# Exercise 1

Consider the following model,

• Prior: $$\theta \sim \text{Beta}(\alpha = 5, \beta = 5)$$
• Likelihood: $$X_1, X_1, \ldots, X_n \sim \text{Bern}(\theta)$$
• Posterior: $$\theta \mid X_1, X_1, \ldots, X_n \sim \ ?$$

and observed data with statistics,

• Sample size: $$n = 20$$
• Number of “successes” $$\sum x_i = 15$$

Use the given prior and the observed data to calculate a Bayes’ estimate of $$\theta$$. (Use the posterior mean.)

### Solution

• Recall that a Beta distribution with parameters $$\alpha$$ and $$\beta$$ has mean $$\frac{\alpha}{\alpha + \beta}$$.
• The posterior distribution for the Beta-Bernoulli model is a Beta distribution with parameters $$a = \alpha + \sum x_i$$ and $$b = \beta + \sum y_i$$.
• Here we have, $$\sum y_i = n - \sum x_i$$.
• Here we are using $$\alpha$$ and $$\beta$$ as the parameters for the prior, and $$a$$ and $$b$$ as the parameters of the posterior.

So, in this case, we have

$\theta \mid X_1, X_1, \ldots, X_n \sim \text{Beta}(a = 20, b = 10)$

Then we have

$\hat{\theta}_B = \frac{a}{a + b} = \frac{20}{20 + 10} = \boxed{\frac{2}{3}}$

# Exercise 2

Consider the following model,

• Prior: $$\theta \sim \text{Beta}(\alpha = 50, \beta = 20)$$
• Likelihood: $$X_1, X_1, \ldots, X_n \sim \text{Bern}(\theta)$$
• Posterior: $$\theta \mid X_1, X_1, \ldots, X_n \sim \ ?$$

and observed data with statistics,

• Sample size: $$n = 40$$
• Number of “successes” $$\sum x_i = 32$$

Use the given prior and the observed data to calculate a Bayes’ estimate of $$\theta$$. (Use the posterior mean.)

### Solution

In this case, we have

$\theta \mid X_1, X_1, \ldots, X_n \sim \text{Beta}(a = 82, b = 28)$

Then we have

$\hat{\theta}_B = \frac{a}{a + b} = \frac{82}{82 + 28} = \boxed{\frac{41}{55}}$

# Exercise 3

Consider the following model,

• Prior: $$\theta \sim \text{Beta}(\alpha = 10, \beta = 60)$$
• Likelihood: $$X_1, X_1, \ldots, X_n \sim \text{Bern}(\theta)$$
• Posterior: $$\theta \mid X_1, X_1, \ldots, X_n \sim \ ?$$

and observed data with statistics,

• Sample size: $$n = 5$$
• Number of “successes” $$\sum x_i = 2$$

Use the given prior and the observed data to calculate a Bayes’ estimate of $$\theta$$. (Use the posterior mean.)

### Solution

In this case, we have

$\theta \mid X_1, X_1, \ldots, X_n \sim \text{Beta}(a = 12, b = 63)$

Then we have

$\hat{\theta}_B = \frac{a}{a + b} = \frac{12}{12 + 63} = \boxed{\frac{4}{25}}$

# Exercise 4

Consider the following model,

• Prior: $$\theta \sim \text{Beta}(\alpha = 10, \beta = 60)$$
• Likelihood: $$X_1, X_1, \ldots, X_n \sim \text{Bern}(\theta)$$
• Posterior: $$\theta \mid X_1, X_1, \ldots, X_n \sim \ ?$$

and observed data with statistics,

• Sample size: $$n = 5$$
• Number of “successes” $$\sum x_i = 2$$

Use the given prior and the observed data to calculate a 99% credible interval

### Solution

qbeta(c(0.005, 0.995), shape1 = 10 + 2, shape2 = 60 + 3)
## [1] 0.06973817931 0.28374038952

# Exercise 5

Consider the following model,

• Prior: $$\theta \sim \text{Beta}(\alpha = 4, \beta = 4)$$
• Likelihood: $$X_1, X_1, \ldots, X_n \sim \text{Bern}(\theta)$$
• Posterior: $$\theta \mid X_1, X_1, \ldots, X_n \sim \ ?$$

and observed data with statistics,

• Sample size: $$n = 5$$
• Number of “successes” $$\sum x_i = 2$$

Use the given prior and the observed data to calculate a 90% credible interval

### Solution

qbeta(c(0.05, 0.95), shape1 = 4 + 2, shape2 = 4 + 3)
## [1] 0.2452998013 0.6847622093

# Exercise 6

Consider the following model,

• Prior: $$\theta \sim \text{Beta}(\alpha = 10, \beta = 4)$$
• Likelihood: $$X_1, X_1, \ldots, X_n \sim \text{Bern}(\theta)$$
• Posterior: $$\theta \mid X_1, X_1, \ldots, X_n \sim \ ?$$

and observed data with statistics,

• Sample size: $$n = 50$$
• Number of “successes” $$\sum x_i = 2$$

Use the given prior and the observed data to calculate a 95% credible interval

### Solution

qbeta(c(0.025, 0.975), shape1 = 10 + 2, shape2 = 4 + 48)
## [1] 0.1024842436 0.2909709491

# Exercise 7

Consider the following model,

• Prior: $$\theta \sim \text{Beta}(\alpha = 10, \beta = 4)$$
• Likelihood: $$X_1, X_1, \ldots, X_n \sim \text{Bern}(\theta)$$
• Posterior: $$\theta \mid X_1, X_1, \ldots, X_n \sim \ ?$$

and observed data with statistics,

• Sample size: $$n = 50$$
• Number of “successes” $$\sum x_i = 20$$

Use the given prior and the observed data to test $$H_0: \theta > 0.50$$ vs $$H_1: \theta \leq 0.50$$

### Solution

pbeta(0.50, shape1 = 10 + 20, shape2 = 4 + 30, lower.tail = FALSE)
## [1] 0.3073275079
pbeta(0.50, shape1 = 10 + 20, shape2 = 4 + 30)
## [1] 0.6926724921

Accept $$H_1$$.

# Exercise 8

Consider the following model,

• Prior: $$\theta \sim \text{Beta}(\alpha = 10, \beta = 10)$$
• Likelihood: $$X_1, X_1, \ldots, X_n \sim \text{Bern}(\theta)$$
• Posterior: $$\theta \mid X_1, X_1, \ldots, X_n \sim \ ?$$

and observed data with statistics,

• Sample size: $$n = 20$$
• Number of “successes” $$\sum x_i = 5$$

Use the given prior and the observed data to test $$H_0: 0.25 < \theta < 0.50$$ vs $$H_1: \theta \leq 0.25, \theta \geq 0.50$$

### Solution

diff(pbeta(c(0.25, 0.50), shape1 = 10 + 5, shape2 = 10 + 15))
## [1] 0.9020703778

Accept $$H_0$$.

# Exercise 9

Consider the following model,

• Prior: $$\theta \sim \text{Beta}(\alpha = 3, \beta = 3)$$
• Likelihood: $$X_1, X_1, \ldots, X_n \sim \text{Bern}(\theta)$$
• Posterior: $$\theta \mid X_1, X_1, \ldots, X_n \sim \ ?$$

and observed data with statistics,

• Sample size: $$n = 100$$
• Number of “successes” $$\sum x_i = 70$$

Use the given prior and the observed data to test $$H_0: \theta > 0.80$$ vs $$H_1: \theta \leq 0.80$$

### Solution

pbeta(0.80, shape1 = 3 + 70, shape2 = 3 + 30, lower.tail = FALSE)
## [1] 0.003745930192
pbeta(0.80, shape1 = 3 + 70, shape2 = 3 + 30)
## [1] 0.9962540698

Accept $$H_1$$.