Bayes Theorem for Discrete Events by Continuous-valued Parameters

2 Bayes Theorem for Continuous-valued Parameters by Discrete Events 

For continuous-valued uncertain parameters 0 = [Oi, O2, , 0n)T, the concern is to update their probability density function. Consider a neighborhood around Oq = [ 10. 20, , that is a hypercube in the parameter space 0  

The event B in Equation (2.5) is defined as the occurrence of the parameter vector falling into this hypercube. For small AOi, AO2. A6ff, the probability of event B is  

This form is applicable to the identification of continuous-valued uncertain parameters with observation of discrete events and p(0 A) is regarded as the updated PDF or posterior PDF of the parameter vector 0. 

An imperfect dice was drawn independently for N times whereas 1 appeared N times. The aim here is to update the probability of the occurrence of 1 in a single draw (denoted as Pi) and this can be achieved by the Bayes theorem. The conditional PDF of the uncertain parameter Pi given the value of N is  

---