Proportionality term in Normal-Gamma distribution I am currently learning from Christopher Bihops's Pattern Recognition and Machine Learning book about posterior distributions for the Normal distribution whenever both $\mu$ and $\tau$ (the precision term) are unknown by using the Normal-Gamma distribution as a conjugate prior.
On these slides (p.47), he defines the NormalGamma distribution $p(\mu, \lambda)$ as
$$
p(\mu, \lambda) = \mathcal{N}(\mu | \mu_0, (\beta\lambda^{-1}))\text{Gam}(\lambda|a,b)
$$
And he continues by implying that
$$
p(\mu, \lambda) \propto \exp\left(-\frac{\beta\lambda}{2}(\mu-\mu_0)^2\right)\lambda^{a-1}\exp\left(-b\lambda\right)
$$
I am confused as to why the proportionality holds true. If we know that
$$
 \mathcal{N}(\mu | \mu_0, (\beta\lambda^{-1})) = \frac{(\beta\lambda)^{1/2}}{(2\pi)^{1/2}} \exp\left(-\frac{\beta\lambda}{2}(\mu-\mu_0)^2\right)
$$
Then $p(\mu, \lambda)$ is clearly a function of two variables, $\lambda$ being one of them. So by removing $(\beta\lambda)^{1/2}$, the proportionality is false.
What am I missing?
 A: If the Normal-Gamma is given by:
$$ p(\mu, \lambda) = \mathcal{N}(\mu | \mu_0, (\beta\lambda^{-1}))\text{Gam}(\lambda|a,b) \tag{1} $$
The two terms separately are
\begin{align}
\mathcal{N}(\mu | \mu_0, (\beta\lambda^{-1}))
&= \frac{\beta^{1/2}}{(2\pi)^{1/2}} \sqrt{\lambda}\exp\left(-\frac{\beta\lambda}{2}(\mu-\mu_0)^2\right)\\
\text{Gam}(\lambda|a,b) &= \Gamma(a)^{-1}b^a \lambda^{a-1}\exp(-b\lambda)
\end{align}
And so together we get
\begin{align}
p(\mu, \lambda) &=
\frac{\beta^{1/2}}{(2\pi)^{1/2}} \sqrt{\lambda}\exp\left(-\frac{\beta\lambda}{2}(\mu-\mu_0)^2\right) \lambda^{a-1}\exp(-b\lambda) \\
&= \frac{b^a}{\Gamma(a)}\sqrt{\frac{\beta}{2\pi}} \exp\left(-\frac{\beta\lambda}{2}(\mu-\mu_0)^2\right) \lambda^{a-1/2}\exp(-b\lambda) \\
&\propto \exp\left(-\frac{\beta\lambda}{2}(\mu-\mu_0)^2\right) \lambda^{a-1/2}\exp(-b\lambda)\tag{2}
\end{align}
which matches the wiki for the Normal-Gamma.
So this is indeed an error as you have found. It is a mistake in the slides you have linked. But, it is also in the book (pg 101).
They show that
$$
p(\mu, \lambda) \propto \exp\left(-\frac{\beta\lambda}{2}(\mu-c/\beta)^2\right) \lambda^{\beta\,/\,2}\exp\left[-\left(d-\frac{c^2}{2\beta}\right)\lambda\right] \tag{3}
$$
and then claim that (1) is true for $\mu_0=c/\beta$, $a = 1 + \beta/2$, and $b=d - c^2/(2\beta)$.
But notice that $$ a - \frac{1}{2} = 1 + \frac{\beta}{2} - \frac{1}{2} = \frac{1}{2} + \frac{\beta}{2} \ne \frac{\beta}{2}$$
so the powers of $\lambda$ do not match between (2) and (3)!
It is corrected in the errata and additional comments of the book, where they note that it should say $a=(1+\beta)/2$. I guess he simply copied his own book for the slides along with the error :) Nice catch.
