Showing existence of a solution for a system of equations involving expectations of random variables Let $f_k(x)$, $k \in \{0,\dots,M\}$, $ M\geq 2$, a set of distinct probability density functions in $\mathbb{R}$. Let $\{\alpha_k\}_{k=1}^M$ a set of positive real numbers such that $\sum_{k=1}^M \alpha_k =1$. Is it generally possible to find a choice of $\{\alpha_k\}_{k=1}^M$ such that
\begin{align}
\int  \left(\log \frac{\sum_{k=1}^M \alpha_m f_k(x)}{f_0(x)} \right) f_s(x) dx= \int \left(\log \frac{\sum_{k=1}^M \alpha_m f_k(x)}{f_0(x)} \right) f_j(x) dx
\end{align}
for all $s,j \in \{1,\dots,M\}$. I believe the result holds for the case that $M=2$ and for general distribution choices, however I am not sure for the case that $M>2$. Is there an example of pdfs such that such a choice of $\{\alpha_k\}_{k=1}^M$ can be found? It looks like we have to find a solution to a system of $M$ nonlinear equations. I am not sure how to proceed with showing that a solution exists.
 A: Here are some cases: 
Case 1:  Suppose $f_0 \in Conv(f_1, ..., f_m)$.
Suppose  $f_0(x) = \sum_{i=1}^m \beta_i f_i(x)$ for some nonnegative constants $\beta_i$ that sum to 1. Then we can choose $\alpha_i=\beta_i$ for all $i \in \{1, ..., m\}$ and we obtain the desired equalities: 
$$ \int f_k(x)\log\left(\frac{\sum_{i=1}^m \alpha_i f_i(x)}{f_0(x)}\right)dx = 0  \quad \forall k \in \{1, ..., m\} $$
Case 2: $f_0$ is uniform over $[a,b]$
Suppose the PDFs all have support over an interval $[a,b]$ and that $f_0$ is uniform over this interval. If $f_0 \notin Conv(f_1, ..., f_m)$, a natural choice is to "project" $f_0$ into $Conv(f_1, ..., f_m)$ (in a KL sense):
Let $P = \{(p_1, ..., p_m): p_i\geq 0, \sum_{i=1}^mp_i=1\}$ denote the probability simplex. Find probabilities $(\alpha_1, ..., \alpha_m) \in P$ that minimize the following expression: 
$$ g(\alpha_1, ..., \alpha_m)=\int_a^b \left(\sum_{i=1}^m \alpha_if_i(x)  \right)\log\left(\frac{\sum_{i=1}^m\alpha_i f_i(x)}{f_0(x)}\right)dx$$
Suppose we are lucky enough to find a minimizer $\alpha^*=(\alpha_1^*, \ldots, \alpha_n^*)$ that has strictly positive entries, so that $\alpha_i^*>0$ for all $i \in \{1, ..., n\}$ (so it is not a boundary point of $P$).  It follows by property of minimizers that there is a constant $c$ such that: 
$$ \left.\frac{\partial g}{\partial \alpha_k}\right|_{\alpha^*} = c \quad \forall k \in \{1, ..., m\}$$
(Otherwise, if the partial with respect to $i$ is larger than the partial with respect to $j$, we can improve $g$ by taking a small amount $\delta$ from $\alpha_i$ and giving it to $\alpha_j$, which contradicts the fact that we are already at a minimum.)
In particular for all $k \in \{1, ..., m\}$ we have
\begin{align}
c &=  \int_a^b f_k(x)\log\left(\frac{\sum_{i=1}^m\alpha_if_i(x)}{f_0(x)}\right)dx + \int_a^b f_k(x)f_0(x)dx\\
&= \int_a^b f_k(x)\log\left(\frac{\sum_{i=1}^m\alpha_if_i(x)}{f_0(x)}\right)dx + \frac{1}{b-a} 
\end{align}
and so the desired integrals are the same for all $k \in \{1, ..., m\}$. 
General case
Let $f_0, f_1, ..., f_m$ be general. Define:
$$ v_k = \int f_k(x)f_0(x)dx \quad \forall k \in \{1, ..., m\} $$
Define the function $h:[0,1]^m\rightarrow\mathbb{R}$ by
$$ h(\alpha_1, ..., \alpha_m) = -\sum_{i=1}^m\alpha_iv_i + \int \left(\sum_{i=1}^m \alpha_i f_i(x)\right)\log\left(\frac{\sum_{i=1}^m\alpha_if_i(x)}{f_0(x)}\right)dx $$
Now minimize $h(\alpha_1, ..., \alpha_m)$ over $(\alpha_1, ..., \alpha_m)\in P$.  Again suppose the minimizer $(\alpha_1^*, ..., \alpha_m^*)$ has  $\alpha_i^*>0$ for all $i\in \{1, ..., m\}$.  Then again we must have a constant $c$ such that 
$$ \left.\frac{\partial h}{\partial \alpha_k}\right|_{\alpha^*} = c \quad \forall k \in \{1, ..., m\}$$
Thus for all $k \in \{1, ..., m\}$ we get: 
\begin{align}
c &=  -v_k +  \int  f_k(x)\log\left(\frac{\sum_{i=1}^m\alpha_if_i(x)}{f_0(x)}\right)dx + \int  f_k(x)f_0(x)dx\\
&= \int f_k(x)\log\left(\frac{\sum_{i=1}^m\alpha_if_i(x)}{f_0(x)}\right)dx 
\end{align}
and so the desired integrals are the same for all $k \in \{1, ..., m\}$. 

Fortunately, the functions $g(\alpha_1, ..., \alpha_m)$ and $h(\alpha_1, ..., \alpha_m)$ are always convex functions!  So the minimization is always a convex minimization. 
In fact, convexity implies that there exist strictly positive values $\alpha_i$ that sum to 1 that satisfy the desired equalities if and only if there is a  minimizer of the function $h$ over the simplex $P$ that is not a boundary point of $P$.
