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Let $X$ and $Y$ be Banach spaces, and let $\varphi : X \to Y$ be a continuous linear map. Suppose that $\mathbb P$ is a probability measure on $X$ which satisfies the continuous disintegration property with respect to $Y$. This means that there exists a measure-valued function $y \mapsto \mathbb P^y(d x)$ which is (1) a regular conditional probability (RCP), and (2) is continuous, where the space of probability distributions $\Delta(X)$ is equipped with the topology of weak-* convergence of measures.

Suppose that, for all $y \in Y$, the probability measure $\mathbb P^y$ is Gaussian with mean $m(y) \in X$ and covariance operator $\hat K : X^* \to X$. It is a consequence of the linear structure that the covariance depends only on the function $\varphi$ and not the actual observed value $y$.

Question: How bad can the original measure $\mathbb P$ be? More precisely, what is the set $G \subseteq \Delta(X)$ of probability measures $\mathbb P$ which result in conditional Gaussian distributions $\mathbb P^y$ of Gaussian type?

Followup question: Does the answer change if the map $y \mapsto \mathbb P^y(dx)$ is simply measurable, and not continuous?

Being a RCP means that $\mathbb P$ satisfies the disintegration equation: for all continuous and bounded functions $f : X \to \mathbb R$, $$\int_X f(x) ~ \mathbb P(dx) = \int_Y \int_X f(x) ~\mathbb P^y(dx) \mathbb P_Y(dy),$$ where $\mathbb P_Y := \mathbb P \circ \varphi^{-1}$ is the push-forward probability measure on $Y$.

Even though the conditional measure is Gaussian, there is no guarantee that the original measure $\mathbb P$ or the push-forward measure $\mathbb P_Y$ are Gaussian. However, it is conceivable that this "non-Gaussianness" cancels out in the conditioning. I am not comfortable enough with the structure of these spaces to know what conditions are necessary for a Gaussian to manifest in the conditioning.

I am asking this question on behalf of an oceanographer friend of mine, who says, "This is, IMO, one of the most important theoretical questions in data assimilation."

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