Background

In a first step, the abn package calculates a cache of scores of the data given each possible model. This cache is then used to estimate the Bayesian network structure ("structure learning") and to estimate the parameters of the model ("parameter learning"). The cache of scores is calculated using the buildScoreCache() function, which is the focus of this vignette.

In abn we distinguish between two approaches: the Bayesian and the information-theoretic score. Only under a frequentist framework, the package supports all possible mixtures of continuous, discrete, and count data (see also vignette("01_quick_start_example.Rmd")). Settings that are specific to the modelling approach are set with the control argument of the buildScoreCache() function.

We will illustrate the model specification process using a simple example data set and the buildScoreCache() function.

Estimate the maximum number of parent nodes

The maximum number of parent nodes for each node in the data set is a crucial parameter to speed up the model estimation in abn. It limits the number of possible combinations and thus the search space for the model estimation. Instead of a wild guess, the maximum number of parent nodes can be set to a reasonable value based on prior knowledge or to the value that maximizes the score of the model given the data.

In the later case, we can estimate the model's score for different maximum numbers of parent nodes and choose the maximum number of parent nodes that maximizes the score of the model given the data.

# Load only a subset of the example data for illustration
mydat <- ex1.dag.data[, c("b1", "p1", "g1", "b2", "p2", "b3", "g2")]
mydists <- list(b1="binomial", 
                p1="poisson", 
                g1="gaussian", 
                b2="binomial", 
                p2="poisson", 
                b3="binomial",
                g2="gaussian")

# Estimate model score for different maximum numbers of parent nodes
num.vars <- ncol(mydat) # number of variables
max.pars <- 1:(num.vars-1) # vector of possible maximum number of parent nodes

npars_scores <- data.frame(max.pars = max.pars, score = rep(NA, length(max.pars))) # data frame to store scores

# loop over maximum number of parent nodes
for (i in max.pars) {
  mycache <- buildScoreCache(data.df = mydat, 
                             data.dists = mydists,
                             method = "bayes", 
                             max.parents = i)
  mp.dag <- mostProbable(mycache)
  myfit <- fitAbn(mp.dag)
  
  npars_scores[i, "score"] <- myfit$mlik # store score
}

# Plot the scores for different maximum numbers of parent nodes
library(ggplot2)
ggplot(npars_scores, aes(x = max.pars, y = score)) +
  geom_point() +
  geom_line() +
  labs(x = "Maximum number of parent nodes", y = "Model score") +
  # set x-axis labels to integers
  scale_x_continuous(breaks = seq(0, num.vars, 1))

We can see that the model score increases with the maximum number of parent nodes up to a certain point and then remains constant. This typical pattern indicates that the maximum number of parent nodes has been reached at the point where the score remains constant.

The value of max.parents can be set to a single value equal for all nodes or to a list with the node names as keys and the maximum number of parent nodes as values as shown in vignette("01_quick_start_example.Rmd").

Include prior domain knowledge

The abn package allows to include prior domain knowledge in the model estimation process by defining edges and their directions as fixed or forbidden.

Arcs that we are certain about can be provided with dag.retained, while arcs that we are certain about not being present can be defined with dag.banned. The dag.retained and dag.banned arguments can be set to an adjacency matrix with the node names as row- and column names. An edge from node i to node j is indicated by a 1 in the i-th row and j-th column of the matrix, while a 0 indicates no edge.

# Load the example data
mydat <- ex1.dag.data
mydists <- list(b1="binomial", 
                p1="poisson", 
                g1="gaussian", 
                b2="binomial", 
                p2="poisson", 
                b3="binomial", 
                g2="gaussian", 
                b4="binomial", 
                b5="binomial", 
                g3="gaussian")

# Define edges and their directions as fixed or forbidden
dag.banned <- matrix(0, nrow = 10, ncol = 10, dimnames = list(names(mydat), names(mydat)))

# Define edges and their directions as forbidden
dag.banned["b1", "b2"] <- 1
dag.banned["b1", "b3"] <- 1
dag.banned["b1", "b4"] <- 1

# Display the matrix
dag.banned

# Plot the forbidden edges
plotAbn(dag = dag.banned, data.dists = mydists)

The plot shows the forbidden edges defined in the dag.banned matrix.