Reasoning in Uncertain Environments

Abstract

 Real-life propositions are neither fully true nor false, also there is always some uncertainty associated with them. Therefore, the reasoning using real-life knowledge should also be in accord. This chapter is aimed to fulfill the above objectives. The chapter presents the prerequisites—foundations of probability theory, conditional probability, Bayes theorem, and Bayesian networks which are graphical representation of conditional probability, propagation of beliefs through these networks, and the limitations of Bayes theorem. Application of Bayesian probability has been demonstrated for specific problem solutions. Another theory—the Dempster–Shafer theory of evidence—which provides better results as the evidences increase is presented, and has been applied on worked examples. Reasoning using fuzzy sets is yet another approach for reasoning in uncertain environments—a theory where membership of sets is partial. Inferencing using fuzzy relations and fuzzy rules has been demonstrated, followed by chapter summary, and a set of exercises at the end.

Exercises

• 1. We assume a domain of 5-card poker hands out of a deck of 52 cards. Answer the following under the assumption that it is a fair deal.

  • a. How many 5-card hands can be there (i.e., number of atomic events in joint probability distribution).

  • b. What is the probability of an atomic event?

  • c. What is the probability that a hand will comprise four cards of the same rank?

• 2. Either prove it is true or give a counterexample in each of the following statements.

  • a. If $P(a \mid b, c) = P(a)$, then show that $P(b \mid c) = P(b)$.

  • b. If $P(a \mid b, c) = P(b \mid a, c)$, then show that $P(a \mid c) = P(b \mid c)$.

  • c. If $P(a \mid b) = P(a)$, then show that $P(a \mid b, c) = P(a \mid c)$.

• 3. Consider that for a coin, the probability that when tossed the head appears up is x and for tails it is $1 - x$. Answer the following:

  • a. Given that the value of x is unknown, are the outcomes of successive flips of this coin independent of each other? Justify your answer in the case of head and tail.

  • b. Given that we know the value of x, are the outcomes of successive flips of the coin independent of each other? Justify.

• 4. Show that following statements of conditional independence,

  $$P(X \mid Z) P(Y | Z) = P(X, Y \mid Z)$$

  are also equivalent to each of the following statements,

  $$ P(Y \mid Z) = P(Y \mid X, Z),$$

  and

  $$P(X \mid Z) = P(X \mid Y, Z).$$

• 5. Out of two new nuclear power stations, one of them will give an alarm when the temperature gauge sensing the core temperature exceeds a given threshold. Let the variables be Boolean types: A = alarm sounds, FA =alarm be faulty, and FG = gauge be faulty. The multivalued nodes are G = reading of gauge, and T = actual core temperature.

  • a. Given that the gauge is more likely to fail when the core temperature goes too high, draw a Bayesian network for this domain.

  • b. Assume that there are just two possible temperatures: actual and measured, normal and high. Let the probability that the gauge gives the correct reading be x when it is working, and y when it is faulty. Find out the conditional probability associated with G.

  • c. Assume that the alarm and gauge are correctly working, and the alarm sounds. Find out an expression for the probability in terms of the various conditional probabilities in the network, that the temperature of the core is too high.

  • d. Let the alarm work correctly unless it is faulty. In case of being faulty, it never sounds. Give the conditional probability table associated with A.

• 6. Compute the graphical representation of the fuzzy membership in the graph 12.19 for following set operations:

  • a. $Small \cap Tall$

  • b. $(Small \cup Medium)$–Tall

• 7. Given the fuzzy sets $A = \{\frac{a}{0.5}, \frac{b}{0.9}, \frac{e}{1}\}$, $B=\{\frac{b}{0.7}, \frac{c}{0.9}, \frac{d}{0.1}\}$, Compute $A \cup B$, $A \cap B, A^\prime , B^\prime$.

• 8. Let $X = \{ a, b, c, d, e\}$, $A = \{\frac{a}{0.5}, \frac{c}{0.3}, \frac{e}{1}\}$. Compute:

  • a. $\bar{A}$

  • b. $\bar{A} \cap A$

  • c. $\bar{A} \cup A$