Plate notation

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. [cite speech
title = Graphical models
author = Ghahramani, Zoubin
first = Zoubin
last = Ghahramani
date = 2007/08
location = Tübingen, Germany
url = http://videolectures.net/mlss07_ghahramani_grafm/
accessdate = 2008-02-21] 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. [cite journal
last = Buntine
first = Wray L.
title = Operations for Learning with Graphical Models
journal = Journal of Artificial Intelligence Research
volume = 2
pages = 159–225
publisher = AI Access Foundation
date = 1994-12
url = http://www.cs.cmu.edu/afs/cs/project/jair/pub/volume2/buntine94a.pdf
format = PDF
issn = 11076-9757
accessdate = 2008-02-21 ]

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 $heta\_i$, 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 for the "j"th 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 $N\_i$ 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".

References