I want to study the set of solutions of a system of non-linear equations involving a parametric multivariate probability distribution. I would like your help to formalise or contradict some of my ideas.
Below $\mathbb{R}^K$ denote the Euclidean space of dimension $K$, $\mathbb{N}$ the set of natural numbers, and $\mathcal{S}\setminus \mathcal{B}$ the complement of $\mathcal{B}$ in $\mathcal{S}$.
Ass 1: Let $\mathcal{R}\subseteq \mathbb{R}^M$ with $M\geq 2$. $\mathcal{R}$ has strictly positive Lebesgue measure. Let $\tau$ be an $M\times 1$ random vector with support $\mathcal{R}$ and distribution absolutely continuous wrto Lebesgue measure with everywhere strictly positive density on $\mathcal{R}$. Let $P_{\tau}(\cdot; \theta)$ denote the probability distribution of $\tau$ which we assume belonging to a known parametric family indexed by the $K\times 1$ parameter $\theta\in \Theta\subseteq \mathbb{R}^K$. Suppose we normalise the variance of $\tau_1$ equal to $1$.
Further notation: Let $\mathcal{X}\subset \mathbb{N}$ with cardinality $L$ and $\mathcal{Y}\equiv \{0,1,...,M\}$.
Let $U\equiv (U_{xy} \text{ }\forall (x,y)\in \mathcal{X}\times \mathcal{Y})$ be an $L\times (M+1)$ vector of real numbers with $U_{x0}\equiv 0$ $\forall x \in \mathcal{X}$.
Let $\mathcal{U}\subset \mathbb{R}^{L\times (M+1)}$ be a finite set of values of $U$.
For any $(U,x,y)\in \mathcal{U}\times \mathcal{X}\times \mathcal{Y}$, define $$ \mathcal{T}_U(y,x)\equiv \{t\equiv(t_1,...,t_M)\in \mathcal{R} \text{ s.t. } U_{xy}+t_y\geq U_{xk}+t_k \text{ }\forall k\in \mathcal{Y}\setminus \{y\}\} $$ with $t_0\equiv 0$.
For example, when $M=2, y=1,x\in \mathcal{X}$ $$ \mathcal{T}_U(1,x)\equiv \{t\equiv(t_1,t_2)\in \mathcal{R} \text{ s.t. } U_{x1}+t_1\geq 0\text{ and }U_{x1}+t_1\geq U_{x2}+t_2 \} $$
System of equations: Consider the system of $L\times (M+1)$ equations $$ P_{\tau}(\mathcal{T}_U(y,x); \theta)=p_{xy} \hspace{1cm}\forall (x,y)\in \mathcal{X}\times \mathcal{Y} $$ with unknowns $(U,\theta)\in \mathcal{U}\times \Theta$ (hence $L\times M+K$ unknowns; indeed remember that $U_{x0}\equiv 0$ $\forall x \in \mathcal{X}$). $p_{xy} $ are known parameters $\text{ }\forall (x,y)\in \mathcal{X}\times \mathcal{Y}$, such that $p_{xy}\in [0,1]$ and $\sum_{y=0}^Mp_{xy}=1$ $\forall (x,y)\in \mathcal{X}\times \mathcal{Y}$.
Question: (A) Does this system has one solution wrto to $U$ for each $\theta\in \Theta$? (B) Does this system has one solution wrto to $\theta$ for each $U\in \mathcal{U}$?
My intuition is that the answer to (A) is YES. For example, assume $M=2$, $\mathcal{R}=\mathbb{R}^2$, $\mathcal{X}=\{\hat{x}, \tilde{x}\}$. Hence, the system of equations becomes $$ \begin{cases} P_{\tau}(\mathcal{T}_U(1,\hat{x}); \theta)=p_{\hat{x}1}\\ P_{\tau}(\mathcal{T}_U(2,\hat{x}); \theta)=p_{\hat{x}2}\\ P_{\tau}(\mathcal{T}_U(0,\hat{x}); \theta)=p_{\hat{x}0}\\ P_{\tau}(\mathcal{T}_U(1,\tilde{x}); \theta)=p_{\tilde{x}1}\\ P_{\tau}(\mathcal{T}_U(2,\tilde{x}); \theta)=p_{\tilde{x}2}\\ P_{\tau}(\mathcal{T}_U(0,\tilde{x}); \theta)=p_{\tilde{x}0}\\ \end{cases} $$ First of all notice that, for a given $(x,U)$, $\{\mathcal{T}_U(0,x), \mathcal{T}_U(1,x), \mathcal{T}_U(2,x)\}$ represents a tri-partition of $\mathcal{R}$ defined by 3 lines with fixed slopes and vertical-horizontal intercepts determined by $U_{x1}, U_{x2}$, as illustrated below for $(U_{x1}, U_{x2})=(-1,1)$.
In the picture: the red points have coordinate in clockwise sense from the top $(0, U_{x1}-U_{x2})$, $(0, -U_{x2})$, $(-U_{x1},0)$, $(U_{x2}-U_{x1},0)$; the oblique line A has slope $1$ and passes thorough $(0, U_{x1}-U_{x2})$ and $(U_{x2}-U_{x1},0)$; the vertical line C passes thorough $(-U_{x1},0)$; the horizontal line B passes thorough $(0, -U_{x2})$; the pink area is $\mathcal{T}_U(1,x)$, the blue area is $\mathcal{T}_U(0,x)$, the orange area is $\mathcal{T}_U(2,x)$.
Secondly, suppose $\tau\sim \mathcal{N}(\mu, \Sigma)$ with $\mu\equiv (\mu_1,\mu_2)$ and $\Sigma\equiv\begin{pmatrix} 1 & \sigma_{12}\\ \sigma_{12} & \sigma^2_2 \end{pmatrix}$. Hence, the pdf of $\tau$ is a bell over $\mathcal{R}$ and $\theta\equiv(\mu_1, \mu_2, \sigma^2_2, \sigma_{12})$.
Thus (A) can be reformulated as: fix $\theta\in \Theta$; in how many ways we can tri-partite $\mathcal{R}$ with 3 lines A,B,C such that
B is horizontal, C is vertical,
B and C intersect at a point $s$ with coordinate $(s_1,s_2)$
A has slope 1 and passes through $s$
the pink, blue, and orange regions are such that the corresponding volumes below the bell are $p_{x1}, p_{x2}, p_{x0}$
?
I think that there exists at least one point $s$ satisfying those conditions (and moreover such a point is unique).
Now, this argument is a very naive and hardly generalisable to $M>2$ and $\mathcal{R}\subseteq \mathbb{R}^M$. If it is correct, could you suggest something more formal? If it is wrong, could you explain why?
Regarding (B), I think the answer is "NOT NECESSARILY" but I can't show why.