Truncated importance sampling

Importance sampling is a fundamental Monte Carlo technique. It involves generating a sample from a proposal distribution in order to estimate some property of a target distribution. Importance sampling can be highly sensitive to the choice of proposal distribution, and fails if the proposal distribution does not sufficiently well approximate the target. Procedures which involve truncation of large importance sampling weights are shown theoretically to improve on standard importance sampling by being less sensitive to the proposal distribution and having lower mean squared estimation error. Consistency is shown under weak conditions, and optimal truncation rates found under more specific conditions. Truncation at rate n is shown to be a good general choice. An adaptive truncation threshold, based on minimizing an unbiased risk estimate, is also presented. As an example, truncation is found to be effective for calculating the likelihood of partially observed multivariate diffusions. It is demonstrated as a component of a sequential importance sampling scheme for a continuous time population disease model. Truncation is most valuable for computationally intensive, multi-dimensional situations in which finding a proposal distribution that is everywhere a good approximation to the target distribution is challenging.