16.322 Stochastic Estimation and Control Professor Vander Velde But if the value of x is given, the statistical properties of m, and m2 depend only on n, and n,, which are independent. Thus the events E, and E,, conditioned on a given value of x, would be independent. Intuitively we would then say P(EE,Ix)=P(E, Ix)P(E, Ix) In general, the statement P(ABJE)= P(AJEP(BJE) is the definition of the conditional independence of the events A and B conditioned on the event e If A and b are conditionally independent, given E, and B is conditioned on E further conditioning on A should not change the probability of the occurrence of B since if e itionally independ BavesTheorem An interesting application of conditional probability. Let A, be mutually exclusive and collectively exhaustive P(AE=P(AE)_ P(AE) P(A)P(EIA) Bayes’rule P(E)∑P(4E)∑P4)P(E|A) This relation results directly from the definition of conditional probability and cannot be questioned. But over the years, the application of this relation to statistical inference has been subjected to a great deal of mathematical and philosophical criticism. Of late the notion has been more generally accepted, and serves as the basis for a well-defined theory of estimation and decision-making The point of the controversy is the use of a probability function to express ones state of imperfect knowledge about something which is itself not probabilistic at Example: Acceptance testing Components are manufactured by a process which is uncertain-some come out good and some bad. a test is devised such that a good component has Lecture16.322 Stochastic Estimation and Control Professor Vander Velde Lecture 2 But if the value of x is given, the statistical properties of m1 and m2 depend only on 1 n and 2 n , which are independent. Thus the events E1 and E2 , conditioned on a given value of x , would be independent. Intuitively we would then say 12 1 2 P EE x P E x P E x ( |) ( |)( |) = In general, the statement P AB E P A E P B E ( | ) ( | )( | ) = is the definition of the conditional independence of the events A and B , conditioned on the event E . If A and B are conditionally independent, given E , and B is conditioned on E , further conditioning on A should not change the probability of the occurrence of B since if E is given, A and B are conditionally independent. Bayes’ Theorem An interesting application of conditional probability. Let Ai be mutually exclusive and collectively exhaustive. ( ) ( ) ( )( | ) ( |) () ( ) ( )( | ) k k kk k i ii i i P AE P AE P A P E A PA E P E P AE P A P E A == = ∑ ∑ Bayes’ rule This relation results directly from the definition of conditional probability and cannot be questioned. But over the years, the application of this relation to statistical inference has been subjected to a great deal of mathematical and philosophical criticism. Of late the notion has been more generally accepted, and serves as the basis for a well-defined theory of estimation and decision-making. The point of the controversy is the use of a probability function to express one’s state of imperfect knowledge about something which is itself not probabilistic at all. Example: Acceptance testing Components are manufactured by a process which is uncertain – some come out good and some bad. A test is devised such that a good component has