S3-SEMINAR ON DATA MINING -BAYESIAN NETWORKS- B. INFERENCE Master Universitario en Inteligencia...

S3-SEMINAR ON DATA MINING-BAYESIAN NETWORKS-

B. INFERENCE

Master Universitario en Inteligencia Artificial

Concha Bielza, Pedro Larrañaga

Computational Intelligence GroupDepartamento de Inteligencia ArtificialUniversidad Politécnica de Madrid

C.Bielza, P.Larrañaga -UPM- 2

Types of queries

Brute-force computation

Probabilistic logic sampling

Variable elimination algorithm

Message passing algorithm

Conceptos básicos

Inference in Bayesian networks

Exact inference:

Approximate inference:


Queries: posterior probabilitiesGiven some evidence e (observations),

Posterior probability of a target variable(s) X :

Other names: probability propagation, belief updating or revision…

Alarm

Earth.Burgl.

WCalls

News

?

Vector

Types of queriesQueries Brute-force VE Message Approx

answer queries about

P


Semantically, for any kind of reasoningPredictive reasoning or deductive (causal inference): predict effects

Alarm

Earth.Burgl.

WCalls

News?

Diagnostic reasoning (diagnostic inference): diagnose the causes

Alarm

Earth.Burgl.

WCalls

News

?

Symptoms|Disease

Disease|Symptoms


Target variable is usually a descendant of the evidence

Target variable is usually an ancestor of the evidence


More queries: maximum a posteriori (MAP)

Most likely configurations (abductive inference): event that best explains the evidence

Total abduction: search for

Partial abduction: search for

K most likely explanations

subset. of unobserved (explanation set)

all the unobserved

Alarm

Earth.Burgl.

WCalls

News

??

Alarm

Earth.Burgl.

WCalls

News

?

??

?


In general, cannot be computed component-wise, with max P(xi|e)


More queries: maximum a posteriori (MAP)


Use MAP for:

Classification: find most likely label, given the evidence

Explanation: what is the most likely scenario, given the evidence


More queries: decision-making

Optimal decisions (of maximum expected utility), with influence diagrams



Brute-force computation of P(X|e)

First, consider P(Xi), without observed evidence e. Conceptually simple but computationally complex

For a BN with n variables, each with its P(Xj|Pa(Xj)):

But this amounts to computing the JPD, often very inefficient and even intractable computationally

CHALLENGE: Without computing the JDP, exploit the factorization encoded by the BN and the distributive law (local computations)

Exact inference [Pearl’88; Lauritzen & Spiegelhalter’88]

Queries Brute-force VE Message Approx

Brute-force approach


Improving brute-forceUse the JPD factorization and the distributive law

Table with 32 inputs (JPD) (if binary variables)

Exact inferenceQueries Brute-force VE Message Approx

?


Improving brute-forceArrange computations effectively, moving some additions

over X5 and X3:

over X4:Biggest table with 8 (like the BN)



Variable elimination algorithmWanted:

A list with all functions of the problemSelect an elimination order of all variables (except i)For each Xk from , if F is the set of functions that involve Xk:

Delete F from the list

Add f’ to the listOutput: combination (multiplication) of all functions in the current list

Eliminate Xk= combine all the functions that contain this variable and marginalize out Xk

Compute

ONE variable





Repeat th

e a

lgorith

m fo

r each

targ

et

varia

ble


Example with Asia network


Visit to Asia (A)

Smoking (S)

Lung Cancer(L)

Tuberculosis(T)

Tub. or Lung Canc (E)

Bronchitis (B)

X-Ray (X) Dyspnea (D)


Brute-force approach

Compute P(D) by brute-force:


x b e l t s a

dxbeltsaPdP ),,,,,,,()(

Complexity is exponential in the size of the graph (number of variables *number of states for each variable)



not necessarily a probability term



4



Size = 8

Local computations (due to moving the additions)

Importance of the elimination ordering, but finding an optimal (minimum cost) is NP-hard [Arnborg et al.’87] (heuristics for good sequences)


Complexity is exponential in the max N. of var. infactors of the summation



Operates passing messages among the nodes of the network. Nodes act as processors that receive, calculate and send information. Called propagation algorithms


Clique tree propagation, based on the same principle as VE but with a sophisticated caching strategy that:Enables to compute the posterior prob. distr. of

all variables in twice the time it takes to compute that of one single variable

Works in an intuitive appealing fashion, namely message propagation


Basic operations for a node

Ask info(i,j): Target node i asks info to node j. Does it for all neighbors j. They do the same until there are no nodes to ask


Send-message(i,j): Each node sends a message to the node that asked him the info… until reaching the target nodeA message is defined over the intersection of domains of fi and fj. It is computed as:

And finally, we calculate locally at each node i:Target combines all received info with his info and marginalize over the target variable


Procedure for X2


Colle

ctE

vid

en

ceAsk


P(X2) as a message passing algorithm


?


VE as a message passing algorithm

Direct correspondence:


?

VE

Mess.


Computing prob. P(Xi|e) of all (unobserved) variables i at a time

We can perform the previous process for each node: but many messages are repeated!


Or, we can use 2 rounds of messages as follows:Select a node as a root (or pivot)Ask or collect evidence from the leaves toward the root (messages in downward direction). As VE.

Distribute evidence from the root toward the leaves (messages in upward direction)

Calculate marginal distributions at each node by local computation, i.e. using its incoming messages

This algorithm never constructs tables larger than those in the BN



X

1 1

12

22

34

56 7

778 8

8

CollectEvidence

Root node


First sweep:

DistributeEvidenceSecond sweep:


Networks with loops

If net is not a polytree, it does not work

Independence assumptions applied in the algorithm cannot be used here (now “any node separates the graph into 2 unconnected parts (polytrees)” does not hold)


Request/messages go in a cycle indefinitely(info goes through 2 paths and is counted twice)

Alternatives??


Complexity


Complexity of propagation algorithms in polytrees (i.e., without loops, cycles in the underlying undirected graph) is linear in the size (nodes+arcs) of the network [brute-force is exponential]Exact inference in multiply-connected BNs is an NP-complete problem [Cooper 1990]


Alternative: clustering methods [Lauritzen & Spiegelhalter’88]

Method implemented in the main BN software packagesTransform the BN into a probabilistically equivalent polytree by merging nodes, removing the multiple paths between two nodes


M

S B

C H

Metastatic cancer (M) is a possible cause of brain tumors (B) and an explanation for increased total serum calcium (S). In turn, either of these could explain a patient falling into a coma (C). Severe headache (H) is also associated with brain tumors.

Create a new node Z, that combines S and B

M

Z=S,B

C H

States of Z: {tt,ft,tf,ff}

P(Z|M)=P(S|M)P(B|M) since they are c.i. given M

P(H|Z)=P(H|B) since H c.i. of S given B


Alternative: clustering methods

Steps for the JUNCTION TREE CLUSTERING ALGORITHM:

1. Moralize the BN2. Triangulate the moral graph and obtain the

cliques3. Create the junction tree and its separators4. Compute new parameters5. Message passing algorithm


Transform BN into a polytree (slow, much memory if dense, but only once)

Belief updating(fast)

CO

MP

ILA

TIO

N


Inferencia aproximada

Why?

Because exact inference is intractable (NP-complete) with large (+40) and densely connected BNs

Both deterministic and stochastic simulation to find approximate answers

the associated cliques for the junction tree algorithm or the intermediate factors in the VE algorithm will grow in size, generating an exponential blowup in the number of computations performed

Approximate inferenceQueries Brute-force VE Message Approx


Stochastic simulation

Uses the network to generate a large number of cases (full instantiations) from the network distribution

Inferencia aproximada Approximate inferenceQueries Brute-force VE Message Approx

P(Xi|e) is estimated using these cases by counting observed frequencies in the samples. By the Law of Large Numbers, estimate converges to the exact probability as more cases are generatedApproximate propagation in BNs within an arbitrary tolerance or accuracy is an NP-complete problemIn practice, if e is not too unlikely, convergence is

quickly


Probabilistic logic sampling [Henrion’88]

2

1

6

4

3

5

When all the nodes have been visited, we have a case, an instantiation of all the nodes in the BN

A forward sampling algorithm

Given an ancestral ordering of the nodes (parents before children), generate from X once we have generated from its parents (i.e. from the root nodes down to the leaves)

Inferencia aproximada Approximate inferenceQueries Brute-force VE Message Approx

Repeat and use the observed frequenciesto estimate P(Xi|e)Use conditional prob.

given the known values of the parents


Software


Software


Software


genie.sis.pitt.edu

Software


http.cs.berkeley.edu/~murphyk/

Software


leo.ugr.es/elvira

Software

C.Bielza, P.Larrañaga -UPM-

S3-SEMINAR ON DATA MINING-BAYESIAN NETWORKS-

B. INFERENCE

Master Universitario en Inteligencia Artificial

Concha Bielza, Pedro Larrañaga

Computational Intelligence GroupDepartamento de Inteligencia ArtificialUniversidad Politécnica de Madrid

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