Ex. A randomly chosen individual has an unknown true IQ \(\theta\). Its prior distr. is \(\theta\sim N(100,\,225)\). This normal distr. describes the whole population with mean IQ of \(m=100\) and sd \(v=15\).
Given a true personal value \(\theta\), the result of an IQ measurement has distr. \(X\sim N(\theta,\,100)\), with no systematic error and a random error \(\sigma=10\). Since
and the posterior is proportional to \(g(\theta)f(x|\theta)\), we find that \(h(\theta|x)\) is proportional to\[h(\theta|x)\propto\exp\left(-\frac{(\theta-m)^{2}}{2v^{2}}-\frac{(x-\theta)^{2}}{2\sigma^{2}}\right)\propto\exp\left(-\frac{(\theta-\gamma m-(1-\gamma)x)^{2}}{2\gamma v^{2}}\right)\]where <- is the so-called shrinkage factor. We conclude that the posterior distr. is normal
with mean \(\gamma m+(1-\gamma)x\) and variance \(\gamma v^{2}\)
