Uncertainty and probabilistic reasoning

Administrivia:

Overview

• Uncertainty is caused by both the complexity and inherent unpredictability of the world.
• In many situations the simplifications possible with deductive inference are not valid.
• Probabilities can be used to describe the agent's degree of belief in a sentence.
• The axioms of probability state:
  1. All probabilities are between 0 and 1.
  2. Valid propositions have probability 1 and unsatisfiable propositions have probability 0.
  3. The probability of a disjunction is

     		P(A OR B) = P(A) + P(B) - P(A AND B)
       

• An agent's beliefs about an uncertain domain can be expressed in terms of a set of random variables.
• The joint probability distribution specifies the probability of each complete assignment of values to random variables.
• The probability of any statement involving the random variables can be calculated from their joint probability distribution.
• In practice, the joint distribution is usually to large to create, learn or use.
• It is often relatively to estimate the conditional probability of an effect given a cause.
• The probability of a cause given an effect can be obtained using Bayes' rule.
• In general, combining many pieces of evidence may require assessing many conditional probabilities.
• Conditional independence is brought about by direct causal relationships in the domain.
• Belief networks are a natural way to represent conditional independence information.
  • The links represent the qualitative cause-effect relationships in the domain.
  • The conditional probability tables represent the quantitative aspects.
• A belief network completely represents the joint distribution but is often exponentially smaller in size.
• Inference in belief networks means computing the probability distribution of a set of query variables, given a set of evidence variables.
• Four types of reasoning are possible using belief networks:
  • causal--from causes to effects
  • diagnostic--from effects to causes
  • intercausal--between causes of a common effect
  • mixed--combining two or more of the above
• Inference can be done in linear time if the belief network is singly connected (at most one undirected path between any two nodes.)
• Inference in multiply connected networks can be intractable for large networks but approximate methods like stochastic simulation can work well in practice.

Uncertainty

• Uncertainty is caused by both the complexity and inherent unpredictability of the world.
• In many situations the simplifications possible with deductive inference are not valid.

  Non-probabilistic rules can be:

  • wrong
    

    	FORALL x Symptom(x, Toothache) => Disease(x, Cavity)
    	FORALL x Disease(x, Toothache) => Symptom(x, Toothache)
    

  • or, too long
    

    	FORALL p Symptom(x, Toothache) => 
    		Disease(x, Cavity) OR
    		Disease(x, GumDisease) OR
    		Disease(x, ImpactedWisdomTooth) OR ...
    

• Probabilities can be used to describe the agent's degree of belief in a sentence.

  The probability that a particular patient has a cavity:

  	p(Cavity) = 0.1
  

• The axioms of probability state:
  1. All probabilities are between 0 and 1.
  2. Valid propositions have probability 1 and unsatisfiable propositions have probability 0.
  3. The probability of a disjunction is

     		p(A OR B) = p(A) + p(B) - p(A AND B)
       

  The last axiom can be remembered easily using a Venn diagram.

• An agent's beliefs about an uncertain domain can be expressed in terms of a set of random variables.

  Random variables (RV's) can assume two or more value. Here is an example where the random variable Weather can assume one of exactly four values:

  	p(Weather=Sunny) = 0.7
  	p(Weather=Rain) = 0.2
  	p(Weather=Cloudy) = 0.08
  	p(Weather=Snow) = 0.02
  

  If the RV's are Boolean, then we can use the normal propositional connectives. For example,

  	p(Cavity AND (NOT Insured)) = 0.06
  

• The joint probability distribution specifies the probability of each complete assignment of values to random variables. The joint probability distribution is written as follows (notice the capital P to distinguish it fromt the probability of an atomic event, the assignment of values to all the variables.)

  	P(X_1, X_2, ... , X_n)
  

  The joint probability can be written as a table:

  	Toothache	NOT Toothache
  Cavity	0.04	0.06
  NOT Cavity	0.01	089.

• The probability of any statement involving the random variables can be calculated from their joint probability distribution.
• In practice, the joint distribution is usually to large to create, learn or use.

  If there are n Boolean RV's, the joint distribution table has 2^n entries!

• It is often relatively to estimate the conditional probability of an effect given a cause.

  Usually your theory is interms of effects:

  	P(Toothache | Cavity)
  

  but you want to diagnose

  	P(Cavity | Toothache)
  

• The probability of a cause given an effect can be obtained using Bayes' rule.

  	P(Y | X) = P(X | Y) P(Y)
  		   -------------
  		      P(X)
  

  For example,

  	p(Cavity | Toothache) = p(Toothache | Cavity) p(Cavity)
  		   		-------------------------------
  		      			p(Toothache)
  

• In general, combining many pieces of evidence may require assessing many conditional probabilities.
• Conditional independence is brought about by direct causal relationships in the domain.

  The dentist is using the little tool that catches on cavities in teeth.

  We say that the RV's Catch and Toothache are conditionally independent given Cavity if Cavity is the direct cause of Catch so that knowing Toothache adds no information if you already know Cavity, and knowing Catch gives no information about Toothache.

  This is expressed as:

  	p(Catch | Cavity AND Toothache) = p(Catch | Cavity
  	p(Toothache | Cavity AND Catch) = p(Toothache | Cavity)
  

• Belief networks are a natural way to represent conditional independence information.
  • The links represent the qualitative cause-effect relationships in the domain.
  • The conditional probability tables represent the quantitative aspects.

  (Fig 15.1 here)

  Burglary	Earthquake P(Alarm | Burglary, Earthquake)
  		True	False
  True	True	0.950	0.050
  True	False	0.950	0.050
  False	True	0.290	0.710
  False	False	0.001	0.999

• A belief network completely represents the joint distribution but is often exponentially smaller in size.

  Because of conditional independence, the an entry in the joint probability table for all the variables is given by:

  	p(x_1, ... ,x_n) = PRODUCT_{i=1}^n p(x_i | Parents(X_i))
  

  For example, the probability of the alram sounding but neither a burglary or earthquake has occured and john and mary both call is:

  	p(J AND M AND A AND (NOT B) AND (NOT E) )
  		= p(J|A) P(M|A) p(A| (NOT B) AND (NOT E)) p(NOT B) p(NOT E)
  		= 0.90 x 0.70 x 0.001 x 0.999 x 0.998 
  		= 0.00062
  

  If we assume n Boolean variables, each of which is influenced by at most k others, then the CPT's will require at most n2^k entries. This is far smaller than the 2^k numbers required for the joint probability distribution.

• Inference in belief networks means computing the probability distribution of a set of query variables, given a set of evidence variables.
• Four types of reasoning are possible using belief networks:
  • causal--from causes to effects
  • diagnostic--from effects to causes
  • intercausal--between causes of a common effect
  • mixed--combining two or more of the above
• Inference can be done in linear time if the belief network is singly connected (at most one undirected path between any two nodes.)
• Inference in multiply connected networks can be intractable for large networks but approximate methods like stochastic simulation can work well in practice.