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In several domains, I came across many popular research papers that do contain statements that should be interpreted informally. If someone interprets them formally, it leads to a totally wrong perception.

The issue is that the authors use the formal terms in-order to make the statements that has to be interpreted informally. The statements I am referring are not the plain English statements, which can be equivocal in nature.

For example, consider the following answer from other stack site

You're right! The generative model f is not the same as the probability density (p.d.f.) function pdata. The kind of phrases you've referred to are to be interpreted informally. You learn f with the hope that sampling a latent vector z from some known distribution (from which it is easy to sample), results in f(z) that has the probability density function pdata. However, merely learning f does not give you the power to estimate what pdata(x) is for some image x. Learning f only gives you the power to sample according to pdata(⋅) (if you've learned an accurate such f).

I interpreted in wrong way and tried to understand for much time and I end up in asking that question (on other site). I came across many such formal phrases from research papers.

In general, there will be no explicit remainders or hints or guidelines or suggestions to know which statements to be interpreted formally and which statements has to be interpreted informally in a formal context.

What is the primary reason behind encouraging such phrases either by journals or authors?

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In the two examples you give, I don't agree the statements need to be "interpreted informally". Rather I would say that the meaning is clear within the context of the problem. For example, suppose you have a function f(a,b,c,d) or an expectation E_{a~p, b~q}[ f(a, b)]. Often authors will just write f or E[f] respectively. There is no mistake, it's just the notation is being left out. It's basically the same thing in the examples you point out. "our generative model learned pdata" is just a shorthand for "we learn a model f(z) which maps a latent space z to an output S so that we can approximate drawing samples from pdata by drawing a sample z". And that's not unclear or "informal", because the latter is just a definition of what generative model means in that context.

Specific problems can have their own language or jargon, and it can be confusing to read papers without knowing it. Part of becoming an expert in a specific problem is learning how people think about that problem.

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