Question

3. Suppose that a proportion parameter $\theta$ has a Beta($a, b$) prior and one observes $y_1, \dots, y_n$ from a Bernoulli distribution with parameter $\theta$, then the posterior density of $\theta$ is Beta($a + y, b + n - y$). Show that the posterior mean of $\theta \mid Y = y \sim$ Beta($a + y, b + n - y$) is a weighted average of the prior mean of $\theta \sim$ Beta($a, b$) and the sample mean $\hat{\theta} = \frac{y}{n}$. Find the two weights and explain their implication for the posterior being a combination of prior and data. Comment how Bayesian inference allows collected data to sharpen one's belief from prior to posterior.

          3. Suppose that a proportion parameter $\theta$ has a Beta($a, b$) prior and one observes
$y_1, \dots, y_n$ from a Bernoulli distribution with parameter $\theta$, then the posterior
density of $\theta$ is Beta($a + y, b + n - y$).
Show that the posterior mean of $\theta \mid Y = y \sim$ Beta($a + y, b + n - y$) is a
weighted average of the prior mean of $\theta \sim$ Beta($a, b$) and the sample mean
$\hat{\theta} = \frac{y}{n}$. Find the two weights and explain their implication for the posterior
being a combination of prior and data. Comment how Bayesian inference
allows collected data to sharpen one's belief from prior to posterior.

3. Suppose that a proportion parameter θ has a Beta(a, b) prior and one observes
y1, …, yn from a Bernoulli distribution with parameter θ, then the posterior
density of θ is Beta(a + y, b + n - y).
Show that the posterior mean of θ| Y = y ∼ Beta(a + y, b + n - y) is a
weighted average of the prior mean of θ∼ Beta(a, b) and the sample mean
θ̂ = (y)/(n). Find the two weights and explain their implication for the posterior
being a combination of prior and data. Comment how Bayesian inference
allows collected data to sharpen one's belief from prior to posterior.

Added by Danielle G.

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