Probabilistic Graphical Models

This notebook was inspired into CITE and its course at Coursera and Stanford University.



In [24]:

    
from nxpd import draw
import networkx as nx



In [25]:

    
def draw_graph(
    graph, labels=None
):
    # create networkx graph
    G = nx.DiGraph()
    G.graph['dpi'] = 120

    G.add_nodes_from(set([
        graph[k1][k2] 
        for k1 in range(len(graph)) 
        for k2 in range(len(graph[k1]))
    ]))
    G.add_edges_from(graph)
    return draw(G, show='ipynb')

Conditional Independence

For events $\alpha,\beta$, $P \vdash \alpha \perp \beta$ if:

\begin{equation} P(\alpha, \beta) = P(\alpha) P(\beta)\\ P(\alpha| \beta) = P(\alpha)\\ P(\beta|\alpha) = P(\beta)\\ \end{equation}

For (sets of) random variable $X, Y, Z$

$P \vdash (X \perp Y | Z) if:$

\begin{equation} P(X, Y|Z) = P(X|Z) P(Y|Z)\\ P(X|Y, Z) = P(X|Z)\\ P(Y|X, Z) = P(Y|Z)\\ P(X, Y, Z) \propto \phi_1(X, Z) \phi_2(Y, Z) \end{equation}



In [29]:

    
# Graphs
graph = [('Coin', 'X1'), ('Coin', 'X2')]
draw_graph(graph)









    Out[29]:

\begin{equation} P \nvdash X_1 \perp X_2\\ P \vdash (X_1 \perp X_2|C)\\ \end{equation}

D-Separation

Factorization -> Independence: BNs

Theorem: If P factorizes over G, and $d\_sep_G(X, Y|Z)$ then $P \vdash (X \perp Y|Z)$

References

https://class.coursera.org/pgm-003/

Table of Contents

Probabilistic Graphical Models

Conditional Independence

D-Separation

References