Bayesian inference with conjugate priors - triplot

Question

I'm trying to make a "triplot" to illustrate Bayesian inference (so I'd like to have prior, likelihood and posterior in the same picture). For likelihood I'm using \begin{equation}\label{eq:lik} f(y|\tau) = \prod_{i=1}^{n}\frac{\tau}{\sqrt{2\pi}}\exp\left(-\frac{\tau(y_{i}-\mu)^{2}}{2}\right) = \frac{\tau}{\sqrt{2\pi}}\exp\left(-\frac{\tau\sum_{i=1}^{n}(y_{i}-\mu)^{2}}{2}\right), \end{equation} i.e., the Gaussian distribution with known mean $\mu$ and unknown precision $\tau=\frac{1}{\sigma}$, where ${\sigma}^{2}$ is the unknown variance.
If we choose the prior on $\tau$ to be a gamma distribution \begin{equation} p(\tau)=\Gamma(\alpha,\beta), \end{equation} with the shape $\alpha$ and the rate $\beta$, we can use the conjugacy theory to find the form of the posterior. The posterior distribution in our example is the following gamma distribution \begin{equation} p(\tau|y)=\Gamma\left(\alpha+\frac{n}{2},\beta+\frac{\sum_{i=1}^{n}(y_{i}-\mu)^{2}}{2}\right). \end{equation} I plotted it with $\alpha=1.5$, $\beta=10.0$, $\mu=0.0$ $n=5$ and the random sample $(y_i)_{i=1\dots 5}$ was generated assuming the "true" value of $\tau=0{.}25$. As you can see, the likelihood is not visible. I tried different configurations of parameters but nothing helped.
I'd be very grateful for any ideas how to choose $\alpha$ and $\beta$ so that I get a nice illustration, something like that.

Answer 1

The method is very easy: I'll rescale the likelihood which is fine because it doesn't have to integrate to 1.