Vertical-likelihood Monte Carlo Integration

We propose the vertical-likelihood Monte Carlo method for approximationg definite integrals of the form $Z = \int L(X) d P(x)$, where $L$ is a target function (often a likelihood) and $P$ a finite measure. This approach can be viewed as a special case of importance sampling with a particular choice of proposal distribution. It exploits a duality between two random variables: the random draw $X$ from the proposal, and the corresponding random likelihood ordinate $Y\equiv L(X)$ of the draw. We address the question: what should the distribution of $L(X)$ be in order to yield a good proposal distribution? This approach turns out to unite seven seemingly disparate classes of algorithms under a single conceptual framework: importance sampling, slice sampling, simulated tempering, the harmonic-mean estimator, the vertical-density sampler, nested sampling, and energy-level sampling methods from statistical physics. Special attention is paid to the connection with nested sampling. An MCMC method for computing the estimator is proposed, and its convergence properties are studied. Two examples demonstrate its excellent performance.