What are the main approaches to the foundation of statistics without probability

The frequentist, likelihood and, to an even greater extent, Bayesian approaches to statistics are all based on probability. Without probability, it seems difficult to use a data sample ("seen" cases), to infer results about a more general population ("unseen" cases), and estimate the uncertainty inherent in such a process. Thus it would appear that all we can do is descriptive statistics, as well as some trivial inequalities (such as, if we observe a person of height 2.0 mt, we know that the maximum height in the population of all living persons is $$\geq$$ 2.0 mt).

Or is it? Are there mathematically rigorous approaches to the foundation of statistics, which do not rely on the theory of probability, and can you point me to references on the subject?

• The main appproaches? I wonder if there is a single approach. Curious to learn more. – Richard Hardy May 26 at 15:37
• For small datasets with few variables, numerical and graphical descriptive summaries can be very useful. For huge datasets with many variables, some algorithmic methods that may rely more on computer power than probability theory are showing promise. But why would anyone want to pursue either path without insights gained from probability theory? – BruceET May 26 at 16:35
• @BruceET there is at least the duality diagram approach: statweb.stanford.edu/~susan/talks/foundations.pdf of which I know very little (basically nothing).As to the "why", I have absolutely no idea. – DeltaIV May 26 at 16:54
• @RichardHardy see my reply to BruceET for one approach (but don't ask me anything about it). – DeltaIV May 26 at 16:55
• I would point out that Cox's method follows from Aristotelian logic. It doesn't define probability. De Finetti's method is doesn't use the word probability. He instead speaks in terms of bets. Probability follows from it and therefore statistics, but it is grounded in gambling. Savage's is grounded in preference theory. Likewise, Kolmogorov's method follows from measurement and not probability. It is a consequence. De Finetti argued that probability, like utility, was a mental construction and did not exist except to help think about things. – Dave Harris May 27 at 2:52