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I am quite confused with the distinction between a latent variable and model parameters.

So say I have two observed variables $x$ and $y$ and they have some unknown relationship between them i.e. $y = f(x)$. Now based on that I have a generative model with $ y = f(x) + e$ where I model $e$ as zero mean independent and identically distributed noise with some variance $\sigma$.

Am I correct in saying that:

1: The relationship $f$ that we try to estimate is the latent variable in this case.

2: $\sigma$ is a model parameter. Now, our estimate of $\sigma$ will also have some uncertainty associated with it. With these model parameters, are we saying that it has a fixed but unknown value that we try and estimate, but due to limited data size our estimates have uncertainty. However, underlying the model is the assumption that the value of a $\sigma$ takes a fixed value or is $\sigma$ also a random variable type entity?

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    $\begingroup$ A model parameter is some parameter we want to learn about to make quantitative claims about the data (e.g., the variance is high for this school versus that school in test scores). A latent parameter is some augmentation we make to the data to include some unobserved variable that usually allows us to make conjugate relationships for computational ease. It's hard to answer (1) because $f$ is not specified. $\endgroup$ – user44764 Jun 28 '14 at 19:05
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    $\begingroup$ For 2: see here. The probability distribution on the model parameter is your own plausibility measure.. see here. $\endgroup$ – user44764 Jun 28 '14 at 19:07

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