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I have thought for a while about this, and I'm not entirely certain what the difference is between a deterministic model and a probabilistic model with error following a degenerate distribution (centered at 0). I'm starting to think that there is a very important difference, especially with respect to analysis and model comparison, but I can't say that I understand it.

So, what is the main difference between a deterministic model and a model that assumes error follows a degenerate distribution (centered at 0)? Is there a difference at all?

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if you are considering models of the sort $Y = f(X)+\varepsilon$ where $\varepsilon \sim F$, then if $F$ is a degenerate distribution you have that $Y = f(X)+\varepsilon$ is just an almost sure translation of $Y = f(X)$. They almost surely coincide when ${\mathbb P}(\varepsilon=0)=1$. –  user10525 Jun 19 '12 at 22:44
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Yes but does degenerate mean the probability distribution is concentrated at 0 or some other value. If it is 0 then they coincide in the sense that the functions are the same with probability 1 as Procrastinator states. But if it is a constant different from 0 then of course the two models always differ by that constant ( a bias existing in the case of the stochastic model with the degenerate distribution). Can anybody tell me what the significance of this question si? I am sorry to say that it looks rather trivial to me. –  Michael Chernick Jun 19 '12 at 23:05
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Yeah, it's a degenerate distribution at 0. Thanks for the suggestion, Michael. Also, would you be willing to make that a full answer, Procrastinator? –  Jonathan Thiele Jun 20 '12 at 0:17
    
@MichaelChernick Sorry, I forgot the second part of your comment. The question isn't particularly significant. I was just a little curious and wanted feedback in case I was incorrect. –  Jonathan Thiele Jun 20 '12 at 14:51

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