Chapter 9

Reasoning in Uncertain Situations

Contents

Uncertain situations Non-monotonic logic and reasoning Certainty Factor algebra Fuzzy logic and reasoning Dempster-Shafer theory of evidence Bayesian belief network Markov models CSC411 Artificial Intelligence 1

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Traditional Logic Based on predicate logic Three important assumptions:

– Predicate descriptions are sufficient w.r.t. to the domain – Information is consistent – Knowledge base grows monotonically Artificial Intelligence 2

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Non-monotonic Logic

Addresses the three assumptions of traditional logic – Knowledge is incomplete No knowledge about p: true or false?

Prolog – closed world assumption – Knowledge is inconsistent Based on how the world usually works Most birds fly, but Ostrich doesn’t – Knowledge base grows non-monotonically New observation may contradict the existing knowledge, thus the existing knowledge may need removal. Inference based on assumptions, how come if the assumptions are later shown to be incorrect Three modal operators are introduced Artificial Intelligence 3

Unless Operator

New information may invalidate previous results Implemented in TMS – Truth Maintenance Systems to keep track of the reasoning steps and preserve the KB consistency Introduce Unless operator – Support inferences based on the belief that its argument is not true – Consider p(X) unless q(X) p(Z) r(W)  s(W) – Monotonocity  r(X) If p(X) is true and not believe q(X) true then r(X) From above, conclude s(X). Later, change believe or find q(X) true, what happens?

Retract r(X) and s(X) – Unless deals with believe, not truth Either unknown or believed false Believed or known true CSC411 Artificial Intelligence 4

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Is-consistent-with Operator M

When reason, make sure the premises are consistent Format: M p – p is consistent with KB Consider –  X good_student(X)  M study_hard(X)  graduates(X) – For all X who is a good student, if the fact that X studies hard is consistent with KB, then X will graduate – Not necessary to prove that X study hard.

How to decide p is consistent with KB – Negation as failure – Heuristic-based and limited search Artificial Intelligence 5

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Default Logic

Introduce a new format of inference rules: – A(Z)  C(Z)  – Similar :B(Z)  C(Z) – If A(Z) is provable, and it is consistent with what we know to assume B(Z), then conclude Compare with is-consistent-with operator – Difference is the reasoning method In default logic, new rules are used to infer sets of plausible extensions – Example: X good_student(X)  :study_hard(X)   graduates(X) Y party(Y)  :not(study_hard(Y))  not(graduates(X)) Artificial Intelligence 6

Stanford Certainty Factor Algebra

Measure of confidence or believe Summation may not be 1 Simple case: – Confidence for: MB(H|E) – Confidence against: MD(H|E) – Properties: 1>MB(H|E)>0 while MD(H|E)=0, or 1>MD(H|E)>0 while MB(H|E)=0 – Put together CF(H|E) = MB(H|E) – MD(H|E) 1 > CF(H|E) > -1 CSC411 Artificial Intelligence 7

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CF Combination

Premises combination – CF( P and Q) = min(CF(P), CF(Q)) – CF( P or Q) = max(CF(P), CF(Q)) Rule CF: each rule has a confidence measure CF propagation – Rule R: P  Q with CF=CF(R) – CF(Q) = CF(P) – Rules R1: P1   CF(R) Rule combination Q: CF1(Q) = CF(P1)xCF(R1) – R2: P2  Q: CF2(Q) = CF(P2)xCF(R2) – CF(Q) = CF1+CF2 – (CF1xCF2) if both positive CF1+CF2 + (CF1xCF2) if both negative (CF1+CF2)/(1-min(|CF1|,|CF2|)) otherwise Artificial Intelligence 8

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Fuzzy Sets

Classic sets – Completeness: x in either A or ¬A – Exclusive: can not be in both A and ¬A Fuzzy sets – Violate the two assumptions – Possibility theory -- measure of confidence or believe – Probability theory – randomness – Process imprecision – Introduce membership function – Believe x  A in some degree between 0 and 1, inclusive Artificial Intelligence 9

The fuzzy set representation for “small integers.” CSC411 Artificial Intelligence 10

A fuzzy set representation for the sets short, medium, and tall males.

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Fuzzy Set Operations

Fuzzy set operations are defined as the operations of membership functions Complement: ¬A = C – mC = 1 – mA Union: A  B =C – mC = max(mA, mB) Intersection: A  B = C – mC = min(mA, mB) Difference: A – B = C – mC = max(0, mA-mB) Artificial Intelligence 12

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Fuzzy Inference Rules Rule format and computation

– If x is A and y is B then z is C mC(z) = min(mA(x), mB(y)) – If x is A or y is B then z is C mC(z) = max(mA(x), mB(y)) – If x is not A then z is C mC(z) = 1 – mA(x) Artificial Intelligence 13

The inverted pendulum and the angle θ and dθ/dt input values.

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The fuzzy regions for the input values θ (a) and dθ/dt (b).

N – Negative, Z – Zero, P – Positive CSC411 Artificial Intelligence 15

The fuzzy regions of the output value u, indicating the movement of the pendulum base: Negative Big, Negative, Zero, Positive, Positive Big .

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The fuzzificzation of the input measures X 1 = 1: mZ(X 1 ) = mP(X 1 ) = 0.5, mN(X 1 ) = 0 X 2 = -4: mZ(X 2 ) = 0.2, mN(X 2 ) = 0.8 , mP(X 2 ) = 0 CSC411 Artificial Intelligence 17

CSC411 The Fuzzy Associative Matrix (FAM) for the pendulum problem. The input values are on the left and top.

Fuzzy Rules: Artificial Intelligence 18

The fuzzy consequents (a) and their union (b). The centroid of the union (-2) is the crisp output.

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Dempster-Shafer Theory

Probability theory limitation – Assign a single number to measure any situation, no matter how it is complex – Cannot deal with missing evidence, heuristics, and limited knowledge Dempster-Shafer theory – Extend probability theory – Consider a set of propositions as a whole – Assign a set of propositions an interval [believe, plausibility] to constraint the degree of belief for each individual propositions in the set – The belief measure bel – The plausibility of p, pl(p) = 1 – bel(not(p)) Range is also in [0,1] is in [0,1] 0 – no support evidence for a set of propositions 1 – full support evidence for a set of propositions Reflect how evidence of not(p) relates to the possibility for belief in p Bel(not(p))=1: full support for not(p), no possibility for p Bel(not(p))=0: no support for not(p), full possibility for p Artificial Intelligence 20

Properties of Dempster-Shafer

Initially, no support evidence for either competing hypotheses, say h1 and h2 – Dempster-Shafer: [bel, pl] = [0, 1] – Probability theory: p(h1)=p(h2)=0.5

Dempster-Shafer belief functions satisfy weaker axioms than probability function Two fundamental ideas: – Obtaining belief degrees for one question from subjective probabilities for related questions – Using Dempster rule to combine these belief degrees when they are based on independent evidence CSC411 Artificial Intelligence 21

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An Example

Two persons Question (Q) M and B with reliabilities detect a computer and claim the result independently. How you believe their claims?

: detection claim Related question (RQ) : detectors’ reliability Dempster-Shafer approach – Obtain belief degrees for Q from subjective (prior) probabilities for RQ for each person – Combine belief degrees from two persons Person M: – reliability 0.9, unreliability 0.1

– Claim h1 – Belief degree of h1 is bel(h1)=0.9

– Belief degree of not(h1) is bel(not(h1))=0.0, different from probability theory, since no evidence supporting not(h1) – pl(h1) = 1 – bel(not(h1)) = 1-0 =1 – Thus belief measure for M claim h1 is [0.9, 1] Person B: – Reliability 0.8, unreliability 0.2

– Claim h2 – bel(h2) =0.8, bel(not(h2))=0, pl(h2)=1-bel(not(h2))=1-0 – Belief measure for B claim h2 is [0.8,1] Artificial Intelligence 22

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Combining Belief Measure

Set of propositions: M claim h1 and B claim h2 – Case 1: h1 = h2 Reliability M and B: 09x0.8=0.72

Unreliability M and B: 0.1x0.2=0.02

The probability that at least one of two is reliable: 1-0.02=0.98

Belief measure for h1=h2 is [0.98,1] – Case 2: h1 = not(h2) Cannot be both correct and reliable At least one is unreliable – Reliable M and unreliable B: 0.9x(1-0.8)=0.18

– Reliable B and unreliable M: 0.8x(1-0.1)=0.08

– Unreliable M and B: (1-0.9)x(1-0.8)=0.02

– At least one is unreliable: 0.18+0.08+0.02=0.28

Given at least one is unreliable, posterior probabilities – Reliable M and unreliable B: 0.18/0.28=0.643

– Reliable B and unreliable M: 0.08/0.28=0.286

Belief measure for h1 – Bel(h1)=0.643, bel(not(h1))=bel(h2)=0.286

– Pl(h1)=1-bel(not(h1))=1-0.286=0.714

– Belief measure: [0.643, 0.714] Belief measure for h2 – Bel(h2)=0.286, bel(not(h2))=bel(h1)=0.683

– Pl(h2)=1-bel(not(h2))=1-0.683=0.317

– Belief measure: [0.286, 0.317] Artificial Intelligence 23

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Dempster’s Rule

– Assumption: probable questions are independent a priori – As new evidence collected and conflicts, independency may disappear Two steps 1.

Sort the uncertainties into a priori independent pieces of evidence – 2.

Carry out Dempster rule Consider the previous example – After M and B claimed, a repair person is called to check the computer, and both M and B witnessed this.

Three independent items of evidence must be combined Not all evidence is directly supportive of individual elements of a set of hypotheses, but often supports different subsets of hypotheses, in favor of some and against others Artificial Intelligence 24

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General Dempster’s Rule

Q – an exhaustive set of mutually exclusive hypotheses Z – a subset of Q M – probability density function to assign a belief measure to Z M n (Z) – belief degree to Z, where n is the number of sources of evidences Artificial Intelligence 25

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Bayesian Belief Network

A computational model for reasoning to the best explanation of a data set in the uncertainty context Motivation – Reduce the number of parameters of the full Bayesian model – Show how the data can partition and focus reasoning – Avoid use of a large joint probability table to compute probabilities for all possible events combination Assumption – Events are either conditionally independent or their correlations are so small that they can be ignored Directed Graphical Model – The events and (cause-effect) relationships form a directed graph, where events are vertices and relationships are links Artificial Intelligence 26

The Traffic Problem

The Bayesian representation of the traffic problem with potential explanations.

The joint probability distribution for the traffic and construction variables Given bad traffic, what is the probability of road construction?

p(C|T)=p(C=t, T=t)/(p(C=t, T=t)+p(C=f, T=t))=.3/(.3+.1)=.75

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An Example

Traffic problem – Events: Road construction C Accident A Orange barrels B Bad traffic T Flashing lights L – Joint probability P(C,A,B,T,L)=p(C)*p(A|C)*p(B|C,A)*p(T|C,A,B)*p(L|C,A,B,T) Number of parameters: 2^5=32 – Reduction Assumption: Parameters are only dependent on parents Calculation of joint probability – P(C,A,B,T,L)=p(C)*p(A)*p(B|C)*p(T|C,A)*p(L|A) – Number of parameters: 2+2+4+8+4=20 CSC411 Artificial Intelligence 28

BBN Definition

CSC411 Links represent conditional probabilities for causal influence These influences are directed: presence of some event causes other events These influences are not circular Thus a BBN is a DAG: Directed Acyclic Graph Artificial Intelligence 29

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Discrete Markov Process

Finite state machine – A graphical representation – State transition depends on input stream – States and transitions reflect properties of a formal language Probabilistic finite state machine – A finite state machine – Transition function represented by a probability distribution on the current state Discrete Markov process (chain, machine) – A specialization of probabilistic finite state machine – Ignores its input values Artificial Intelligence 30

A Markov state machine or Markov chain with four states, s 1 , ..., s 4 At any time the system is in one of distinct states The system undergoes state change or remain Divide time into discrete intervals: t1, t2, …, tn Change state according to the probability distribution of each state S(t) – the actual state at time t p(S(t)) = p(S(t)|S(t-1), s(t-2), s(t-3), …) First-order markov chain – Only depends on the direct predecessor state – P(S(t)) = p(S(t)|S(t-1)) CSC411 Artificial Intelligence 31

Observable Markov Model

Assume p(S(t)|S(t-1)) is time invariant, that is, transition between specific states retains the same probabilistic relationship State transition probability a ij between s i and s j : – a ij =p(S(t)=s – Properties: a i |S(t-1)=s ij >=0,  i a j ), 1<=i,j<=N – If i=j, no transition (remain the same state) ij =1 CSC411 Artificial Intelligence 32

CSC411 Artificial Intelligence S1 – sun S2 – cloudy S3 – fog S4 – precipitation Time intervals: noon to noon Question: suppose that today is sunny, what is the probability of the next five days being sunny, sunny, cloudy, cloudy, precipitation?

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