Bayesian Inferences and Developing Information
As a poker coach, I frequently get asked questions with potential answers that lie on a continuum, from people who want a binary response. How big of a sample size of games do I need to have before I am sure that I am a winner? How many hands does my opponent need to open before I should start adjusting against it? When should I start to assume my opponent likes to 3-bet bluff, rather than just getting a string of good hands? In truth, after the very first time you win a HUSNG, see your opponent open a button, or get 3-bet, you already have information that you should be working into your best guess of those frequencies.
To be clear, I think it is important to not be a nit about terminology, and that doing so can lead to some errors. Here's an example: In psychology literature, there is a somewhat famous study about how “illogical” people can be. The study describes a woman named Linda, who “is 31 years old, single, outspoken, and very bright. She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in anti-nuclear demonstrations.” The experiment then asks which is more probable:
A) Linda is a bank teller.
B) Linda is a bank teller and is active in the feminist movement.
Despite the fact that if Linda is a bank teller who is active in the feminist movement, she must also be a bank teller (which makes A clearly more likely) 85% of respondents chose B. This is often cited as a robust example of how illogical people can be and the massive cognitive errors our brains have, but I completely disagree with that interpretation. In my view, people (consciously or otherwise) decided to interpret the question as different than the literal meaning, given that this question is rarely asked with that response in mind in the real world. Typically, a question like this would be asked far more often as a way of trying to get a better holistic sense of Linda, willing to give up some accuracy for a more complete description. Additionally, when presented with these choices, many people will figure that it is quite possible that letter A is meant to indicate “Linda is a bank teller and not active in the feminist movement”. Thus, many respondents are actually employing fantastic Bayesian thinking by deciding to give what is likely to be the most relevant answer to the question that is most likely being asked. As a reward, subjects are chided for their errors in Psychological Review. Seems harsh to me.
Similarly, when players ask how big of a sample size they need to be sure they are a winner, we should give them a little more credit and assume they are asking around how big of a sample size do they need to be reasonably sure they are a winner. If you are a logic and numbers nerd like I am, always make sure you give people enough credit for what they mean to say rather than taking every word literally.
To introduce the concept of a Bayesian Inference, I would like to take you to Chicago for an interview with the finance company I work for. My preference is toward simple, easily accessible problems, and here's one I might ask you to test your ability to think probabilistically:
You leave your apartment groggily one morning, closing the door behind you. Suddenly, you are hit by a terrifying question: Do you have your keys, or are you now locked out? You stand there thinking about it for a few seconds, before deciding that yes, you probably have your keys, further estimating that 80% of the time, you have them. You also decide that there is an equal chance of your keys either being in your left pocket or your right pocket, and if they aren't in either pocket then you don't have them at all. Slowly, perversely enjoying the sweat, you slide your hand into your right pocket, and find that your keys are not there. What should you now think is the probability that your keys are in your left pocket?
If the answer does not seem clear to you, don't feel bad – the answer was not immediately clear to many of my co-workers, either. On one hand, it seems like the answer should be 40% - you checked one out of the two pockets, and now half your chance is gone. Maybe it should still be 80% - you thought that this was the probability before, and you're not done checking, so the probability doesn't go down until you've actually completely looked. Or maybe it should be somewhere in the middle – but if so, how would we come up with a number?
This situation is analogous to a lot of poker thinking. When your opponent raises his first three buttons when he gets down to 10-14bb deep, you have to decide – is he really that aggressive, or did he just get two or three inducing hands in quick succession? When your opponent 3-bets two out of your first four button opens, do we have reads now that our opponent likes to 3-bet with a wide range? The basic gist of the answer to these questions, and the keys question, is that we have to learn to think about the potential worlds we could be in and how those change over time. It is now more likely that we do not have the keys than before, but we very easily could be in the world where we have them. It is now more likely that our opponent likes to raise a high percentage of hands (or 3-bet a high frequency), but we cannot be sure. To do the best overall given all of the situations we could potentially be in, we have to learn to appropriately weight new information with what we knew before.
For the keys example, one easy way to think of it is that there are five possible worlds we could be in after closing the door, with us having the keys 80% of the time and there being an equal chance of them being in each pocket. In two worlds, the keys are in the left pocket, in two more, they are in the right pocket, and in one oh so cruel world, the keys are sitting on your desk inside. Once we check the right pocket and find no keys, there are only three possible worlds we can be in: The two where we will happily find them in the left pocket, and the one that will cause us to bang our head against the wall. From this, we learn that the best approximation is now a 2/3 chance that we have the keys and with a bit of relief we can go on with our day.
This type of probabilistic reasoning is called a Bayesian Inference, which is a fancy way of saying we are using Bayes' Rule, which is a fancy way of saying we're taking into account all the possible worlds we could be in given new evidence (both concepts are worth looking up on wikipedia, for those interested). In poker, you start with a general sense of what people do on average, and then face an opponent who has tendencies that are somewhere in that vast distribution. The very first hand you play gives you information, but just because your opponent raised preflop and continuation bet does not mean your opponent frequently raises and continuation bets – it is just now slighly more likely. Your optimal strategy readless will be to play against what your general opponent pool does on average, and then make increasing adjustments as you get more and more data about tendencies that are different than the norm. Some of those tendencies you can learn more quickly about: If your opponent quickly folds when presented three straight opportunities to check/raise bluff on a dry board, we already know that our opponent is significantly less likely to check/raise a wide range, and it becomes a significant error to check behind with hands that might be borderline otherwise.
As you play against new opponents, consciously think about that process of obtaining new information and hedging it against what you know about what most players do. Realize that as your sample size increases, so do your adjustments to unusual frequencies, but that there is no single point where it starts to be correct to make a drastic adjustment, as you should be making those over time as you get more and more information. Especially considering it is impossible to know for sure what your opponent is doing, optimal poker embraces the uncertainty, tries to make the guess that will have the best expectation on average, and constantly updates that guess when given new information.