In Bayesian inference, plate notation is a method of representing variables that repeat in a graphical model. Instead of drawing each repeated variable individually, a plate or rectangle is used to group variables into a subgraph that repeat together, and a number is drawn on the plate to represent the number of repetitions of the subgraph in the plate. The assumptions are that the subgraph is duplicated that many times, the variables in the subgraph are indexed by the repetition number, and any links that cross a plate boundary are replicated once for each subgraph repetition.

Example

In this example, we consider Latent Dirichlet allocation, a Bayesian network that models how documents in a corpus are topically related. There are two variables not in any plate; α is the parameter of the uniform Dirichlet prior on the per-document topic distributions, and β is the parameter of the uniform Dirichlet prior on the per-topic word distribution. The outermost plate represents all the variables related to a specific document, including \theta_i, the topic distribution for document i. The M in the corner of the plate indicates that the variables inside are repeated M times, once for each document. The inner plate represents the variables associated with each of the N_i words in document i: z_{ij} is the topic distribution for the jth word in document i, and w_{ij} is the actual word used. The N in the corner represents the repetition of the variables in the inner plate N_j times, once for each word in document i. The circle representing the individual words is shaded, indicating that each w_{ij} is observable, and the other circles are empty, indicating that the other variables are latent variables. The directed edges between variables indicate dependencies between the variables: for example, each w_{ij} depends on z_{ij} and β.

Extensions

A number of extensions have been created by various authors to express more information than simply the conditional relationships. However, few of these have become standard. Perhaps the most commonly used extension is to use rectangles in place of circles to indicate non-random variables—either parameters to be computed, hyperparameters given a fixed value (or computed through empirical Bayes), or variables whose values are computed deterministically from a random variable. The diagram on the right shows a few more non-standard conventions used in some articles in Wikipedia (e.g. variational Bayes):

Software implementation

Plate notation has been implemented in various TeX/LaTeX drawing packages, but also as part of graphical user interfaces to Bayesian statistics programs such as BUGS and BayesiaLab and PyMC.