Probabilistic Reasoning Chapter 14 Capturing uncertain knowledge Probabilistic

Probabilistic Reasoning Chapter 14 • Capturing uncertain knowledge • Probabilistic inference CSE 471/598, CBS 598 by H. Liu Copyright, 1996 © Dale Carnegie & Associates, Inc.

Knowledge representation Joint probability distribution n can answer any question about the domain can become intractably large as #RV grows can be difficult to specify P for atomic events Conditional independence can simplify P assign’t A data structure - a belief network or Bayesian network that represents the dependence between variables and gives a concise specification of the joint. CSE 471/598, CBS 598 by H. Liu 2

A Bayesian network is a graph: n n A set of random variables A set of directed links connects pairs of nodes Each node has a conditional P table that quantifies the effects that the parents have on the node The graph has no directed cycles (DAG) It is usually much easier for an expert to decide conditional dependence relationships than specifying probabilities n Sometimes, experts can have very different opinions CSE 471/598, CBS 598 by H. Liu 3

Once the network is specified, we need only specify conditional probabilities for the nodes that participate in direct dependencies, and use those to compute any other probabilities. A simple Bayesian network (Fig 14. 1) An example of burglary-alarm-call (Fig 14. 2) The topology of the network can be thought of as the general structure of the causal process. Many details (Mary listening to loud music, or phone ringing and confusing John) are summarized in the uncertainty associated with the links from Alarm to John. Calls and Mary. Calls. CSE 471/598, CBS 598 by H. Liu 4

The probabilities actually summarize a potentially infinite set of possible circumstances Specifying the CPT for each node (Fig 14. 2) n n n A conditioning case - a possible combination of values for the parent nodes (2^n) Each row in a CPT must sum to 1 A node with no parents has only one row (priors) CSE 471/598, CBS 598 by H. Liu 5

The semantics of Bayesian networks Two equivalent views of a Bayesian network n n Representing the JPD - helpful in understanding how to construct networks Representing conditional independence relations helpful in designing inference procedures CSE 471/598, CBS 598 by H. Liu 6

1. Representing JPD - constructing a BN A Bayesian network provides a complete description of the domain. Every entry in the JPD can be calculated from the info in the network. A generic entry in the joint is the probability of a conjunction of particular assignments to each variable. P(x 1, …, xn)= P(xi|Parents(xi)) (14. 1) What’s the probability of the event of J^M^A^!B^!E? n n =P(j|a)P(m|a)P(a|!b^!e)P(!b)P(!e) Find the values in Figure 14. 2 and done CSE 471/598, CBS 598 by H. Liu 7

A method for constructing Bayesian networks Eq 14. 1 defines what a given BN means but implies certain conditional independence relationships that can be used to guide the construction. P(x 1, …, xn)=P(xn|xn-1, …, x 1)P(xn-1, …, x 1) P(Xi|Xi-1, …, X 1)=P(Xi|Parents(Xi)) (14. 2) The BN is a correct representation of the domain only if each node is C-indep’t of its predecessors in the node ordering, given its parents. n E. g. , P(M|J, A, E, B)=P(M|A) CSE 471/598, CBS 598 by H. Liu 8

Incremental network construction Choose relevant variables describing the domain Choose an ordering for the variables While there are variables left: n n n Pick a var and add a node to the network Set its parents to some minimal set of nodes already in the net to satisfy Eq. 14. 2 Define the CPT for the var. CSE 471/598, CBS 598 by H. Liu 9

Compactness A Bayesian network can often be far more compact than the full joint. In a locally structured system, each subcomponent interacts directly with only a bounded number of other components. Local structure is usually associated with linear rather than exponential growth in complexity. With 30 (n) nodes, if a node is directly influenced by 5 (k) nodes, what’s the difference between BN & joint? n 30*2^5 vs. 2^30, or n*2^k vs. 2^n CSE 471/598, CBS 598 by H. Liu 10

Node ordering The correct order to add nodes is to add the “root causes” first, then the variables they influence, and so on until we reach the leaves that have no direct causal influence on the other variables. What happens if we happen to choose the wrong order? Fig 14. 3 shows an example. If we stick to a true causal model, we end up having to specify fewer numbers, and the numbers will often be easier to come up with. CSE 471/598, CBS 598 by H. Liu 11

2. Conditional independence relations In design inference algorithms, we need to know if more general conditional independences hold. Given a network, can we know if a set of nodes X is independent of another set Y, given a set of evidence nodes E? It boils down non-descendants. n As in Fig 14. 2, John. Calls is indept of Burglary and Earthquake, given Alarm. A node is cond independent of all other nodes in the network, given its parents, children, and children’s parents (its Markov blanket). n Burglary is indept of John. Calls and Mary. Calls, given Alarm and Earthquake CSE 471/598, CBS 598 by H. Liu 12

Representation of CPTs Given canonical distributions, the complete table can be specified by naming the distribution with some parameters. A deterministic node has its value specified exactly by the values of its parents. Uncertain relationships can often be characterized by “noisy” logical relationships. An example for determine cond probabilities starting with P(!Fever) on page 501 given the individual inhibition probabilities as n P(!fe|c, !f, !m) = 0. 6, P(!fe|!c, f, !m) = 0. 2, and P(!fe|!c, !f, m) = 0. 1 CSE 471/598, CBS 598 by H. Liu 13

Inference in Bayesian networks Exact inference n n Inference by enumeration The variable elimination algorithm The complexity of exact inference Clustering algorithms Approximate inference n Direct sampling methods w Rejection sampling w Likelihood weighting n Inference by Markov chain simulation CSE 471/598, CBS 598 by H. Liu 14

Knowledge engineering for uncertain reasoning Decide what to talk about Decide on a vocabulary of random variables Encode general knowledge about the dependence Encode a description of the specific problem instance Pose queries to the inference procedure and get answers CSE 471/598, CBS 598 by H. Liu 15

Other approaches to uncertain reasoning Different generations of expert systems n n n Strict logic reasoning (ignore uncertainty) Probabilistic techniques using the full Joint Default reasoning - believed until a better reason is found to believe something else Rules with certainty factors Handling ignorance - Dempster-Shafer theory Vagueness - something is sort of true (fuzzy logic) Probability makes the same ontological commitment as logic: the event is true or false CSE 471/598, CBS 598 by H. Liu 16

Default reasoning The four-wheel car conclusion is reached by default. New evidence can cause the conclusion retracted, while FOL is strictly monotonic. Representatives are default logic, nonmonotonic logic, circumscription There are problematic issues n Details in Chapter 10 CSE 471/598, CBS 598 by H. Liu 17

Rule-based methods Logical reasoning systems have properties like: n n Monotonicity Locality Detachment Truth-functionality These properties are good for obvious computational advantages; bad as they’re inappropriate for uncertain reasoning. CSE 471/598, CBS 598 by H. Liu 18

Summary Reasoning properly n n In FOL, it means conclusions follow from premises In probability, it means having beliefs that allow an agent to act rationally Conditional independence info is vital A Bayesian network is a complete representation for the JPD, but exponentially smaller in size Bayesian networks can reason causally, diagnostically, intercausally, or combining two or more of the three. For polytrees, the computational time is linear in network size. CSE 471/598, CBS 598 by H. Liu 19