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The wikipage of Bayesian Network says

"Formally, Bayesian networks are directed acyclic graphs whose nodes represent random variables in the Bayesian sense"

But in the model I need to build, cyclic structure of constraint is necessary. For example, A influences B, B influences C, C influences A.

What should I use then(instead of Bayesian Network)?

ps. I saw "Markov Network", but it says that the variables should have the "Markov property"(Memorylessness), which is not necessarily true in my intended application, in the sense that some variable is influenced by the history of others as well as its own.

Thank You!


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1 Answer 1

You might be interested in this paper on discovering cyclic causal models: http://arxiv.org/abs/1206.3273

While cycles can be introduced into directed graphical models, it makes it significantly more complicated to compute the probability of some configuration. For the graphical model $X\rightarrow Y$, if we know the marginal probability of $X$ and the conditional probability of $Y$ given $X$, then the joint probability of $X=x$ and $Y=y$ is just $Pr(X=x,Y=y) = Pr(X=x)Pr(Y=y|X=x)$, and in general any probability can be computed by a combination of multiplication and marginalization. If by contrast we have a cyclic graphical model $X \leftrightarrow Y$, then this is best viewed as a Markov chain $\ldots X\rightarrow Y \rightarrow X \rightarrow Y \ldots$ where, even if we know the transition probabilities $p(x|y)$ and $p(y|x)$, we still have to solve for the stationary state of a Markov chain to compute various probabilities. For cyclic models with larger graphs the situation becomes even more complicated.

I realize this doesn't solve your problem exactly though. Spelling out the problem you're trying to solve in more detail might help. For instance, if the influences between A, B and C are symmetric (A influences B the same way B influences A) then an undirected graphical model would make more sense. If there is a temporal aspect to the influence of one variable on another, a Markov chain would make more sense (and in some sense a cyclic causal model can be seen as a kind of Markov chain, where we are trying to reason about properties of the stationary distribution).

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