Normal-Normal Model

Objective:

• By the end of this lesson, students will have reviewed the following topics:

  • When to implement the normal-normal model

  • How to implement the normal-normal model

  • How to disseminate analysis results

Duration:

• 75 minutes

Materials:

• Handouts with exercises and problems related to basic probability
• Computer, projector, and screen

Introduction:

Introduce the lesson’s topic:

Main Content:

The Normal Model

Recall the normal model,

\[f(y) = \frac{1}{\sqrt{2\pi\sigma^2}} \exp\left\{-\frac{\left(y - \mu\right)^2}{2\sigma^2} \right\}\]

where \(y\) is a continuous random variable that can take any value in \(\mathbb{R}\); i.e., \(Y \in \left(-\infty, \infty\right)\).

Suppose we have \(n\) observations, then the joint distribution is given by

\[f(\overset{\to}{y}|\mu) = \prod_{i=1}^n f(y_i|\mu) = \prod_{i=1}^n \frac{1}{\sqrt{2\pi\sigma^2}} \exp\left\{-\frac{\left(y_i - \mu\right)^2}{2\sigma^2} \right\}\]

Then the likelihood is given by,

\[ \begin{align*} L(\mu| \overset{\to}{y}) &\propto \prod_{i=1}^n \exp\left\{-\frac{\left(y_i - \mu\right)^2}{2\sigma^2} \right\} = \exp\left\{-\frac{\sum_{i=1}^n \left(y_i - \mu\right)^2}{2\sigma^2} \right\} \\ &\propto \exp\left\{-\frac{\left(\bar{y} - \mu\right)^2}{2\sigma^2/n} \right\} \end{align*} \]

The Normal-Normal Model

• Let \(\mu \in (-\infty, \infty)\) be an unknown mean parameter and \((Y_1, Y_2, ..., Y_n)\) be an independent \(N(\mu, \sigma^2)\), where \(\sigma\) is assumed to be known.

• The Normal-Normal Bayesian model has Normal distributions for both prior and data. The Normal prior is on the unknown mean, \(\mu\).

\[Y_i | \mu \overset{\text{ind}}{\sim} N(\mu, \sigma^2)\]

\[\mu \sim N(\theta, \tau^2)\]

• When we have data \(\overset{\to}{y} = (y_1, ..., y_n)\) with mean \(\bar{y}\), the posterior distribution for \(\mu\) is also Normal with updated parameters,

\[\mu|\overset{\to}{y} \sim N\left( \theta \frac{\sigma^2}{n\tau^2 + \sigma^2} + \bar{y} \frac{n\tau^2}{n\tau^2+\sigma^2}, \frac{\tau^2 \sigma^2}{n\tau^2 + \sigma^2} \right)\]

Calculation and Practice:

• Example 1: concussions

• Example 2: stock prices

Discussion and Wrap-Up:

• Facilitate a class discussion to review the example problems, reinforce key concepts, and answer any questions the students have.

Homework:

• Assign additional problems to practice the basic probability rules.

Formative Assessment:

• Evaluate students based on their participation in discussions, their ability to solve example problems, and their performance on the assigned homework.

Conclusion:

• Our goal is to analyze in the best way possible.