Part BCSIR NET June 2025a-flat-improper-prior-adds-no-information-so-the-posterior-variance-does-not-shrink

A flat improper prior adds no information so the posterior variance does not shrink

Suppose the distribution of X given $θ$ is normal with mean $θ$ and variance 15. Further, let the prior (improper) distribution of $θ$ be proportional to $1, - \infty < θ < \infty$ . If the observed value of X is 13, then which of the following statements is true?

A.Posterior mean = Maximum likelihood estimate of $θ$ , Posterior variance = Var(X| $θ)$ ✓
B.Posterior mean = Maximum likelihood estimate of $θ$ , Posterior variance < Var(X| $θ)$
C.Posterior mean > Maximum likelihood estimate of $θ$ , Posterior variance = Var(X| $θ)$
D.Posterior mean > Maximum likelihood estimate of $θ$ , Posterior variance < Var(X| $θ)$

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward, and why this one tests hypothesis dropped.

See pricing

50 are analysed free — try those first.

The trap it tests

Hypothesis dropped

A theorem applied without checking one of its preconditions.

Drill statements like this

Related counterexample: Pairwise independent events are mutually independent

A flat improper prior adds no information so the posterior variance does not shrink

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