Reasoning Under Uncertainty in Expert System

Situations often arise wherein a decision must be made when the results of each possible choice are uncertain.

What is Uncertainty? Uncertainty is a situation which involves imperfect or unknown information. Uncertainty is essentially lack of information to formulate a decision. Uncertainty may result in making poor or bad decisions. As living creatures, we are accustomed to dealing with uncertainty – that’s how we survive.

How to Expert Systems Deal with Uncertainty? Expert systems provide an advantage when dealing with uncertainty as compared to decision trees. With decision trees, all the facts must be known to arrive at an outcome. Probability theory is devoted to dealing with theories of uncertainty. There are many theories of probability – each with advantages and disadvantages.

Decision tree

Theories to Deal with Uncertainty Bayesian Probability Hartley Theory Shannon Theory Dempster-Shafer Theory Markov Models Zadeh’s Fuzzy Theory

Dealing with Uncertainty

What is Reasoning? The action of thinking about something in a logical, sensible way. When we require any knowledge system to do something it has not been explicitly told how to do it must reason. Reasoning is the process of machining inferences from a body of information.

Example:- “zeno is a spider” Given the information that ”zeno is a spider”, it is reasonable to conclude that Zeno ha eight legs. zeno is a spider reasoning Therefore,zeno has eight legs.

Type of Reasoning Deductive Reasoning Top down logic. Deals with exact facts and exact conclusions. “ uses patterns to arrive at a conclusion ” Inductive Reasoning Bottom up logic. Not as strong as deductive – premises support the conclusion but do not guarantee it. “ uses facts, rules, definition or properties to arrive at a conclusion” There are a number of methods to pick the best solution in light of uncertainty. When dealing with uncertainty, we may have to settle for just a good solution.

Top down logic Bottom up logic.

Errors Related to Hypothesis Many types of errors contribute to uncertainty. Type I Error – accepting a hypothesis when it is not true – False Positive. Type II Error – Rejecting a hypothesis when it is true – False Negative

Errors Related to Measurement Errors of precision – how well the truth is known Errors of accuracy – whether something is true or not Unreliability -(data)

Types of Errors

Examples of Common Types of Errors

Classical Probability First proposed by Pascal and Fermat in 1654 Also called a priori probability because it deals with ideal games or systems: -Assumes all possible events are known -Each event is equally likely to happen Fundamental theorem for classical probability is P = W / N, where W is the number of wins and N is the number of equally possible events.

Deterministic vs. Nondeterministic Systems When repeated trials give the exact same results, the system is deterministic. Otherwise, the system is nondeterministic. Nondeterministic does not necessarily mean random – could just be more than one way to meet one of the goals given the same input.

Sample Space of Intersecting Events The sample space of an experiment is the set of all possible outcomes o f that experiment. the range of values of a random variable.

Experimental Probabilities Experimental probability is the ratio of the number of times an event occurs to the total number of trials or times the activity is performed.

Subjective probability Subjective probability measure a person’s belief that an event will occur. These probabilities vary from person to person. They also change as one learns new in formation. OR It is the probability assign to an event based on subjective judgment, experience, information and belief.

Example:- A doctor may feel a patient has a 90% chance of a full recovery.

Compound Probabilities Compound probabilities can be expressed by: S is the sample space and A and B are events. independent events are events that do not affect each other. For pairwise independent events

Conditional Probabilities The probability of an event A occurring, given that event B has already occurred is called conditional probability:

Example:- Ali took two tests Example:- Ali took two tests. The probability of her passing both tests is 0.6. The probability of her passing the first test is 0.8. What is the probability of her passing the second test given that she has passed the first test? Passing both test A or B is=0.6 First test pass=0.8 Second test pass=? formula: P(A/B)=P(A/B)/P(B) p(second test B)=P(Passing both test A or B )/P(First test pass) =o.6/0.8 =0.75 So, second test pass =0.75

Baye’s Theorem Bayes Theorem is used heavily in statistics, analysis of data sets. Bayes' theorem (also known as Baye's rule) is a useful tool for calculating conditional probabilities. This is the inverse of conditional probability. Bayes’ Theorem is commonly used for decision tree analysis of business and social sciences.

Temporal Reasoning Reasoning about events that depend on time Expert systems designed to do temporal reasoning to explore multiple hypotheses in real time are difficult to build. One approach to temporal reasoning is with probabilities – a system moving from one state to another over time.

Markov Chain Characteristics The process has a finite number of possible states. The process can be in one and only one state at any one time. The process moves or steps successively from one state to another over time.

State Diagram Interpretation of a Transition Matrix Transition matrix – represents the probabilities that the system in one state will move to another.