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Problem from Bayesian Data Analysis (Bayesian interpretation of non-Bayesian estimates): Consider the following estimation procedure, which is based on classical hypothesis testing. A matched pairs experiment is done, and the differences $y_1, ... , y_n$ are recorded and modeled as independent draws from $\text{N}(\theta, \sigma^2)$ where $\sigma$ is known. The parameter $\theta$ is estimated as the average observed difference if it is ‘statistically significant’ and zero otherwise:

$$\hat{\theta} \equiv \begin{cases} \bar{y} & & |\bar{y}| \geqslant z_{0.025} \cdot \sigma / \sqrt{n}, \\[6pt] 0 & & \text{otherwise}, \end{cases}$$

where $z_{0.025} \approx 1.96$ is the normal quantile for a 5% significant test. I am hoping to find a prior distribution for $\theta$ that gives this estimator as a Bayesian estimator. I was hoping that an improper prior parameterised by $\alpha>0$ with the following form would work:

$$\pi(\theta) \propto \begin{cases} 1+\alpha & & \theta = 0, \\[6pt] 1 & & \theta \neq 0. \end{cases}$$

My hope was that the posterior would be a normal distribution centered at $\bar{y}$ with variance $\sigma^2 / n$ except at $\theta = 0$ where there would be a probability spike. Depending on the parameter $\alpha$, the posterior mode would be either at 0 or $\bar{y}$. Would appreciate feedback.

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