"It was the correct play even though I lost" *sorry if this isn't the right SE community, maybe it's more philosophical*
You often hear this refrain in games like Poker or Hearthstone. The idea is that making play A this game resulted in a loss, but always making play A in the long run/limit is the best odds/EV.
My question is: Why does this idea seem to require a frequentist approach, yet at the same time, even if this is the ONLY game played, the same play is still "correct". Are there any situations in the physical world were frequentism and bayesianism make separate predictions? (I know QM interpretations get into objective vs subjective nature of probability, but that won't be settled anytime soon). How can I reassure myself taking a frequentist approach is always the best for right here and now?
 A: "The Correct Play is the one that should have won" is a mantra in professional poker. Hearthstone players are probably borrowing it. From the top result of "Poker correct play" I found it expressed as: If you’ve won money, it doesn’t mean you played the hand well. If you’ve lost money, it doesn’t mean that you played the hand badly.
A few results down I found a poker forum dedicated to this. The multiple deeply jargony replies to a "was this correct" hits it home how a whole culture thinks that can be determined regardless of actual results or number of hands played later. It's also interesting since they talk about the known probabilities of cards, but also guesses as to what other players were to likely do.
Annie Duke's Thinking in Bets is all over this idea. A person flipped one or two houses for a profit and assumes they're good at it, then goes on to lose their shirt. For one thing, too small a sample. For another, if they reviewed things they'd have noticed how much luck they required both times and realized that was proof they were terrible at flipping houses.
Poker players actually mock considering a series of hands. If some duffer won with a lucky inside straight two hands ago, you know they're going to go for it now (inside straights are "hot") and you can raise a little higher to take more money from them. But I couldn't say if that's more Bayesian or Frequentist.
A: I don't think either that this is a question about frequentist vs bayesian.
There is someone, in fact, that argue that frequentist approach to the case of once-only experiments is not solid enough: what interest do I have on what happens to an experiment if I repeat it indefinitevely, if I actually don't have the possibility or the intention to repeat it any more time?
And of course a lot of people think bayesian view of probability is more natural to most people. In the frequentist point of view, if you are going to win or lose a hand is a fixed fact: there are your cards, there are the other players cards, there is no misty randomness about them, and whoever is going to win is written in clear letters in the book of nature, that you can only try to read with a bit of uncertainty. Instead, more bayesian-fond statisticians will tell you that, since you don't know the other players hands, what you have, when you consider your prior knowledge of them and of the game in general, and after looking of your cards, is an informed and revised distribution of your odds of winning. In fact, if you win or not is indeed random to bayesian philosophy.
After you play and lose one game, I think that there is similarly little consolation in knowing that if you had made the same bets in some infinite loop you would have won most of the times, or in recalling that according to your bayesian posterior information you had better chances of winning. What's the point anyway?
The point is that if you have a method that maximizes your chances of winning (if we talk about future experiments, it's probability in both frameworks, and it works the same), you stick to it. Because it maximizes your chances, no further reason needed. The play was correct because it was what the method suggested.
A: As others are saying, the problem has nothing to do with frequentist VS bayesian. The problem is that at the point of making the decision you don't have any information about whether it will be a win or a loss.
If you introduce that information into your framework, then you are leaving yourself open to hindsight bias (which IMO it is the elephant in the room here and hasn't really been acknowledged in the other answers).
Hence, if you don't consider the end result, you need to rely in your model/computed odds/whatever information you had available at decision time. And sadly, that means that sometimes you'll lose, even when making the correct play.
A: I do not believe that this is a question of Bayesian vs. frequentist frameworks. It is a question of having the correct (predictive) distribution and minimizing the expected loss with respect to this distribution and a specified loss function. Whether the predictive distribution is delivered by a Bayesian or by a frequentist is irrelevant - all that matters is how far it diverges from reality. (Of course, getting only a single realization makes it hard to assess this, but again, that is orthogonal.)
A: "The correct play" is the play from the strategy that you believe works out the best for you, calculated through some kind of loss function. If you have that strategy and stick with it, the math says you will do well.
If you get into "but but but..." then you no longer follow the winning strategy you've developed.
Your winning strategy will result in you getting burned sometimes, perhaps often enough that an unlucky streak will bankrupt you before you get back to making money. However, if you allow for "but but but" you no longer follow your winning strategy and no longer use the strategy optimized for the least loss.
