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An introduction to Bayesian networks Stochastic Processes Course Hossein Amirkhani Spring 2011. Outline. Introduction, Bayesian Networks , Probabilistic Graphical Models, Conditional Independence, I-equivalence. Introduction.

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an introduction to bayesian networks stochastic processes course hossein amirkhani spring 2011
An introduction to Bayesian networks

Stochastic Processes Course

Hossein Amirkhani

Spring 2011

outline
Outline
  • Introduction,
  • Bayesian Networks,
  • Probabilistic Graphical Models,
  • Conditional Independence,
  • I-equivalence.
introduction
Introduction
  • Our goal is to represent a joint distribution over some set of random variables .
  • Even in the simplest case where these variables are binary-valued, a joint distribution requires the specification of numbers.
  • The explicit representation of the joint distribution is unmanageable from every perspective:
    • Computationally, Cognitively, and Statistically.
bayesian networks
Bayesian Networks
  • Bayesian networks exploit conditional independenceproperties of the distribution in order to allow a compact and natural representation.
  • They are a specific type of probabilistic graphical models.
    • BNs are directed acyclic graphs (DAG).
probabilistic graphical models
Probabilistic Graphical Models
  • Nodes are the random variables in our domain.
  • Edges correspond, intuitively, to direct influence of one node on another.
probabilistic graphical models1
Probabilistic Graphical Models
  • Graphs are an intuitive way of representing and visualising the relationships between many variables.
  • A graph allows us to abstract out the conditional independence relationships between the variables from the details of their parametric forms.
    • Thus we can answer questions like: “Is A dependent on B given that we know the value of C ?” just by looking at the graph.
  • Graphical models allow us to define general message-passing algorithms that implement probabilistic inference efficiently.

Graphical models = statistics × graph theory × computer science.

conditional independence example 12
Conditional Independence: Example 1

Smoking

Lung Cancer

Yellow Teeth

conditional independence example 22
Conditional Independence: Example 2

Type of Car

Speed

Amount of speeding Fine

conditional independence example 3
Conditional Independence: Example 3

head-to-head at c

v-structure

conditional independence example 32
Conditional Independence: Example 3

Ability of team B

Ability of team A

Outcome of A vs. B game

d separation
D-separation
  • A, B, and C are non-intersecting subsets of nodes in a directed graph.
  • A path from A to B is blocked if it contains a node such that either
    • the arrows on the path meet either head-to-tail or tail-to-tail at the node, and the node is in the set C, or
    • the arrows meet head-to-head at the node, and neither the node, nor any of its descendants, are in the set C.
  • If all paths from A to B are blocked, A is said to be d-separated from B by C.
  • If A is d-separated from B by C, the joint distribution over all variables in the graph satisfies .
i equivalence
I-equivalence
  • Let be a distribution over . We define to be the set of independence assertions that hold in .
  • Two graph structures and over are I-equivalent if .
  • The set of all graphs over X is partitioned into a set of mutually exclusive and exhaustive I-equivalence classes.
the skeleton of a bayesian network
The skeleton of a Bayesian network
  • The skeleton of a Bayesian network graph over is an undirected graph over that contains an edge for every edge in .
immorality
Immorality
  • A v-structure is an immorality if there is no direct edge between X and Y.
relationship between immorality skeleton and i equivalence
Relationship between immorality, skeleton and I-equivalence
  • Let and be two graphs over . Then and have the same skeleton and the same set of immoralitiesif and only if they are I-equivalent.
  • We can use this theorem to recognize that whether two BNs are I-equivalent or not.
  • In addition, this theorem can be used for learning the structure of the Bayesian network related to a distribution.
    • We can construct the I-equivalence class for a distribution by determining its skeleton and its immoralities from the independence properties of the given distribution.
    • We then use both of these components to build a representation of the equivalence class.
identifying the undirected skeleton
Identifying the Undirected Skeleton
  • The basic idea is to use independence queries of the form for different sets of variables .
  • If and are adjacent in , we cannot separate them with any set of variables.
  • Conversely, if and are not adjacent in , we would hope to be able to find a set of variables that makes these two variables conditionally independent: we call this set a witness of their independence.
identifying the undirected skeleton1
Identifying the Undirected Skeleton
  • Let be an I-map of a distribution , and let and be two variables that are not adjacent in . Then either or .
  • Thus, if and are not adjacent in , then we can find a witness of bounded size.
  • Thus, if we assume that has bounded indegree, say less than or equal to d, then we do not need to consider witness sets larger than d.
identifying immoralities
Identifying Immoralities
  • At this stage we have reconstructed the undirected skeleton. Now, we want to reconstruct edge direction.
  • Our goal is to consider potential immoralitiesin the skeleton and for each one determine whether it is indeed an immorality.
  • A triplet of variables X, Z, Y is a potential immoralityif the skeleton contains but does not contain an edge between X and Y.
  • A potential immorality is an immorality if and only ifZ is not in the witness set(s) for X and Y.
representing equivalence classes
Representing Equivalence Classes
  • An acyclic graph containing both directed and undirected edges is called a partially directed acyclic graphor PDAG.
representing equivalence classes1
Representing Equivalence Classes
  • Let be a DAG. A chain graph is a class PDAG of the equivalence class of if shares the same skeleton as , and contains a directed edge if and only if all that are I-equivalent to contain the edge .
  • If the edge is directed, then all the members of the equivalence class agree on the orientation of the edge.
  • If the edge is undirected, there are two DAGs in the equivalence class that disagree with the orientation of the edge.
representing equivalence classes2
Representing Equivalence Classes
  • Is the output of Mark-Immoralities the class PDAG?
  • Clearly, edges involved in immoralities must be directed in K.
  • The obvious question is whether K can contain directed edges that are not involved in immoralities.
  • In other words, can there be additional edges whose direction is necessarily the same in every member of the equivalence class?
references
References
  • D. Koller and N. Friedman: Probabilistic Graphical Models. MIT Press, 2009.
  • C. M. Bishop: Pattern Recognition and Machine Learning. Springer, 2006.