Assume $z \in \mathbb{R}^m$ and $x \in \mathbb{R}^n$. Assume we have proper density function $P(z)$ and proper conditional density function $P(x|z)$. We give the definition $$ T(x_1,\ldots,x_n) := \int_{\mathbb{R}^m}P(z)\prod_{i=1}^nP(x|z)dz $$ Therefore we have a group of functions $T(x_1),T(x_1,x_2),\ldots$.

What are the behaviors of such a group of functions? I mean, if we only have a group of functions $T(x_1),T(x_1,x_2),\ldots$ without knowing $P(z)$ and $P(x|z)$, can we judge if these functions can be obtained by some $P(z)$ and $P(x|z)$?

Are there any papers that can help?

Assume $z \in \mathbb{R}^m$ and $x \in \mathbb{R}^n$. Assume we have proper density function $P(z)$ and proper conditional density function $P(x|z)$. We give the definition $$ T(x_1,\ldots,x_n) := \int_{\mathbb{R}^m}P(z)\prod_{i=1}^nP(x_i|z)dz $$ Therefore we have a group of functions $T(x_1),T(x_1,x_2),\ldots$.

What are the behaviors of such a group of functions? I mean, if we only have a group of functions $T(x_1),T(x_1,x_2),\ldots$ without knowing $P(z)$ and $P(x|z)$, can we judge if these functions can be obtained by some $P(z)$ and $P(x|z)$?

Are there any papers that can help?