Input Data

The FactorGraph package requires knowledge about the joint probability density function $g(\mathcal{X})$ of the set of random variables $\mathcal{X} = \{x_1,\dots,x_n\}$ that can be factorised proportionally ($\propto$) to a product of local functions:

\[ g(\mathcal{X}) \propto \prod_{i=1}^m \psi_i(\mathcal{X}_i).\]

The FactorGraph package supports disrecte random variables, where each random variable $x_i$ having $k_i$ possible states, while local function $\psi_i(\mathcal{X}_i)$ is defined as the conditional probability distribution:

\[ p_i(x_i|\mathcal{X}_i \setminus x_i),\]

Hence, the local function is associated with the conditional distribution $p_i(x_i|\mathcal{X}_i \setminus x_i)$ and the set of random variables $\mathcal{X}_i$. The conditional distribution takes probability values over all possible state combinations of the random variables from the set $\mathcal{X}_i$ forming the conditional probability table. To describe the joint probability density function $g(\mathcal{X})$, it is enough to define the container that saves indices of discrete variables and conditional probability tables.

Thus, the parameters that describe the factor graph structure are represented by variable probability containing indices of random variables, while variable table saves conditional probability tables. The function discreteTreeModel() accepts variables probability and table.

Build the graphical model

Let us observe the following joint probability density function:

\[ g(\mathcal{X}) \propto p_1(x_1)p_2(x_1|x_2)p_3(x_1|x_2,x_3),\]

where all variables have two states, state $1$ and state $2$. The conditional probability tables can be written in the compact form:

Let us define the variable probability that contains discrete variable indices:

probability1 = [1]
probability2 = [1; 2]
probability3 = [1; 2; 3]

probability = [probability1, probability2, probability3]

Further, we define the variable table that holds conditional probability tables:

table1 = zeros(2)
table1[1] = 0.6; table1[1] = 0.4

table2 = zeros(2, 2)
table2[1, 1] = 0.8; table2[2, 1] = 0.5; table2[1, 2] = 0.2; table2[2, 2] = 0.5

table3 = zeros(2, 2, 2)
table3[1, 1, 1] = 1.0; table3[2, 1, 1] = 0.1; table3[1, 2, 1] = 0.5; table3[2, 2, 1] = 0.6
table3[1, 1, 2] = 0.0; table3[2, 1, 2] = 0.9; table3[1, 2, 2] = 0.5; table3[2, 2, 2] = 0.4

table = [table1, table2, table3]

Passing data directly via command-line arguments supports the above format, where the function discreteTreeModel() accepts variables probability and table:

bp = discreteTreeModel(probability, table)

Here, the variable bp holds the main composite type related to the discrete model. We can also pack the input data as a dictionary and pass to the function discreteTreeModel():

inputData = Dict("probability" => (probability1, probability2, probability3),
                 "table"  => (table1, table2, table3))

bp = discreteTreeModel(inputData)