Chapter 13 Uncertainty

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Chapter 13 Uncertainty
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Outline

• Uncertainty
• Probability theory
• Probability notation and axioms
• Inference using full joint distributions
• Independence and Bayes Rule
• Probabilistic reasoning in wumpus world

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Uncertainty
Let action At leave for airport t minutes
before flight Will At get me there on time?

• Problems
• Partial observability (road state, other drivers
  plans, etc.)
• uncertainty in action outcomes (flat tire, run
  out of gas, etc.)
• immense complexity of modeling and predicting
  traffic

• Hence a purely logical approach either
• risks falsehood A25 will get me there on time
  ?
• or leads to conclusions that are too weak for
  decision making ?
• A25 will get me there on time if there is no
  accident on the bridge and it does not rain and
  my tires remain intact and .

A1440 might be said to get me there on time but
Id have to stay overnight in the airport ?

• The rational decision (the right thing to do)
  depends on
• both the relative importance of various goals
• and the likelihood that, and degree to which,
  they will be achieved.

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Handling Uncertainty

• diagnostic rules in medical diagnosis domain
  using FOL?

• causal rules in medical diagnosis domain using
  FOL?

• The connection between toothaches and cavities is
  just not a logical consequence in either
  direction. FOL fails because of

• Laziness
• too much work to list the complete set of
  premises and conclusions
• Ignorance
• Theoretical ignorance medical science has no
  complete theory for the domain
• Practical ignorance know all rules, but might be
  uncertain about a particular patient since not
  all the necessary tests have been or can be run.
• Other judgmental domains
• laws, business, design, automobile repair,

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Probability Theory

• Probability provides a way of summarizing the
  uncertainty that comes from our laziness and
  ignorance.
• We say there is an 80 chance, i.e., a
  probability of 0.8, that the patient has a cavity
  if he/she has a toothache.
• The missing 20 summarizes all the other possible
  causes of toothache that we are too lazy or
  ignorant to confirm or deny.

• Probability theory deals with degrees of belief,
  which is different from a degree of truth (fuzzy
  logic)
• The sentence, e.g., The patient has a cavity.
  itself is in fact either true or false.
• The probability that the patient has a cavity is
  0.8. is about the agents beliefs, not about the
  fact.
• These beliefs depend on the percepts (evidence)
  of the agent has received and probabilities can
  change when more evidence is acquired.
• prior (unconditional) probability vs. posterior
  (conditional) probability

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Utility Theory and Decision Theory

• Considering A90 plan vs. A1440 plan for getting
  to the airport, the agent must have preferences
  between different possible outcomes.
• A particular outcome is a completely specified
  state, including factors arrives on time, length
  of wait, etc.
• Utility the quality of being useful
• Utility theory every state has a degree of
  usefulness (utility) to an agent and that the
  agent will prefer states with higher utility.

• Decision theory combines utilities with
  probabilities in general theory of rational
  decisions

Decision theory probability theory utility
theory

• Maximum Expected Utility (MEU) An agent is
  rational if and only if it chooses the action
  that yields the highest expected utility,
  averaged over all the possible outcomes of the
  action.

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Probability notation

• Random variable referring to a part of the
  world whose status is initially unknown.

or
or
e.g.
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Probability notation

• Atomic events (sample points)
• a complete specification of the state of the
  world about which the agent is uncertain.
• an assignment of particular values to all the
  variables of which the world is composed

e.g.

• Properties of atomic events
• mutually exclusive at most one can actually be
  the case

e.g.
and
cannot both be the case

• exhaustive at least one must be the case

• any particular atomic event entails the truth or
  falsehood of every proposition

e.g.
entails the truth of
and the falsehood of

• any proposition is logically equivalent to the
  disjunction of all atomic events that entails the
  truth of the proposition.

e.g.
is equivalent to
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Prior Probability
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Probability for continuous variables
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Conditional probability
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Probability axioms

• Kolmogorovs axioms

• Any probability distribution on a single variable
  must sum to 1.

• The probability of a proposition is equal to the
  sum of the probabilities of the atomic events in
  which it holds

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Inference using full joint distributions

• Probabilistic inference the computation from
  observed evidence of posterior probabilities for
  query propositions

• Marginal probability (maginalization summing out)

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Inference using full joint distributions

• calculate the probability of any proposition,
  simple or complex

• compute conditional probabilities

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Inference using full joint distributions

• Normalization

ensures the distribution P (Cavity toothache)
adds up to 1.
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Inference using full joint distributions

• General inference procedure

• Suppose X be the query variable, e.g., Cavity,
• let E be the set of evidence variables, e.g.,
  Toothache,
• let e be the observed values for them, and
• let Y be the remaining unobserved variables,
  e.g., Catch,
• the query P (X e) can be evaluated as

where the summation is over all possible ys
(i.e., all possible combinations of values of the
unobserved variables Y) Inference by enumeration
X, E, and Y constitute the complete set of
variables P (X, e, y) is a subset of
probabilities from the full joint distribution.
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Independence
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Conditional Independence
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Conditional Independence

• Fill in the table of full joint distribution

• How many independent numbers we need?

2 2 1 5

• In most cases, the use of conditional
  independence reduces the size of the
  representation of joint distribution from
  exponential in n to linear in n.

• Conditional independence is our most basic and
  robust form of knowledge about uncertain
  environments.

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Bayess Rule
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Bayess Rule

• A single cause directly influences a number of
  effects, all of which are conditionally
  independent, given the cause.

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Probabilistic reasoning in wumpus world

• Specifying the probability model

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Probabilistic reasoning in wumpus world

• Observations and Queries

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Probabilistic reasoning in wumpus world

• Using conditional independence
• Observations are conditionally independent of
  other hidden squares given neighboring hidden
  squares.

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Probabilistic reasoning in wumpus world
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Probabilistic reasoning in wumpus world