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Introduction

Most of the discussion in Chapter 2 treats spectral fitting as a maximum-likelihood problem: find the single parameter vector $\hat{\bm{\theta}}$ that minimizes the fit statistic ($-2\ln L$), and report confidence intervals from the curvature of the statistic around that minimum. This chapter covers the alternative Bayesian view, in which the goal is the full posterior distribution

$\displaystyle p(\bm{\theta}\vert D) \;\propto\; p(D\vert\bm{\theta})\,p(\bm{\theta}),$ (3.1)

where $p(D\vert\bm{\theta})$ is the likelihood given by the chosen fit statistic, $p(\bm{\theta})$ is the prior on the parameters, and $D$ is the observed data. Once the posterior is in hand, every downstream quantity — best-fit point, credible interval, derived-quantity distribution, evidence for model comparison — is just an expectation or quantile against $p(\bm{\theta}\vert D)$. The work shifts from minimizing a function to sampling from a distribution.

XSPEC has built up a complete Bayesian toolchain over the last several releases. The user supplies priors with the bayes command and then chooses one of four samplers: Metropolis–Hastings, Goodman–Weare, Hamiltonian Monte Carlo, or nested sampling. The first two are accessed through the chain command; the third and fourth have their own hmc and nest top-level commands. The four are not interchangeable: each is well-suited to a different regime, and the choice has a much larger effect on elapsed time and convergence reliability than any single tuning knob within a given sampler. This chapter explains what each one does, when to reach for it, and which mistakes to avoid.

next up previous contents
Next: Priors Up: Bayesian Methods Previous: Bayesian Methods   Contents