Artificial Intelligence A Modern Approach Uncertainty

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Artificial Intelligence A Modern
ApproachUncertainty
2005-2006 Winter Break Seminar

• Hyoseok Yoon
• GIST U-VR Lab.
• Gwangju 500-712
• http//uvr.gist.ac.kr

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Outline

• Acting Under Uncertainty
• Basic Probability Notation
• The Axioms of Probability
• Bayes Rule and Its Use
• Where Do Probabilities Come From?
• Relations to Interpreter

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Acting Under Uncertainty

• Problem with the logical-agent approach
• Agents almost never have access to the whole
  truth about their environment
• There will be cases to which the agent cannot
  find a categorical answer
• Qualification problem
• Many rules about the domain will be incomplete,
  because there are too many conditions to be
  explicitly enumerated, or because some of the
  conditions are unknown
• The right thing to do, the rational decision,
  therefore, 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 uncertain knowledge

• Using first-order logic, we have to add an almost
  unlimited list of possible causes
• Trying to use first-order to scope with a domain
  like medical diagnosis thus fails for three main
  reasons
• Laziness It is too much work to list the
  complete set of antecedents or consequents needed
  to ensure an exceptionless rule, and too hard to
  use the enormous rules that result
• Theoretical ignorance Medical science has no
  complete theory for the domain
• Practical ignorance Even if we know all the
  rules, we may be uncertain about a particular
  patient because all the necessary tests have not
  or cannot be run

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Handling uncertain knowledge (contd)

• The agents knowledge can at best provide only a
  degree of belief in the relevant sentences
• Probability provides a way of summarizing the
  uncertainty that comes from our laziness and
  ignorance
• Evidence the percepts the agent has received to
  date
• An assignment of probability to a proposition is
  analogous to saying whether or not a given
  logical sentence (or its negation) is entailed by
  the knowledge base
• As the agent receives new percepts, its
  probability assessments are updated to reflect
  the new evidence.
• Before the evidence is obtained, we talk about
  prior or unconditional probability
• After the evidence is obtained, we talk about
  posterior or conditional probability

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Uncertainty and rational decisions

• An agent must first have preferences between the
  different possible outcomes of the various plans
• Use utility theory to represent and reason with
  preferences. The term utility is used here in the
  sense of the quality of being useful
• Utility theory says that every state has a degree
  of usefulness, or utility, to an agent, and that
  the agent will prefer states with higher utility
• Preferences, as expressed by utilities, are
  combined with probabilities in the general theory
  of rational decisions called decision theory
• Decision theory probability theory utility
  theory
• The fundamental idea of decision theory
• 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 the principle of Maximum Expected
  Utility (MEU)

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Design for a decision-theoretic agent

• The structure of an agent that uses decision
  theory to select actions is identical, at an
  abstract level, to that of the logical agent

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Basic Probability Notation

• Prior probability
• The notation P(A) is used for the unconditional
  or prior probability that the proposition A is
  true
• Ex) P(Cavity) 0.1 , means that in the absence
  of any other information, the agent will assign a
  probability of 0.1 (a 10 chance) to the event of
  the patients having a cavity
• Propositions can also include equalities
  involving so-called random variables
• Ex) P(Weather Sunny) 0.7
• Each random variable X has a domain of possible
  values that it can take on, usually deals with
  discrete sets of values
• Use an expression such as P(Weather) to denote a
  vector of values for the probabilities of each
  individual state of the weather
• Probability distribution for the random variables
• P(Weather) lt 0.7,0.2,0.08,0.02gt

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Basic Probability Notation (contd)

• Conditional or posterior probability
• Notation P(AB) is used and read as the
  probability of A given that all we know is B
• P(CavityToothache) 0.8 indicates that if a
  patient is observed to have a toothache, and no
  other information is yet available, then the
  probability of the patient having a cavity will
  be 0.8
• In general, if we are interested in the
  probability of a proposition A, and we have
  accumulated evidence B, then the quality we must
  calculate is P(AB)
• When conditional probability is not available
  directly in the knowledge base, we must resort to
  probabilistic inference

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The axioms of probability

• It is normal to use a small set of axioms that
  constrain the probability assignments that an
  agent can make to a set of propositions
• 1. All probabilities are between 0 and 1
• 0 lt P(A) lt 1
• 2. Necessarily true (i.e., valid) propositions
  have probability 1, and necessarily false (i.e.,
  unsatisfiable) propositions have probability 0
• P(True) 1, P(False) 0
• 3. The probability of a disjunction is given by
• P(A V B) P(A) P(B) P(A B)

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Why the axioms of probability are reasonable

• One argument for the axioms of probability, first
  stated in 1931 by Bruno de Finetti
• The key to de Finettis argument is the
  connection between degree of belief and actions,
  a game between two agents
• If agent 1 expresses a set of degrees of belief
  that violate the axioms of probability theory
  then there is a betting strategy for Agent 2 that
  guarantees that Agent 1 will lose money

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The joint probability distribution

• Completely specifies an agents probability
  assignments to all propositions in the domain
• A probabilistic model of a domain consists of a
  set of random variables that can take on
  particular values with certain probabilities, X1
  Xn
• An atomic event is an assignment of particular
  values to all the variables
• a complete specification of the state of the
  domain
• Used to compute any probabilistic statement we
  care to know about the domain, by expressing the
  statement as a disjunction of atomic events and
  adding up their probabilities

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Bayes Rule and Its Use

• Applying Bayes rule The simple case
• Probability that disease meningitis causes the
  patient to have a stiff neck 50 of the time
• Unconditional facts the prior probability of a
  patient having meningitis is 1/50,000
• The prior priority of any patient having a stiff
  neck is 1/20
• Let S be the proposition that the patient has a
  stiff neck
• Let M be the proposition that the patient has
  meningitis

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Normalization

• Avoid direct assessment by considering an
  exhaustive set of cases
• This process is called normalization, because it
  treats 1/P(S) as a normalizing constant that
  allows the conditional terms to sum to 1
• In general
• Where a is the normalization constant needed to
  make the entries in the table P(YX) sum to 1

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Using Bayes rule Combining evidence

• The process of Bayesian updating incorporates
  evidence one piece at a time, modifying the
  previously held belief in the unknown variable
• Beginning with Toothache, we have
• When Catch is observed, we can apply Bayes rule
  with Toothache as the constant conditioning
  context
• Thus, in Bayesian updating, as each new piece of
  evidence is observed, the belief in the unknown
  variable is multiplied by a factor that depends
  on the new evidence

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Conditional independence

• In the cavity example, the cavity is the direct
  cause of both the toothache and the probe
  catching in the tooth
• Mathematically,
• To say that X and Y are independent given Z, we
  write
• P(XY,Z) P(XZ)
• Bayes rule for multiple evidence is
• Where a is a normalization constant such that
  entries in P(ZX,Y) sum to 1

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Where Do Probabilities Come From?

• Source and status of probability numbers
• Frequentist the numbers can come from
  experiements
• Ex) test 100 people and find 10 people with
  cavity, probability of a cavity is then about 0.1
• Objectivist the probabilities are real aspects
  of the universe
• Subjectivist a way of characterizing an agents
  beliefs, rather than having any external physical
  significance
• Ex) allow doctor or analyst to make up numbers

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Summary

• Uncertainty arises because of both laziness and
  ignorance. It is inescapable in complex, dynamic,
  or inaccessible worlds
• Uncertainty means that many of the
  simplifications that are possible with deductive
  inference are no longer valid
• Probabilities express the agents inability to
  reach a definite decision regarding the truth of
  a sentence, and summarize the agents beliefs
• Basic probability statements include prior
  probabilities and conditional probabilities over
  simple and complex propositions

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Summary (contd)

• The axioms of probability specifies constraints
  on reasonable assignments of probabilities to
  propositions. An agent that violates the axioms
  will behave irrationally in some circumstances
• The joint probability distribution specifies the
  probability of each complete assignment of values
  to random variables. It is usually far too large
  to create or use
• Bayes rule allows unknown probabilities to be
  computed from known, stable ones
• In the general case, combining many pieces of
  evidence may require assessing a large number of
  conditional probabilities
• Conditional independence brought about by direct
  causal relationships in the domain allows
  Bayesian updating to work effectively even with
  multiple pieces of evidence

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Relations to interpreter

• Interpreter only has limited view on the
  environment
• Through input context
• Source context may not map categorically to
  target context
• Different properties, characteristics,
  structures, ontology
• Bayesian approach in SOCAM
• The supports for using probability-annotated
  context ontology (OWL) and BN have been built in
  SOCAM. The context ontology with additional
  dependency markups is created and stored in a
  context database. SOCAM can translate this
  ontology into a BN
• Prob(Status(John,Sleeping)) 0.8

T. Gu, H. K. Pung, D. Q. Zhang. A Bayesian
Approach for Dealing with Uncertain Contexts. In
Proc of the 2nd Intl Conf on Pervasive Computing
(Pervasive 2004), in book "Advances in Pervasive
Computing" , vol. 176, Austria, Apr 2004.
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Dealing with uncertainty

• T. Gu, H. Pung, D. Zhang, "A Bayesian approach
  for dealing with uncertain contexts", Proceedings
  of the Second International Conference on
  Pervasive Computing , April 2004
• Binh An Truong, Young-Koo Lee, Sung-Young Lee,
  Modeling and Reasoning about Uncertainty in
  Context-Aware Systems, e-Business Engineering,
  2005. ICEBE 2005. pp. 102-109 2005.
• A. Ranganathan, J. Al-Muhtadi, R. H. Campbell,
  "Reasoning about Uncertain Contexts in Pervasive
  Computing Environments", IEEE Pervasive
  Computing, pp 62-70, Volume 3, Issue 2, Apr-June
  2004.

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QA

• Discussions More information
• Hyoseok Yoon
• GIST U-VR Lab, Gwangju 500-712, S. Korea
• Tel. [phone]
• Fax. [phone]
• mailto hyoon_at_gist.ac.kr
• Web http//uvr.gist.ac.kr

Thank You!