1. AIMA 3e Chapter 13: Quantifying Uncertainty
CSC 450
AIMA 3e Chapter 13:
Quantifying Uncertainty

2. OUTLINE Overview 1. rationale for a new representational language
what logical representations can't do
2. utilities & decision theory
3. possible worlds & propositions
4. unconditional & conditional probabilities
5. random variables
6. probability distributions
7. using the Joint Probability Distribution
for inference by enumeration
for unconditional & conditional probabilities
Quantifying Uncertainty

3. Quantifying Uncertainty
consider our approach so far
we've handled limited observability &/or non-determinism using belief states that capture all possible world states
but the representation can become large, as can corresponding contingent plans, and it's possible that no plan can be guaranteed to reach the goal, yet the agent must act
agents should behave rationally
this rationality depends both on the importance of goals and on the chances of & degree to which they'll be reached
Quantifying Uncertainty

4. A Visit to the Dentist
we'll use medical/dental diagnosis examples extensively
our new prototype problem relates to whether a dental patient has a cavity or not
the process of diagnosis always involves uncertainty & this leads to difficulty with logical representations (propositional logic examples)
(1) toothache  cavity
(2) toothache  cavity  gumDisease  ...
(3) cavity  toothache
(1) is just wrong since other things cause toothaches
(2) will need to list all possible causes
(3) tries a causal rule but it's not always the case that cavities cause toothaches & fixing the rule requires making it logically exhaustive
Quantifying Uncertainty

5. Representations for Diagnosis
logic is not sufficient for medical diagnosis, due to
our Laziness: it's too hard to list all possible antecedents or consequents to make the rule have no exceptions
our Theoretical Ignorance: generally, there is no complete theory of the domain, no complete model
our Practical Ignorance: even if the rules were complete, in any particular case it's impractical or impossible to do all the necessary tests, to have all relevant evidence
the example relationship between toothache & cavities is not a logical consequence in either direction
instead, knowledge of the domain provides a degree of belief in diagnostic sentences & the way to represent this is with probability theory
next slide: recall our discussion of ontological & epistemological commitments from 352
Quantifying Uncertainty

6. Epistemological Commitment
ontological commitment
what a representational language assumes about the nature of reality - logic & probability theory agree in this, that facts do or do not hold
epistemological commitment
the possible states of knowledge
for logic, sentences are true/false/unknown
for probability theory, there's a numerical degree of belief in sentences, between 0 (certainly false) and 1 (certainly true)
Quantifying Uncertainty

7. The Qualification Problem
for a logical representation
the success of a plan can't be inferred because of all the conditions that could interfere but can't be deduced not to happen (this is the qualification problem)
probability is a way of dealing with the qualification problem by numerically summarizing the uncertainty that derives from laziness &/or ignorance
returning to the toothache & cavity problem
in the real world, the patient either does or does not have a cavity
a probabilistic agent makes statements with respect to the knowledge state, & these may change as the state of knowledge changes
for example, an agent initially may believe there's an 80% chance (probability 0.8) that the patient with the toothache has a cavity, but subsequently revises that as additional evidence is available

8. Rational Decisions
making choices among plans/actions when the probabilities of their success differ
this requires additional knowledge of preferences among outcomes
this is the domain of utility theory: every state has a degree of utility/usefulness to the agent & the agent will prefer those with higher utility
utilities are specific to an agent, to the extent that they can even encompass perverse or altruistic preferences
Quantifying Uncertainty

9. Rational Decisions
making choices among plans/actions when the probabilities of their success differ
we can combine preferences (utilities) + probabilities to get a general theory of rational decisions: Decision Theory
a rational agent chooses actions to yield the highest expected utility averaged over all possible outcomes of the action
this is the Maximum Expected Utility (MEU) principle
expected = average of the possible outcomes of an action weighted by their probabilities
choice of action = the one with highest expected utility
Quantifying Uncertainty

10. Revising Belief States
in addition to the possible world states that we included before, belief states now include probabilities
the agent incorporates probabilistic predictions of action outcomes, selecting the one with the highest expected utility
AIMA3e chapters 13 through 17 address various aspects of using probabilistic representations
an algorithmic description of the Decision Theoretic Agent
function DT-AGENT (percept) returns an action
persistent: belief-state, probabilistic beliefs about the current state of the world
action, the agent's action
update belief-state based on action and percept
calculate outcome probabilities for actions,
given action descriptions and current belief state
select action with the highest expected utility,
given probabilities of outcomes and utility information
return action

11. Notation & Basics we should interpret probabilities as describing
possible worlds and their likelihoods
the sample space is the set of all possible worlds
note that possible worlds are mutually exclusive & exhaustive
for example, a roll of a pair of dice has 36 possible worlds
we use the Greek letter omega to refer to possible worlds
 refers to the sample space,  to its elements (particular possible worlds)
a basic axiom for probability theory
(13.1) 0  P()  1,
as an example, for the dice rolls, each possible world is a pair
(1, 1), (1, 2), ..., (6, 6)
each with a probability of 1/36, all summing to 1
Quantifying Uncertainty

12. Notation & Basics assertions & queries in probabilistic reasoning
these are usually about sets of possible worlds
these are termed events in probability theory
for AI, the sets of possible worlds are described by propositions in a formal language
the set of possible worlds corresponding to a proposition contains those in which the proposition holds
the probability of the proposition is the sum over those possible worlds
Quantifying Uncertainty

13. Propositions propositions
another axiom of probability theory, using the Greek letter phi () for proposition
(13.2)
so for a fair pair of dice P(total = 7) = P((1+6))+P((6+1))+P((2+5))+P((5+2))+P(((3+4))+P((4+3))
=1/36+1/36+1/36+1/36+1/36+1/36 = 1/6
asserting the probability of a proposition constrains the underlying probability model without fully determining it

14. Propositions propositions: unconditional & conditional probabilities
P(total = 7) from the previous slide & similar probabilities are called unconditional or prior probabilities, sometimes abbreviated as priors
they indicate the degree of belief in propositions without any other information, though in most cases, we do have other information, or evidence
when we have evidence, the probabilities are conditional or posterior, given the evidence

15. Conditional Probability
P(A | B) is the probability of A given B
Assumes that B is the only info known.
Defined by: A B
AB
True

16. Random Variables & Values
propositions in our probability notation
by convention, the values for random variables use lower case letters, for example Weather = rain
each random variable has a domain, its set of possible values
for a Boolean random variable the domain is {true, false}
also by convention, A = true is written as simply a, A = false as ¬a
domains also may be arbitrary sets of tokens, like the {red, green, blue} of the map coloring CSP or {juvenile, teen, adult} for Age
when it's unambiguous, a value by itself may represent the proposition that a variable has that value
for example, using just sunny for Weather = sunny

17. Distribution Notation
bold is used as a notational coding
for the probabilities of all possible values of a random variable
we may list the propositions or we may abbreviate, given an ordering on the domain
as in the ordering (sunny, rain, cloudy, snow) for Weather
then P(Weather) = <0.6, 0.1, 0.29, 0.01>, where bold indicates there's a vector of values
this defines a probability distribution for the random variable Weather
we can use a similar shorthand for conditional distributions, for example:
P(X|Y) lists the values for P(X=xi | Y=yj) for all i,j pairs
Quantifying Uncertainty

18. Continuous Variables distributions are
the probabilities of all possible values of a random variable
there's alternative notation for continuous variables where there cannot be an explicit list: instead, express the distribution as a parameterized function of value
for example, P(NoonTemp=x) = Uniform[18C,26C] (x) specifies a probability density function (pdf) that defines density function values for intervals of the NoonTemp variable values

19. Distribution Notation
for distributions on multiple variables
we use commas between the variables: so P(Weather, Cavity) denotes the probabilities of all combinations of values of the 2 variables
for discrete random variables we can use a tabular representation, in this case yielding a 4x2 table of probabilities
this gives the joint probability distribution of Weather & Cavity
tabulates the probabilities for all combinations

20. Full Joint Distribution
semantics of a proposition
the probability model is determined by the joint distribution for all the random variables: the full joint probability distribution
for the Cavity, Toothache, Weather domain, the notation is:
P(Cavity, Toothache, Weather)
this can be represented as a 2x2x4 table
given the definition of the probability of a proposition as a sum over possible worlds, the full joint distribution allows calculating the probability of any proposition over its variables by summing entries in the FJD

21. Inference With Probability
using the full joint distributions for inference
here's the FJD for the Toothache, Cavity, Catch domain of 3 Boolean variables
as required by the axioms, the probabilities sum to 1.0
when available, the FJD gives a direct means of calculating the probability of any proposition
just sum the probabilities for all the possible worlds in which the proposition is true
Quantifying Uncertainty

22. Inference With Probability
P(toothache)=
= .20 or 20%

23. Inference With Probability
P(toothachecavity) =
.20 + ??
.28

24. Inference With Probability

25. Inference for Probability
given the full joint distribution & 13.9
we can answer all probability queries for discrete variables
are we left with any unresolved issues?
well, given n variables, and d as an upper bound on the number of values then the full joint distribution table size & corresponding processing of it are O(dn), exponential in n
since n might be 100 or more for real problems, this is often simply not practical
as a result, the FJD is not the implementation of choice for real systems, but functions more as the theoretical reference point (analogous to role of truth tables for propositional logic)
the next sections we look at are foundational for developing practical systems
Quantifying Uncertainty

26. End of Lecture
Quantifying Uncertainty