>There are cases where irreducibility is trivially fulfilled,
>e.g., when all prior and conditional probabilities are non-zero.
>This is called the positivity condition.
But even when these conditions are satisfied, convergence, although
linear, can be exceedingly slow. The convergence rate depends on the
modulus of the largest non-unit eigenvalue of the transition matrix.
For complex graphical models the eigenvalues of the transition matrix
can be very difficult to find or even obtain bounds for.
Convergence diagnostics recommended in the literature may indicate
convergence when the sampler has not in fact converged, but is
sampling in a local basin of attraction.
Adaptive sampling methods (samplers for which the transition
probabilities change with sampling history, of which annealing is a
commonly used example) may have all non-zero priors and conditional
probabilities, but may not even converge to a unique stationary
distribution, let alone converge at a linear rate.
Kathy Laskey
