Model Format

We use a simple text-file format specified below to describe the graphical model neworks (Markov networks) for problem instances. The format is a generalization of the Ergo file format initially developed by Noetic systems Ergo software. We use the suffix .uai to denote the model files.

Structure

A file in the UAI format consists of the following two parts, in order:

<Preamble>
<Function tables>

which are described below.

Preamble

The preamble consists of the following portions, in order:

<Graph Type>
<Variables and Domains>
<Function Scopes>

which we now describe…

Graph Type

The Graph Type is a metalabel signifying the type of network being encoded. Generally, this can be either BAYES for a Bayesian network or MARKOV for a Markov network. For our competition, the label will always be:

MARKOV

Variables and Domains

This portion of the preamble contains two lines.

The first line consists of a single number, N, specifying the number of variables in the network. Each variable is implicitly labled as 0, 1, … , N.

The second line specifies the cardinalities (or domain size) of each variable separated by a whitespace (in order from 0, 1, … , N).

An example of a network with three variables, the first two having a domain size of 2 and the last variable having a domain size of 3, would look like:

3
2 2 3

Function Scopes

The last section of the preamble states the number of functions in the network as well as their respective scopes.

First is a single number specifying the number of functions in the network.

Next, there will be a line for each function. Each successive line (which is for a single funciton) states the number of variables that belong to the function’s scope followed by the index identity of each of the variables in the function’s scope.

For a network containing three functions, the first of which is a unary factor on variable index 0, the second a factor over variables index 0 and index 1, and the third of which has a scope consisting of variables index 1 and index 2, the file would show:

3
1 0
2 0 1
2 1 2

(Note: The order of the list of varaibles in a function’s scope is not restricted. When defining the function values, the ordering of variables within a factor will follow the order provided here).

Preamble Summary

The preamble for our simple Markov network with three variables and two functions described above would look like:

MARKOV
3
2 2 3
3
1 0
2 0 1
2 1 2

In the example above, the first line denotes a Markov network, the second line tells us the problem consists of three variables, let’s refer to them as X, Y, and Z (which will implicitly have indexes 0, 1, and 2, respectively). The third line tells us that X, Y, and Z’s cardinalities are 2, 2, and 3 respectively. Line four specifies that there are 3 functions. Based on the final three lines, we know that the first function is a unary function on X, the second function is defined over X and Y, while the second is defined over Y and Z.

Function Tables

Under the preamble, each function is specified - one-by-one, separated by a blank line, in the order introduced in the preamble - by giving its full table. Each function table is presented as follows:

<blank line>
<Number of Table Values>
<Table Values>

Before each function specification, there is a separating blank line.

Next, a single integer representing the number of table values for the function is presented. (This number corresponds to product of the cardinalities (ie. domain sizes) of each variable within its scope).

The next line(s) enumerate the function values corresponding to each assignment of the variables within the function’s scope, separated by a whitespace (the whitespace optionally being a newline).

An example of a function specification is as follows:


6
0.210 0.333 0.457 0.811 0.000 0.189

The 6 indicates that there are a total of six possible assignments to the variables in the functions scope, and so the table has 6 values.

The next line contains the function values for the six different assignments possible to the variables within the function’s scope. Tuples acting as assignments to the variables are implicitly assumed in ascending order, with the first variable in its scope (as presented in the preamble) being the ‘most significant’ and the last variable in the scope as the ‘least significant’. For example, assuming our example function above has a scope of two variables Y and Z with cardinalities 2 and 3 respectively, the function values are presented in order for the assignments: (Y=0,Z=0), (Y=0,Z=1), (Y=0,Z=2), (Y=1,Z=0), (Y=1,Z=1), (Y=1,Z=2).

To illustrate this more comprehensively, we continue with our Markov network example from the preamble description assuming the following conditional probability tables for the functions outlined in our preamble (in order):

X P(X)
0 0.436
1 0.564
X Y P(Y,X)
0 0 0.128
0 1 0.872
1 0 0.920
1 1 0.080
Y Z P(Z,Y)
0 0 0.210
0 1 0.333
0 2 0.457
1 0 0.811
1 1 0.000
1 2 0.189

The associated function tables can look as follows:

2
 0.436 0.564

4
 0.128 0.872
 0.920 0.080

6
 0.210 0.333 0.457
 0.811 0.000 0.189

(Note that line breaks and empty lines act as a whitespace, exactly like plain spaces “ ”. They are used here to improve readability).

Summary

In summary, a problem file consists of two sections (with associated subsections):

  1. Preamble
    • Graph Type
    • Variables and Domains
    • Function Scopes
  2. Function Tables
    • blank line
    • Number of Table Values
    • Table Values

For our Markov network example above, the full .uai file will look like:

MARKOV
3
2 2 3
3
1 0
2 0 1
2 1 2

2
 0.436 0.564

4
 0.128 0.872
 0.920 0.080

6
 0.210 0.333 0.457
 0.811 0.000 0.189