Proposal of a Score Based Approach to Sampling Using Monte Carlo Estimation of Score and Oracle Access to Target Density

Score based approaches to sampling have shown much success as a generative algorithm to produce new samples from a target density given a pool of initial samples. In this work, we consider if we have no initial samples from the target density, but rather $0^{th}$ and $1^{st}$ order oracle access to the log likelihood. Such problems may arise in Bayesian posterior sampling, or in approximate minimization of non-convex functions. Using this knowledge alone, we propose a Monte Carlo method to estimate the score empirically as a particular expectation of a random variable. Using this estimator, we can then run a discrete version of the backward flow SDE to produce samples from the target density. This approach has the benefit of not relying on a pool of initial samples from the target density, and it does not rely on a neural network or other black box model to estimate the score.

Bayesian Properties of Normalized Maximum Likelihood and its Fast Computation

The normalized maximized likelihood (NML) provides the minimax regret solution in universal data compression, gambling, and prediction, and it plays an essential role in the minimum description length (MDL) method of statistical modeling and estimation. Here we show that the normalized maximum likelihood has a Bayes-like representation as a mixture of the component models, even in finite samples, though the weights of linear combination may be both positive and negative. This representation addresses in part the relationship between MDL and Bayes modeling. This representation has the advantage of speeding the calculation of marginals and conditionals required for coding and prediction applications.