This is the second post in a sequence. For the introduction post, see here.

Graphical World Models

A world model is a mathematical model of a particular world. This can be our real world, or an imaginary world. To make a mathematical model into a model of a particular world, we need to specify how some of the variables in the model relate to observable phenomena in that world.

We introduce our graphical notation for building world models by creating an example graphical model of a game world. In the game world, a simple game of dice is being played. The player throws a green die and a red die, and then computes their score by adding the two numbers thrown.

We create the graphical game world model in thee steps:

We introduce three random variables and relate them to observations we can make when the game is played once in the game world. The variable $X$ represents the observed number of the green die, $Y$ is the red die, and $S$ is the score.
We draw a diagram:

We define the two functions that appear in the annotations above the nodes in the diagram: $\begin{matrix} D (d) = (i f d \in {1, 2, 3, 4, 5, 6} t h e n 1 / 6 e l s e 0), s u m (a, b) = a + b . \end{matrix}$

Informal interpretation of the graphical model

We can read the above graphical model as a description of how we might build a game world simulator, a computer program that generates random examples of game play. To compute one run of the game, the simulator would traverse the diagram, writing an appropriate observed value into each node, as determined by the function written above the node. Here are three possible simulator runs:

We can interpret the mathematical expression $P (S = 12)$ , the probability that $S$ equals 12, as being the exact probability that the next simulator run puts the number 12 into node $S$ .

We can interpret the expression $E (S)$ , the expected value of $S$ , as the average of the values that the simulator will put into $S$ , averaged over an infinite number of runs.

The similarity between what happens in the above drawings and what happens in a spreadsheet calculation is not entirely coincidental. Spreadsheets can be used to create models and simulations without having to write a full computer program from scratch.

Formal interpretation of the graphical model

In section 2.4 of the paper, I define the exact formal semantics of graphical world models. These formal definitions allow one to calculate the exact valu...

Counterfactual Planning