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I am having troubles understanding the plate notation being used in LDA and GMM. In specific the class-variable deciding which parameters that generates the observation in GMM's and the topic-assignment in LDA's. Why does it not have a an edge directed to the actual parameters? Lets have a look att GMM's:

enter image description here

Notice that Z_n governs which mu and sigma should be used for that specific observation. In my mind Z_n should have two outgoing edges pointing to the parameters which in turn have directed edged pointing to our observations. Why is this not the case?

Notice that the structure that i proposed generates a different independence structure, there must be some vital intuition in d-separation that i missed when modelling DGM's.

Notice that in the plate notation for GMM's the class variable Z_n and the corresponding parameters gets conditionally dependent given the observations X_n. In my proposed solution they are not.

This same mindfuck occurs in me when i watch the plate notation for LDA's: enter image description here

In my mind, z should point to b..

can someone shine some light on my confusion?

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I'm not an expert but my understanding of arrows in plate diagrams is "the node pointed to depends on the origin node". That is to say, there is an arrow only if the value of random variable is influenced by the origin node.

For example in LDA the value of $w$ depends the value of $z$ and the value of $\beta$, but the value of $\beta$ does not depend on $z$. The fact that picking the value of $w$ requires to know $z$ in order to select the correct distribution for $\beta$ is just not represented.

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  • $\begingroup$ yeah but why is it done that way and not the way i proposed? it has to do with the resulting independence structure and i am missing out on something there $\endgroup$
    – richard
    Dec 5, 2022 at 19:34
  • $\begingroup$ @richard I think that the way you propose would cause some confusion: for a given variable like $\beta$, it would receive arrows not only from the nodes its value actually depends on ($\eta$) but also from nodes ($z$) that are used in order to find the value of another variable which depends on $\beta$ ($w$)... I think I see your logic, but I suspect that practically it would actually make things more messy. If you think about the arrow as a dependency which impacts the value of a variable, the standard way makes sense imho: $\beta$ itself doesn't depend on $z$. $\endgroup$
    – Erwan
    Dec 5, 2022 at 21:14

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