Given that \(\theta\) is the true proportion of people over age 40 in my community to have hypertension, I would assume \(\theta\sim Beta(\alpha,\beta)\) Encoding my prior belief about \(\theta\) (and it’s point estimate such as mean), let’s assume \(\hat{\theta}=0.6\). From \(Beta\) distribution

\begin{align}
\int\theta p(\theta)d\theta=\frac{\alpha}{\alpha+\beta}=0.6\\
\end{align}

Let’s choose \(\alpha=6,\beta=4\)

\begin{equation}
\begin{split}\displaystyle p(\theta|x)&\displaystyle\propto p(x|\theta)p(\theta)\end{split}\\
\end{equation}

01b

Now after the initial survey (set of bernoulli trials; bionomial distribution), need to update belief about \(\theta\). We know that with \(Beta\) prior, the posterior is also a conjugate \(Beta\) for such distribution. \(\theta|y\sim Beta(\alpha+y,n+\beta-y)\)

\begin{equation}
\begin{split}\displaystyle p(\theta|y)&\displaystyle\propto p(y|\theta)p(\theta)\\
&\displaystyle\propto\theta^{y}(1-\theta)^{n-y}\theta^{\alpha-1}(1-\theta)^{\beta-1}\end{split}\\
\end{equation}

where \(n\) = total number of people selected for the survey and \(y\) = total number of hypertensive people observed. To calculate the point estimate

\begin{align}
\overline{\theta}_{post}=\int\theta p(\theta|y)d\theta=E(\theta|y)=\frac{\alpha+y}{\alpha+\beta+n}\\
\end{align}

Plugging \(y=4,n=5,\alpha=6,\beta=4,\overline{\theta}_{post}=0.66\). A \(10\%\) improvement in my belief about \(\theta\). The prior dominates.

01c

Similarly, plugging \(y=400,n=1000,\alpha=6,\beta=4,\overline{\theta}_{post}=0.40\). A 33% reduction in my belief about \(\theta\). The data dominates as opposed to the prior.

So with smaller data-sets, incorrect prior beliefs can be reinforced. The more data we gather, the more accurate our beliefs about \(\theta\) approaches to the real one.