Multitree Monte Carlo
Largeradius npoint Correlations
 
This extends our work on exact algorithms for computing npoint
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 ntree exact algorithm to
generate strata for stratified sampling. Formulating npoint
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
Highdimensional 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 largedataset inferences. After some more experimental studies,
this work should be submitted soon.

