When inferring the probability for a random variable to outcome a particular value in a Bayesian Network, one may assign a set of evidence. A set of evidence is basically a set of random variables which we know the outcome. My question is very simple but keeps me puzzled. To what extent can you set evidence? Is there any purpose of working with Bayesian Networks if one knows the outcome of every random variable in the network expect one?

Consider two random variables $X$ and $Y$. Both can take two values: $\{true, false \}$ and $Y$ outcome is conditionally dependent of $X$ outcome. We can express this relationship through $P(Y | X)$ (probability of $Y$ given $X$). This can be represented with a very simplistic Bayesian Network: $X \rightarrow Y$. One can see that if we know the current outcome of $X$, there is no interest in doing inference on $Y$, because there is nothing to infer. Indeed, the Bayesian Network contains the conditional probability table of $P(X)$ and $P(Y | X)$. If we know that $X = true$, computing $P(Y|X=true)$ is straightforward (one just need to read the corresponding cell in $Y$ conditional probability table).

Mathematically, there is no restriction about the knowledge of the evidence. However, is there any concrete case where it is realistic to consider that everything is known, except the probability of the random variable we want to infer? If yes, would it be possible to simplify every class of problems using Bayesian Networks into a Bayesian Network such that everything is known except the probability we want to infer?


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