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Bayesian Estimation

Bayesian estimation uses subjective judgment in an engineering design.

For discrete case, let the parameter θ takes the values θi; i = 1, 2, ... , n with the probabilities Pi = P [θ = θi]. Let θ0 be the observed outcome of the experiment. Then by Bayes' theorem

we obtain,

Then the expected value of 8 is called Bayesian estimator of the parameter, i.e.,

Using this we can calculate

For continuous case, let 8 be a random variable of the parameter of the distribution given by the density function f '(θ). Then

P[θi < θ < θi + ∆θ] = f'(θi) . ∆θ,                                 i = 1, 2, ... , n

If θ0 is an observed experimental outcome, then

i = 1, 2, ... , n

 

In the limit we obtain, f" (θ) =

Then the Bayesian estimator is

Using this we can calculate