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I am trying to find a rigorous mathematical definition of a data generating process (DGP) under a well-defined probability space.

The closest source I have found on Cross Validated is this one, and it seems to come from a Evans and Rosenthal textbook (see the post).

Are there other rigorous definitions of a DGP or a reference textbook that has the definition you can share?

  • Added Remark: Maybe just the definition of a data generating process is a joint probability distribution of a set of stochastic processes?
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  • $\begingroup$ There's nothing here that would distinguish a DGP from a random variable with values in a vector space. Do you have a reference or context suggesting the concept of DGP is any different than that? $\endgroup$
    – whuber ♦
    Commented Aug 17, 2022 at 14:11

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The DGP may not have a rigorous definition. It is just whatever happens in the real world that gives rise to the data. We can attempt to model it, but we should not "confuse the map for the territory".

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Maybe just the definition of a data generating process is a joint probability distribution of a set of stochastic processes?

This is a common misunderstanding in causal inference, it produce any sort of ambiguities and contradictions. In causal inference the DGP is not a synonym for joint distribution, it is a different thing. For some details read here: What's the DGP in causal inference?

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  • $\begingroup$ Hi Markowitz, thanks for the response. But a data generating process is a stochastic process or a set of stochastic processes that produces the evolution of the random variables of interests. It is "us" humans who rope them together into joint distributions. This is my interpretation. $\endgroup$
    – Frank Swanton
    Commented Aug 18, 2022 at 17:42
  • $\begingroup$ Note that the word "produce" vehicles causal meanings. Indeed I suggest causal meaning for the DGP. It is true that what DGP produce can be represented in a joint distribution, those are related but different things. I suggest this point of view. $\endgroup$
    – markowitz
    Commented Aug 19, 2022 at 12:45
  • $\begingroup$ They are the same in my opinion. When you lump these stochastic processes together, then they become the joint distribution. Social scientists choose which variables interest them and arbitrarily decide what should be in the joint distribution. $\endgroup$
    – Frank Swanton
    Commented Aug 21, 2022 at 0:18
  • $\begingroup$ I the view I suggest scientists can collect arbitrarily some variable in a joint distribution but not in the (presumed) DGP. $\endgroup$
    – markowitz
    Commented Aug 22, 2022 at 8:09

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