Bayesian inference

To perform Bayesian inference for the \(\Lambda CDM\) model using Type Ia Supernovae and cosmic chronometers, in the ini file we must set LCDM and SN+HD:

[custom]
...

model = LCDM

datasets = SN+HD

...

There are three options to make parameter estimation through Bayesian inference in SimpleMC:

• With Metropolis-Hastings algorithm

• With EMCEE algorithm

• With nested sampling

• Notebook example

Once setting the ini file according to the selected sampler, we can run SimpleMC as in the python script example.

In both cases, the output is a summary with the parameter estimation and an output file with the Markov Chain (see the section of outputs for details).

With Metropolis-Hastings algorithm

The model keyword for using the Metropolis-Hastings algorithm is mcmc as the analyzer. The basic keys in the [mcmc] section are the number of samples nsamp, the burn-in steps skip and the Gelman-Rubin stopping criterion GRstop.

[custom]
...
...
analyzer = mcmc
...

[mcmc]
nsamp = 10000
skip    = 100
GRstop  = 0.01
...

With EMCEE algorithm

To use the EMCEE algorithm we use emcee library. The basic keys in the [emcee] section of the ini file are the number of walkers of the ensemble walkers, the number of samples for each walker [nsamp] and the burn-in steps burnin.

The number of walkers must be at least twice the number of free parameters.

[custom]
...
...
analyzer = emcee
...

[emcee]
walkers = 10
nsamp = 200
burnin = 0
...

With nested sampling

To perform nested sampling we use the dynesty library. In this case, in the ini file the most important keys in the [nested] section are the number of live points nlivepoints and the difference between the Bayesian evidence of two consecutive steps (accuracy).

[custom]
...
...
analyzer = nested
...

[nested]
nlivepoints = 100
accuracy = 0.02
...

Notebook example

In the following notebook there is an example of Bayesian inference to the LCDM model, using SNIa and cosmic chronometers, with the three samplers available in SimpleMC.