Probabilistic Graphical Model Family
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A Probabilistic Graphical Model Family is a statistical model family that uses a graph structure to represent probability distributions between random variables.
- AKA: Statistical Graphical Models, GMs, PGMs.
- Context:
- It can be associated with a Graphical Statistical Model Instance.
- It can range from being a Directed Probabilistic Graphical Model Family to being an Undirected Probabilistic Graphical Model Family.
- It can range from being a Conditional Probabilistic Graphical Family to being a Joint Probabilistic Graphical Family
- It can range from being a Generative Probabilistic Graphical Metamodel Model to being a Discriminative Probabilistic Graphical Metamodel.
- It can be used for a Structured Data Modeling Task.
- Example(s):
- a Factor Graph Model Family, for a Factor Graph network.
- a … for a Markov Probabilistic Graphical Model.
- a Bayesian Graph Model Family, for Bayesian networks.
- a Conditional Random Fields, for a CRF network.
- a … for a Markov Random Field.
- a … for a Maximum-Margin Markov Network.
- a Latent Dirichlet Allocation Model ? Metamodel.
- …
- Counter-Example(s):
- See: Conceptual Graph Model.
References
2020
- (Wikipedia, 2020) ⇒ https://en.wikipedia.org/wiki/Graphical_model Retrieved:2020-2-1.
- 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. (...)
Generally, probabilistic graphical models use a graph-based representation as the foundation for encoding a distribution over a multi-dimensional space and a graph that is a compact or factorized representation of a set of independences that hold in the specific distribution. Two branches of graphical representations of distributions are commonly used, namely, Bayesian networks and Markov random fields. Both families encompass the properties of factorization and independences, but they differ in the set of independences they can encode and the factorization of the distribution that they induce.[1]
- 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.
- ↑ Koller, D.; Friedman, N. (2009). Probabilistic Graphical Models. Massachusetts: MIT Press. p. 1208. ISBN 978-0-262-01319-2. Archived from the original on 2014-04-27.
2014
- (FACTORIE Website, 2014) ⇒ http://factorie.cs.umass.edu/usersguide/UsersGuide030Overview.html
- Graphical models are a formalism in which a graph denotes the conditional dependence structure between random variables. The formalism is the marriage between probability theory and graph theory. It provides an elegant framework that combines uncertainty (probabilities) and logical structure (independence constraints) such that complex joint probability distributions over multiple variables that would have otherwise been intractable to represent or manipulate can instead be represented compactly and often manipulated efficiently. Since graphical models can straightforwardly express so many different probabilistic models, they have become a lingua-franca for statistics, machine learning, and data mining.
In graphical models, variables are depicted by the nodes a graph, drawn as circles, and dependencies among variables are depicted by edges, drawn either as directed (with arrows), or undirected (without arrows).
There are two main types of graphical models. .. Directed graphical models … Undirected graphical models.
- Graphical models are a formalism in which a graph denotes the conditional dependence structure between random variables. The formalism is the marriage between probability theory and graph theory. It provides an elegant framework that combines uncertainty (probabilities) and logical structure (independence constraints) such that complex joint probability distributions over multiple variables that would have otherwise been intractable to represent or manipulate can instead be represented compactly and often manipulated efficiently. Since graphical models can straightforwardly express so many different probabilistic models, they have become a lingua-franca for statistics, machine learning, and data mining.
2009
- (Nielsen & Jensen, 2009) ⇒ Thomas Dyhre Nielsen, and Finn Verner Jensen (2009). "Bayesian Networks and Decision Graphs". Springer Science & Business Media. ISBN: 0-387-68281-3, 0-387-68282-1, 978-0-387-68281-5, 978-0-387-68282-2.
- QUOTE: Probabilistic graphical models and decision graphs are powerful modeling tools for reasoning and decision making under uncertainty. As modeling languages they allow a natural specification of problem domains with inherent uncertainty, and from a computational perspective they support efficient algorithms for automatic construction and query answering. This includes belief updating, finding the most probable explanation for the observed evidence, detecting conflicts in the evidence entered into the network, determining optimal strategies, analyzing for relevance, and performing sensitivity analysis.
The book introduces probabilistic graphical models and decision graphs, including Bayesian networks and influence diagrams. The reader is introduced to the two types of frameworks through examples and exercises, which also instruct the reader on how to build these models.
- QUOTE: Probabilistic graphical models and decision graphs are powerful modeling tools for reasoning and decision making under uncertainty. As modeling languages they allow a natural specification of problem domains with inherent uncertainty, and from a computational perspective they support efficient algorithms for automatic construction and query answering. This includes belief updating, finding the most probable explanation for the observed evidence, detecting conflicts in the evidence entered into the network, determining optimal strategies, analyzing for relevance, and performing sensitivity analysis.
2008
- (Blei, 2008) ⇒ David M. Blei. (2008). “Modeling Science." Presentation. April 17, 2008
2006
- (Bishop, 2006) ⇒ Christopher M. Bishop. (2006). “Chapter 8: Graphical Models”. In: "Pattern Recognition and Machine Learning". New York, NY : Springer, 2006. DOI:10.1117/1.2819119 ISBN:0-387-31073-8, 1-493-93843-6, 978-0387-31073-2, 978-1493-93843-8.
- QUOTE: A graph comprises nodes (also called vertices) connected by links (also known as edges or arcs). In a probabilistic graphical model, each node represents a random variable (or group of random variables), and the links express probabilistic relationships between these variables. The graph then captures the way in which the joint distribution over all of the random variables can be decomposed into a product of factors each depending only on a subset of the variables. We shall begin by discussing Bayesian networks, also known as directed graphical models, in which the links of the graphs have a particular directionality indicated by arrows. The other major class of graphical models are Markov random fields, also known as undirected graphical models, in which the links do not carry arrows and have no directional significance. Directed graphs are useful for expressing causal relationships between random variables, whereas undirected graphs are better suited to expressing soft constraints between random variables. For the purposes of solving inference problems, it is often convenient to convert both directed and undirected graphs into a different representation called a factor graph.
2000
- (Valpola, 2000) ⇒ Harri Valpola. (2000). “Bayesian Ensemble Learning for Nonlinear Factor Analysis." PhD Dissertation, Helsinki University of Technology.
- QUOTE: graphical model: A graphical representation of the causal structure of a probabilistic model. Variables are denoted by circles and arrows are used for representing the conditional dependences.
1999
- (Cowell et al., 1999) ⇒ Robert Cowell, A. Philip Dawid, Steffen Lauritzen, and David Spiegelhalter. (1999). “Probabilistic Networks and Expert Systems." Springer. ISBN:978-0-387-98767-5
- QUOTE: Over the last ten years there has been a steady shift in the focus of attention from algorithms for propagating evidence towards methods for learning parameters and structure from data. This has been accompanied by a broadening of scope, into the general area of graphical modelling: a term with its roots in Statistics, but which also incorporates neural networks, hidden Markov models, and many other techniques that exploit conditional independence properties for modelling, display, and computation.
1998a
- (Jordan, 1998) ⇒ Michael I. Jordan (ed). (1998). “Learning in Graphical Models." MIT Press. ISBN:0-262-60032-3
1998b
- (Murphy, 1998) ⇒ Kevin Murphy. (1998). “A Brief Introduction to Graphical Models and Bayesian Networks."
1997a
- (Jordan, 1997) ⇒ Michael I. Jordan. (1997). “An Introduction to Graphical Models." Tutorial at NIPS-1997.
- Graphical models are a marriage between graph theory and probability theory
- They clarify the relationship between neural networks and related network-based models such as HMMs, MRFs, and Kalman lters
- Indeed, they can be used to give a fully probabilistic interpretation to many neural network architectures
- Some advantages of the graphical model point of view
- inference and learning are treated together
- supervised and unsupervised learning are merged seamlessly
- missing data handled nicely
- a focus on conditional independence and computational issues
- interpretability (if desired)
1997b
- (Mitchell, 1997) ⇒ Tom M. Mitchell. (1997). “Machine Learning." McGraw-Hill.
1996
- (Lauritzen, 1996) ⇒ S. Lauritzen. (1996). “Graphical Models.” Oxford.
- mathematical exposition of the theory of graphical models.
1988
- (Pearl, 1988) ⇒ Judea Pearl. (1988). “Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference." Morgan Kaufmann. ISBN:1558604790