Torkel Erhardsson: Non-parametric Bayesian inference for integrals with respect to an unknown finite measure

We consider the problem of estimating a finite number of integrals with respect to a common unknown finite measure from noisy observations of some of the integrals. A new method to carry out Bayesian inference for the integrals is proposed. We use a Dirichlet or Gamma process as a prior for the measure, and construct an approximation to the posterior distribution of the integrals using the SIR algorithm and samples from a new multidimensional version of a Markov chain introduced by Feigin and Tweedie. We prove that the Markov chain is positive Harris recurrent, and that the approximating distribution converges weakly to the posterior as the number of samples increases, under a mild integrability condition. Applications to polymer chemistry and mathematical finance are given.