# Why do we say Bayesian statistics is suited for probability of one-time events?

People say frequentist statistics is not suited for one-time events, with Bayesian statistics being better suited. I wonder what the reason for this is. I also wonder what "one-time event" means, because (for example) a coin toss also seems to be a one-time event because we cannot reproduce the exact same environment whenever we toss a coin.

This is all philosophical (as it must be):

When a probability is a frequency you will have difficulty applying it to something that can't happen more than once, like Thursday's match between Liverpool and Arsenal. The problem is that a frequency is of the form $$k$$ out of every $$n$$. You could conceive of a class of Thursdays sufficiently similar to this one (weather, &c.) where Liverpool plays Arsenal in a sufficiently similar way (team compositions, &c.) that you would consider them (almost) repetitions of the same event and so have an idea of how this would go on average, though finding the data might be difficult. But never for a given match, which can only happen this once. Similarly, you can't assign a probability to the next coin toss, because there is only one, but you can say something about the class of sufficiently similar coin tosses.

If a probability is a degree of belief, you can assign it to anything that you can have an opinion on if you follow the proper rules of such assignments. It is my belief that the next coin toss is about as probable to come up heads as it is to come up tails.

• What do you mean by "class"? Nov 13, 2020 at 12:05
• A collection of events Nov 13, 2020 at 12:07
• Thank you for the answer and can you please also explain the Presidential Election case? Nov 13, 2020 at 12:11
• That would be your classical example a one-time event. There is only one 2020 United States presidential election. Perhaps you could think of a collection of US presidential elections where the state of the nation and the candidates are sufficiently similar that you could use this as a reference class to build a frequency from. Note also that although you can assign a degree of beilef with greater philosophical ease, this must follow certain logical/cosistency rules. You will probably still end up considering such a reference class as in a frequency analysis. Nov 13, 2020 at 12:17
• @StoryMay To the comment's second question: you can do both, but it'll only give you a meaningful answer if there's an underlying pattern. Nov 14, 2020 at 15:36

People say frequentist statistics is not suited for one-time events, with Bayesian statistics being better suited.

You are stating "People say...", but there are actually only very few people who say this. I doubt that this 'frequentist is not for one time events' is something that a lot of people say.

I wonder what the reason for this is. I also wonder what "one-time event" means

These 'one-time' events do not truly exist. You can 'isolate' single events. But there is no application that uses statistics and applies it to only one single event. There are always relations with multiple events.

### Statistics is never about a one-time event.

There is a lot of philosophical talk around probability, and this divides 'people'. But, I believe that this is often a matter of semantics and a matter of having different points of view about what the problem actually is.

In the end, in the case of a clear practical example and problem setup, with a clear goal, there is little disagreement.

For instance: when you do a coin toss, then the predictions or inference that we relate to it (whether it is frequentist or Bayesian) are never isolated to a single coin toss. We place it in a wider perspective of

• 1: more available information and knowledge

Knowledge-based on more events and not a one-time event. For instance, in the case of the coin toss, a Bayesian approach might put a sharp prior on the coin placing high odds on the coin to be fair (because we have experience with coins being typically fair).

• 2: applications in the future

Doing inference or predictions on a single event is rather useless. Observing the past without predicting the future is useless.

So our 'statistical' thing that we do with the "one-time" event, is always placed in the background of a wider scheme. We apply it to another future event, or we relate it to another past event.

Statistics is about studying historical numbers/data (and taking into consideration the randomness in these figures when we want to apply knowledge of these historical numbers/data to do inference or prediction). Statistics is not about 'one-time events'. Making predictions or inference related to one-time events (without any other data) is not statistics but instead is abracadabra.

### Probability interpretations

I must agree that you have this discrepancy between different interpretations of probability and you have this subtle difference between frequentist probability and propensity probability. Personally I find the difference between these two of no practical use. But well, my current nickname on this forum is not for nothing S.E.

In a brief search I could find this article "Diversity in Interpretations of Probability: Implications for Weather Forecasting" by Ramón de Elía and René Laprise where it is stated

A way to solve the problem created by an irregular die is to estimate the probabilities of each face by throwing the die a large number of times and afterward studying the relative frequency of occurrence f of each face (that in a regular die would be f = ⅙ for each one).

For the holders of the frequentist interpretation, this is the only correct way of understanding a probability, since one cannot be sure of the fairness of a die until it is thrown a large number of times.

But I have never met these holders of the frequentist interpretation. I believe that this idea of people having this "frequentist interpretation" is a false idea.

Or maybe people occasionally employ this idea, for instance when advocating a Popperian attitude towards scientific theories and stating that they need to be tested based on multiple experiments/repetitions, but it is wrong to say that people are employing only one kind of interpretation of probability. I believe that it is wrong to say that there are 'holders of the frequentist interpretation'. These interpretations do not depend on the person but instead depend on the problem. (In a similar sense it is wrong to speak about Bayesians and Frequentists as if this relates to specific people with specific believe systems.)

• Nice answer with much more nuance than I put into mine (+1). I only allude to the fact that you basically end up with a reference class problem anyway in a comment. Nov 13, 2020 at 14:32