Plate notation

Last updated January 06, 2024

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.^[1] 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.^[2]

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

Bayesian multivariate Gaussian mixture model using plate notation. Smaller squares indicate fixed parameters; larger circles indicate random variables. Filled-in shapes indicate known values. The indication [K] means a vector of size K; [D,D] means a matrix of size DxD; K alone means a categorical variable with K outcomes. The squiggly line coming from z ending in a crossbar indicates a switch -- the value of this variable selects, for the other incoming variables, which value to use out of the size-K array of possible values. Bayesian-gaussian-mixture-vb.svg — Bayesian multivariate Gaussian mixture model using plate notation. Smaller squares indicate fixed parameters; larger circles indicate random variables. Filled-in shapes indicate known values. The indication [K] means a vector of size K; [D,D] means a matrix of size D×D; K alone means a categorical variable with K outcomes. The squiggly line coming from z ending in a crossbar indicates a *switch* — the value of this variable selects, for the other incoming variables, which value to use out of the size-K array of possible values.

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):

Variables that are actually random vectors are indicated by putting the vector size in brackets in the middle of the node.
Variables that are actually random matrices are similarly indicated by putting the matrix size in brackets in the middle of the node, with commas separating row size from column size.
Categorical variables are indicated by placing their size (without a bracket) in the middle of the node.
Categorical variables that act as "switches", and which pick one or more other random variables to condition on from a large set of such variables (e.g. mixture components), are indicated with a special type of arrow containing a squiggly line and ending in a T junction.
Boldface is consistently used for vector or matrix nodes (but not categorical nodes).

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.

Related Research Articles

A hidden Markov model (HMM) is a Markov model in which the observations are dependent on a latent Markov process. An HMM requires that there be an observable process $whose outcomes depend on the outcomes of in a known way. Since cannot be observed directly, the goal is to learn about state of by observing By definition of being a Markov model, an HMM has an additional requirement that the outcome of at time must be "influenced" exclusively by the outcome of at and that the outcomes of and at must be conditionally independent of at given at time Estimation of the parameters in an HMM can be performed using maximum likelihood. For linear chain HMMs, the Baum-Welch algorithm can be used to estimate the parameters.$

A Bayesian network is a probabilistic graphical model that represents a set of variables and their conditional dependencies via a directed acyclic graph (DAG). While it is one of several forms of causal notation, causal networks are special cases of Bayesian networks. Bayesian networks are ideal for taking an event that occurred and predicting the likelihood that any one of several possible known causes was the contributing factor. For example, a Bayesian network could represent the probabilistic relationships between diseases and symptoms. Given symptoms, the network can be used to compute the probabilities of the presence of various diseases.

A graphical model or probabilistic graphical model (PGM) or structured probabilistic model is a probabilistic model for which a graph expresses the conditional dependence structure between random variables. They are commonly used in probability theory, statistics—particularly Bayesian statistics—and machine learning.

In statistics, Gibbs sampling or a Gibbs sampler is a Markov chain Monte Carlo (MCMC) algorithm for sampling from a specified multivariate probability distribution when direct sampling from the joint distribution is difficult, but sampling from the conditional distribution is more practical. This sequence can be used to approximate the joint distribution ; to approximate the marginal distribution of one of the variables, or some subset of the variables ; or to compute an integral. Typically, some of the variables correspond to observations whose values are known, and hence do not need to be sampled.

Belief propagation, also known as sum–product message passing, is a message-passing algorithm for performing inference on graphical models, such as Bayesian networks and Markov random fields. It calculates the marginal distribution for each unobserved node, conditional on any observed nodes. Belief propagation is commonly used in artificial intelligence and information theory, and has demonstrated empirical success in numerous applications, including low-density parity-check codes, turbo codes, free energy approximation, and satisfiability.

In statistics, a mixture model is a probabilistic model for representing the presence of subpopulations within an overall population, without requiring that an observed data set should identify the sub-population to which an individual observation belongs. Formally a mixture model corresponds to the mixture distribution that represents the probability distribution of observations in the overall population. However, while problems associated with "mixture distributions" relate to deriving the properties of the overall population from those of the sub-populations, "mixture models" are used to make statistical inferences about the properties of the sub-populations given only observations on the pooled population, without sub-population identity information.

<span class="mw-page-title-main">Dirichlet distribution</span> Probability distribution

In probability and statistics, the Dirichlet distribution (after Peter Gustav Lejeune Dirichlet), often denoted $, is a family of continuous multivariate probability distributions parameterized by a vector of positive reals. It is a multivariate generalization of the beta distribution, hence its alternative name of multivariate beta distribution (MBD) . Dirichlet distributions are commonly used as prior distributions in Bayesian statistics, and in fact, the Dirichlet distribution is the conjugate prior of the categorical distribution and multinomial distribution.$

<span class="mw-page-title-main">Markov random field</span> Set of random variables

In the domain of physics and probability, a Markov random field (MRF), Markov network or undirected graphical model is a set of random variables having a Markov property described by an undirected graph. In other words, a random field is said to be a Markov random field if it satisfies Markov properties. The concept originates from the Sherrington–Kirkpatrick model.

This glossary of statistics and probability is a list of definitions of terms and concepts used in the mathematical sciences of statistics and probability, their sub-disciplines, and related fields. For additional related terms, see Glossary of mathematics and Glossary of experimental design.

Variational message passing (VMP) is an approximate inference technique for continuous- or discrete-valued Bayesian networks, with conjugate-exponential parents, developed by John Winn. VMP was developed as a means of generalizing the approximate variational methods used by such techniques as latent Dirichlet allocation, and works by updating an approximate distribution at each node through messages in the node's Markov blanket.

Multilevel models are statistical models of parameters that vary at more than one level. An example could be a model of student performance that contains measures for individual students as well as measures for classrooms within which the students are grouped. These models can be seen as generalizations of linear models, although they can also extend to non-linear models. These models became much more popular after sufficient computing power and software became available.

In natural language processing, latent Dirichlet allocation (LDA) is a Bayesian network for modeling automatically extracted topics in textual corpora. The LDA is an example of a Bayesian topic model. In this, observations are collected into documents, and each word's presence is attributable to one of the document's topics. Each document will contain a small number of topics.

In probability theory, Dirichlet processes are a family of stochastic processes whose realizations are probability distributions. In other words, a Dirichlet process is a probability distribution whose range is itself a set of probability distributions. It is often used in Bayesian inference to describe the prior knowledge about the distribution of random variables—how likely it is that the random variables are distributed according to one or another particular distribution.

In probability theory and statistics, the Dirichlet-multinomial distribution is a family of discrete multivariate probability distributions on a finite support of non-negative integers. It is also called the Dirichlet compound multinomial distribution (DCM) or multivariate Pólya distribution. It is a compound probability distribution, where a probability vector p is drawn from a Dirichlet distribution with parameter vector $, and an observation drawn from a multinomial distribution with probability vector p and number of trials n . The Dirichlet parameter vector captures the prior belief about the situation and can be seen as a pseudocount: observations of each outcome that occur before the actual data is collected. The compounding corresponds to a Pólya urn scheme. It is frequently encountered in Bayesian statistics, machine learning, empirical Bayes methods and classical statistics as an overdispersed multinomial distribution.$

In probability theory and statistics, a categorical distribution is a discrete probability distribution that describes the possible results of a random variable that can take on one of K possible categories, with the probability of each category separately specified. There is no innate underlying ordering of these outcomes, but numerical labels are often attached for convenience in describing the distribution,. The K-dimensional categorical distribution is the most general distribution over a K-way event; any other discrete distribution over a size-K sample space is a special case. The parameters specifying the probabilities of each possible outcome are constrained only by the fact that each must be in the range 0 to 1, and all must sum to 1.

In computer vision, the problem of object categorization from image search is the problem of training a classifier to recognize categories of objects, using only the images retrieved automatically with an Internet search engine. Ideally, automatic image collection would allow classifiers to be trained with nothing but the category names as input. This problem is closely related to that of content-based image retrieval (CBIR), where the goal is to return better image search results rather than training a classifier for image recognition.

<span class="mw-page-title-main">Maximum cut</span> Problem of finding a maximum cut in a graph

In a graph, a maximum cut is a cut whose size is at least the size of any other cut. That is, it is a partition of the graph's vertices into two complementary sets $S$ and $T$ , such that the number of edges between $S$ and $T$ is as large as possible. Finding such a cut is known as the max-cut problem.

Bootstrapping populations in statistics and mathematics starts with a sample $observed from a random variable.$

Exponential Random Graph Models (ERGMs) are a family of statistical models for analyzing data from social and other networks. Examples of networks examined using ERGM include knowledge networks, organizational networks, colleague networks, social media networks, networks of scientific development, and others.

In statistics and machine learning, the hierarchical Dirichlet process (HDP) is a nonparametric Bayesian approach to clustering grouped data. It uses a Dirichlet process for each group of data, with the Dirichlet processes for all groups sharing a base distribution which is itself drawn from a Dirichlet process. This method allows groups to share statistical strength via sharing of clusters across groups. The base distribution being drawn from a Dirichlet process is important, because draws from a Dirichlet process are atomic probability measures, and the atoms will appear in all group-level Dirichlet processes. Since each atom corresponds to a cluster, clusters are shared across all groups. It was developed by Yee Whye Teh, Michael I. Jordan, Matthew J. Beal and David Blei and published in the Journal of the American Statistical Association in 2006, as a formalization and generalization of the infinite hidden Markov model published in 2002.

References

↑ Ghahramani, Zoubin (August 2007). Graphical models (Speech). Tübingen, Germany. Retrieved 21 February 2008.
↑ Buntine, Wray L. (December 1994). "Operations for Learning with Graphical Models" (PDF). Journal of Artificial Intelligence Research . AI Access Foundation. 2: 159–225. arXiv: cs/9412102 . Bibcode:1994cs.......12102B. doi:10.1613/jair.62. ISSN 1076-9757. S2CID 11672931 . Retrieved 21 February 2008.

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[1] Ghahramani, Zoubin (August 2007). Graphical models (Speech). Tübingen, Germany. Retrieved 21 February 2008.

[2] Buntine, Wray L. (December 1994). "Operations for Learning with Graphical Models" (PDF). Journal of Artificial Intelligence Research . AI Access Foundation. 2: 159–225. arXiv: cs/9412102 . Bibcode:1994cs.......12102B. doi:10.1613/jair.62. ISSN 1076-9757. S2CID 11672931 . Retrieved 21 February 2008.

[1]

[2]