Introduction to Causal Bayesian Networks, based on Judea Pearl's book.

1. Causal
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Oswaldo Cruz Foundation
November 1, 2006

2. Graphs
Causal
Bayesian Sets of elements called vertices, V , that may or may not be
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connected to other vertices in the same set by a set of edges,E
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Basic graph the set of vertices, e.g.E = {W , X , Y , Z }:
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G : E = {(W , Z ), (Z , Y ), (Y , X ), (X , Z )}
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by running the above code, you’ll get the following output:
[ Y , X , Z , W ][( Y , X ), ( Y , Z ), ( X , Z ), ( Z , W )]
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https://networkx.lanl.gov/

3. Some properties of graphs
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Basic graph The order of a graph corresponds to its number of vertices;
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The size of a graph corresponds to its number of edges;
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Vertices connected by an edge are neighbors or adjacent;
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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.

4. Visualizing the graph
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From the code above we get the following picture:
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5. Directed Acyclic Graph (DAG)
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In directed acyclic graphs we use arrows to represent edges.
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The output:
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6. DAG properties
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Parents,children,descendants,ancestors, etc.
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Root node
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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.

7. Bayesian Networks
Causal
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:
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theory 1 Convenient means of expressing assumptions
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2 economical representation of Joint probabilit functions
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Why Bayesian?
1 Subjective nature of input information
2 Reliance on Bayes conditioning for updating information
3 The distinction between causal and evidential reasoning

8. Definitions
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Markovian parents (PAj ) P(xj | paj ) = P(xj | x1 , . . . , xj−1 )
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such that no subset of PAj satisfies the above
Bayesian equation.
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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 .

9. Theorems
Causal
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Coelho Probabilistic Implications of d-separation
If sets X and Y are d-separated by Z in a DAG G , then X is
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theory independent of Y conditional on Z every distribution
Bayesian compatible with G . Conversely, if X and Y are not d-separated
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by Z in a DAG G , then X and Y are dependent conditional on
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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 .

10. Theorems, cont.
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Parental Markov Condition
Basic graph A necessary and sufficient condition for a probability
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distribution P to be markovian relative a DAG G is that every
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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.

11. Inference with Bayesian Networks
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Basic graph
theory In the presence of a set of observations X the posterior
Bayesian probability:
s P(y , x, s)
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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

12. Causal Bayesian Networks
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DAGs constructed around Causal, instead of associational
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Coelho information is mor intuitive and more reliable.
Basic graph Causal relationships are a direct representations of our
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beliefs
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Direct representation of mechanisms
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Simple to represent interventions thanks to modularity in
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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.

13. Causal Bayesian Networks
Causal
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Denote by P∗ the set of all interventional distributions
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Px (v ), X ⊆ V , including P(v ), which represents no
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intervention (i.e., X = ∅). A DAG G is said to be a causal
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bayesian network compatible with P∗ if and only if the
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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.

14. Causal
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Thank you!
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