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*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?

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  • $\begingroup$ You should also take into consideration that the event in question may happen thousands or millions of times, but to different players, in different occasions, on different days. Some specific poker hand may happen to you only once in your life but you have to consider that it will happen again and again to other people throughout time. So the expression "It was the correct play even though I lost" makes sense entirely, because in reality, the situation doesn't happen only once. $\endgroup$ – Victor S. Sep 15 at 3:18
  • $\begingroup$ @VictorS.I feel like I could say something like the universe never passes through the same state twice so every event is novel and will never repeat. Who knows what's more correct. I guess we just follow the model that matches observations best $\endgroup$ – J Kusin Sep 15 at 3:22
  • $\begingroup$ "I could say something like the universe never passes through the same state twice so every event is novel and will never repeat." You want to express a probability model for, AHEM, every event in the state space of the universe? $\endgroup$ – Alexis Sep 15 at 3:35
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    $\begingroup$ @JKusin : Poincare recurrence disagrees with you and seems to apply to this universe (although there is still room for debate on this point). $\endgroup$ – Eric Towers Sep 15 at 15:32
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    $\begingroup$ If anyone's looking for a lively example, a key plot-point of the film Molly's Game (2017) turns on this concept. $\endgroup$ – Sycorax Sep 16 at 4:14
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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.)

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  • $\begingroup$ I probably need to do some reading, but don't the distributions assume something about the nature of probability though, like fundamentally subjective vs objective? Or can I just assign probabilities according to either framework because they work, and leave subjective vs objective to the physicists? And always be confident my personal probabilities are correct in either framework regardless of the fundamental nature of probability. $\endgroup$ – J Kusin Sep 14 at 16:19
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    $\begingroup$ I think this answer might be sidestepping the issues in the question. In particular, why, in a one-off problem, is minimizing expected loss the best procedure or even a good one? $\endgroup$ – whuber Sep 14 at 16:44
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    $\begingroup$ @whuber: we are going into decision theory there, and I'll very happily admit that I'm not an expert here. I would be interested in whether there are reasonable decision rules that cannot be cast as minimizing the expectation of an appropriate loss function. But for now, I have not found an example in which this framework would not be sufficient. $\endgroup$ – Stephan Kolassa Sep 14 at 17:40
  • $\begingroup$ @JKusin: I wouldn't put it as "assigning" a probability, but rather as understanding our data generating process well enough to describe it in terms of a distribution. Which sounds objective enough to me. Even if the data our DGP generates consists of a single point. I personally would rather worry about the inevitable deviation between the true DGP and our distribution than about subjective/objective. I may not be sufficiently philosophical about this. $\endgroup$ – Stephan Kolassa Sep 14 at 17:42
  • $\begingroup$ I think loss (or utility) has to play a role, certainly. But game theory suggests that minimax considerations can be useful. $\endgroup$ – whuber Sep 14 at 17:50
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"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.

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    $\begingroup$ What is the relationship you're trying to imply here between frequentists and people who believe in lucky streaks? $\endgroup$ – duckmayr Sep 15 at 13:37
  • $\begingroup$ @duckmayr I'm probably misunderstanding the OP's use of "frequentist approach". I also suppose more bad players believe luck has to even-uot -- future results must contradict expected ones. I don't know what you formal approach one could hammer that into. $\endgroup$ – Owen Reynolds Sep 15 at 16:58
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    $\begingroup$ OP is referring to one of the main types of statistical inference, Frequentist inference (with the other main type being Bayesian inference) $\endgroup$ – duckmayr Sep 15 at 17:09
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    $\begingroup$ The last paragraph, as duckmayr pointed out, doesn't make any sense because that's not what frequentist means. $\endgroup$ – eps Sep 16 at 15:05
  • $\begingroup$ I'm trying to reply somewhat to OP's "this idea seems to require a frequentist approach". But that doesn't seem to have gone anywhere. They only teach us ComSci majors basic Bayesian conditional probability on the way to Kolmogorov complexity, so I can't say for sure "running experiments != frequentist" $\endgroup$ – Owen Reynolds Sep 18 at 16:13
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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.

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  • $\begingroup$ Ya I think both answers are right, my question maybe put the cart before the horse. Both freq and bays are mathematically sound, and we justify applying them because they match our observations. We don't need to bring in the deeper nature of where probability comes from or if the models are perfect fits for reality to justify using freq and bays. $\endgroup$ – J Kusin Sep 14 at 19:48
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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.

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"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.

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