DeepAI

# A Generalization of Spatial Monte Carlo Integration

Spatial Monte Carlo integration (SMCI) is an extension of standard Monte Carlo integration and can approximate expectations on Markov random fields with high accuracy. SMCI was applied to pairwise Boltzmann machine (PBM) learning, with superior results to those from some existing methods. The approximation level of SMCI can be changed, and it was proved that a higher-order approximation of SMCI is statistically more accurate than a lower-order approximation. However, SMCI as proposed in the previous studies suffers from a limitation that prevents the application of a higher-order method to dense systems. This study makes two different contributions as follows. A generalization of SMCI (called generalized SMCI (GSMCI)) is proposed, which allows relaxation of the above-mentioned limitation; moreover, a statistical accuracy bound of GSMCI is proved. This is the first contribution of this study. A new PBM learning method based on SMCI is proposed, which is obtained by combining SMCI and the persistent contrastive divergence. The proposed learning method greatly improves the accuracy of learning. This is the second contribution of this study.

• 21 publications
• 3 publications
12/21/2020

### Spatial Monte Carlo Integration with Annealed Importance Sampling

Evaluating expectations on a pairwise Boltzmann machine (PBM) (or Ising ...
04/07/2022

### Composite Spatial Monte Carlo Integration Based on Generalized Least Squares

Although evaluation of the expectations on the Ising model is essential ...
03/01/2022

### Integration of bounded monotone functions: Revisiting the nonsequential case, with a focus on unbiased Monte Carlo (randomized) methods

In this article we revisit the problem of numerical integration for mono...
03/15/2022

### A novel sampler for Gauss-Hermite determinantal point processes with application to Monte Carlo integration

Determinantal points processes are a promising but relatively under-deve...
03/25/2020

### Exploiting Low Rank Covariance Structures for Computing High-Dimensional Normal and Student-t Probabilities

We present a preconditioned Monte Carlo method for computing high-dimens...
10/16/2019

### Weighted Monte Carlo with least squares and randomized extended Kaczmarz for option pricing

We propose a methodology for computing single and multi-asset European o...
01/22/2020

### Saddlepoint approximations for spatial panel data models

We develop new higher-order asymptotic techniques for the Gaussian maxim...