CausalBayesianNetworks

Flavio CodecoCoelho

Basic graphtheory

BayesianNetworks

Inference

Causal Bayesian Networks

Flavio Codeco Coelho

Oswaldo Cruz Foundation

November 1, 2006

CausalBayesianNetworks

Flavio CodecoCoelho

Basic graphtheory

BayesianNetworks

Inference

Graphs

Sets of elements called vertices, V , that may or may not beconnected to other vertices in the same set by a set of edges,EA graph may be defined uniquely by its set of edges, wich implythe set of vertices, e.g.E = {W ,X ,Y ,Z}:

G : E = {(W ,Z ), (Z ,Y ), (Y ,X ), (X ,Z )}

1

by running the above code, you’ll get the following output:

[′Y ′,′ X ′,′ Z ′,′ W ′][(′Y ′,′ X ′), (′Y ′,′ Z ′), (′X ′,′ Z ′), (′Z ′,′ W ′)]

1https://networkx.lanl.gov/

CausalBayesianNetworks

Flavio CodecoCoelho

Basic graphtheory

BayesianNetworks

Inference

Some properties of graphs

Graphs can be directed or undirected;

The order of a graph corresponds to its number of vertices;

The size of a graph corresponds to its number of edges;

Vertices connected by an edge are neighbors or adjacent;

The order of a vertex corresponds to its number ofneighbors;

A path is a list of edges connecting two vertices;

A cycle is a path starting and ending in the same vertex;

A graph with no cycles is termed acyclic.

CausalBayesianNetworks

Flavio CodecoCoelho

Basic graphtheory

BayesianNetworks

Inference

Visualizing the graph

From the code above we get the following picture:

CausalBayesianNetworks

Flavio CodecoCoelho

Basic graphtheory

BayesianNetworks

Inference

Directed Acyclic Graph (DAG)

In directed acyclic graphs we use arrows to represent edges.

The output:

CausalBayesianNetworks

Flavio CodecoCoelho

Basic graphtheory

BayesianNetworks

Inference

DAG properties

Parents,children,descendants,ancestors, etc.

Root node

sink node

Every DAG has at least one root and one sink

Tree graph: every node has at most one parent

Chain graph: every node has at most on child

Complete graph: All possible edges exist.

CausalBayesianNetworks

Flavio CodecoCoelho

Basic graphtheory

BayesianNetworks

Inference

Bayesian Networks

Advantages:

1 Convenient means of expressing assumptions

2 economical representation of Joint probabilit functions

3 Facilitate efficient inferences from observations

Why Bayesian?

1 Subjective nature of input information

2 Reliance on Bayes conditioning for updating information

3 The distinction between causal and evidential reasoning

CausalBayesianNetworks

Flavio CodecoCoelho

Basic graphtheory

BayesianNetworks

Inference

Definitions

Markovian parents (PAj) P(xj | paj) = P(xj | x1, . . . , xj−1)such that no subset of PAj satisfies the aboveequation.

Markov compatibility If a probability function admits thefactorization P(xi | x1, . . . , xn) = P(xi | pai )relative to a DAG G we say that G and P arecompatible or that P is Markov relative to G .

d-separation Z d-separates X and Y iff Z blocks every pathfrom a node in X to a node in Y .

CausalBayesianNetworks

Flavio CodecoCoelho

Basic graphtheory

BayesianNetworks

Inference

Theorems

Probabilistic Implications of d-separationIf sets X and Y are d-separated by Z in a DAG G , then X isindependent of Y conditional on Z every distributioncompatible with G . Conversely, if X and Y are not d-separatedby Z in a DAG G , then X and Y are dependent conditional onZ in at least one dist. compatible with G .

Ordered Markov ConditionA necessary and sufficient condition for a probabilitydistribution P to be markovian relative a DAG G is that everyvariable be independent of all its predecessors in someoredering of the variables that agrees with the arrows of G .

CausalBayesianNetworks

Flavio CodecoCoelho

Basic graphtheory

BayesianNetworks

Inference

Theorems, cont.

Parental Markov ConditionA necessary and sufficient condition for a probabilitydistribution P to be markovian relative a DAG G is that everyvariable be independent of all its nondescendants (in G ),conditional on its parents.

Observational EquivalenceTwo DAGs are observationally equivalent if and only if theyhave the same skeletons and the same sets of v-structures, thatis, two converging arrows whose tails are not connected by anarrow.

CausalBayesianNetworks

Flavio CodecoCoelho

Basic graphtheory

BayesianNetworks

Inference

Inference with Bayesian Networks

In the presence of a set of observations X the posteriorprobability:

P(y | x) =

∑s P(y , x , s)∑

y ,s P(y , x , s)

can be calculated from a DAG G and the conditionalprobabilities P(xi | pai ) defined on the families of G

CausalBayesianNetworks

Flavio CodecoCoelho

Basic graphtheory

BayesianNetworks

Inference

Causal Bayesian Networks

DAGs constructed around Causal, instead of associationalinformation is mor intuitive and more reliable.

Causal relationships are a direct representations of ourbeliefs

Direct representation of mechanisms

Simple to represent interventions thanks to modularity inthe network

Definition: Causal bayesian networkLet P(v) be a probability distribution on a set of V variables,and let Px(v) denote the distributionresulting from theintervention do(X = x) that sets a subset X of variables toconstants x.

CausalBayesianNetworks

Flavio CodecoCoelho

Basic graphtheory

BayesianNetworks

Inference

Causal Bayesian Networks

Definition (cont.): Causal bayesian networkDenote by P∗ the set of all interventional distributionsPx(v), X ⊆ V , including P(v), which represents nointervention (i.e., X = ∅). A DAG G is said to be a causalbayesian network compatible with P∗ if and only if thefollowing three conditions hold for every Px ∈ P∗:

1 Px(v) is Markov relative to G;

2 Px(vi ) = 1 for all Vi ∈ X whenever vi is consistent withX = x ;

3 Px(vi | pai ) = P(vi | pai ) for all Vi ∈ X whenever pai isconsistent with X = x .

CausalBayesianNetworks

Flavio CodecoCoelho

Basic graphtheory

BayesianNetworks

InferenceThank you!