# What are the hyperparameters and base measure in the conjugate prior for the exponential family?

## Setup

Suppose we have an exponential family model $$\{P_{\theta} : \theta \in \Theta\}$$. Let the density function of a random variable $$X$$ and the prior on $$\theta$$ have following forms:

\begin{align} p_{\theta}(x) &= h_1(x) \exp\left\{\theta^{\top} u(x) - a(\theta)\right\}, \tag{1} \\ p_{\chi, \nu}(\theta) &= h_2(\chi, \nu) \exp\left\{\theta^{\top} \chi - \nu a(\theta) \right\}. \tag{2} \end{align}

Here, $$h_1$$ and $$h_2$$ are base measures, $$u(x)$$ are sufficient statistics, and $$a(\theta)$$ is the log-normalizer for $$p_{\theta}(x)$$. It's a standard result that the posterior is

$$\pi_{\chi, \nu}(\theta) \prod_{n=1}^N p_{\theta}(x) \propto \exp\left\{ \theta^{\top}\left(\chi + \sum_{n=1}^N u(x_n) \right) - \left(\nu + N \right) a(\theta) \right\} \propto \pi_{\chi_N, \nu_N}(\theta), \tag{3}$$

where $$\chi_N = \chi + \sum u(x_n)$$ and $$\nu_N = \nu + N$$. This demonstrates conjugacy.

## My question

I've seen the general formulation of equations $$1$$ and $$2$$ many times, but I couldn't tell you concretely what $$h_2$$, $$\chi$$, and $$\nu$$ are in specific cases. For example, consider this Gaussian model:

\begin{align} X &\sim \mathcal{N}(\theta, \sigma_x^2), \tag{4} \\ \theta &\sim \mathcal{N}(\mu_0, \sigma_0^2), \tag{5} \end{align}

I can easily derive various quantities, e.g. the posterior predictive directly, using basic properties of the Gaussian. But I'm not sure how to do this using a general formula for the posterior predictive (see Wikipedia). While both densities (equations $$4$$ and $$5$$) are Gaussian, they have slightly different functional forms in equation $$1$$ and $$2$$. I can compute that $$h_1(x) = 1 / \sqrt{2 \pi}$$, the normalizer, for example, but then what is $$h_2(\chi, \nu)$$?