Adaptive truncation of infinite sums

applications to statistics

Resumo

It is often the case in Statistics that one needs to compute sums of infinite series, especially in marginalising over discrete latent variables. This has become more relevant with the popularization of gradient-based techniques (e.g. Hamiltonian Monte Carlo) in the Bayesian inference context, for which discrete latent variables are hard or impossible to deal with. For many major infinite series, custom algorithms have been developed which exploit specific features of each problem. In contrast, here we employ basic results from the theory of infinite series to investigate general, problem-agnostic algorithms to approximate (truncate) infinite sums within an arbitrary tolerance ε > 0 and provide robust computational implementations with provable guarantees.