Introduction to Reasoning under Uncertainty

1
Introduction to Reasoning under Uncertainty
MINDLab. Seminars on Reasoning and Planning under
Uncertainty

• Ugur Kuter
• MIND Lab.
• 8400 Baltimore Avenue, Ste. 200
• College Park, Maryland, 20742
• Web Site for the seminars http//www.cs.umd.edu/u
  sers/ukuter/uncertainty/

2
From Classical Logic to Uncertainty

• Logical reasoning is the process of deriving
  previously-unknown facts based on the known ones
• However, it is not always possible to have access
  to the entire set of facts to do the reasoning
• What we lose under uncertainty
• Deduction (reasoning in stages)
• Incrementality
• Modality Locality Detachment

3
Existing Approaches

• Semantic Models for Uncertainty
• Rule-based systems (i.e., expert systems)
• Organizes the knowledge in terms of if-then
  rules, each associated with a numeric certainty
  measure
• Suffer from most of the problems with classical
  logic under uncertainty
• Declarative Systems
• Organizes knowledge based on likelihood,
  relevance, and causation among events
• Flexible to uncertainty

4
Basic Terminology

• Random variables describe the features of the
  world whose status may or may not be known
• Propositions are bi-modal (i.e., Boolean) random
  variables
• We will assume a finite set of propositional
  symbols, denoted as A,B,
• Sometimes, we will use English sentences to
  denote propositions, e.g.,
• Without loss of generality, we will assume that
  the world can be described by a finite set of
  propositions
• An event in the world is an occurrence in the
  world that is modeled by assigning truth values
  to one or more propositions
• e.g., the outcomes of rolling two dice are the
  same

5
Probability Theory

• Probabilities as a way for representing the
  structure of knowledge and reasoning over that
  knowledge
• Notation P(A) is the probability of the
  proposition A being true in the knowledge base
  (or alternatively, it is the probability of the
  event A occurs in the world)
• The Axioms of Probability Theory
• 0 ? P(A) ? 1
• P(certain event) 1
• P(A or B) P(A) P(B) - P(A and B)

6
Joint Probability Distributions

• A joint event describes two occurrences at the
  same time
• e.g., (A and B) specifies that both propositions
  A and B is true in the world
• A joint probability distribution over a set of
  random variables specifies a probability for each
  possible combinations of values for those
  variables
• e.g., a joint probability distribution for
  boolean variables X and Y specifies a probability
  for four cases
• (X and Y), (X and ?Y), (?X and Y), and finally
  (?X and ?Y)
• The sum of the joint probability of each case
  must be equal to 1

7
Absolute Independence

• Suppose two events in a joint probability
  distribution are independent from each other
  (i.e., they are mutually-exclusive)
• Then we can decompose the joint probability
  distribution into smaller distributions
• The joint probability of two events under
  absolute independence
• P(A and B) P(A) P(B)
• However in most problems, absolute independence
  does not hold

8
Set-theoretic Interpretation of Probabilities
W The set of all possible worlds described by
the knowledge base
WA The set of possible worlds in which A is true

• The probability of A being true is the proportion
  of WA to W
• Set-theoretic operations correspond logical
  connectives i.e.,
• A and B ? WA ? WB
• A or B ? WA ? WB
• ?A ? W - WA

9
Revisiting the Third Axiom

• P(A or B) P(A) P(B) - P(A and B)
• that is, the proportion of WA ? WB to W
• If A and B are mutually-exclusive events (i.e.,
  WA ? WB ?) , then we have
• P(A or B) P(A) P(B)

WA
WB
10
Using the Axioms of Probability

• Rule via the union of joint events
• P(A) P(A and B) P(A and ?B),
• since
• A and B ? WA ? WB
• A and ?B ? WA - WB
• More generally,
• P(A) ?i P(A and Bi),
• where B1, B2, , Bk is a set of exhaustive and
  mutually-exclusive set of events

11
Using the Axioms of Probability

• Rule for absolute falsity
• P(A) P(A and A) P(A and ?A), by rule of the
  union of joint events
• P(A) P(false), by logical
  equivalence
• P(false) 0, by algebra
• Rule for Negation
• P( A or ?A) P(A) P(?A) - P(A and ?A), by
  axiom 3
• P(true) P(A) P(?A) - P(false), by
  logical equivalence
• 1 P(A) P(?A) - 0, by axiom
  2
• P(A) 1 - P(?A), by algebra

12
Conditional Probabilities

• A conditional probability, P(A B), describes
  the belief in the event A, under the assumption
  that another event B is known with absolute
  certainty
• Formal Definition
• P(A B) P(A and B) / P(B)
• That is, the proportion of
• WA ? WB to WB

13
The Bayesians (1763 -- present)

• The basis of the Bayesian Theory is conditional
  probabilities
• Bayesian Theory sees a conditional probability as
  a way to describe the structure and organization
  of human knowledge
• In this view, A B stands for the event A in the
  context of the event B
• E.g., the symptom A in the context of a disease B

14
Mathematicians vs. Bayesians, i.e.,

• P(A B) P(A and B) / P(B) vs. P(A and B)
  P(A B) P(B)
• Example
• The probability of the event A ? the outcomes of
  two dice are equal
• Mathematician would compute the rule for joint
  events A and Bi, where
• each Bi is the event the outcome of first dice
  is i
• P(A) ?i P(A and Bi),
• 6 x (1 / 36)
• 1 / 6

15
Mathematicians vs. Bayesians, contd

• The Bayesian Mindset for Dice Rolling
• P( Equality) ?i P (Outcome of the second dice
  is i Bi) P(Bi)
• 6 x (1/6) (1/6)
• 1 /6
• Since this is more natural in the
  assumption-based mental processes of human
  reasoning, i.e.
• given that I know

16
The CHAIN Rule

• The probability that a joint event (A1, , Ak)
  occurs can be computed via the conditional
  probabilities
• P (A1, , Ak) P(Ak A1, , Ak-1)
• P(Ak-1 A1, , Ak-2)
• P(A2 A1) P(A1)

17
Evidential Reasoning The Inversion Rule

• Reasoning about hypotheses and evidences
  that do/do not support those hypothesis is the
  main venue for Bayesian Inference
• P(H e) given that I know about an evidence e,
  the probability that my hypothesis H is true
• P(H e) P(e H) P(H) / P(e),
• where P(e H) is the probability that evidence
  e will actually be observed in the world, if the
  hypothesis H is true.

18
Pooling of Evidence

• Suppose the hypothesis H is that a patient has a
  particular disease
• Let e1, , ek be the possible symptoms of that
  disease
• If we observed some of the symptoms but not all
  of them, then the combined belief that the
  hypothesis is true can be computed as
• P(e1, , ek H)P(H) / P(e1, , ek
  ?H)P(?H)
• This will require an exponential number of
  conditional probabilities to specify

19
Naïve Bayes

• Use conditional independence assumption
• e.g., the event that whether we observe a symptom
  or not depends only on whether the patient has
  the disease, not on other symptoms
• Then, the conditional probability P(e1, , ek
  H) can be computed as
• P(e1, , ek H) ?i P(ei H)

20
Incremental Bayesian Updating

• Let H be an hypothesis, E e1, , ek be the
  past data (evidence) observed for the hypothesis
  H
• Suppose we observed a new data (evidence) e
• What is the probability that the hypothesis is
  true, given the past and the new evidence?
• Computing this probability is in efficient
  because it requires to store all the past
  evidence and combine it with the new evidence
• Bayesian Theory allows us to reformulate the
  question as follows
• How do we update our belief in H given the new
  evidence and our past belief in H?

21
Incremental Bayesian Updating (contd)

• Combining prior beliefs with new evidence
• P(H E and e) P(H E) P(e E,H) / P(e E)
  
• Assuming condition independence between the new
  evidence and the old ones given the hypothesis
  (i.e., P(e E,H) P(e H))

Normalization constant
Updated belief in H
Probability that the new evidence will be
observed, given the old evidence and the
hypothesis
Prior belief in H
P(H E and e) ? P(H E) P(e H)
22
Hierarchical Models

• So far, we assumed that the evidence from the
  world is directly linked to the hypothesis we are
  evaluating
• E.g., a disease and its symptoms
• In many problems, this not the case e.g.,

e1
e2
ek
H
23
1. Cascading Inference

• What is the probability that a burglary took
  place, given the two neighbors testimonies?
• P(B G,W) ? P(G, W B) P(B)
• ? P(H) ( P(G,W B, S true)
  P(Strue B)
• P(G,W B, S false) P(Sfalse B)
  )
• From conditional independence, we have
• P(G,W B, S true, false) P(G S) P(W
  S)
• Thus, we have
• P(B G,W) ? P(H) ?Strue,false P(G S)
  P(W S) P(S B)

24
2. Predicting Future

• The daughter will call with some probability, if
  she hears the alarm sound
• What is the probability that the daughter will
  call, given the testimonies of the neighbors?

25
Predicting Future

• The probability that
• the daughter will call is
• P(D e) ?Strue,false P(D S) P(S e)
• By inversion rule,
• P(Strue,false e) ? P(e Strue,false)
  P(Strue,false)
• where P(e S ) P(G S) P(W S) as before, and
• P(S) ?Btrue,false P(S B) P(B)

Known
Unknown
26
3. Explaining Away

• If an event has multiple causes, sometimes the
  occurrence of one cause reduces our belief in the
  occurrence of the other
• If the Alarm is sensitive enough to go off when
  an earthquake occurs, the occurrence of the
  earthquake explains away the burglary hypothesis

27
Interactions between Multiple Causes

• Conditional Probability Tables
• (CPTs)
• P(SB,E) P(B) P(E)
• P(S?B,E)P(?B)P(E)
• P(SB, ?E)P(B)P(?E)
• P(S?B, ?E)P(?B)P(?E)
• The size of a CPT is exponential in the size of
  the number of causes
• i.e., if there are k causes of an event X, then
  the CPT for X has 2k entries
• This creates serious efficiency problems for
  exact reasoning algorithms
• Approximation techniques are developed for
  reasoning over incomplete CPTs
• E.g., Noisy-OR (NOR), Generalized-NOR,
  Recursive-NOR
• We will discuss these techniques later

28
Summary

• Reasoning with classical logic has problems with
  plausible reasoning under uncertainty
• Probability theory provides a machinery for
  uncertain reasoning, but often times, it is very
  inefficient to depend on solely probability
  theory
• Exponential-sized CPTs
• No structural reasoning over the knowledge
• Bayesians use probability theory in order to
  describe the organization (i.e., the structure)
  of human knowledge and develop reasoning
  machinery over such structures
• Hierarchical Modeling
• Next Week
• Structural Models of the Bayesians Networks of
  Plausible Reasoning