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In an engineering design, Bayesian estimation uses subjective judgement. The estimation is an estimated value that decreases the posterior expected value of a loss function. Also, it increases the posterior expected value of utility function.

For Bayesian estimation, for discrete cases, if the parameter takes a value of, where i=1, 2,….,n, with probability  and let be the resultant outcome.

Therefore, by Bayes’ theorem we get,
Bayesian Estimation 1
P where i=1, 2, ….., n.

Thus, as a result, the expected value of 8 is known as Bayesian estimator of the parameter, that is,
Bayesian Estimation 2

With this, the following can be obtained,

P

If the case be continuous, then let 8 be the random variable of the parameter of distribution denoted by the density function f’(.

Therefore, P [< θ <  + ∆θ] = f'() . ∆θ, where i=1, 2, …, n.
Bayesian Estimation 3

If θ0 is the observed experimental outcome, then

f” () ∆θwhere i=1, 2, 3,…., n.

Bayesian Estimation 4

In the limit, f”(

Bayesian Estimation 5

Therefore, the Bayesian estimator is

Bayesian Estimation 6

This can be used to calculate P

Bayesian Estimation 7