dist.ContinuousRelaxation {LaplacesDemon} R Documentation

## Continuous Relaxation of a Markov Random Field Distribution

### Description

This is the density function and random generation from the continuous relaxation of a Markov random field (MRF) distribution.

### Usage

dcrmrf(x, alpha, Omega, log=FALSE)
rcrmrf(n, alpha, Omega)


### Arguments

 x This is a vector of length k. n This is the number of random deviates to generate. alpha This is a vector of length k of shape parameters. Omega This is the k \times k precision matrix \Omega. log Logical. If log=TRUE, then the logarithm of the density is returned.

### Details

• Application: Continuous Multivariate

• Density:

p(\theta) \propto \exp(-\frac{1}{2} \theta^T \Omega^{-1} \theta) \prod_i (1 + \exp(\theta_i + alpha_i))

• Inventor: Zhang et al. (2012)

• Notation 1: \theta \sim \mathcal{CRMRF}(\alpha, \Omega)

• Notation 2: p(\theta) = \mathcal{CRMRF}(\theta | \alpha, \Omega)

• Parameter 1: shape vector \alpha

• Parameter 2: positive-definite k \times k matrix \Omega

• Mean: E(\theta)

• Variance: var(\theta)

• Mode: mode(\theta)

It is often easier to solve or optimize a problem with continuous variables rather than a problem that involves discrete variables. A continuous variable may also have a gradient, contour, and curvature that may be useful for optimization or sampling. Continuous MCMC samplers are far more common.

Zhang et al. (2012) introduced a generalized form of the Gaussian integral trick from statistical physics to transform a discrete variable so that it may be estimated with continuous variables. An auxiliary Gaussian variable is added to a discrete Markov random field (MRF) so that discrete dependencies cancel out, allowing the discrete variable to be summed away, and leaving a continuous problem. The resulting continuous representation of the problem allows the model to be updated with a continuous MCMC sampler, and may benefit from a MCMC sampler that uses derivatives. Another advantage of continuous MCMC is that stationarity of discrete Markov chains is problematic to assess.

A disadvantage of solving a discrete problem with continuous parameters is that the continuous solution requires more parameters.

### Value

dcrmrf gives the density and rcrmrf generates random deviates.

### References

Zhang, Y., Ghahramani, Z., Storkey, A.J., and Sutton, C.A. (2012). "Continuous Relaxations for Discrete Hamiltonian Monte Carlo". Advances in Neural Information Processing Systems, 25, p. 3203–3211.

### See Also

dmvn

### Examples

library(LaplacesDemon)
x <- dcrmrf(rnorm(5), rnorm(5), diag(5))
x <- rcrmrf(10, rnorm(5), diag(5))


[Package LaplacesDemon version 16.1.6 Index]