Causal Bayesian Networks

Causal Bayesian Networks Fl´vio Code¸o a c Coelho Basic graph theory Causal Bayesian Networks Bayesian Networks Inference Fl´vio Code¸o Coelho a c Oswaldo Cruz Foundation November 1, 2006

Graphs Causal Bayesian Sets of elements called vertices, V , that may or may not be Networks connected to other vertices in the same set by a set of edges,E Fl´vio Code¸o a c Coelho A graph may be deﬁned uniquely by its set of edges, wich imply Basic graph the set of vertices, e.g.E = {W , X , Y , Z }: theory Bayesian Networks G : E = {(W , Z ), (Z , Y ), (Y , X ), (X , Z )} Inference 1 by running the above code, you’ll get the following output: [ Y , X , Z , W ][( Y , X ), ( Y , Z ), ( X , Z ), ( Z , W )] 1 https://networkx.lanl.gov/

Some properties of graphs Causal Bayesian Networks Fl´vio Code¸o a c Coelho Graphs can be directed or undirected; Basic graph The order of a graph corresponds to its number of vertices; theory Bayesian The size of a graph corresponds to its number of edges; Networks Vertices connected by an edge are neighbors or adjacent; Inference The order of a vertex corresponds to its number of neighbors; 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.

Visualizing the graph Causal Bayesian Networks Fl´vio Code¸o a c Coelho Basic graph theory Bayesian Networks From the code above we get the following picture: Inference

Directed Acyclic Graph (DAG) Causal Bayesian Networks In directed acyclic graphs we use arrows to represent edges. Fl´vio Code¸o a c Coelho Basic graph theory Bayesian Networks The output: Inference

DAG properties Causal Bayesian Networks Fl´vio Code¸o a c Coelho Basic graph Parents,children,descendants,ancestors, etc. theory Root node Bayesian Networks sink node Inference 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.

Bayesian Networks Causal Bayesian Networks Fl´vio Code¸o a Coelho c Advantages : Basic graph theory 1 Convenient means of expressing assumptions Bayesian Networks 2 economical representation of Joint probabilit functions Inference 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

Definitions Causal Bayesian Networks Fl´vio Code¸o a c Coelho Markovian parents (PAj ) P(xj | paj ) = P(xj | x1 , . . . , xj−1 ) Basic graph theory such that no subset of PAj satisfies the above Bayesian equation. Networks Inference Markov compatibility If a probability function admits the factorization P(xi | x1 , . . . , xn ) = P(xi | pai ) relative to a DAG G we say that G and P are compatible or that P is Markov relative to G . d-separation Z d-separates X and Y iff Z blocks every path from a node in X to a node in Y .

Theorems Causal Bayesian Networks Fl´vio Code¸o a c Coelho Probabilistic Implications of d-separation If sets X and Y are d-separated by Z in a DAG G , then X is Basic graph theory independent of Y conditional on Z every distribution Bayesian compatible with G . Conversely, if X and Y are not d-separated Networks by Z in a DAG G , then X and Y are dependent conditional on Inference Z in at least one dist. compatible with G . Ordered Markov Condition A necessary and suﬃcient condition for a probability distribution P to be markovian relative a DAG G is that every variable be independent of all its predecessors in some oredering of the variables that agrees with the arrows of G .

Theorems, cont. Causal Bayesian Networks Fl´vio Code¸o a c Coelho Parental Markov Condition Basic graph A necessary and suﬃcient condition for a probability theory distribution P to be markovian relative a DAG G is that every Bayesian Networks variable be independent of all its nondescendants (in G ), Inference conditional on its parents. Observational Equivalence Two DAGs are observationally equivalent if and only if they have the same skeletons and the same sets of v-structures, that is, two converging arrows whose tails are not connected by an arrow.

Inference with Bayesian Networks Causal Bayesian Networks Fl´vio Code¸o a c Coelho Basic graph theory In the presence of a set of observations X the posterior Bayesian probability: s P(y , x, s) Networks Inference P(y | x) = y ,s P(y , x, s) can be calculated from a DAG G and the conditional probabilities P(xi | pai ) deﬁned on the families of G

Causal Bayesian Networks Causal Bayesian Networks DAGs constructed around Causal, instead of associational Fl´vio Code¸o a c Coelho information is mor intuitive and more reliable. Basic graph Causal relationships are a direct representations of our theory beliefs Bayesian Networks Direct representation of mechanisms Inference Simple to represent interventions thanks to modularity in the network Deﬁnition: Causal bayesian network Let P(v ) be a probability distribution on a set of V variables, and let Px (v ) denote the distributionresulting from the intervention do(X = x) that sets a subset X of variables to constants x.

Causal Bayesian Networks Causal Bayesian Networks Fl´vio Code¸o a c Deﬁnition (cont.): Causal bayesian network Coelho Denote by P∗ the set of all interventional distributions Basic graph theory Px (v ), X ⊆ V , including P(v ), which represents no Bayesian intervention (i.e., X = ∅). A DAG G is said to be a causal Networks bayesian network compatible with P∗ if and only if the Inference following 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 with X = x; 3 Px (vi | pai ) = P(vi | pai ) for all Vi ∈ X whenever pai is consistent with X = x.

Causal Bayesian Networks Fl´vio Code¸o a c Coelho Basic graph theory Bayesian Networks Thank you! Inference

Causal Bayesian Networks

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