Bayesian Methods for Astrophysics and Particle Physics

1.6 Numerical Methods
1.6 Numerical Methods
The calculation of the posterior probability and the evaluation of the Bayesian
evidence is generally not a simple task. Priors and posteriors are often complex
distributions which may not be easily expressed in analytical formulae. The
simplest approach is to evaluate the posterior probability density on a discrete
parameter grid but this approach become impractical even for problems with
very few ( 5) dimensions as the number of posterior probability evaluations
required to produce a fine grid grows exponentially with the dimensionality of
the problem. In the rest of this section, we describe the workhorse of Bayesian
modelling, the Metropolis–Hastings algorithm based on Markov Chain Monte
Carlo (MCMC) sampling and the standard method of evaluating the Bayesian
evidence, “Thermodynamic Integration”.
1.6.1 Markov Chain Monte Carlo (MCMC) Sampling
As discussed in the previous section, the calculation of posterior probability and
the Bayesian evidence present a particularly challenging problems even for mod-
erate dimensional parameter spaces. Rather than evaluating the posterior prob-
ability density on a fine grid, a much more efficient approach is to draw Monte
Carlo samples from the posterior probability distribution such that the number
density of these Monte Carlo samples is equal to the posterior probability density.
Once enough posterior samples have been drawn, a smoothed histogram of these
samples would be a good representation of the posterior density. Another ad-
vantage of the Monte Carlo sampling of the posterior is that the marginalization
is straightforward: simply disregard the “uninteresting” parameters and make a
histogram of the “interesting” parameter sample values.
The Metropolis–Hastings ((Metropolis et al. 1953) provides an elegant solution
to the problem of drawing Monte Carlo samples from the posterior distribution.
Metropolis–Hastings algorithm works by creating a Markov Chain: a series of
random samples (points in the parameter space) whose values are determined
only by the values of previous points in the series. The probability distribution
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