6. Example: Jeremy’s IQ

This example shows how to use the Normal-Normal conjugate pair.

Jeremy, a Georgia Tech student, is modeling his IQ as \(\mathcal{N}(\theta, 80)\). Our prior on a GT student’s mean IQ, \(\theta\), is \(\mathcal{N}(110, 120)\). Jeremy takes an IQ test and scores 98.

Here’s our model:

\[\begin{split} \begin{align*} X | \theta &\sim \mathcal{N}(\theta, 80) \\ \theta &\sim \mathcal{N}(110, 120) \end{align*} \end{split}\]

According to the conjugate table, we can use the Normal-Normal pair as follows:

\[\theta\mid X\sim \mathcal{N}\left(\frac{\tau^2}{\tau^2 +\sigma^2/n}\bar{X} + \frac{\sigma^2/n}{\tau^2 + \sigma^2/n}\mu, \frac{\tau^2\sigma^2/n}{\tau^2+ \sigma^2/n}\right)\]

We know the following from the problem description:

\(\mu = 110, \tau^2 = 120, \sigma^2 = 80, n = 1, \bar{X} = 98\)

So:

\[\begin{split} \begin{align*} \theta\mid X &\sim \mathcal{N}\left(\frac{120}{120 + 80/1}(98) + \frac{80/1}{120 + 80/1}(110), \frac{(120)(80)/1}{120 + 80/1}\right) \\ &\sim \mathcal{N}(102.8, 48) \end{align*} \end{split}\]

If we increase \(n\) to 5 with the same mean, the posterior mean and variance both go down as the data starts to overwhelm the prior: \(\mathcal{N}(99.4, 14.1)\).