4. Linear Models

Easy.

4E1. In the model definition below, which line is the likelihood?

$$\begin{align} y_i &\sim Normal(\mu, \sigma) \\ \mu &\sim Normal(0, 10) \\ \sigma &\sim Uniform(0, 10) \end{align}$$

$y_i \sim Normal(\mu, \sigma)$ is the likelihood.

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4E2. In the model definition just above, how many parameters are in the posterior distribution?

There are two parameters in the posterior distribution: $\mu$ and $\sigma$.

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4E3. Using the model definition above, write down the appropriate form of Bayes’ theorem that includes the proper likelihood and priors.

$$Pr(\mu, \sigma \mid y) = \frac{\prod_i Normal(y_i \mid \mu, \sigma) \, Normal(\mu \mid 0, 10) \, Uniform(\sigma \mid 0, 10)}{\iint \prod_i Normal(y_i \mid \mu, \sigma) \, Normal(\mu \mid 0, 10) \, Uniform(\sigma \mid 0, 10) \, d \mu d \sigma}$$

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4E4. In the model definition below, which line is the linear model?

$$\begin{align} y_i &\sim Normal(\mu, \sigma) \\ \mu_i &= \alpha + \beta x_i \\ \alpha &\sim Normal(0, 10) \\ \beta &\sim Normal(0, 1) \\ \sigma &\sim Uniform(0, 10) \end{align}$$

$\mu_i = \alpha + \beta x_i$ is the linear model.

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4E5. In the model definition just above, how many parameters are in the posterior distribution?

There are three parameters in the posterior distribution: $\alpha$, $\beta$ and $\sigma$.