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Metropolis–Hastings (chain type mh)
Metropolis–Hastings (MH) is the classical MCMC algorithm: at each
step a candidate parameter vector is drawn from a proposal
distribution centered on the current point, the candidate is accepted
with probability
,
and either the candidate or a repeat of the current point is added to
the chain. XSPEC's implementation lets the user choose the proposal
shape (Gaussian, Cauchy, or uniform) and the source of its covariance
matrix (the fit covariance, an explicit diagonal, or a previously
loaded chain).
Strengths. MH is robust and well-understood. Every acceptance step costs one likelihood evaluation, so the per-iteration cost is the cheapest of the four samplers. It handles arbitrarily shaped posteriors and is the only sampler that lets the user supply a fully custom proposal class through the initpackage mechanism (see Appendix G).
Weaknesses. The proposal scale and shape must roughly
match the posterior or mixing collapses. Acceptance rates well below
or well above 
indicate the scale is off. For
highly correlated parameters a Gaussian proposal derived from the fit
covariance is essential; a diagonal proposal will accept almost
nothing in narrow valleys.
Tips. Always run a fit before the chain so the proposal
covariance is drawn from a meaningful Hessian. Examine
chain stat for each free parameter — the
Rubin–Gelman statistic should be below 
and the Geweke
statistic should be within 
before the chain is trusted. The
fraction-of-repeats line is the easiest way to spot a proposal that is
much too wide. When proposals from the Hessian are rejected because
the covariance matrix has temporarily-frozen parameters, fall back to
proposal gaussian deltas <value> with
<value> of order unity.