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Comment by salty_biscuits | Hacker News Reader

"Vanilla deep learning models are statistical models (a la linear regression) and not probabilistic models (a la Gaussian mixture). It is important to maintain the distinction."

Isn't this just a matter of interpretation of the models? You can interpret linear regression in a Bayesian way and say that the prediction of the linear model is the MAP of the mean, you can also calculate the variance, the l2 norm objective is saying the distribution of errors is normally distributed, l2 regularisation is a normal prior on the coefficients, etc, etc? All the same stuff can be applied to deep learning models.

Maybe I don't understand your distinction between statistical and probabilistic though?

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> Isn't this just a matter of interpretation of the models?

Not really. This is the classic frequentist vs Bayesian debate. In frequentist-land, you are computing point estimates of the model parameters. In Bayesian-land, you are computing distribution estimates of the model parameters. It is true that there is a difference in interpretation of the generative process but the two choices demand fundamentally different models because of the decision about which of the parameters or data are considered "real" and which are considered "generated".

I think a more abstract/general way to put it is: "statistics" is concerned with statistical summary values (i.e. mean-field estimates over measures) while "probability" is concerned more with distributions (i.e., topologies of measures). I'm not sure this is a rigorously correct way to characterize it, but it illustrates the intuition I'm trying to convey.

Statistics as practiced today (1930s until now?) consists almost entirely of making inferences about unobserved probability distributions. That includes nonparametric statistics, and frequentist versus Bayesian has nothing to do with it.

There are some probability models that are not really statistical models, but there are few or no statistical models that are not also probability models.

Least-squares regression is a probability model. Even if you don't particularly care about the error distribution, you are still estimating a conditional expectation and setting a conditional independence assumption on the residuals. If that's not a probability model, then I don't know what it is!

Maybe the more intuitively obvious way to put it is:

probability : distributions :: statistics : expected values

I don’t think I have read anything more false on the internet. XD

I apologize. ^This comment was to harsh.

Statistics can be summarizes as one thing n -> N. Does ‘little n’ represent ‘big N’. In other words, does the sample generalize to the population. Statistics means something like “description of the state”. It was born out of census samples where larger population samples had to be estimated. “n” could be a handful of fish in a “N” lake. “n” could also be the parameter estimated in a linear regression with the sample of data collected while “N” is the true parameter of the relationship if we had all the data. Point estimation is about finding the needle in the haystack, but much more often statistics is about finding the haystack given the needle. One tool statistics uses to get to the haystack is probability.

A point estimate of distribution parameters describing a population is frequentist. A point estimate of distribution parameters describing another distribution's parameter is Bayesian.

How the parameters are estimated it not the message.

In statistics there are latin letters and greek letters. When you see a symbol denoted as a greek letter then that is a population parameter. When you see a latin letter that is a sample estimate. It could be Frequentist, Bayesian, Likelihoodist, Fiducial, Empirical Bayes, etc. Theoretical population greeks or sample calculated latins.

Now that's a fascinating and deliciously crackpot theory.

Don't take my word for it. https://ocw.smithw.org/csunstatreview/statisticalsymbols.pdf

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