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I am looking for a citable reference (books, research papers, PhD theses, not websites, internal reports, etc.) about the heuristic interpretation of self-conjugate priors as "eigenfunctions for the conditioning operator". This is best explained (though without reference) in the Wikipedia page Conjugate prior#Interpretations.
In a nutshell: A statistical model is any family $\{P_\theta\}_\theta$ of probability distributions, parametrized by $\theta\in T$ (e.g., $T\subset \mathbb R$ or $T\subset \mathbb R^k$). Some data $X:=(X_1,\dotsc, X_k)$ is sampled from the model in two stages:
- firstly, we let $\Theta\in T$ be a $Q$-distributed random variable;
- secondly, we draw $X_1,\dotsc, X_k$ in such a way that $X_1,\dotsc, X_k\mid \Theta\sim P_\Theta$ conditionally i.i.d..
The posterior distribution of $Q$ given $X$ is then defined as $Q^X:=Q[\Theta\in \cdot\mid X]$.
A family of distributions $\{Q_\alpha\}_{\alpha\in A}$ (again $A\subset \mathbb R$ or $\mathbb R^k$) is self-conjugate (for a fixed model) if the posterior distribution $Q_\alpha^X$ satisfies $Q_\alpha^X=Q_{\alpha'}$ for some $\alpha'\in A$.
Heuristically, $Q_\alpha$ is (sort of) an "eigenfunction" for the conditioning operator $\mathbf E[\cdot \mid X]$.