An approximation scheme for quasi-stationary distributions of killed diffusions

In this paper we study the asymptotic behavior of the normalized weighted empirical occupation measures of a diffusion process on a compact manifold which is killed at a smooth rate and then regenerated at a random location, distributed according to the weighted empirical occupation measure. We show that the weighted occupation measures almost surely comprise an asymptotic pseudo-trajectory for a certain deterministic measure-valued semiflow, after suitably rescaling the time, and that with probability one they converge to the quasi-stationary distribution of the killed diffusion. These results provide theoretical justification for a scalable quasi-stationary Monte Carlo method for sampling from Bayesian posterior distributions.Comment: v2: revised version, 29 pages, 1 figur

Improved Weighted Random Forest for Classification Problems

Several studies have shown that combining machine learning models in an appropriate way will introduce improvements in the individual predictions made by the base models. The key to make well-performing ensemble model is in the diversity of the base models. Of the most common solutions for introducing diversity into the decision trees are bagging and random forest. Bagging enhances the diversity by sampling with replacement and generating many training data sets, while random forest adds selecting a random number of features as well. This has made the random forest a winning candidate for many machine learning applications. However, assuming equal weights for all base decision trees does not seem reasonable as the randomization of sampling and input feature selection may lead to different levels of decision-making abilities across base decision trees. Therefore, we propose several algorithms that intend to modify the weighting strategy of regular random forest and consequently make better predictions. The designed weighting frameworks include optimal weighted random forest based on ac-curacy, optimal weighted random forest based on the area under the curve (AUC), performance-based weighted random forest, and several stacking-based weighted random forest models. The numerical results show that the proposed models are able to introduce significant improvements compared to regular random forest