: +1.0e-12| +3.9e-11 ******************* +7.6e-11 | +1.0e-10
Z=-29.5(0.65%) | Like=-24.15..-19.16 [-24.1478..-24.1233] | it/evals=1073/1744 eff=74.3075% N=300
Z=-29.3(0.73%) | Like=-24.04..-19.16 [-24.0364..-24.0269]*| it/evals=1080/1754 eff=74.2779% N=300
Z=-28.9(1.12%) | Like=-23.73..-19.16 [-23.7674..-23.7334] | it/evals=1105/1789 eff=74.2109% N=300
Z=-28.9(1.20%) | Like=-23.66..-19.16 [-23.6769..-23.6608] | it/evals=1110/1796 eff=74.1979% N=300
Z=-28.5(1.61%) | Like=-23.33..-19.16 [-23.3336..-23.3265]*| it/evals=1133/1831 eff=74.0039% N=300
Mono-modal Volume: ~exp(-7.67) * Expected Volume: exp(-3.80) Quality: ok
index : +1.0| +2.4 ************ +3.3 | +5.0
amplitude: +1.0e-12| +4.0e-11 ***************** +7.3e-11 | +1.0e-10
Z=-28.4(1.74%) | Like=-23.28..-19.16 [-23.2943..-23.2828] | it/evals=1139/1841 eff=73.9130% N=300
Z=-28.4(1.77%) | Like=-23.28..-19.16 [-23.2794..-23.2584] | it/evals=1140/1842 eff=73.9300% N=300
Z=-28.1(2.49%) | Like=-22.93..-19.16 [-22.9258..-22.9085] | it/evals=1169/1877 eff=74.1281% N=300
Z=-28.1(2.52%) | Like=-22.91..-19.16 [-22.9258..-22.9085] | it/evals=1170/1878 eff=74.1445% N=300
Z=-27.8(3.46%) | Like=-22.55..-19.16 [-22.5703..-22.5480] | it/evals=1196/1913 eff=74.1476% N=300
Z=-27.7(3.63%) | Like=-22.53..-19.16 [-22.5347..-22.5252]*| it/evals=1200/1924 eff=73.8916% N=300
Mono-modal Volume: ~exp(-8.03) * Expected Volume: exp(-4.02) Quality: ok
index : +1.0| +2.4 *********** +3.3 | +5.0
amplitude: +1.0e-12| +4.2e-11 *************** +7.1e-11 | +1.0e-10
Z=-27.6(3.90%) | Like=-22.46..-19.16 [-22.4567..-22.4564]*| it/evals=1206/1932 eff=73.8971% N=300
Z=-27.4(4.91%) | Like=-22.21..-19.16 [-22.2204..-22.2088] | it/evals=1230/1959 eff=74.1410% N=300
Z=-27.2(6.20%) | Like=-21.97..-19.16 [-21.9708..-21.9563] | it/evals=1255/1994 eff=74.0850% N=300
Z=-27.1(6.44%) | Like=-21.92..-19.16 [-21.9223..-21.9215]*| it/evals=1260/2000 eff=74.1176% N=300
Mono-modal Volume: ~exp(-8.29) * Expected Volume: exp(-4.24) 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=-27.0(7.24%) | Like=-21.84..-19.16 [-21.8597..-21.8359] | it/evals=1273/2017 eff=74.1409% N=300
Z=-26.9(8.39%) | Like=-21.71..-19.16 [-21.7052..-21.6965]*| it/evals=1290/2040 eff=74.1379% N=300
Z=-26.7(10.40%) | Like=-21.50..-19.16 [-21.5049..-21.5046]*| it/evals=1317/2075 eff=74.1972% N=300
Z=-26.6(10.68%) | Like=-21.48..-19.16 [-21.4805..-21.4789]*| it/evals=1320/2081 eff=74.1157% N=300
Mono-modal Volume: ~exp(-8.29) Expected Volume: exp(-4.47) Quality: ok
index : +1.0| +2.5 ********** +3.2 | +5.0
amplitude: +1.0e-12| +4.4e-11 ************* +6.8e-11 | +1.0e-10
Z=-26.5(12.32%) | Like=-21.36..-19.16 [-21.3553..-21.3515]*| it/evals=1341/2113 eff=73.9658% N=300
Z=-26.4(13.09%) | Like=-21.30..-19.16 [-21.3128..-21.2997] | it/evals=1350/2127 eff=73.8916% N=300
Z=-26.3(15.20%) | Like=-21.14..-19.16 [-21.1449..-21.1391]*| it/evals=1374/2162 eff=73.7916% N=300
Z=-26.3(15.70%) | Like=-21.10..-19.16 [-21.1231..-21.1038] | it/evals=1380/2170 eff=73.7968% N=300
Z=-26.1(17.87%) | Like=-20.95..-19.16 [-20.9524..-20.9507]*| it/evals=1403/2205 eff=73.6483% N=300
Mono-modal Volume: ~exp(-8.68) * Expected Volume: exp(-4.69) Quality: ok
index : +1.0| +2.5 ******** +3.1 | +5.0
amplitude: +1.0e-12| +4.5e-11 *********** +6.7e-11 | +1.0e-10
Z=-26.1(18.25%) | Like=-20.94..-19.16 [-20.9444..-20.9334] | it/evals=1407/2211 eff=73.6264% N=300
Z=-26.1(18.51%) | Like=-20.89..-19.16 [-20.9273..-20.8942] | it/evals=1410/2215 eff=73.6292% N=300
Z=-25.9(21.32%) | Like=-20.74..-19.16 [-20.7403..-20.7316]*| it/evals=1438/2250 eff=73.7436% N=300
Z=-25.9(21.52%) | Like=-20.73..-19.16 [-20.7299..-20.7224]*| it/evals=1440/2252 eff=73.7705% N=300
Z=-25.8(24.16%) | Like=-20.60..-19.16 [-20.6005..-20.5975]*| it/evals=1465/2288 eff=73.6922% N=300
Z=-25.8(24.76%) | Like=-20.58..-19.16 [-20.5767..-20.5704]*| it/evals=1470/2295 eff=73.6842% N=300
Mono-modal Volume: ~exp(-9.04) * Expected Volume: exp(-4.91) Quality: ok
index : +1.0| +2.6 ******** +3.1 | +5.0
amplitude: +1.0e-12| +4.6e-11 *********** +6.6e-11 | +1.0e-10
Z=-25.8(25.33%) | Like=-20.56..-19.16 [-20.5648..-20.5582]*| it/evals=1474/2302 eff=73.6264% N=300
Z=-25.7(28.18%) | Like=-20.47..-19.16 [-20.4664..-20.4574]*| it/evals=1500/2334 eff=73.7463% N=300
Z=-25.6(31.33%) | Like=-20.33..-19.16 [-20.3267..-20.3236]*| it/evals=1528/2369 eff=73.8521% N=300
Z=-25.5(31.61%) | Like=-20.32..-19.16 [-20.3190..-20.3179]*| it/evals=1530/2373 eff=73.8061% N=300
Mono-modal Volume: ~exp(-9.21) * Expected Volume: exp(-5.14) Quality: ok
index : +1.0| +2.6 ******* +3.1 | +5.0
amplitude: +1.0e-12| +4.7e-11 ********* +6.3e-11 | +1.0e-10
Z=-25.5(32.97%) | Like=-20.26..-19.16 [-20.2633..-20.2625]*| it/evals=1541/2391 eff=73.6968% N=300
Z=-25.4(35.27%) | Like=-20.20..-19.16 [-20.2017..-20.1965]*| it/evals=1560/2417 eff=73.6892% N=300
Z=-25.4(38.23%) | Like=-20.11..-19.16 [-20.1139..-20.1115]*| it/evals=1585/2452 eff=73.6524% N=300
Z=-25.3(38.75%) | Like=-20.10..-19.16 [-20.1023..-20.0942]*| it/evals=1590/2457 eff=73.7135% N=300
Mono-modal Volume: ~exp(-9.21) 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(41.02%) | Like=-20.05..-19.16 [-20.0511..-20.0486]*| it/evals=1608/2489 eff=73.4582% N=300
Z=-25.2(42.51%) | Like=-20.01..-19.16 [-20.0219..-20.0118] | it/evals=1620/2503 eff=73.5361% N=300
Z=-25.2(45.57%) | Like=-19.92..-19.16 [-19.9373..-19.9249] | it/evals=1645/2538 eff=73.5031% N=300
Z=-25.2(46.19%) | Like=-19.91..-19.16 [-19.9120..-19.9032]*| it/evals=1650/2546 eff=73.4639% N=300
Z=-25.1(48.72%) | Like=-19.86..-19.16 [-19.8552..-19.8537]*| it/evals=1673/2581 eff=73.3450% N=300
Mono-modal Volume: ~exp(-9.57) * 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(48.93%) | Like=-19.85..-19.16 [-19.8531..-19.8520]*| it/evals=1675/2583 eff=73.3684% N=300
Z=-25.1(49.54%) | Like=-19.84..-19.16 [-19.8405..-19.8386]*| it/evals=1680/2590 eff=73.3624% N=300
Z=-25.0(52.33%) | Like=-19.80..-19.16 [-19.8034..-19.8020]*| it/evals=1704/2625 eff=73.2903% N=300
Z=-25.0(52.96%) | Like=-19.79..-19.16 [-19.7924..-19.7917]*| it/evals=1710/2634 eff=73.2648% N=300
Z=-25.0(55.93%) | Like=-19.71..-19.16 [-19.7129..-19.7128]*| it/evals=1738/2669 eff=73.3643% N=300
Z=-25.0(56.13%) | Like=-19.71..-19.16 [-19.7111..-19.7108]*| it/evals=1740/2671 eff=73.3868% N=300
Mono-modal Volume: ~exp(-9.62) * Expected Volume: exp(-5.81) Quality: ok
index : +1.0| +2.6 ***** +3.0 | +5.0
amplitude: +1.0e-12| +4.9e-11 ******* +6.1e-11 | +1.0e-10
Z=-25.0(56.36%) | Like=-19.71..-19.16 [-19.7107..-19.7076]*| it/evals=1742/2673 eff=73.4092% N=300
Z=-24.9(59.19%) | Like=-19.66..-19.16 [-19.6554..-19.6547]*| it/evals=1768/2710 eff=73.3610% N=300
Z=-24.9(59.40%) | Like=-19.65..-19.16 [-19.6541..-19.6536]*| it/evals=1770/2713 eff=73.3527% N=300
Z=-24.9(61.70%) | Like=-19.62..-19.16 [-19.6200..-19.6115]*| it/evals=1793/2751 eff=73.1538% N=300
Z=-24.9(62.40%) | Like=-19.60..-19.16 [-19.5969..-19.5968]*| it/evals=1800/2761 eff=73.1410% N=300
Mono-modal Volume: ~exp(-10.27) * Expected Volume: exp(-6.03) Quality: ok
index : +1.0| +2.7 ***** +3.0 | +5.0
amplitude: +1.0e-12| +5.0e-11 ****** +6.1e-11 | +1.0e-10
Z=-24.8(63.25%) | Like=-19.59..-19.16 [-19.5878..-19.5871]*| it/evals=1809/2776 eff=73.0614% N=300
Z=-24.8(65.23%) | Like=-19.55..-19.16 [-19.5543..-19.5528]*| it/evals=1830/2802 eff=73.1415% N=300
Z=-24.8(67.42%) | Like=-19.52..-19.16 [-19.5158..-19.5158]*| it/evals=1854/2838 eff=73.0496% N=300
Z=-24.8(67.97%) | Like=-19.51..-19.16 [-19.5121..-19.5077]*| it/evals=1860/2844 eff=73.1132% N=300
Mono-modal Volume: ~exp(-10.33) * Expected Volume: exp(-6.25) Quality: ok
index : +1.0| +2.7 **** +2.9 | +5.0
amplitude: +1.0e-12| +5.1e-11 ***** +6.0e-11 | +1.0e-10
Z=-24.8(69.38%) | Like=-19.48..-19.16 [-19.4834..-19.4813]*| it/evals=1876/2863 eff=73.1955% N=300
[ultranest] Explored until L=-2e+01
[ultranest] Likelihood function evaluations: 2871
[ultranest] logZ = -24.38 +- 0.08883
[ultranest] Effective samples strategy satisfied (ESS = 1010.0, 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=0.28, 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 = -24.389 +- 0.307
single instance: logZ = -24.389 +- 0.119
bootstrapped : logZ = -24.382 +- 0.159
tail : logZ = +- 0.262
insert order U test : converged: True correlation: inf iterations
index : 2.24 │ ▁ ▁ ▁▁▁▂▂▃▃▄▆▅▆▇▆▇▇▇▅▅▄▃▃▂▂▁▁▁▁▁▁▁▁▁▁ │3.46 2.82 +- 0.17
amplitude : 0.0000000000345│ ▁ ▁▁▁▁▂▂▂▄▄▄▅▇▇▆▆▇▆▆▄▃▃▂▂▂▁▁▁▁▁▁ ▁ ▁▁ │0.0000000000805 0.0000000000549 +- 0.0000000000059
[ultranest] Sampling 300 live points from prior ...
Mono-modal Volume: ~exp(-4.17) * Expected Volume: exp(0.00) Quality: ok
index : +1.0|************************************************| +5.0
amplitude: +1.0e-12|******************* ************* ************ *| +1.0e-10
Z=-inf(0.00%) | Like=-980.72..-13.44 [-980.7186..-85.9354] | it/evals=0/301 eff=0.0000% N=300
Z=-153.4(0.00%) | Like=-148.48..-13.44 [-980.7186..-85.9354] | it/evals=30/333 eff=90.9091% N=300
Z=-141.4(0.00%) | Like=-136.57..-13.44 [-980.7186..-85.9354] | it/evals=60/365 eff=92.3077% 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=-134.1(0.00%) | Like=-129.53..-13.44 [-980.7186..-85.9354] | it/evals=87/397 eff=89.6907% N=300
Z=-133.3(0.00%) | Like=-128.57..-13.44 [-980.7186..-85.9354] | it/evals=90/401 eff=89.1089% N=300
Z=-121.9(0.00%) | Like=-117.50..-13.44 [-980.7186..-85.9354] | it/evals=118/437 eff=86.1314% N=300
Z=-121.6(0.00%) | Like=-117.35..-13.44 [-980.7186..-85.9354] | it/evals=120/439 eff=86.3309% N=300
Mono-modal Volume: ~exp(-4.17) Expected Volume: exp(-0.45) Quality: ok
index : +1.0| ***********************************************| +5.0
amplitude: +1.0e-12|********************************* ************ *| +1.0e-10
Z=-114.4(0.00%) | Like=-109.32..-13.42 [-980.7186..-85.9354] | it/evals=149/471 eff=87.1345% N=300
Z=-114.1(0.00%) | Like=-109.29..-13.42 [-980.7186..-85.9354] | it/evals=150/472 eff=87.2093% N=300
Z=-102.6(0.00%) | Like=-97.26..-13.42 [-980.7186..-85.9354] | it/evals=176/507 eff=85.0242% N=300
Z=-100.8(0.00%) | Like=-95.26..-13.42 [-980.7186..-85.9354] | it/evals=180/511 eff=85.3081% N=300
Mono-modal Volume: ~exp(-4.75) * Expected Volume: exp(-0.67) Quality: ok
index : +1.0| *********************************************| +5.0
amplitude: +1.0e-12| ******************************** ************ *| +1.0e-10
Z=-95.3(0.00%) | Like=-90.71..-13.42 [-980.7186..-85.9354] | it/evals=201/545 eff=82.0408% N=300
Z=-92.6(0.00%) | Like=-87.10..-13.42 [-980.7186..-85.9354] | it/evals=210/554 eff=82.6772% N=300
Z=-85.7(0.00%) | Like=-80.59..-13.42 [-85.9207..-42.8796] | it/evals=231/589 eff=79.9308% N=300
Z=-83.3(0.00%) | Like=-78.40..-13.42 [-85.9207..-42.8796] | it/evals=240/598 eff=80.5369% N=300
Z=-76.0(0.00%) | Like=-71.09..-13.42 [-85.9207..-42.8796] | it/evals=267/633 eff=80.1802% N=300
Mono-modal Volume: ~exp(-4.84) * Expected Volume: exp(-0.89) Quality: ok
index : +1.0| ********************************************| +5.0
amplitude: +1.0e-12| ******************************* **************| +1.0e-10
Z=-75.9(0.00%) | Like=-70.75..-13.42 [-85.9207..-42.8796] | it/evals=268/634 eff=80.2395% N=300
Z=-75.5(0.00%) | Like=-70.11..-13.42 [-85.9207..-42.8796] | it/evals=270/638 eff=79.8817% N=300
Z=-68.1(0.00%) | Like=-63.37..-13.42 [-85.9207..-42.8796] | it/evals=300/673 eff=80.4290% N=300
Z=-64.1(0.00%) | Like=-58.67..-13.42 [-85.9207..-42.8796] | it/evals=326/709 eff=79.7066% N=300
Z=-63.3(0.00%) | Like=-58.24..-13.42 [-85.9207..-42.8796] | it/evals=330/713 eff=79.9031% N=300
Mono-modal Volume: ~exp(-4.84) Expected Volume: exp(-1.12) Quality: ok
index : +1.0| ******************************************| +5.0
amplitude: +1.0e-12| *********************************************| +1.0e-10
Z=-59.1(0.00%) | Like=-54.22..-13.42 [-85.9207..-42.8796] | it/evals=351/746 eff=78.6996% N=300
Z=-58.0(0.00%) | Like=-52.84..-13.42 [-85.9207..-42.8796] | it/evals=360/757 eff=78.7746% N=300
Z=-53.9(0.00%) | Like=-48.17..-13.42 [-85.9207..-42.8796] | it/evals=389/796 eff=78.4274% N=300
Z=-53.7(0.00%) | Like=-48.13..-13.42 [-85.9207..-42.8796] | it/evals=390/797 eff=78.4708% N=300
Mono-modal Volume: ~exp(-5.16) * Expected Volume: exp(-1.34) Quality: ok
index : +1.0| * ***************************************| +5.0
amplitude: +1.0e-12| ******************************************| +1.0e-10
Z=-52.0(0.00%) | Like=-47.23..-13.42 [-85.9207..-42.8796] | it/evals=402/813 eff=78.3626% N=300
Z=-49.6(0.00%) | Like=-44.44..-13.42 [-85.9207..-42.8796] | it/evals=420/837 eff=78.2123% N=300
Z=-45.9(0.00%) | Like=-40.80..-13.42 [-42.8767..-27.5593] | it/evals=448/872 eff=78.3217% N=300
Z=-45.6(0.00%) | Like=-40.49..-13.42 [-42.8767..-27.5593] | it/evals=450/877 eff=77.9896% N=300
Mono-modal Volume: ~exp(-5.34) * Expected Volume: exp(-1.56) Quality: ok
index : +1.0| ********************************* | +5.0
amplitude: +1.0e-12| *********************************** *****| +1.0e-10
Z=-43.7(0.00%) | Like=-39.02..-13.42 [-42.8767..-27.5593] | it/evals=469/906 eff=77.3927% N=300
Z=-42.8(0.00%) | Like=-37.73..-13.36 [-42.8767..-27.5593] | it/evals=480/918 eff=77.6699% N=300
Z=-40.1(0.00%) | Like=-35.37..-13.30 [-42.8767..-27.5593] | it/evals=507/953 eff=77.6417% N=300
Z=-39.8(0.00%) | Like=-34.85..-13.30 [-42.8767..-27.5593] | it/evals=510/956 eff=77.7439% N=300
Z=-38.2(0.00%) | Like=-33.41..-13.30 [-42.8767..-27.5593] | it/evals=530/993 eff=76.4791% N=300
Mono-modal Volume: ~exp(-5.83) * Expected Volume: exp(-1.79) Quality: ok
index : +1.0| **************************** +4.2 | +5.0
amplitude: +1.0e-12| ****************************************| +1.0e-10
Z=-37.8(0.00%) | Like=-33.16..-13.30 [-42.8767..-27.5593] | it/evals=536/1003 eff=76.2447% N=300
Z=-37.6(0.00%) | Like=-32.86..-13.30 [-42.8767..-27.5593] | it/evals=540/1009 eff=76.1636% N=300
Z=-35.9(0.00%) | Like=-31.15..-13.30 [-42.8767..-27.5593] | it/evals=568/1045 eff=76.2416% N=300
Z=-35.8(0.00%) | Like=-30.95..-13.30 [-42.8767..-27.5593] | it/evals=570/1047 eff=76.3052% N=300
Z=-34.2(0.00%) | Like=-29.31..-13.30 [-42.8767..-27.5593] | it/evals=595/1084 eff=75.8929% N=300
Z=-33.9(0.00%) | Like=-28.93..-13.30 [-42.8767..-27.5593] | it/evals=600/1090 eff=75.9494% N=300
Mono-modal Volume: ~exp(-5.83) Expected Volume: exp(-2.01) Quality: ok
index : +1.0| +2.0 ************************ +3.9 | +5.0
amplitude: +1.0e-12| ********************************* *****| +1.0e-10
Z=-32.6(0.00%) | Like=-27.81..-13.30 [-42.8767..-27.5593] | it/evals=624/1122 eff=75.9124% N=300
Z=-32.3(0.00%) | Like=-27.41..-13.30 [-27.5529..-20.4068] | it/evals=630/1133 eff=75.6303% N=300
Z=-31.2(0.00%) | Like=-26.33..-13.30 [-27.5529..-20.4068] | it/evals=652/1169 eff=75.0288% N=300
Z=-30.8(0.00%) | Like=-26.13..-13.30 [-27.5529..-20.4068] | it/evals=660/1180 eff=75.0000% N=300
Mono-modal Volume: ~exp(-6.23) * Expected Volume: exp(-2.23) Quality: ok
index : +1.0| +2.0 ********************* +3.7 | +5.0
amplitude: +1.0e-12| +2.4e-11 *************************************| +1.0e-10
Z=-30.4(0.00%) | Like=-25.70..-13.30 [-27.5529..-20.4068] | it/evals=670/1193 eff=75.0280% N=300
Z=-29.6(0.00%) | Like=-24.79..-13.30 [-27.5529..-20.4068] | it/evals=690/1217 eff=75.2454% N=300
Z=-28.5(0.00%) | Like=-23.72..-13.30 [-27.5529..-20.4068] | it/evals=718/1253 eff=75.3410% N=300
Z=-28.4(0.00%) | Like=-23.60..-13.30 [-27.5529..-20.4068] | it/evals=720/1255 eff=75.3927% N=300
Mono-modal Volume: ~exp(-6.79) * Expected Volume: exp(-2.46) Quality: ok
index : +1.0| +2.1 ******************* +3.6 | +5.0
amplitude: +1.0e-12| +2.6e-11 ********************************** | +1.0e-10
Z=-27.8(0.01%) | Like=-23.16..-13.30 [-27.5529..-20.4068] | it/evals=737/1278 eff=75.3579% N=300
Z=-27.4(0.01%) | Like=-22.66..-13.30 [-27.5529..-20.4068] | it/evals=750/1293 eff=75.5287% N=300
Z=-26.7(0.02%) | Like=-21.75..-13.30 [-27.5529..-20.4068] | it/evals=772/1329 eff=75.0243% N=300
Z=-26.4(0.02%) | Like=-21.41..-13.30 [-27.5529..-20.4068] | it/evals=780/1340 eff=75.0000% N=300
Z=-25.8(0.04%) | Like=-21.10..-13.30 [-27.5529..-20.4068] | it/evals=799/1377 eff=74.1876% N=300
Mono-modal Volume: ~exp(-6.84) * Expected Volume: exp(-2.68) Quality: ok
index : +1.0| +2.2 **************** +3.5 | +5.0
amplitude: +1.0e-12| +2.9e-11 ***************************** | +1.0e-10
Z=-25.7(0.05%) | Like=-20.95..-13.30 [-27.5529..-20.4068] | it/evals=804/1383 eff=74.2382% N=300
Z=-25.5(0.05%) | Like=-20.82..-13.30 [-27.5529..-20.4068] | it/evals=810/1389 eff=74.3802% N=300
Z=-25.1(0.09%) | Like=-20.38..-13.30 [-20.3760..-19.7283] | it/evals=830/1425 eff=73.7778% N=300
Z=-24.8(0.11%) | Like=-20.00..-13.30 [-20.3760..-19.7283] | it/evals=840/1439 eff=73.7489% N=300
Z=-24.3(0.19%) | Like=-19.61..-13.30 [-19.6369..-19.6071] | it/evals=864/1479 eff=73.2824% N=300
Z=-24.2(0.21%) | Like=-19.57..-13.30 [-19.5668..-19.5075] | it/evals=870/1486 eff=73.3558% N=300
Mono-modal Volume: ~exp(-7.15) * Expected Volume: exp(-2.90) Quality: ok
index : +1.0| +2.2 *************** +3.4 | +5.0
amplitude: +1.0e-12| +3.0e-11 *************************** | +1.0e-10
Z=-24.2(0.21%) | Like=-19.51..-13.30 [-19.5668..-19.5075] | it/evals=871/1487 eff=73.3783% N=300
Z=-23.7(0.34%) | Like=-19.13..-13.30 [-19.1403..-19.1278] | it/evals=897/1522 eff=73.4043% N=300
Z=-23.6(0.35%) | Like=-19.09..-13.30 [-19.0904..-19.0896]*| it/evals=900/1525 eff=73.4694% N=300
Z=-23.2(0.55%) | Like=-18.67..-13.30 [-18.6973..-18.6721] | it/evals=929/1560 eff=73.7302% N=300
Z=-23.2(0.56%) | Like=-18.65..-13.30 [-18.6504..-18.6465]*| it/evals=930/1561 eff=73.7510% N=300
Mono-modal Volume: ~exp(-7.31) * Expected Volume: exp(-3.13) Quality: ok
index : +1.0| +2.2 *************** +3.3 | +5.0
amplitude: +1.0e-12| +3.2e-11 *********************** ** | +1.0e-10
Z=-23.1(0.61%) | Like=-18.54..-13.30 [-18.5421..-18.5359]*| it/evals=938/1574 eff=73.6264% N=300
Z=-22.8(0.86%) | Like=-18.03..-13.30 [-18.0263..-18.0215]*| it/evals=960/1600 eff=73.8462% N=300
Z=-22.4(1.29%) | Like=-17.63..-13.30 [-17.6762..-17.6256] | it/evals=986/1635 eff=73.8577% N=300
Z=-22.3(1.35%) | Like=-17.52..-13.30 [-17.5248..-17.5029] | it/evals=990/1639 eff=73.9358% N=300
Mono-modal Volume: ~exp(-7.61) * Expected Volume: exp(-3.35) 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=-22.1(1.70%) | Like=-17.30..-13.30 [-17.2982..-17.2874] | it/evals=1005/1662 eff=73.7885% N=300
Z=-21.9(2.11%) | Like=-17.12..-13.30 [-17.1167..-17.1128]*| it/evals=1020/1685 eff=73.6462% N=300
Z=-21.5(2.94%) | Like=-16.79..-13.30 [-16.8037..-16.7884] | it/evals=1049/1720 eff=73.8732% N=300
Z=-21.5(2.98%) | Like=-16.79..-13.30 [-16.7881..-16.7768] | it/evals=1050/1721 eff=73.8916% N=300
Mono-modal Volume: ~exp(-7.75) * Expected Volume: exp(-3.57) Quality: ok
index : +1.0| +2.4 *********** +3.2 | +5.0
amplitude: +1.0e-12| +3.6e-11 ******************** +7.4e-11 | +1.0e-10
Z=-21.3(3.88%) | Like=-16.54..-13.30 [-16.5508..-16.5372] | it/evals=1072/1750 eff=73.9310% N=300
Z=-21.2(4.20%) | Like=-16.45..-13.30 [-16.4609..-16.4482] | it/evals=1080/1760 eff=73.9726% N=300
Z=-20.9(5.40%) | Like=-16.19..-13.30 [-16.2207..-16.1916] | it/evals=1106/1795 eff=73.9799% N=300
Z=-20.9(5.60%) | Like=-16.17..-13.30 [-16.1682..-16.1533] | it/evals=1110/1799 eff=74.0494% N=300
Z=-20.7(6.81%) | Like=-15.97..-13.30 [-15.9858..-15.9678] | it/evals=1131/1836 eff=73.6328% N=300
Mono-modal Volume: ~exp(-8.17) * Expected Volume: exp(-3.80) Quality: ok
index : +1.0| +2.4 ********** +3.1 | +5.0
amplitude: +1.0e-12| +3.7e-11 ****************** +7.3e-11 | +1.0e-10
Z=-20.6(7.31%) | Like=-15.90..-13.30 [-15.8978..-15.8939]*| it/evals=1139/1848 eff=73.5788% N=300
Z=-20.6(7.39%) | Like=-15.89..-13.30 [-15.8939..-15.8918]*| it/evals=1140/1849 eff=73.5959% N=300
Z=-20.4(9.29%) | Like=-15.63..-13.30 [-15.6337..-15.6275]*| it/evals=1168/1884 eff=73.7374% N=300
Z=-20.4(9.45%) | Like=-15.63..-13.30 [-15.6265..-15.6237]*| it/evals=1170/1886 eff=73.7705% N=300
Z=-20.2(11.52%) | Like=-15.41..-13.30 [-15.4086..-15.4031]*| it/evals=1195/1921 eff=73.7199% N=300
Z=-20.2(11.89%) | Like=-15.33..-13.30 [-15.3329..-15.3165] | it/evals=1200/1927 eff=73.7554% N=300
Mono-modal Volume: ~exp(-8.45) * Expected Volume: exp(-4.02) Quality: ok
index : +1.0| +2.4 ********* +3.1 | +5.0
amplitude: +1.0e-12| +3.9e-11 **************** +6.9e-11 | +1.0e-10
Z=-20.1(12.40%) | Like=-15.24..-13.30 [-15.2401..-15.2386]*| it/evals=1206/1935 eff=73.7615% N=300
Z=-19.9(14.97%) | Like=-15.06..-13.30 [-15.0621..-15.0484] | it/evals=1230/1973 eff=73.5206% N=300
Z=-19.7(17.93%) | Like=-14.91..-13.30 [-14.9130..-14.9130]*| it/evals=1257/2008 eff=73.5948% N=300
Z=-19.7(18.27%) | Like=-14.91..-13.30 [-14.9108..-14.9086]*| it/evals=1260/2013 eff=73.5552% N=300
Mono-modal Volume: ~exp(-8.95) * Expected Volume: exp(-4.24) 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(19.87%) | Like=-14.84..-13.30 [-14.8449..-14.8442]*| it/evals=1273/2030 eff=73.5838% N=300
Z=-19.6(21.83%) | Like=-14.76..-13.30 [-14.7564..-14.7514]*| it/evals=1290/2049 eff=73.7564% N=300
Z=-19.4(24.92%) | Like=-14.62..-13.30 [-14.6215..-14.6161]*| it/evals=1317/2084 eff=73.8229% N=300
Z=-19.4(25.21%) | Like=-14.61..-13.30 [-14.6125..-14.6123]*| it/evals=1320/2087 eff=73.8668% N=300
Mono-modal Volume: ~exp(-8.96) * Expected Volume: exp(-4.47) 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=-19.3(27.58%) | Like=-14.55..-13.30 [-14.5474..-14.5390]*| it/evals=1340/2114 eff=73.8699% N=300
Z=-19.3(28.81%) | Like=-14.51..-13.30 [-14.5147..-14.5145]*| it/evals=1350/2125 eff=73.9726% N=300
Z=-19.2(31.93%) | Like=-14.41..-13.30 [-14.4092..-14.4085]*| it/evals=1376/2160 eff=73.9785% N=300
Z=-19.1(32.40%) | Like=-14.40..-13.30 [-14.4025..-14.4025]*| it/evals=1380/2164 eff=74.0343% N=300
Z=-19.1(35.24%) | Like=-14.34..-13.30 [-14.3428..-14.3408]*| it/evals=1403/2202 eff=73.7645% N=300
Mono-modal Volume: ~exp(-8.98) * Expected Volume: exp(-4.69) 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=-19.0(35.77%) | Like=-14.33..-13.30 [-14.3302..-14.3266]*| it/evals=1407/2206 eff=73.8195% N=300
Z=-19.0(36.21%) | Like=-14.32..-13.30 [-14.3198..-14.3188]*| it/evals=1410/2211 eff=73.7834% N=300
Z=-19.0(39.05%) | Like=-14.25..-13.30 [-14.2460..-14.2447]*| it/evals=1433/2246 eff=73.6382% N=300
Z=-18.9(39.86%) | Like=-14.23..-13.30 [-14.2341..-14.2323]*| it/evals=1440/2254 eff=73.6950% N=300
Z=-18.9(43.04%) | Like=-14.16..-13.30 [-14.1615..-14.1587]*| it/evals=1465/2290 eff=73.6181% N=300
Z=-18.9(43.65%) | Like=-14.15..-13.30 [-14.1483..-14.1406]*| it/evals=1470/2296 eff=73.6473% N=300
Mono-modal Volume: ~exp(-9.04) * Expected Volume: exp(-4.91) Quality: ok
index : +1.0| +2.5 ****** +3.0 | +5.0
amplitude: +1.0e-12| +4.4e-11 *********** +6.3e-11 | +1.0e-10
Z=-18.8(44.11%) | Like=-14.13..-13.30 [-14.1261..-14.1256]*| it/evals=1474/2301 eff=73.6632% N=300
Z=-18.8(47.31%) | Like=-14.07..-13.30 [-14.0744..-14.0674]*| it/evals=1500/2331 eff=73.8552% N=300
Z=-18.7(50.64%) | Like=-14.02..-13.30 [-14.0174..-14.0146]*| it/evals=1528/2366 eff=73.9593% N=300
Z=-18.7(50.85%) | Like=-14.01..-13.30 [-14.0143..-14.0141]*| it/evals=1530/2368 eff=73.9845% N=300
Mono-modal Volume: ~exp(-9.24) * Expected Volume: exp(-5.14) Quality: ok
index : +1.0| +2.5 ****** +2.9 | +5.0
amplitude: +1.0e-12| +4.5e-11 ********* +6.2e-11 | +1.0e-10
Z=-18.7(52.02%) | Like=-13.99..-13.30 [-13.9861..-13.9850]*| it/evals=1541/2385 eff=73.9089% N=300
Z=-18.6(54.06%) | Like=-13.95..-13.30 [-13.9472..-13.9381]*| it/evals=1560/2414 eff=73.7938% N=300
Z=-18.6(56.64%) | Like=-13.88..-13.30 [-13.8888..-13.8786] | it/evals=1585/2449 eff=73.7552% N=300
Z=-18.6(57.20%) | Like=-13.87..-13.30 [-13.8655..-13.8652]*| it/evals=1590/2459 eff=73.6452% N=300
Mono-modal Volume: ~exp(-9.61) * Expected Volume: exp(-5.36) Quality: ok
index : +1.0| +2.6 ***** +2.9 | +5.0
amplitude: +1.0e-12| +4.5e-11 ********* +6.1e-11 | +1.0e-10
Z=-18.6(59.11%) | Like=-13.84..-13.30 [-13.8373..-13.8345]*| it/evals=1608/2490 eff=73.4247% N=300
Z=-18.5(60.33%) | Like=-13.81..-13.30 [-13.8082..-13.8058]*| it/evals=1620/2508 eff=73.3696% N=300
Z=-18.5(62.92%) | Like=-13.77..-13.30 [-13.7695..-13.7644]*| it/evals=1645/2543 eff=73.3393% N=300
Z=-18.5(63.41%) | Like=-13.76..-13.30 [-13.7623..-13.7599]*| it/evals=1650/2551 eff=73.3008% N=300
Mono-modal Volume: ~exp(-9.92) * Expected Volume: exp(-5.58) Quality: ok
index : +1.0| +2.6 ***** +2.9 | +5.0
amplitude: +1.0e-12| +4.6e-11 ******** +6.0e-11 | +1.0e-10
Z=-18.4(65.76%) | Like=-13.73..-13.30 [-13.7288..-13.7286]*| it/evals=1675/2585 eff=73.3042% N=300
Z=-18.4(66.19%) | Like=-13.73..-13.30 [-13.7251..-13.7235]*| it/evals=1680/2591 eff=73.3304% N=300
Z=-18.4(68.49%) | Like=-13.69..-13.30 [-13.6869..-13.6865]*| it/evals=1706/2626 eff=73.3448% N=300
Z=-18.4(68.84%) | Like=-13.68..-13.30 [-13.6814..-13.6814]*| it/evals=1710/2631 eff=73.3591% N=300
[ultranest] Explored until L=-1e+01
[ultranest] Likelihood function evaluations: 2652
[ultranest] logZ = -18.03 +- 0.08518
[ultranest] Effective samples strategy satisfied (ESS = 984.1, 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: 0.11 bs:0.09 tail:0.26 total:0.28 required:<0.50
[ultranest] done iterating.
logZ = -18.025 +- 0.298
single instance: logZ = -18.025 +- 0.112
bootstrapped : logZ = -18.026 +- 0.142
tail : logZ = +- 0.262
insert order U test : converged: True correlation: inf iterations
index : 2.15 │ ▁ ▁▁▁▂▂▄▄▅▅▆▇▆▇▇▇▇▆▄▄▄▂▂▂▁▁▁▁▁▁ ▁ ▁ │3.47 2.75 +- 0.17
amplitude : 0.0000000000284│ ▁▁▁▁▁▁▂▂▄▄▆▆▇▇▇▆▇▇▅▅▅▄▃▂▂▁▁▁▁▁▁ ▁ ▁ │0.0000000000886 0.0000000000535 +- 0.0000000000077
Comparing the posterior distribution of all runs#
For a comparison of different posterior distributions, we can use the package chainconsumer. As this is not a Gammapy dependency, you’ll need to install it. More info here : https://samreay.github.io/ChainConsumer/
# Uncomment this if you have installed `chainconsumer`.
# from chainconsumer import Chain, ChainConfig, ChainConsumer, PlotConfig, Truth, make_sample
# from pandas import DataFrame
# c = ChainConsumer()
# def create_chain(result, name, color="k"):
# return Chain(
# samples=DataFrame(result, columns=["index", "amplitude"]),
# name=name,
# color=color,
# smooth=7,
# shade=False,
# linewidth=1.0,
# cmap="magma",
# show_contour_labels=True,
# kde= True
# )
# c.add_chain(create_chain(result_joint.samples, "joint"))
# c.add_chain(create_chain(result_0.samples, "run0", "g"))
# c.add_chain(create_chain(result_1.samples, "run1", "b"))
# c.add_chain(create_chain(result_2.samples, "run2", "y"))
# fig = c.plotter.plot()
# plt.show()
Corner plot comparison#
Corner plot comparing the three Crab runs.#
We can see the joint analysis allows to better constrain the parameters than the individual runs (more observation time is of course better). One can note as well that one of the run has a notably different amplitude (possibly due to calibrations or/and atmospheric issues).
Highest density intervals#
Given the samples, one can also compute the highest density interval (HDI) which is also known as the smallest credible interval (SCI). See more details here. This is the smallest interval in which a given probability (e.g. 68%) is contained.
For unimodal distributions, the HDI is a single continuous interval containing the mode whereas for multimodal distributions, the HDI can be a set of disconnected intervals. The HDI can be particularly helpful with multimodal distributions as opposed to the mean and quantiles approaches which will not report the important information. Here, we showcase the HDI using the Arviz package. Check out the many possibilities offered by Arviz, a package to analyze the samples posterior distributions.
from arviz import hdi
import scipy.stats as stats
# Multi-modal samples example
weight = 0.3
n_samples = 10000
mu1 = 5.5e-11
sigma1 = 0.7e-11
mu2 = 3.5e-11
sigma2 = 0.3e-11
weight = 0.7
rng = np.random.default_rng(42)
component_mask = rng.uniform(size=n_samples) < weight
samples = np.empty(n_samples)
samples[component_mask] = rng.normal(mu1, sigma1, component_mask.sum())
samples[~component_mask] = rng.normal(mu2, sigma2, (~component_mask).sum())
fig, (ax1, ax2) = plt.subplots(
2, 1, sharex=True, figsize=(9, 7), gridspec_kw={"height_ratios": [5, 2]}
)
# Highest density intervals
hdis = hdi(samples, prob=0.68, method="multimodal")
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 47.166 seconds)
