mgm (version 1.1-7)

mgmsampler:

Description

Samples from a pairwise Mixed Graphical Model

Usage

mgmsampler(n, type, lev, graph, thresh, parmatrix = NA, nIter = 250, varadj = .2)

Arguments

n

Number of samples

type

x 1 vector specifying the type of distribution for each variable ("g" = Gaussian, "p" = Poisson, "e" = Exponential, "c" = Categorical)

lev

p x 1 vector specifying the number of levels for each variables (for continuous variables: 1)

graph

A p x p symmetric (weighted) adjacency matrix

thresh

A list in which each entry corresponds to a variable; categorical variables have as many thresholds as categories

parmatrix

Optional; A matrix specifying all parameters in the model. This provides a possibility to exactly specify all parameters instead of using the default mapping from the weighted adjacency matrix to the model parameter matrix as described below (Details). If provided, graph will be ignored.

nIter

The number of iterations in the Gibbs sampler

varadj

Additive constant to conditional Gaussian variances before normalization. This avoids partial correlations close to 1.

Value

n x p data matrix

Details

For interactions involving categorical variables with more than m = 2 categories, an interaction is comprised by more than one parameter. Specifically, in the overcomplete representation used here, in case of an interaction between two categorical variables with each m categories, we have m^2 parameters, whereas in case of an interaction between a cateogircal and a continuous variable we have m parameters.

We use the following (arbitrary) mapping from provided edge-weights to interaction parameters: for catogorical-categorical interactions, we use the parameterization of the Potts model (see e.g. Wainwright & Jordan, 2008) extended to the case of interactions between categorical variables with different number of categories. All non-zero elements in the parameter matrix are assigned the value specified in the weighted adjacency matrix. In case of Categorical-Continous interactions, parameters for categories m > |m|/2 have a nonzero parameter with the value is given by the weighted adjacency matrix. All categories m >= |m|/2 have zero parameters. A customized parameterization of interactions including categorical variables can be specified using the parmatrix argument.

The specification of the mixed graphical model has to satisfy a number of contraints. For details see Yang et al. (2014) or Haslbeck and Waldorp (2015).

References

Haslbeck, J., & Waldorp, L. J. (2015). Structure estimation for mixed graphical models in high-dimensional data. arXiv preprint arXiv:1510.06871.

Yang, E., Baker, Y., Ravikumar, P., Allen, G., & Liu, Z. (2014). Mixed graphical models via exponential families. In Proceedings of the Seventeenth International Conference on Artificial Intelligence and Statistics (pp. 1042-1050).

Wainwright, M. J., & Jordan, M. I. (2008). Graphical models, exponential families, and variational inference. Foundations and Trends in Machine Learning, 1(1-2), 1-305.

Examples

n <- 10 # number of samples
type <- c("g", "c") # one Gaussian, one categorical
lev <- c(1, 3) # the categorical variable has 3 categories
graph <- matrix(0, 2, 2) 
graph[1, 2] <- graph[2, 1] <- .5 # we have an edge with weight .5 between the two nodes
thresh <- list(c(0), c(0, 0, 0)) # all thresholds are zero 

data <- mgmsampler(n, type, lev, graph, thresh, parmatrix = NA, nIter = 1000)

head(data)