# Does conjugate prior for natural exponential family needs jacobian to transform natural parameter back to original parameter?

From bayesian theory, we have that if $$f(x|\eta) \propto \exp(\eta \cdot T(x)- A(\eta))$$ - a natural exponential family, then the prior conjugate of $$\eta$$ is $$\pi^*(\eta | \mu, \lambda) \propto \exp(\mu \cdot x- \lambda A(\eta))$$ and the posterior is $$\pi^*(\eta|x) = \pi(\eta | \mu + T(x), \lambda + 1)$$ [1].

I am confused by my bayesian professor (in a undergrad class) with respect of findind these parameters, $$\mu$$ and $$\lambda$$.

My question is: when we start with "standard" parameterization $$f(x|\theta) \propto \exp(B(\theta) \cdot T(x) - A(\theta))$$ and then reparameterize to natural parameter $$\eta$$ as above with $$\eta = B(\theta)$$, does $$\pi(\theta | \mu, \lambda) = |B'(\theta)| \pi^*(B(\theta) | \mu, \lambda)$$? Or does $$\pi(\theta | \mu, \lambda) \propto \pi^*(B(\theta) | \mu, \lambda)$$?

In other words, we can only substitute $$B(\theta)$$ on $$\eta$$? Or do we need to consider the jacobian, since $$B(\theta)$$ is a transformation of the random variable $$\theta$$?

If only substitution is enough, then we conclude that jacobian does not change the family of $$\pi$$. And in that case, the "correct" hyperparameters $$\mu$$ and $$\lambda$$ are not of importance to bayesian inference. But in that case, the table from the reference book [1] does not hold.

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For example, if $$x \sim Poisson(\theta)$$ then $$f(x | \theta) \propto \exp\{B(\theta)x - \theta\}$$, where $$B(\theta) = \ln(\theta)$$. Hence $$\eta = \ln(\theta)$$ and $$f(x | \eta) \propto \exp\{\eta x - e^{\eta}\}$$. Therefore $$\pi^*(\eta | \mu, \lambda) \propto \exp\{\mu \eta - \lambda e^{\eta}\}$$.

Now we have $$|B'(\theta)| = \theta^{-1}$$ and $$\pi(\theta | \mu, \lambda) \propto \theta^{\mu-1} e^{-\lambda \theta}$$, which is Gamma$$(\mu, \lambda)$$. If we do not consider the jacobian, $$\pi(\theta | \mu, \lambda) \propto \theta^{\mu} e^{-\lambda \theta}$$, which is Gamma$$(\mu+1, \lambda)$$.

So there is a correct parameterization? Notice that both of them follow the posterior of this table (under a reparameterization to $$\alpha$$ and $$\beta$$).

[1] Robert, Christian. The Bayesian choice: from decision-theoretic foundations to computational implementation. Springer Science & Business Media, 2007.