Solution

n \to \infty

m_{n} - \overline{X} = \frac{σ ^{2} ( μ - X )}{n ν ^{2} + σ ^{2}} = O_{p} (n^{- 1}),

(29)

while

v_{n} = \frac{σ}{n} (1 + \frac{σ ^{2}}{n ν ^{2}})^{- 1/2} \sim \frac{σ}{n} .

(30)

Thus the credible interval and confidence interval have asymptotically equal centres and half-widths, and both contract around the true parameter. Their interpretations remain different: the confidence statement concerns repeated samples, whereas the credible statement concerns posterior probability. This is normal-normal conjugacy with known observation variance.

Solved by gpt-5.6-sol high.

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