We investigate Bayesian alternatives to classical Monte Carlo methods for evaluating integrals. Bayesian Monte Carlo (BMC) allows the incorporation ofprior knowledge, such as smoothness of the integrand, into the estimation. In a simple problem we show that this outperforms any classical importance sampling method. We also attempt more challenging multidimensionalintegrals involved in computing marginal likelihoods ofstatistical models (a.k.a.
This paper introduces Tree-Pyramidal Adaptive Importance Sampling (TP-AIS), a novel iterated sampling method that outperforms current state-of-the-art approaches. TP-AIS iteratively builds a proposal distribution parameterized by a tree pyramid, where each tree leaf spans a convex subspace and represents it's importance density. After each new sample operation, a set of tree leaves are subdivided improving the approximation of the proposal distribution to the target density. Unlike the rest of the methods in the literature, TP-AIS is parameter free and requires zero manual tuning to achieve its best performance. Our proposed method is evaluated with different complexity randomized target probability density functions and also analyze its application to different dimensions. The results are compared to state-of-the-art iterative importance sampling approaches and other baseline MCMC approaches using Normalized Effective Sample Size (N-ESS), Jensen-Shannon Divergence to the target posterior, and time complexity.
Localization is a fundamental challenge for autonomous robotics. Although accurate and efficient techniques now exist for solving this problem, they require explicit probabilistic models of the robot's motion and sensors. These models are usually obtained from time-consuming and error-prone measurement or tedious manual tuning. In this paper we examine automatic calibration of sensor and motion models from a Bayesian perspective. We introduce an efficient MCMC procedure for sampling from the posterior distribution of the model parameters. We also present a novel extension of particle filters to make use of our posterior parameter samples. Finally, we demonstrate our approach both in simulation and on a physical robot. Our results demonstrate effective inference of model parameters as well as a paradoxical result that using posterior parameter samples can produce more accurate position estimates than the true parameters.