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Hello and apologies if this is a basic question (I am very new to information theory).
Suppose that $X$, $Y$, and $Z$ are discrete random variables with joint probability mass function, $p_{_{XYZ}}$ that has full support on $\mathcal{X}\times\mathcal{Y}\times\mathcal{Z}$, for some finite sets $\mathcal{X}$, $\mathcal{Y}$, and $\mathcal{Z}$. Consider a Bayesian agent who has a prior belief $p_{_{XYZ}}$ over $(X,Y,Z)$. After observing the realization $(x,y)$ of $(X,Y)$ -- but not the realization of $Z$ -- the agent forms a posterior belief $q_{_{Z|XY}}(\cdot|x,y)$ about $Z$ using Bayes' rule, i.e. $$ q_{_{Z|XY}}(z|x,y) = \frac{p_{_{XYZ}}(x,y,z)}{\sum\limits_{\tilde{x}\in\mathcal{X}}\sum\limits_{\tilde{y}\in\mathcal{Y}} p_{_{XYZ}}(\tilde{x},\tilde{y},z)},\quad \forall (x,y,z)\in\mathcal{X}\times\mathcal{Y}\times \mathcal{Z}. $$
My question: Is there a precise, mathematical way to compare the relative influence of $X$ and $Y$ on the agent's posterior beliefs? That is, I am interested in saying the following in a precise, (rigorous, mathematically formal, etc.) way:
"Given realization $(x,y)$ of $(X,Y)$, $X$ had more influence on the agent's posterior beliefs than did $Y$"
It would be even better if there is a way to make a cardinal statement like
"Given realization $(x,y)$ of $(X,Y)$, $X$ had $\alpha\%$ more influence on the agent's posterior beliefs than did $Y$"
Are there any commonly used/standard notions of either of the above? Apologies that this question is not fully precise (I usually try to make sure I am being as precise as possible). But indeed, my question is essentially on how to make either of the above statements fully precise.
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