  We investigate the use of a certain class of functional inequalities known as
weak Poincar\'e inequalities to bound convergence of Markov chains to
equilibrium. We show that this enables the straightforward and transparent
derivation of subgeometric convergence bounds for methods such as the
Independent Metropolis--Hastings sampler and pseudo-marginal methods for
intractable likelihoods, the latter being subgeometric in many practical
settings. These results rely on novel quantitative comparison theorems between
Markov chains. Associated proofs are simpler than those relying on
drift/minorization conditions and the tools developed allow us to recover and
further extend known results as particular cases. We are then able to provide
new insights into the practical use of pseudo-marginal algorithms, analyse the
effect of averaging in Approximate Bayesian Computation (ABC) and the use of
products of independent averages, and also to study the case of lognormal
weights relevant to particle marginal Metropolis--Hastings (PMMH).
