[Actuarial] Normal Mean with Normal Prior

If the claim size X follows a normal distribution, that is to say 

    X ~ N(theta, v),

where theta is the mean and v is the variance, and theta has a normal prior,

    theta ~ N(mu, a),

where mu is the mean of the prior distribution and a is the variance, the following method can be used to find the posterior mean and variance of theta and X.

Assume that we observe n claims, and each has a size of x_i ( i in [1,n] ). Then we have a credibility factor

    Z = n*a / (n*a + v),

and the posterior mean of theta, denoted mu_new, is therefore 

    mu_new = Z * avg(x_i) + (1-Z) * mu = (a*sum(x_i) + v*mu) / (n*a + v).

The posterior variance of theta, a_new, can be computed using 

    a_new = v*a / (n*a + v).

The posterior variance of X, v_new, is the sum of the variance of theta and its own variance v, in other words,

    v_new = a_new + v.