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The principle of indifference states that,

In the absence of any relevant evidence, agents should distribute their credence (or 'degrees of belief') equally among all the possible outcomes under consideration.

This means that two individuals with different amounts of "evidence" can come up with two different probabilities - that is, probabilities can be subjective. I can't think of a case where subjective probabilities might be useful in predicting the future. For example, say I have a biased coin that's biased towards heads but I do not know that it's biased and so I assign the probability of heads and tails to be both $\frac{1}{2}$. But this probability distribution will be useless in predicting the future as in the long run, I will end up with more heads than tails since the coin's biased towards heads. I was wondering what are some examples of when subjective probabilities might be useful?

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    $\begingroup$ Prior implies a model based approach, so subjective probability can be used to test correctness of one's "degree of belief". In your example, one could figure out that initial prior may not converge. $\endgroup$
    – patagonicus
    Commented Aug 17, 2021 at 5:48

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To take your coin flipping example,

But this probability distribution will be useless in predicting the future as in the long run, I will end up with more heads than tails since the coin's biased towards heads.

This is missing the key point:

This means that two individuals with different amounts of "evidence" can come up with two different probabilities

Each time you observe the outcome of a coin flip you have more evidence about the bias of the coin than the "past you" that started from an uninformative prior, so of course the "current you" will have a different probability. For subjective Bayesianism, you can update your prior using Bayes rule to give the new prior for the next observation using Bayes rule.

Now before you observe the first coin flip it is not the case that you are completely ignorant of the bias of the coin. It is easy to make a two-headed coin or a two-tailed coin, but it is very difficult to make a coin that is biased, but still has heads and tails, whilst still being symmetrical enough not to be obviously biased. So a subjectivist Bayesian could construct a prior with a spikes at 0 and 1 for the probability of a head and the rest of the probability distributed fairly close to 0.5. In practice, this prior will give better predictions than a uniform prior, because it has more information about the problem. Of course as the number of flips observed grows large, the results from both initial priors will converge.

Another example where subjective probability is "useful" would be betting on horse races (betting was originally a strong motivation for research on probability). Horse races only occur once, so they don't really have long run frequencies, so we can't apply frequentist probabilities to the outcome of particular horse races. If we thought each horse was equally likely to win a-priori the bookies would make a very large profit. Instead, punters use their expertise to judge the subjective probability that a particular horse will win (based on its physiology, past record, the conditions etc.). The more expertise a punter has, the more likely they will win their bets. The bookies of course are likely to be very expert, and also have the evidence from the punter's bets that they can use, which is why the bookies will make a profit.

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    $\begingroup$ Thanks for the great answer. So when we use the principle of indifference to assign a probability distribution, can we think of this uniform probability distribution as the "starting point" - what you called the uninformative prior in your answer? And for the horse race example, you're basically saying that the past evidence that the expert has observed is what gives them an edge over assuming a uniform distribution...is that correct? $\endgroup$
    – sl2outnow
    Commented Aug 17, 2021 at 9:38
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    $\begingroup$ @sl2outnow Also see stats.stackexchange.com/a/442587/77222 for more details on how you may go about this kind of Bayesian modelling. $\endgroup$
    – Jarle Tufto
    Commented Aug 17, 2021 at 10:19
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    $\begingroup$ Read Probability is Logic by ET Jaynes who convincingly portrays probability as information. Or IJ Good who gave an example where one poker player knew that a certain card was scuffed and the other players did not know this. The one player assigns different probabilities than the others. $\endgroup$
    – Frank Harrell
    Commented Aug 17, 2021 at 12:23
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    $\begingroup$ Got it. Thank you $\endgroup$
    – sl2outnow
    Commented Aug 18, 2021 at 9:54
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    $\begingroup$ Presumably the horses in a given race were capable of running other races and so they have a track record of performance. This could be used to form long-run probability statements about the performance of a particular horse and therefore the outcome of a given race (or estimated long-run probability statements with a margin of error). $\endgroup$
    – Geoffrey Johnson
    Commented Aug 18, 2021 at 14:14
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A subjective probability is useful for the experimenter to quantify his feelings. To the Bayesian probability is about updating his/her personal experience with the coin. Probability measures the experimenter, though it can appear as if it measures the parameter. To the frequentist probability statements must be falsifiable and so these statements only concern the long-run behavior of the coin and the experiment. Here are some threads that discuss these differences in approach and philosophy (1) (2) (3)

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    $\begingroup$ Re "There is no concern about being verifiable ...": That reads like a straw man argument. Gelman et al. strongly disagree. Or see S. James Press, Subjective and Objective Bayesian Statistics, 2nd Ed. $\endgroup$
    – whuber ♦
    Commented Aug 18, 2021 at 15:00
  • $\begingroup$ A posterior probability is not a verifiable statement about the actual parameter, the hypothesis, nor the experiment even if the unknown true parameter under investigation was indeed sampled from the prior distribution. Can you elaborate on "that reads like a straw man argument"? $\endgroup$
    – Geoffrey Johnson
    Commented Aug 18, 2021 at 15:09
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    $\begingroup$ By attributing to "Bayesians" an opinion that few (if any) have, you are erecting a contrafactual proposition whose analysis does little to help understand the subject of this thread. $\endgroup$
    – whuber ♦
    Commented Aug 18, 2021 at 15:13
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    $\begingroup$ You seem to conflate statements of the form "X is such-and-such" and "I believe X is such-and-such." $\endgroup$
    – whuber ♦
    Commented Aug 18, 2021 at 16:58
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    $\begingroup$ That's right--but you continually write answers, like this one, that appear to conflate beliefs with facts! In particular, a subjective probability is not generally used to quantify "feelings." Your statement "probability measures the experimenter" could, if we replaced "probability" by a sensation to create a familiar analogy, be rendered "color measures the experimenter." Yes, perhaps the scientist sees the blue glow of Cherenkov radiation: but when other people and other instruments agree it's a blue glow, then denying the objectivity of "blue" amounts to solipsism. $\endgroup$
    – whuber ♦
    Commented Aug 18, 2021 at 18:17

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