Multi-tree Monte Carlo
Large-radius n-point Correlations
This extends our work on exact algorithms for computing n-point correlations. The weakness of that approach is that it is slow for large radii (though for very large radii it becomes fast again). We have developed a new form of Monte Carlo method which utilizes the recursion tree of the n-tree exact algorithm to generate strata for stratified sampling. Formulating n-point correlations in terms of binomial sampling yields a confidence band for the whole procedure. This unique combination of geometry and sampling results in a dramatic speedups over the exact algorithm for large search radii, and is undergoing validation by our astrophysicist collaborators. This will be submitted very soon.

Adaptive Monte Carlo Methods
I have developed algorithms which combine diverse ideas to achieve unique new kinds of Monte Carlo strategies. In an unbiased sampling method, the error due to computational approximation is given exactly by the variance. Thus such approximate methods can yield speedups in cases where our usual exact algorithms are not efficient, yet still yield rigorous error bounds.

FIRE: Function Integration by Reconstruction
High-dimensional Integrals ∫ f(x) dx
Bayesian inference is severely bridled by its need to compute difficult integrals. Markov Chain Monte Carlo is the main viable tool but can be slow and problematic in practice, and no effective general method exists for some integrals such as the Bayesian normalization constant. I have derived a new alternative method for general integration. From `first principles' (of Monte Carlo variance reduction), I cast the integration problem as one of statistical estimation (nonparametric regression), using a new risk functional which directly measures integration error. This method treats the normalization constant problem and exhibits favorable performance compared to existing methods where function evalution is expensive, as in large-dataset inferences. After some more experimental studies, this work should be submitted soon.