The coalescent process is an important and widely used model for inferring the dynamics of biological populations from samples of genetic diversity. Coalescent analysis typically involves applying statistical methods to either samples of genetic sequences or an estimated genealogy in order to estimate the demographic history of the population from which the samples originated. Several parametric and non-parametric estimation techniques, employing diverse methods, such as Gaussian processes and Monte Carlo particle filtering, already exist. However, these techniques often trade estimation accuracy and sophistication for methodological flexibility and ease of use. Thus, there is room for new coalescent estimation techniques that can be easily implemented for a range of inference problems while still maintaining some sense of statistical optimality. Here we introduce the Bayesian Snyder filter as a natural, easily implementable and flexible minimum mean square error estimator for parametric demographic functions. By reinterpreting the coalescent as a self-correcting inhomogeneous Poisson process, we show that the Snyder filter can be applied to both isochronous (sampled at one time point) and heterochronous (serially sampled) estimation problems. We test the estimation performance of the filter on both standard, simulated demographic models and on a well-studied empirical dataset comprising hepatitis C virus sequences from Egypt. Additionally, we provide some analytical insight into the relationship between the Snyder filter and popular maximum likelihood and skyline plot techniques for coalescent inference. The Snyder filter is an exact and direct Bayesian estimation method that provides optimal mean square error estimates. It has the potential to become as a useful, alternative technique for coalescent inference.