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Is there a formal difference between the terms "Probabilistic Model" and "Statistical Model"? Is there a methodological difference between the two or just a preference of terminology? I see the former used most often with graphical models and Bayesian models, while "statistical" seems to be anything specifically Frequentist.

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    $\begingroup$ I wouldn't read too much into this. I haven't noticed an aversion among Bayesians to talking about statistical modeling. My impression is that probabilistic modeling is the more likely phrase if authors think of themselves as probabilists and have little or no concern with estimation or inference using real data, and conversely for statisticians and data-sensitive texts. Thus a probability-oriented text might emphasise the theory of Markov chains, Markov processes, etc., etc. but say little about confrontations with real data. $\endgroup$
    – Nick Cox
    Aug 11 '13 at 8:25
  • $\begingroup$ I thought probabilistic models are those that output probability (can be 'statistical' or 'machine learning'), so frequentist logistic regression would be probabilistic and bayesian linear regression would not. But I don't know actually if this is correct interpretation $\endgroup$
    – rep_ho
    Jan 25 '19 at 10:16
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Stats textbook are sometimes divided into two sections: 1) Probability and 2) Statistics.

Probability was about permutation, combination, conditional probability. Probability was often explained with dice, coins, colored marbles and other discrete artifacts. Probability is the measure of the likelihood that an event will occur. Although probability can be calculated using statistical models, and probability does not have to be from countable events and a rational number on 0-1, that would be its first meaning. Statistics was about mostly real number measurements, like the t-, z-, W- and U- statistics. A statistic (singular) or sample statistic is a single measure of some attribute of a sample (e.g., its arithmetic mean value).

There is an overlap or gray zone for probability for the discrete distributions including the binomial distribution, the multinomial distribution, and the Poisson distribution, which are still finger counting, i.e., literally countable probable events, for which statistics are parameters and for statistics for continuous distributions that are real number models of probability.

Probability models as a first meaning imply countability, for example, the likelihood of getting exactly 5 heads from 10 coin tosses. Statistical models, as a first meaning contain some statistic. However, one can say that "Marginal probability is a statistic" so that using which phrase is used when depends on what connotation one is making.

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