Introduction to Artificial Intelligence Inference in Bayesian - - PowerPoint PPT Presentation

Introduction to Artificial Intelligence Inference in Bayesian networks

Janyl Jumadinova September 28-30, 2016

Inference tasks

◮ Simple queries: compute posterior probabilities ◮ Optimal decisions: decision networks include utility

information; probabilistic inference required for P(outcome|action, evidence)

◮ Value of information: which evidence to seek next? ◮ Sensitivity analysis: which probability values are most critical? 2/19

Bayesian Networks

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T: The lecture started by 10 : 35

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L: The lecturer arrives late

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R: The lecture concerns robots

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M: The lecturer is Masha

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S: It is sunny

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Bayesian Networks

Let’s say we want to find: P(R|T, S)?

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Bayesian Networks

Let’s say we want to find: P(R|T, S)?

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Bayesian Networks

Let’s say we want to find: P(RT, S)?

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Joint Probability: General Case

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Conditional probabilities by enumerating all matching entries in the joint are expensive:

Exponential in the number of variables

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Winter? (A) Sprinkler? (B) WetGrass? (D) Rain? (C) SlipperyRoad? (E) A ΘA true .6 false .4 A B ΘB|A true true .2 true false .8 false true .75 false false .25 A C ΘC|A true true .8 true false .2 false true .1 false false .9 B C D ΘD|B,C true true true .95 true true false .05 true false true .9 true false false .1 false true true .8 false true false .2 false false true false false false 1 C E ΘE|C true true .7 true false .3 false true false false 1

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Variable Elimination

Pr(D, E) ?

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Variable Elimination

Pr(D, E) ?

◮ We can sum out variables without having to construct the joint

probability distribution explicitly.

◮ Variables can be summed out while keeping the original

distribution, and all successive distributions, in factored form ( θ

• prior probability)

P(d, e) =

• a,b,c

θe|cθd|bcθc|aθb|aθa

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Variable Elimination

Pr(D, E) ?

◮ We can sum out variables without having to construct the joint

probability distribution explicitly.

◮ Variables can be summed out while keeping the original

distribution, and all successive distributions, in factored form ( θ

• prior probability)

P(d, e) =

• a,b,c

θe|cθd|bcθc|aθb|aθa

◮ This allows the procedure to sometimes escape the exponential

complexity of the brute-force method.

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Variable elimination: carry out summations right-to-left, storing intermediate results (factors) to avoid recomputation Factors are matrices indexed by the values of its argument variables

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Variable Elimination

Summing out a variable from a product of factors f :

• move any constant factors outside the summation
• add up submatrices in pointwise product of remaining factors
• x f1 × · · · × fk = f1 × · · · × fi
• x fi+1 × · · · × fk = f1 × · · · × fi × f ¯

X

assuming f1, . . . , fi do not depend on X Pointwise product of factors f1 and f2: f1(x1, . . . , xj, y1, . . . , yk) × f2(y1, . . . , yk, z1, . . . , zl) = f (x1, . . . , xj, y1, . . . , yk, z1, . . . , zl) E.g., f1(a, b) × f2(b, c) = f (a, b, c)

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Review:

Bayesian Inference

Bayesian inference is about the quantification and propagation of uncertainty, defined via a probability, in light of observations of the system. From Prior → Posterior

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Review:

Bayesian Inference

Bayesian inference is about the quantification and propagation of uncertainty, defined via a probability, in light of observations of the system. From Prior → Posterior

Reminder: A posterior probability is the probability of the event’s outcome given the data (observation). A prior probability is the probability of the event’s outcome before you collect the data (make observations).

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Bayesian Inference: determining posterior distributions in belief networks

◮ Exact inference by enumeration ◮ Exact inference by variable elimination ◮ Approximate inference by stochastic simulation 14/19

Bayesian Inference

◮ Exact inference by variable elimination:

• Exploit the structure of the network to eliminate (sum out)

the non-observed, non-query variables one at a time

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Bayesian Inference

◮ Exact inference by variable elimination:

• Exploit the structure of the network to eliminate (sum out)

the non-observed, non-query variables one at a time

• Finding an elimination ordering that results in the smallest

tree-width is NP-hard

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Bayesian Inference

◮ Exact inference by variable elimination:

• Exploit the structure of the network to eliminate (sum out)

the non-observed, non-query variables one at a time

• Finding an elimination ordering that results in the smallest

tree-width is NP-hard

◮ Approximate inference by stochastic simulation 15/19

Inference by stochastic simulation

Basic Idea:

• 1. Draw N samples from a sampling distribution
• 2. Compute an approximate posterior probability
• 3. Show this converges to the true probability

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Inference by stochastic simulation

◮ Rejection sampling: reject samples disagreeing with evidence 18/19

Inference by stochastic simulation

◮ Rejection sampling: reject samples disagreeing with evidence ◮ Likelihood weighting: use evidence to weight samples 18/19

Inference by stochastic simulation

◮ Rejection sampling: reject samples disagreeing with evidence ◮ Likelihood weighting: use evidence to weight samples ◮ Markov chain Monte Carlo (MCMC): sample from a

stochastic process whose stationary distribution is the true posterior

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Bayesian Inference

Class Exercise

• 1. Find Netlogo models titled Bayes1D, Drift, Spatial and

Monte Carlo Pi in the “cs370f2016-share/in-class/sep30 BaysianNetlogoModels” directory, and explore them in this order.

• 2. For the first three models, (in the Google form) in your own

words give a 1-2 sentence description of the model and comment on how it uses Bayesian inference.

• 3. For the last model (Monte Carlo), complete the documentation

inside the Netlogo’s “Info” tab and submit the updated model to your own repository.

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