Reasoning Under Uncertainty

1
Reasoning Under Uncertainty

• Artificial Intelligence
• Chapter 9

2
Part 2
Reasoning
3
Notation

• Random variable (RV) a variable (uppercase)that
  takes on values (lowercase) from a domainof
  mutually exclusive and exhaustive values
• Aa a proposition, world state, event, effect,
  etc.
• abbreviate P(Atrue) to P(a)
• abbreviate P(Afalse) to P(Øa)
• abbreviate P(Avalue) to P(value)
• abbreviate P(A¹value) to P(Øvalue)
• Atomic event a complete specification of the
  stateof the world about which the agent is
  uncertain

4
Notation

• P(a) a prior probability of RV Aawhich is the
  degree of belief proposition ain absence of any
  other relevant information
• P(ae) conditional probability of RV Aa given
  Eewhich is the degree of belief in proposition
  awhen all that is known is evidence e
• P(A) probability distribution, i.e. set of P(ai)
  for all i
• Joint probabilities are for conjunctions of
  propositions

5
Reasoning under Uncertainty

• Rather than reasoning about the truth or
  falsityof a proposition, instead reason about
  the belief that a proposition is true.
• Use knowledge base of known probabilities to
  determine probabilities for query propositions.

6
Reasoning under Uncertaintyusing Full Joint
Distributions

• Assume a simplified Clue game havingtwo
  characters, two weapons and two rooms

• each row is an atomic event- one of these must
  be true
• - list must be mutually exclusive
• - list must be exhaustive

Who What Where
plum rope hall
plum rope kitchen
plum pipe hall
plum pipe kitchen
green rope hall
green rope kitchen
green pipe hall
green pipe kitchen
Probability
1/8
1/8
1/8
1/8
1/8
1/8
1/8
1/8

• prior probability for each is 1/8
• - each equally likely
• - e.g. P(plum,rope,hall) 1/8

? P(atomic_eventi) 1 since each RV's domain
isexhaustive mutually exclusive
7
Determining Marginal Probabilitiesusing Full
Joint Distributions

• The probability of any proposition is equal
  tothe sum of the probabilities of the atomic
  eventsin which it holds, which is called the set
  e(a).
• P(a) ? P(ei)where ei is an element of e(a)
• its the disjunction of atomic events in set e(a)
• recall this property of atomic eventsany
  proposition is logically equivalent to the
  disjunctionof all atomic events that entail the
  truth of that proposition

8
Determining Marginal Probabilitiesusing Full
Joint Distributions

• Assume a simplified Clue game havingtwo
  characters, two weapons and two rooms

P(a) ? P(ei)where ei is an element of e(a)
Who What Where
plum rope hall
plum rope kitchen
plum pipe hall
plum pipe kitchen
green rope hall
green rope kitchen
green pipe hall
green pipe kitchen
Probability
1/8
1/8
1/8
1/8
1/8
1/8
1/8
1/8
P(plum) ?
1/81/81/81/8 1/2
when obtained in this manner it is called a
marginal probability can be just a prior
probability (shown) or more complex (next) this
process is called marginalization or summing out
9
Reasoning under Uncertaintyusing Full Joint
Distributions

• Assume a simplified Clue game havingtwo
  characters, two weapons and two rooms

Who What Where
plum rope hall
plum rope kitchen
plum pipe hall
plum pipe kitchen
green rope hall
green rope kitchen
green pipe hall
green pipe kitchen
Probability
1/8
1/8
1/8
1/8
1/8
1/8
1/8
1/8
P(green,pipe)
P(rope, Øhall)
P(rope Ú hall)
1/8
10
Independence

• Using the game cluefor an example is
  uninteresting! Why?
• Because the random variablesWho, What, Where are
  independent.
• Does picking the murder from the deck of cards
  affect which weapon is chosen? Location?
• No! Each is randomly selected.

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Independence

• Unconditional (absolute) IndependenceRVs have
  no affect on each other's probabilities
• 1. P(XY) P(X)
• 2. P(YX) P(Y)
• 3. P(X,Y) P(X) P(Y)
• Example (full clue)
• P(green hall) P(green, hall) / P(hall)
  6/324 / 1/9 P(green) 1/6
• P(hall green) P(hall) 1/9
• P(green, hall) P(green) P(hall) 1/54
• We need a more interesting example!

12
Independence

• Conditional IndependenceRVs (X, Y) are
  dependent on another RV (Z)but are independent
  of each other
• 1. P(XY,Z) P(XZ)
• 2. P(YX,Z) P(YZ)
• 3. P(X,YZ) P(XZ) P(YZ)
• Ideasneezing (x) and itchy eyes (y)are both
  directly caused by hayfever (z)
• but neither sneezing nor itchy eyeshas a direct
  effect on each other

13
Reasoning under Uncertaintyusing Full Joint
Distributions

• Assume three boolean RVs Hayfever HF, Sneeze SN,
  ItchyEyes IE

and fictional probabilities
P(a) ? P(ei)where ei is an element of e(a)
HF SN IE
false false false
false false true
false true false
false true true
true false false
true false true
true true false
true true true
Probability
0.5
0.09
0.1
0.1
0.01
0.06
0.04
0.1
P(sn) 0.1 0.1 0.04 0.10.34
P(hf) 0.01 0.06 0.04 0.10.21
P(sn,ie) 0.1 0.10.20
P(hf,sn) 0.04 0.10.14
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Reasoning under Uncertaintyusing Full Joint
Distributions

• Assume three boolean RVs Hayfever HF, Sneeze SN,
  ItchyEyes IE
• and fictional probabilities

HF SN IE
false false false
false false true
false true false
false true true
true false false
true false true
true true false
true true true
Probability
0.5
0.09
0.1
0.1
0.01
0.06
0.04
0.1
P(ae) P(a, e) / P(e)
P(hf sn) P(hf,sn) / P(sn) 0.14 /
0.34 0.41
P(hf ie) P(hf,ie) / P(ie) 0.16 /
0.35 0.46
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Reasoning under Uncertaintyusing Full Joint
Distributions

• Assume three boolean RVs Hayfever HF, Sneeze SN,
  ItchyEyes IE
• and fictional probabilities

HF SN IE
false false false
false false true
false true false
false true true
true false false
true false true
true true false
true true true
Probability
0.5
0.09
0.1
0.1
0.01
0.06
0.04
0.1
P(ae) P(a, e) / P(e)
Instead of computing P(e),could use normalization
P(hf sn) 0.14 / P(sn)
also computeP(Øhf sn) 0.20 / P(sn) since
P(hf sn) P(Øhf sn) 1 substituting and
solving gives P(sn) 0.34 !
16
Combining Multiple Evidence

• As evidence describing the state of the worldis
  accumulated, we'd like to be able to easily
  update the degree of belief in a conclusion.
• Using the Full Joint Prob. Dist. Table
• P(v1,...,vkvk1,...,vn) ?P(V1v1,...,Vnvn) /
  ?P(Vk1vk1,...,Vnvn)
• sum of all entries in the table, where V1v1,
  ..., Vnvn
• divided by the sum of all entries in the
  tablecorresponding to the evidence, where
  Vk1vk1, ..., Vnvn

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Combining Multiple Evidenceusing Full Joint
Distributions

• Assume three boolean RVs and fictional
  probabilities Hayfever HF, Sneeze SN, ItchyEyes
  IE

HF SN IE
false false false
false false true
false true false
false true true
true false false
true false true
true true false
true true true
Probability
0.5
0.09
0.1
0.1
0.01
0.06
0.04
0.1
P(ab, c) P(a,b,c) / ? P(b,c) as described in
prior slide
P(hf sn, ie) P(hf,sn,ie) / ? P(sn,ie)
0.10 / (0.10.1) 0.5
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Combining Multiple Evidence (cont.)

• FJDT techniques are intractable in general
  because the table size grows exponentially.
• Independence assertions can help reducethe size
  of the domain and the complexityof the inference
  problem.
• Independence assertions are usually basedon the
  knowledge of the domain enablingFJD table to be
  factored in to separate JD tables.
• it's a good thing that problem domains are
  independent
• but typically subsets of dependent RVs are quite
  large

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Probability Rulesfor Multi-valued Variables

• Summing Out P(Y) ? P(Y, z) sum over all
  values z of RV Z
• Conditioning P(Y) ? P(Yz) P(z) sum over
  all values z of RV Z
• Product Rule P(X, Y) P(XY) P(Y) P(YX) P(X)
• Chain Rule P(X, Y, Z) P(XY, Z) P(YZ) P(Z)
• this is a generalization of product rule with Y
  Y,Z
• order of RVs doesn't matter, i.e. gives same
  result
• Conditionalized Chain Rule(let YAB) P(X,
  AB) P(XA, B) P(AB)(order doesn't matter)
  P(AX, B) P(XB)

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Bayes' Rule

• Bayes' RuleP(ba) (P(ab) P(b)) / P(a)
• derived from P(a Ù b) P(ba) P(a) P(ab)
  P(b)just divide both sides of equation by P(a)
• basis of AI systems using probabilistic reasoning
• For Example
• ahappy, bsun a sneeze, b fall
• P(sunhappy) ? P(fallsneeze) ?
• P(happysun) 0.95 P(sneezefall)
  0.85P(sun) 0.5 P(fall)
  0.25P(happy)
  0.75 P(sneeze) 0.3
• (0.95 0.5)/0.75 0.63 (0.85 0.25)/0.3
  0.71

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Bayes' Rule

• P(ba) (P(ab) P(b)) / P(a)What's the benefit
  of being able to calculateP(ba) from the three
  probabilities on the right?
• Usefulness of Bayes' Rule
• many problems have good estimates of
  probabilities on right
• P(ba) needed to identify cause, classification,
  diagnosis, etc
• typical use is to calculate diagnostic
  knowledgefrom causal knowledge

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Bayes' Rule

• Causal knowledge from causes to effects
• e.g. P(sneezecold) probability of effect sneeze
  given cause common cold
• this probability the doctors obtains from
  experiencetreating patients and understanding
  the disease process
• Diagnostic knowledge from effects to causes
• e.g. P(coldsneeze)probability of cause common
  cold given effect sneeze
• knowing this probability helps a doctor make
  adisease diagnosis based on a patient's symptoms
• diagnostic knowledge is more fragile that causal
  knowledgesince it can change significantly over
  time given variationsin rate of occurrence of
  its causes (due to epidemics, etc.)

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Bayes' Rule

• Using Bayes' Rule with causal knowledge
• want to determine diagnostic knowledge
  (diagnostic reasoning)that is difficult to
  obtain from a general population
• e.g. symptom is sstiffNeck, disease is
  mmeningitis
• P(sm) 1/2 the casual knowledge
• P(m) 1/50000, P(s) 1/20 prior probabilities
• P(ms) ? desired diagnostic knowledge
• (1/2 1/50000)/ (1/20) 1/5000
• doctor can now use P(ms) to guide diagnosis

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Combining Multiple Evidenceusing Bayes' Rule

• How do you update conditional probabilityof Y
  given two pieces of evidence A and B?
• General Bayes' Rule for multi-valued RVsP(YX)
  (P(XY) P(Y)) / P(X)
• let XA,B
• P(YA,B) (P(A,BY) P(Y) ) / P(A,B) (P(Y)
  (P(BA,Y) P(AY)) / (P(BA) P(A))
• P(Y)(P(AY)/P(A))(P(BA,Y)/P(BA))
• conditionalized chain rule used, product rule
  used
• Problems
• P(BA,Y) generally hard to compute or obtain
• doesn't scale well for n evidence RVs, table size
  grows O(2n)

25
Combining Multiple Evidenceusing Bayes' Rule

• Problems can be circumvented
• If A and B are conditionally independent given
  Ythen P(A,BY) P(AY)P(BY) and for P(A,B) use
  product rule
• P(YA,B) (P(Y) P(A,BY) ) / P(A,B) Bayes' Rule
  Multi-E
• P(YA,B) P(Y) (P(AY)/P(A))
  (P(BY)/P(BA))
• No joint probabilities, representation grows O(n)
• If A is unconditionally independent of Bthen
  P(A,BY) P(AY)P(BY) and P(A,B) P(A)P(B)
• P(YA,B) (P(Y) P(A,BY) ) / P(A,B) Bayes' Rule
  Multi-E P(YA,B) P(Y) (P(AY)/P(A))
  (P(BY)/P(B))
• This equation used to define a naïve Bayes
  classifier.

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Combining Multiple Evidenceusing Bayes' Rule

• Example
• What is the likelihood that a patient has
  sclerosis colangitis?
• doctor's initial belief P(sc) 1/1,000,000
• examine reveals jaundice P(j)
  1/10,000 P(jsc) 1/5
• doctor's belief given test result P(scj)
  P(sc)P(jsc)/P(j) 2/1000
• tests reveal fibrosis of bile ducts P(fsc)
  4/5 P(f) 1/100
• doctor naïvely assumes jaundice and fibrosis are
  independent
• doctor's belief now rises P(scj,f) 16/100
  P(scj,f) P(sc)(P(j sc)/P(j)) (P(f
  sc)/P(f )) P(YA,B) P(Y) (P(AY)/P(A))(P(BY
  ) /P(B))

27
Naïve Bayes Classifier

• Naïve Bayes Classifierused where single class is
  based on a number of featuresor where single
  cause influences a number of effects
• based on P(YA,B) P(Y) (P(AY)/P(A))
  (P(BY)/P(B))
• given RV C
• domain is possible classifications say c1,c2,c3
• classifies input example of features F1, , Fn
• compute
• P(c1F1, , Fn), P(c2F1, , Fn), P(c3F1, , Fn)
• naïvely assume features are independent
• choose value for C that gives maximum probability
• works surprising well in practice evenwhen
  independence assumption aren't true

28
Bayesian Networks

• AKA Bayes Nets, Belief Nets, Causal Nets, etc.
• Encodes the full joint probability distribution
  (FJPD) for the set of RVs defining a problem
  domain
• Uses a space-efficient data structure by
  exploiting
• fact that dependencies between RVs are generally
  local
• which results in lots of conditionally
  independent RVs
• Captures both qualitative and quantitative
  relationships between RVs

29
Bayesian Networks

• Can be used to compute any value in FJPD
• Can be used to reason
• predictive/causal reasoningforward (top-down)
  from causes to effects
• diagnostic reasoningbackward (bottom-up) from
  effects to causes

30
Bayesian Network Representation

• Is an augmented DAG (i.e. directed, acyclic
  graph)
• Represented by V,E where
• V is a set of vertices
• E is a set of directed edges joining vertices, no
  loops
• Each vertex contains
• the RV's name
• either a prior probability distribution ora
  conditional probability distribution table
  (CDT)that quantifies the effects of the parents
  on this RV
• Each directed arc
• is from cause (parent) to its immediate effects
  (children)
• represents direct causal relationship between RVs

31
Bayesian Network Representation

• Example in class
• each row in conditional probability tables must
  sum to 1
• columns don't need to sum to 1
• values obtained from experts
• Number of probabilities required is typicallyfar
  fewer than the number required for a FJDT
• Quantitative information is usually givenby an
  expert or determined empirically from data

32
Conditional Independence

• Assume effects are conditionally independentof
  each other given their common cause
• The net is constructed so that given its
  parents,a node is conditionally independent of
  its non-descendant RVs in the net
• P(X1x1, ..., Xnxn) P(xi parents(Xi)) ...
  P(xn parents(Xn))
• Note the full joint probability distribution
  isn't needed, only need conditionals relative to
  their parent RVs

33
Algorithm for ConstructingBayesian Networks

• Choose a set of relevant random variables
• Choose an ordering for them
• Assume they're X1 .. Xm where X1 is first, X2 is
  second, etc.
• For i 1 to m
• add a new node for Xi to the network
• set Parents(Xi) to be a minimal subset of X1 ..
  Xi-1such that we have conditional independence
  of Xiand all other members of X1 ..Xi-1 given
  Parents(Xi)
• add directed arc from each node in Parents(Xi) to
  Xi
• non-root nodes define a conditional probability
  table P(Xi x combinations of
  Parents(Xi)) root nodes define prior
  probability distribution at Xi P(Xi)

34
Algorithm for ConstructingBayesian Networks

• For a given set of random variables (RVs)there
  is not, in general, a unique Bayesian Netbut all
  of them represent the same information
• For the best net, topologically sort RVs in step
  2
• each RV comes before all of its children
• first nodes are roots, then nodes they directly
  influence
• Best Bayesian Network for a problem has
• fewest number of probabilities and arcs
• easy to determine probabilities for the CDT
• Algorithm won't construct a net that violatesthe
  rules of probability

35
Computing Joint Probabilitiesusing a Bayesian
Network

• Use product rule
• Simplify using independence
• For Example

• Compute P(a,b,c,d) P(d,c,b,a)
• order RVS in joint probability bottom up D,C,B,A
• P(dc,b,a) P(c,b,a) Product Rule P(d,c,b,a)
  
• P(dc) P(c,b,a) Conditional Independ. of D
  given C
• P(dc) P(cb,a) P(b,a) Product Rule
  P(c,b,a)
• P(dc) P(cb,a) P(ba) P(a) Product Rule
  P(b,a)
• P(dc) P(cb,a) P(b ) P(a) Independence of
  B and A given no evidence

36
Computing Joint Probabilitiesusing a Bayesian
Network

• Any entry in the full joint dist. table(i.e.
  atomic event) can be computed!
• P(v1,...,vn) PP(viParents(Vi)) over i from 1
  to n
• e.g. given boolean RVs what is P(a,..,h,k,..,p)?

• P(a)P(b)P(c)P(da,b)P(eb,c)P(f)P(gd,e)P(h)
  P(kf,g)P(lgh)P(mk)P(nk)P(ok,l)P(pl)
• Note this is fast, i.e. linearin the number of
  nodes in the net!

37
Computing Joint Probabilitiesusing a Bayesian
Network

• How is any joint probability computed?
• sum the relevant joint probabilities
• e.g. Compute P(a,b)

• P(a,b,c,d) P(a,b,c,Ød) P(a,b,Øc,d)
  P(a,b,Øc,Ød)
• e.g. Compute P(c)
• P(a,b,c,d) P(a,Øb,c,d) PØa,b,c,d)
  PØa,Øb,c,d) P(a,b,c,Ød) P(a,Øb,c,Ød)
  P(Øa,b,c,Ød) P(Øa,Øb,c,Ød)
• A BN can answer any query (i.e. probability)
  about the domain by summing the relevant joint
  probs.
• Enumeration can require many computations!

38
Computing Conditional Probabilitiesusing a
Bayesian Network

• Basic task of probabilistic systemis to compute
  conditional probabilities.
• Any conditional probability can be computed
• P(v1,...,vkvk1,...,vn) ?P(V1v1,...,Vnvn) /
  ?P(Vk1vk1,...,Vnvn)
• Key problem is that the technique of
  enumeratingjoint probabilities can make the
  computations intractable (exponential in the
  number of RVs).

39
Computing Conditional Probabilitiesusing a
Bayesian Network

• These computations generally relyon the
  simplifications resulting fromthe independence
  of the RVs.
• Every variable that isn't an ancestorof a query
  variable or an evidence variableis irrelevant to
  the query.
• What ancestors are irrelevant?

40
Independence in a Bayesian Network

• Given a Bayesian Networkhow is independence
  established?

• A node is conditionally independent (CI)of its
  non-descendants, given its parents.
• e.g. Given D and E, G is CI of ?

41
Independence in a Bayesian Network

• Given a Bayesian Networkhow is independence
  established?

• A node is conditionally independent (CI)of its
  non-descendants, given its parents.
• e.g. Given D and E, G is CI of ?

A, B, C, F, H
e.g. Given F and G, K is CI of ?
42
Independence in a Bayesian Network

• Given a Bayesian Networkhow is independence
  established?

• A node is conditionally independentof all other
  nodes in the network givenits parents, children,
  and children'sparents, which is called a Markov
  blanket
• e.g. What is the Markov blanket for G?

G
Given this blanket G is CI of ? A, B, C, M,
N , O, P
What about absolute independence?
43
Computing Conditional Probabilitiesusing a
Bayesian Network

• The general algorithm for computingconditional
  probabilities is complicated.
• It is easy if the query involves nodesthat are
  directly connected to each other.
• examples assumed to use boolean RVs
• Simple causal inference P(EC)
• conditional prob. dist. of effect E given cause C
  as evidence
• reasoning in same direction as arc, e.g. disease
  to symptom
• Simple diagnostic inference P(QE)
• conditional prob. dist. of query Q given effect E
  as evidence
• reasoning in direction opposite of arc, e.g.
  symptom to disease

44
Computing Conditional ProbabilitiesCausal
(Top-Down) Inference

• Compute P(ec)
• conditional probability of effect Ee given cause
  Cc as evidence
• assume arc exists to E from C and C2

1. Rewrite conditional probability of e in termsof
   e and all of its parents (that aren't
   evidence)given evidence c
2. Re-express each joint probability backto the
   probability of e given all of its parents
3. Simplify using independence and Look Uprequired
   values in the Bayesian Network

45
Computing Conditional ProbabilitiesCausal
(Top-Down) Inference

• Compute P(ec)
• P(e,c) / P(c) product rule
• (P(e,c,c2) P(e,c,Øc2)) / P(c)
  marginalizing
• P(e,c,c2) / P(c) P(e,c,Øc2) / P(c)
  algebra
• P(e,c2c) P(e,Øc2c) product
  rule, e.g. Xe,c2
• P(ec2,c) P(c2c) P(eØc2,c) P(Øc2c) cond.
  chain rule
• Simplify given C and C2 are independent
• P(c2c) P(c2)
• P(Øc2c) P(Øc2)
• P(ec2,c) P(c2) P(eØc2,c) P(Øc2)
  algebra
• now look up values to finish computation

46
Computing Conditional ProbabilitiesDiagnostic
(Bottom-Up) Inference

• Compute P(ce)
• conditional probability of cause Cc given effect
  Ee as evidence
• assume arc exists from C to E
• idea convert to casual inference using Bayes'
  rule
• Use Bayes' rule P(ce) P(ec) P(c) / P(e)
• Compute P(ec) using causal inference method
• Look up value of P(c) in Bayesian Net
• Use normalization to avoid computing P(e)
• requires computing P(Øce)
• using steps as in 1 3 above

47
Summary the Good News

• Bayesian Nets are the bread and butter
  ofAI-uncertainty community (like resolution to
  AI-logic)
• Bayesian Nets are a compact representation
• don't require exponential storage to holdall of
  the info in the full joint probability
  distribution (FJPD) table
• are a decomposed representation of the FJPD table
• conditional probability distribution tables in
  non-root nodes are only exponential in the max
  number of parents of any node
• Bayesian Nets are fast at computing joint
  probs P(V1, ..., Vk) i.e. prior probability of
  V1, ..., Vk
• computing the probability of an atomic event can
  be donein linear time with the number of nodes
  in the net

48
Summary the Bad News

• Conditional probabilities can also be computed
• P(QE1, ..., Ek)posterior probability of query
  Q given multiple evidence E1, ..., Ek
• requires enumerating all of the matching
  entries,which takes exponential time in the
  number of variables
• in special cases it can be done faster, lt
  polynomial timee.g. polytree linear time for
  nets structured like trees
• In general, inference in Bayesian Networks
  (BN)is NP-hard. ?
• but BNs are well studied so there exists many
  efficient exact solution methods as well as a
  variety of approximation techniques