Wikipedia:Reference desk/Archives/Mathematics/2023 February 7

= February 7 =

Term for probabilities
How do you refer to things like fair dice or coin tossings that have a well-defined probability vs things like sport outcomes, wars, the next pandemic, that cannot be defined so graciously? And what about things like Knightian uncertainty or Black Swan events? Bumptump. Do mathematicians have an own term to refer to these? (talk)` Bumptump (talk) 21:11, 7 February 2023 (UTC)


 * I don't think there is a generic term covering the concept of "hardly definable" probabilities. There is some connection to one's preferred interpretation of probability, a hotly debated topic with no clear way of determining a winner. The prevailing approach is that of frequentism, not because it is philosophically appealing, but because of its conceptual simplicity when dealing with neat problems. One issue that should always be kept in mind when applying statistics to the real world, or interpreting the results of such applications, is that any such application critically depends on a mathematical model of the relevant aspects of the real world. If the model is not adequate, and there are innumerable ways it can be inadequate, the results are worthless. A statistical model dealing with probabilities needs to define an event space. Usually, defining the set of all possible outcomes is the easy part. Often, instead of defining one specific probability function, the statistician will consider a family of probability distributions, whose members can be described with just a few parameters, and part of the process is estimating these parameters, based on observed data. The frequentist approach assumes that observed occurrences in the real world are due to randomness with possibly unknown, yet definite probabilities. If that assumption is unwarranted, the basis drops out from under the approach. From an epistemic point of view, the situation in which the assumption is unwarranted should be the default assumption unless there is some specific good reason (such as the symmetry of a standard die) to assume definite probabilities. So Knightian uncertainty is the default, and Black Swan events are only unexpected for people who cling to unwarranted assumptions and forget that they should expect the unexpected. Some forms of the Bayesian approach offer some moral relief, acknowledging that our uncertainty is due to a lack of knowledge, but have little concrete to offer when it comes to dealing in an objective way with that uncertainty. See also Uncertainty quantification. --Lambiam 23:02, 7 February 2023 (UTC)
 * You might be thinking of the distinction between probability as a branch of mathematics and statistics which is a discipline which uses a lot of such mathematics in the real world. NadVolum (talk) 23:39, 7 February 2023 (UTC)
 * According to Wiktionary and most major dictionaries, probability is a number in the range from 0 to 1, and not a branch of mathematics. The term statistics can refer to descriptive statistics (collecting, summarizing and presenting statistical data), but as used in my reply above it refers to mathematical statistics, that is, the use of probability theory (a branch of mathematics) to analyze and draw conclusions from statistical data. Probability theory and mathematical statistics are as intertwined as quantum theory and quantum physics. --Lambiam 00:53, 8 February 2023 (UTC)