Transcript Lecture 11: Uncertainty

Intro to AI
Uncertainty
Ruth Bergman
Fall 2002
Why Not Use Logic?
• Suppose I want to write down rules about medical diagnosis:
Diagnostic rules: A x has(x,sorethroat)  has(x, cold)
Causal rules:
A x has(x,cold)  has(x, sorethroat)
• Clearly, this isn’t right:
Diagnostic case:
• we may not know exactly which collections of symptoms
or tests allow us to infer a diagnosis (qualification problem)
• even if we did we may not have that information
• even if we do, how do we know it is correct?
Causal rules:
• Symptoms don’t usually appear guaranteed; note logical
case would use contrapositive
• There are lots of causes for symptoms; if we miss one we
might get an incorrect inference
• How do we reason backwards?
Uncertainty
• The problem with pure FOL is that it deals with black
and write
• The world isn’t black and write because of uncertainty:
1. Uncertainty due to imprecision or noise
2. Uncertainty because we don’t know everything about the
domain
3. Uncertainty because in practice we often cannot acquire all
the information we’d like.
– As a result, we’d like to assign a degree of belief (or
plausibility or possibility) to any statement we make
– note this is different than a degree of truth!
Ways of Handling
Uncertainty
• MYCIN: operationalize uncertainty with the rules:
• a  b with certainty 0.7
• we know a with certainty 1
– ergo, we know b with 0.7
– but, we if we also know
• a  c with certainty 0.6
• b v c  d with certainty 1
– do we know d with certainty .7, .6, .88, 1, ....?
• suppose a ~e and ~e  ~d ....
– In a rule-based system, such non-local
dependencies are hard to catch
Probability
• Problems such as this have led people to invent lots
of calculi for uncertainty; probability still dominates
• Basic idea:
– I have some DoB (a prior probability) about some proposition
p
– I receive evidence about p; the evidence is related to p by a
conditional probability
– From these two quantities, I can compute an updated DoB
about p --- a posterior probability
Probability Review
• Basic probability is on propositions or propositional
statements:
– P(A) (A is a proposition)
• P(Accident), P(phonecall), P(Cold)
– P(X = v) (X is a random variable; v a value)
• P(card = JackofClubs), P(weather=sunny), ....
– P(A v B), P(A ^ B), P(~A) ...
– Referred to as the prior or unconditional probability
• The conditional probability of A given B
P(A | B) = P(A,B)/P(B)
– the product rule P(A,B) = P(A | B) * P(B)
• Conditional independence P(A | B) = P(A)
– A is conditionally independent of B
Probability Review
• The joint distribution of A and B
– P(A,B) = x ( equivalent to P(A ^ B) = x)
A=1 A=2 A=3
B = T .1
.1
.2
.4
B = F .2
.1
.3
.6
.3
.2
.5
1
P(A=1,B) = .1
P(A=1) = .1 + .2 = .3
P(A =1 | B) = .1/.4 = .25
Bayes Theorem
• P(A,B) = P(A | B) P(B) = P(B | A) P(A)
P(A|B) = P(B | A) P(A) / P(B)
• Example: what is the probability of meningitis when a patient
has a stiff neck?
P(S|M) = 0.5
P(M) = 1/50000
P(S) = 1/20
P(M|S) = P(S|M)P(M)/P(S) = 0.5 * 1/50000 / 1/20 = 0.0002
• More general
P(A | B , E) = P(B | A , E) P(A | E)/ P(B | E)
Alarm System Example
• A burglary alarm system is fairly reliable at detecting
burglary
• It may also respond to minor earthquakes
• Neighbors John and Mary will call when they hear the
alarm
• John always calls when he hears the alarm
• He sometimes confuses the telephone with the alarm
and calls
• Mary sometimes misses the alarm
• Given the evidence of who has or has not called, we
would like to estimate the probability of a burglary.
Alarm System Example
• P(Alarm|Burglary) A burglary alarm system is fairly reliable at
detecting burglary
• P(Alarm|Earthquake) It may also respond to minor earthquakes
• P(JohnCalls|Alarm), P(MaryCalls|Alarm) Neighbors John and
Mary will call when they hear the alarm
• John always calls when he hears the alarm
• P(JohnCalls|~Alarm) He sometimes confuses the telephone with
the alarm and calls
• Mary sometimes misses the alarm
• Given the evidence of who has or has not called, we would like
to estimate the probability of a burglary.
P(Burglary|JohnCalls,MaryCalls)
Influence Diagrams
• Another way to present this information is an
influence diagram
earthquake
burglary
alarm
John calls
Mary calls
Influence Diagrams
1. A set of random variables.
2. A set of directed arcs
An arc from X to Y means that X has influence on Y.
3. Each node has an associated conditional probability table.
4. The graph has no directed cycle.
earthquake
burglary
alarm
John calls
Mary calls
Conditional Probability
Tables
• Each row contains the conditional probability for a possible
combination of values of the parent nodes
• Each row must sum to 1
earthquake B E P(Alarm|B, E)
T
F
burglary
alarm
John calls
Mary calls
T T 0.95
0.05
T F 0.94
0.06
F T 0.29
0.71
F F 0.001
0.999
Belief Network for the Alarm
P(B)
0.001
earthquake
burglary
alarm
A P(A
)
T 0.90
F 0.05
John calls
B E
P(A)
T
T
F
F
0.95
0.94
0.29
0.001
T
F
T
F
Mary calls
P(E)
0.002
A P(A
)
T 0.70
F 0.01
The Semanics of Belief
Networks
• The probability that the alarm sounded but neither a burglary
nor an earthquake has occurred and both John and Mary call
– P(J ^ M ^ A ^ ~B ^ ~E) =
P(J | A) P(M | A) P(A | ~B ^ ~E) P(~B) P(~E) =
0.9 * 0.7 * 0.001 * 0.999* 0.998 = 0.00062
• More generally, we can write this as
– P(x1, ... xn) = πi P(xi | Parents(Xi))
Constructing Belief
Networks
1. Choose the set of variables Xi that describe the
domain
2. Choose an ordering for the variables
1. Ideally, work backward from observables to root causes
3. While there are variables left:
1. Pick a variable Xi and add it to the network
2. Set Parents{Xi} to the minimal set of nodes such that
conditional independence holds
3. Define the conditional probability table for Xi
• Once you’re done, its likely you’ll realize you need to
fiddle a little bit!
Node Ordering
• The correct order to add nodes is
– Add the “root causes” first
– Then the variables they
influence
– And so on…
John calls
Mary calls
• Alarm example: consider the
ordering
– MaryCalls, JohnCalls, Alarm,
Burglary, Earthquake
earthquake
– MaryCalls, JohnCalls,
Earthquake, Burglary, Alarm
burglary
alarm
Probabilistic Inference
• Diagnostic inference (from effets to causes)
– Given that JohnCalls, infer that P(B|J) = 0.016
• Causal inference (from causes to effects)
– Given Burglary, P(J|B) = 0.86 and P(M|B) = 0.67
• Intercausal inference (between causes of a common
effect)
– Given Alarm, P(B|A) = 0.376
– If Earthquake is also true, P(B|A^E) = 0.003
• Mixed inference (combining two or more of the
above)
– P(A|J ^ ~E) = 0.03
– P(B|J ^ ~E) = 0.017
Conditional Independence
D-separation
• if every undirected path from a set of nodes X to a set of nodes
Y is d-separated by E, then X and Y are conditionally
independent given E
• a set of nodes E d-separates two sets of nodes X and Y if every
undirected path from a node in X to a node in Y is blocked
given E
X
Z
Z
Z
E
Y
Conditional Independence
•
An undirected path from X to Y is blocked given E if there is a
node Z s.t.
1. Z is in E and there is one arrow leading in and one arrow
leading out
2. Z is in E and Z has both arrows leading out
3. Neither Z nor any descendant of Z is in E and both path
arrows lead into Z
X
Z
Z
Z
E
Y
An Inference Algorithm for
Belief Networks
• In order to develop an algorithm, we will assume our networks are
singly connected
– A network is singly connected if there is at most a single
undirected path between nodes in the network
• note this means that any two nodes can be d-separated by
removing a single node
– These are also known as polytrees.
• We will then consider a generic node X with parents U1...Um,
and children Y1 ... Yn.
– parents of Yi are Zi,j
– Evidence above X is Ex+; below is Ex-
Singly Connected Network
Ex+
…
U1
Um
X
Ex-
Z1j
Z1j
Y1
…
Y1
Inference in Belief
Networks
• P(X|Ex) = P(X | Ex+, Ex-) = k P(Ex- | X, Ex+) P(X | Ex+)
k P(Ex- | X) P(X | Ex+)
– the last follows by noting that X d-separates its parents and
children
• Now, we note that we can apply the product rule to the second
term
i
P(X | Ex+) = Σu P(X | u, Ex+) P(u | Ex+)
= Σu P(X | u) πi P(ui | EU/X)
again, these last facts follow from conditional independence
• Note that we now have a recursive algorithm: the first term in the
sum is just a table lookup; the second is what we started with on
a smaller set of nodes.
Inference in Belief
Networks
• P(X|E) = k P(Ex- | X) P(X | Ex+)
• The evaluation for the first expression is similar, but
more involved, yielding
P(X | Ex+) = k2 πi Σy P(Ex-| yi) Σz P(yi | X, zi ) πj P(zij | EZij/Yi)
• P(Ex-| yi) is a recursive instance of P(Ex- | X)
• P(yi | X, zi ) is a conditional probability table entry for Yi
• P(zij | EZij/Yi) is a recursive instance of the P(X|E)
calculation
The Algorithm
Support-Except(X,V) return P(X| Ex/v)
if EVIDENCE(X) then return point dist for X
else
calculate P(E-x/v| X) = evidence-except(X,V)
U  parents(X)
if U is empty
then return normalized P(E-x/v| X) P(X)
else
for each Ui in U
calculate and store P(Ui|Eui/X) = support-except(Ui,X)
return k P(Ex- | X) Σu P(X | u) πi P(ui | EU/X)
The Algorithm
Evidence-Except(X,V) return P(E-X\V| X )
Y  children[X] – V
if Y is empty
then return a uniform distribution
else
for each Yi in Y do
calculate P(E-Yi|yi) = Evidence-Except(Yi, null)
Zi = PARENTS(Yi) – X
foreach Zij in Zi
calculate P(Zij | Ezij\Yi) = Support-Except(Zij,Yi)
return k2 πi Σy P(Ex-| yi) Σz P(yi | X, zi ) πj P(zij | EZij/Yi)
The Call
• For a node X, call Support-Except(X,null)
PathFinder
• Diagnostic system for lymph node disease
• Pathfinder IV a Bayesian model
–
–
–
–
8 hrs devising vocabulary
35 hrs defining topology
40 hrs to make 14000 probability assessments
most recent version appears to outperform the
experts who designed it!
Other Uncertainty
Calculi
• Dempster-Shafer Theory
– Ignorance: there are sets which have no probability
– In this case, the best you can do, in some cases, is bound
the probability
– D-S theory is one way of doing this
• Fuzzy Logic
– Suppose we introduce a fuzzy membership function (a
degree of membership
– Logical semantics are based on set membership
– Thus, we get a logic with degrees of truth
• e.g. John is a big man  bigman(John) w. truth value 0.