Intelligent Systems 2II40 C7

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Intelligent Systems (2II40)C7

• Alexandra I. Cristea

October 2003
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VI. Uncertainty

• VI.1. Decision theory basics
• Uncertainty
• Probability
• Syntax
• Semantics
• Inference Rules
• VI.2. Probabilistic reasoning
• Conditional independence
• Bayesian networks syntax and semantics
• Exact inference
• Approximate inference

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VI.2.B. Belief networks (Bayesian networks)
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Return to belief network example

• Neighbors John and Mary promised to call if the
  alarm goes off sometimes it starts because of
  earthquakes. Is there a burglar?
• Variables Burglary, Earthquake, Alarm,
  JohnCalls, MaryCalls (n variables)

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Belief network example cont.
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Constructing belief networks

• Choose an ordering of variables X1,,Xn
• For i1 to n
• add Xi to the network
• select parents from X1,,Xi-1 such
  that
• P(XiParents(Xi))
  P(XiX1,,Xi-1)

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Constructing belief networks example
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Constructing belief networks example
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Constructing belief networks example
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Constructing belief networks example
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Constructing belief networks example
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Example car diagnosis

• Initial evidence engine wont start
• Testable variables (thin ovals), diagnosis
  variables (thick ovals), hidden variables
  (shaded) ensure sparse structure, reduce
  parameters

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Example car insurance

• Predict claim (medical, liability, property)
• Given data on application form (other unshaded
  nodes)

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Efficient conditional distributions

• CPT grows exponentially w. no. of parents
• CPT becomes infinite w. continuous variables
• Other, more compact methods are needed

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Compact conditional distributions - cont.

• Noisy-OR distributions model multiple
  noninteracting causes
• Parents U1 Uk include all causes (can add leak
  node)
• Independent failure probability qi for each cause
  alone ? P(XU1Uj,?Uj1, ?Uk)1 - ?ji1qi

Number of parameters linear in number of parents
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Hybrid (discrete continuous) networks

• Discrete (Subsidy? and Buys?)
• Continuous (Harvest and Cost)

• How to deal with this?

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Probability density functions

• Instead of probability distributions
• For continuous variables
• Ex. let X denote tomorrows maximum temperature
  in the summer in Eindhoven
• Belief that X is distributed uniformly between 18
  and 26 degree Celsius
• P(Xx) U18,26(x)
• P(X20,5) U18,26(20,5)0,125/C

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Hybrid (discrete continuous) networks

• Discrete (Subsidy? and Buys?)
• Continuous (Harvest and Cost)

• How to deal with this?

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Hybrid (discrete continuous) networks

• Discrete (Subsidy? and Buys?)
• Continuous (Harvest and Cost)

• Option 1 discretization
• possibly large errors, large CPTs
• Option 2 finitely parameterized canonical
  families
• Continuous variable, discrete continuous
  parents (e.g., Cost)
• Discrete variable, continuous parents (e.g.,
  Buys?)

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a) Continuous child variables

• Need one conditional density function for child
  variable given continuous parents, for each
  possible assignment to discrete parents
• Most common is the linear Gaussian model, e.g.
• Mean Cost varies linearly w. Harvest, variance is
  fixed
• Linear variation is unreasonable over the full
  range, but works OK if the likely range of
  Harvest is narrow

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Continuous child variables ex.

• All-continuous network w. LG distribution ?
  full joint is a multivariate Gaussian
• Discrete continuous LG network is a conditional
  Gaussian network, i.e., a multivariate Gaussian
  over all continuous variables for each
  combination of discrete variable values

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b) Discrete child, continuous parent

• P(buysCostc) ??((-c ??) / ?)
• with ? - threshold for buying
• Probit distribution
• ? - the integral on the standard normal
  distribution
• Logit distribution
• Uses the sigmoid function

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VI.2. Probabilistic reasoning

• Conditional independence
• Bayesian networks syntax and semantics
• Exact inference
• Exact inference by enumeration
• Exact inference by variable elimination
• Approximate inference
• Approximate inference by stochastic simulation
• Approximate inference by Markov chain Monte Carlo

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Exact inference w. enumeration
ndn n2n
dn 2n
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Enumeration algorithm

• Exhaustive depth-first enumeration O(n) space,
  O(dn) time

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Inference by variable elimination

• Enumeration is inefficient repeated computation
• e.g., computes P(Jtruea)P(Mtruea) for each
  value of e
• Variable elimination summation from right to
  left, storing intermediate results (factors) to
  avoid recomputation

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Variable elimination basic operations

• 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)
• Summing out a variable from a product of factors
• move any constant factors outside the summation
• ?xf1??fk f1??fi ?xfi1??fk f1? ?fi fX
• assuming f1,fi do not depend on X

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Example pointwise product
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Example pointwise product
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Variable elimination algorithm
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Complexity of exact inference

• Polytrees (singly connected network) network in
  which there is at most one undirected path
  between any two nodes
• Time, space complexity of exact inference on
  polytrees linear in size of network
• Multiply connected networks ?polytrees
• Variable elimination can have exponential time
  and space complexity
• inference in Bayesian networks is NP-hard
• includes inference in propositional logics as
  special case

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• VI.2. D. Approximate inference

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Inference by stochastic simulation

• Basic idea
• Draw N samples from a sampling distribution S
• Compute approximate posterior probability P
• Show it converges to the true probability P

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VI.2. D. Approximate inference

• Sampling from an empty network
• Rejection sampling reject samples disagreeing w.
  evidence
• Likelihood weighting use evidence to weight
  samples
• MCMC sample from a stochastic process whose
  stationary distribution is the true posterior

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i. Sampling from an empty network

• function PRIOR-SAMPLE(bn) returns an
  event sampled from the prior specified by bn
• x ? an event w. n elements
• for i1 to n do
• xi ? a random sample from
  P(Xiparents(Xi))
• return x
• P(Cloudy) lt0.5,0.5gt

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i. Sampling from an empty network cont.

• Probability that PRIOR-SAMPLE generates a
  particular event
• SPS(x1, xn) ? n i1P(XiParents(Xi))P(x1,xn)
  
• NPS (Yy) no. of samples generated for which Yy
  for any set of variables Y.
• Then, P(Yy) NPS(Yy)/N and
• lim N??? P(Yy) ?h SPS(Yy,Hh)
• ?h P(Yy,Hh)
• P(Yy)
• ? estimates derived from PRIOR-SAMPLE are
  consistent

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ii. Rejection sampling example

• Estimate P(RainSprinklertrue) ? using
• 100 samples
• 27 samples have Sprinklertrue out of these,
• 8 have Raintrue and
• 19 have Rainfalse.
• P(RainSprinklertrue)
• NORMALIZE(lt8,19gt) lt0.296,0.704gt
• Similar to a basic real-world empirical
  estimation procedure

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ii. Rejection sampling

• P(Xe) is estimated from samples agreeing with
  evidence e

PROBLEM a lot of collected samples are thrown
away!!
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iii. Likelihood weighting

• Idea
• fix evidence variables E,
• sample only nonevidence var., X, Y
• weight ?? sample by likelihood it accords to
  evidence E

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iii. Likelihood weighting example

• Estimate P(RainSprinklertrue,WetGrasstrue)

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iii. Likelihood weighting example

• Sample generation process
• w ? 1.0
• Sample P(Cloudy)lt0.5,0.5gt say true
• Sprinkler has value true, so
• w ? w ? P(Sprinklertrue Cloudytrue) 0.1
• Sample P(RainCloudytrue)lt0.8,0.2gt say true
• WetGrass has value true, so
• w ? w ?P(WetGrasstrueSprinklertrue,Raintrue)
  0.099

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iii. Likelihood weighting function
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iii. Likelihood weighting analysis

• Sampling probability for WEIGHTED-SAMPLE is
• SWS(y,e) ? l i1P(yiParents(Yi))
• Note pays attention to evidence in ancestors
  only ? somewhere in between prior and posterior
  distribution
• Weight for a given sample y,e, is
• w(y,e) ? n i1P(eiParents(Ei))
• Weighted sampling probability is
• SWS(y,e) w(y,e) ? l i1P(yiParents(Yi)) ? m
  i1P(eiParents(Ei)) P(y,e) by
  standard global semantics of network
• Hence, likelihood weighting is consistent
• But performance still degrades w. many evidence
  variables

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iv. MCMC inference

• State of network current assignment to all
  variables
• Generate next state by sampling one variable
  given Markov blanket
• Sample each variable in turn, keeping evidence
  fixed
• Approaches stationary distribution long-run
  fraction of time spent in each state is exactly
  proportional to its posterior probability

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Markov blanket - reminder

• Each node is conditionally independent of all
  others given its Markov blanket parents
• children childrens parents

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MCMC algorithm
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Homework 7

• Continue till step 8 with your project.