Description Usage Arguments Details Value References See Also Examples

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

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`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 |

Application: Continuous Multivariate

Density:

*p(theta) = exp(-0.5 theta^T Omega^(-1) theta) prod i=1 (1 + exp(theta[i] + alpha[i]))*Inventor: Zhang et al. (2012)

Notation 1:

*theta ~ CRMRF(alpha, Omega)*Notation 2:

*p(theta) = CRMRF(theta | alpha, Omega)*Parameter 1: shape vector

*alpha*Parameter 2: positive-definite

*k x 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.

`dcrmrf`

gives the density and
`rcrmrf`

generates random deviates.

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.

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