Downloaded from https://www.everyspec.com AMCP 706-1M then check it out by simulation, revise it 3-9.7 CONJUeSATE PRIOR DISTRIBU- and check again; repeat the process until you TIONS are satisfied that you believe what you've recorded. When the events and probabi'ities are in the form of a probability distribution, it is Bayesian analysis is tedious and time con- mathematically possible to find a form of suming when it is done right, When it is prior distribution such that the after-test done wrong (sloppily), it can be very mis- (posterior) distribution has the same para- leading. metric form (from the same family of distribu- tions) as the prior; see Ref. 25 or other Bayesian textbook.
3-9.6 BAYES FORMULAS, CONTINUOUS In this cae the new distribution is the RANDOM VARIABLES same as the old, except for different values of the parameters. As an example, for the usual failure rate, one can hypothesize pdf JO I c Pd/ •}pd{ 1,} (3-40) prior distribution with parameters r and r , ,Pd/, I - 1 (3-41) where r - number of failures and T' total test time, The estimate of X, usually used Is simply PIT. The new estimate of X is made by adding the new Increments in r and 7" to where the old values-just as if a classical test were being run.
But, in Bayesian statistics, the Svalue of the continuous ran- old r and r don't have to be actual test re- dom variable suits, they can be numbers that are equivalent to your prior degree-of-belief. nestiate of w made froTh Q 'rh calulational simplicity of this ap- new test proach has much appeal, and its use is pop- pd€ (0 prior pdf assigned to 0 (before ularized in books and articles. Before using the test) it, compare it with the discrete method de- scribed earlier in this paragraph. Use simu- pO"f 00 1 likelihood of getting the test lation to find the consequences of your result • given that 4' is the assumptions.
But remember, computational resvalue aimplicity is not the purpose of a Bayesian analysis. The real purpose is to find a pdf f10•}1 new pdf assigned to (h after reasoiable way to put your prior knowledge seeing the test restilts to work. implies the integral over the Example No, 30 illustrates the procedure. domain of 3.9.8 LIFE-TESTING Other forms of the equations are possible for Ref. 29 shows how Bayesian techniques other combinations of' discrete and random can be applied to accept/reject tests, Be very careful in using them. Simulate exttn- variables for the unknown variable and the sively before applying any in an important prior information1, situation.
The mcthematlce is deceptively I' mll