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 |

`n` |
This is the number of random deviates to generate. |

`alpha` |
This is a vector of length |

`Omega` |
This is the |

`log` |
Logical. If |

### 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

### Examples

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

*LaplacesDemon*version 16.1.6 Index]