Does Mathematica allow testing abstract algebraic relationships with probabilities? For example, checking a relationship between probabilities without expressing the form (e.g. binomial etc.) of these probabilities but under generic assumptions?
Like assuming a generic discrete distribution or a Markov chain relationship and then test/simplify/modify relationships between them. Simple example, multivariate discrete distribution in which part of the variables are related via Markov chain and one want to know the expression of the Kullback Leiber divergence with a product of marginals.
Example, thest that the following relationships are correct \begin{align*} & I \big(A_1 {:} \dots {:} A_K ; \mathcal{Y} \big) \\ &\triangleq \sum_{\ell=1}^L \ \sum_{y^\ell \in \mathbb{Y}^\ell} p \big(y^\ell\big) \cdot \min_{k=1,\dots, K} \left \{ I\big(A_k;Y^{\ell-1} \mid y^\ell\big) \right \} \\ &= \sum_{\ell=1}^L \ \sum_{y^\ell} p \big( y^\ell \big) \ I \big(A_{k(y^\ell)} ;Y^{\ell-1} \mid y^\ell \big) \\ &= \sum_{\ell=1}^L \ \sum_{y^\ell} \ \sum_{a_{k(y^\ell)}, y^{\ell-1}} p\big(a_{k(y^\ell)}, y^{\ell-1}, y^\ell \big) \ \log \frac{p \big(a_{k(y^\ell)}, y^{\ell-1} \mid y^\ell \big)}{p \big(a_{k(y^\ell)} \mid y^\ell \big) \ p(y^{\ell-1} \mid y^\ell)} \\ &= \sum_{\ell=1}^L \ \sum_{\substack{a_1, \dots, a_K \\ y^0, \dots, y^L}} p\big(a_1, \dots, a_K, y^0, \dots, y^L \big) \ \log \frac{p \big(a_{k(y^\ell)}, y^{\ell-1} \mid y^\ell \big)}{p \big(a_{k(y^\ell)} \mid y^\ell \big) \ p(y^{\ell-1} \mid y^\ell)} \\ &= \sum_{\substack{a_1, \dots, a_K \\ y^0, \dots, y^L}} p\big(a_1, \dots, a_K, y^0, \dots, y^L \big) \ \sum_{\ell=1}^L \log \frac{p \big(a_{k(y^\ell)}, y^{\ell-1} \mid y^\ell \big)}{p \big(a_{k(y^\ell)} \mid y^\ell \big) \ p(y^{\ell-1} \mid y^\ell)} \\ \end{align*} where $k(y^\ell) \triangleq \arg \min_{k=1,\dots, K} \left \{ I\big(A_k;Y^{\ell-1} \mid y^\ell\big) \right \}$ and the $Y^\ell$ are linked by deterministic functions. All variables descrete.
Can Mathematica handle this kind of symbolic problems?
