bayesian networks using random variables to represent objects and events in the world –various...
TRANSCRIPT
Bayesian Networks• Using random variables to represent
objects and events in the world– Various instantiations to these variables can
model the current state of the world• Computing joint probabilities over these variables
• Estimating joint probabilities over all random variables often leads to a combinatorial explosion– Estimating probabilities of every combination
of values for the variables involved
Conditional independence• Use the chain rule to replace joint
probabilities with conditional ones:P(A1, A2, A3, A4, A5) = P(A1 | A2, A3, A4,A5) · P(A2 | A3, A4, A5) · P(A3 | A4, A5) · P(A5)
• Bayesian Networks allow to reduce the terms even further, by taking into account conditional independence:– P(A|C1, …, Cn, U) = P(A|C1, …, Cn) for some
set U of random variables– Given {C1, …, Cn}, A is independent of U
Conditional independence (cont’d)• The structure of a Bayesian Network
reflects a number of independence assumptions (d-separation criterion)
• Directed arcs between random variables represent conditional dependencies
• Combined with the chain rule, conditional independencies allow us to manipulate smaller conditional probability tables to estimate the joint probabilities
Ind( Xi ; {X1,…,Xi-1}\Pai | Pai )
where Pai are parents of Xi
Formal notation• Each random variable A is conditionally
independent of all other random variables that are not descendants of A, given A’s parents
n
iiin xpxxp
11 )|(),,( pa
n
iiin xxxpxxp
1111 ),|(),,(
Example: modeling real world• “Mary walks outside and finds that the street and
lawn are wet. She concludes that it has just rained recently. Furthermore, she decides that she doesn’t need to water her roses.”
• Mary’s logic:– rain or sprinklers street = wet– rain or sprinklers lawn = wet– lawn = wet soil = moist– soil = moist roses = OK
Mary’s world (cont’d)• Let’s compute the probability of the
following (intuitively unlikely) world event:– The roses are OK– The soil is dry– The lawn is wet– The street is wet– The sprinklers are off– It’s raining
Where are the missing probabilities, e.g., P(sprinklers=F) or P(street=dry|rain=T, sprinklers=T) ?
Computing the probability of a complex event
• P(sprinklers = F, rain = T, street = wet, lawn = wet, soil = dry, roses = OK) =
P(roses = OK | soil = dry) *
P(soil = dry | lawn = wet) *
P(lawn = wet | rain = T, sprinklers = F) * P(street = wet | rain = T, sprinklers = F) * P(sprinklers = F) * P(rain = T) =
0.2 * 0.1 * 1.0 * 1.0 * 0.6 * 0.7 = 0.0084
The event is quite unlikely !
Usage scenarios
• There are 2 types of computations performed with Bayesian Networks– Belief updating
• Computation of probabilities over random variables
– Belief revision• Finding maximally probably global assignment• Given some evidence or observation, our task is to
come up with a set of hypotheses (variable assignments) that together constitute the most satisfactory explanation/interpretation of the evidence at hand.
Belief revision• Let W be a set of all random variables in
the network
• Let e be the evidence (e is a subset of W)
• Any instantiation of all the variables in W that is consistent with e is called an explanation or interpretation of e
• The problem is to find an explanation w* such that P(w* | e) = max P(w | e)– w* is called the most probable explanation
Belief updating• Computing marginal probabilities of a
subset of random variables given the evidence
• Typically, the task is to determine the best instantiation of a single random variable given the evidence
Belief revision - example• Evidence: e = {roses = OK}
• Goal: determine the assignment to all the random variables that maximizes P(W|e)– P(e) is constant, and e is a subset of W
it’s sufficient to maximize P(W)
– Intuitively, non-evidence random variables in W can be viewed as possible hypotheses for e
• Solution: P(sprinklers=F, rain=T, street=wet, lawn=wet, soil=wet, roses=OK) = 0.2646
Belief updating - example• Evidence: e = {roses = OK}
• Goal: determine the probability that the lawn is either wet or dry given this observation– P(lawn=dry | roses = OK) = 0.1190– P(lawn=wet | roses = OK) = 0.8810