Minimizing Kullback-Leibler divergence This paper is showing how KL divergence can be minimized by matching the expected values of the sufficient statistics.
More precisely,
For any distribution p of the exponential family with pdf:
$$ p_{\theta}(x) = \frac{1} {Z(\theta)}\exp(\theta^T \phi(x))$$
the distribution $$p_{θ∗} $$ which minimises the Kullback-Leibler divergence,
$$ KL (p||p_{θ∗})$$ over the exponential family with natural statistic $\phi$ is implicitly given by:
$$E_{p_{\theta}*(x)} [\phi(x)]  = E_{p_{(x)}} [\phi(x)]$$
I need help to show that this is true for the particular case of a normal-gamma distribution. So I need to make the same proof that is on the paper but for the special case of Normal gamma.

We already have that the sufficient statistics are
$$
T_1 = \ln T,\; T_2 = T,\; T_3 = TX,\; T_4=TX^2.
$$
You should select the approximating density $q(X,T)$. Then, by Theorem 1, you simply equate the true expectations with the expectations under the approximating distribution. That is,
$$
\begin{align}
E_q(\ln T) &= \psi(\alpha) - \ln(\beta)\\
E_q(T) &= \frac{\alpha}{\beta},\\
E_q(TX) &= \frac{\alpha\mu}{\beta},\\
E_q(TX^2) &= \frac{\alpha\mu^2}{\beta} + \frac{1}{\lambda}.
\end{align}
$$
Until you specify what your approximating density is, this is all I can say for this problem. The cited note is explaining assumed density filtering where the posterior distribution is the true distribution and the approximating density $q$ is an exponential family.
