Photo courtesy of Henning Buchholz
Fact is, you're already doing it. What we describe below isn't a new concept that you never thought of. We're all doing it all the time, even when not playing poker.
Every time we see something happen, we adjust the hypotheses for which it is relevant. It's a natural, unconscious process.
But if you can bring the process from your spine into the conscious part of your brain, you might be able to increase your precision quite a bit.
In a game of incomplete information, precision equals money.
If you're not interested in money, you can stop reading here.
Prior knowledge meets observation
Before we make an observation, we always have some kind of idea about the world around us.
Before the guy across the table looked at the flop and let out a happy shriek, he looked like he was going to die from boredom.
Or else, he had been shrieking merrily all the time because he just got married (or divorced).
Did you notice how the change of prior knowledge changes the information you retrieve from that happy shriek of his?
First it was full of meaning, then it meant nothing.
In the first case you modified your hypothesis about his hand. In the second you didn't.
To sum up, every observation you make at the poker table affects your view of things in interaction with what you thought before the observation - your prior knowledge.
The math for updating a hypothesis
According to probability theory, this is how you update a hypothesis H upon observing new data D:
Odds( H | D ) = Odds(H) * P( D | H ) / P( D | ~H )
Expressed in words, it goes like: "The (odds of the hypothesis given the observation) equals the (odds of the hypothesis before the observation) times the (probability of the data given the hypothesis) divided by the (probability of the data given that the hypothesis is false)."
Before your brain catches fire, let's look at an example.
Example: Villain starts talking
You're on the river facing an all in bet. You can only beat a bluff, and from the way the hand played out you think he's bluffing 25% of the time. His bet is the size of the pot, so 25% isn't good enough for you to motivate a call; you need 33%.
Then he tells you to call the bet and get it over with, claiming he's bluffing. This doesn't give his hand away entirely, but in your experience he'd do this more often if he's bluffing than if he's got a winner. Say twice as often.
Using the notation above, we then have the following numbers:
- Odds(H) = 0.25 / 0.75 = 1/3 - the odds that he's bluffing, before the observation
- P( D|H ) / P( D|~H ) = 2 - he talks twice as often if he's bluffing than if he isn't
The updated odds of the hypothesis given the observation are:
Odds( H|D ) = Odds(H) * P( D|H ) / P( D|~H ) = 1/3 * 2 = 2/3
Thanks to the observation (his talk), the chance that he's bluffing goes from 25% to 40% (odds 2:3). Now you can call his bet with a positive expectation.
Example: A king on the river
You're in a hand of Texas holdem. You pick up two aces and the board comes as follows.
After the turn betting, you feel 80% certain that the opponent has two kings in his hand and hit a set on the flop. You're getting ready to fold your pocket rockets on the river.
A second king comes on the river! How should you update your hypothesis upon this observation? How likely is he to be holding two kings in the hole now? Should you really be folding your aces?
You have seen six cards and if the hypothesis is correct, he has two kings in his hand. There are 44 unknown cards, one of which is the case king.
If villain doesn't have two kings, we'll assume he doesn't have one king either. Then there are three kings left in the deck and 46 unseen cards. (This is a simplification, but we'll motivate it by how the betting went.)
With these assumptions we have the following numbers:
- Odds(H) = 0.8/0.2 = 4 - the odds that he has a set of kings, before the observation (king on the river)
- P( D|H ) = 1/44 - the probability of the observation given the hypothesis (one king left in the deck)
- P(D|~H) = 3/46 - the probability of the observation given the hypothesis is wrong (three kings left)
The updated hypothesis given the observation becomes:
Odds( H|O ) = 4*(1/44) / (3/46) = 1.39
Which is close to 7/4 and corresponds to a probability of 58%.
So, the king on the river lowers the probability that villain holds two kings from 80% to 58%. You may have to call a bet with your unimproved pair of aces after all.
Comparing two unlikely events
It's interesting to note that the unexpected king on the river didn't make the hypothesis completely unlikely, as you may have thought. Why is that?
The case king falling on the river certainly is an unlikely event, but the thing is: even with three kings left in the deck, a king on the river isn't very likely. We're comparing two unlikely outcomes.
The king on the river is a fact, and we're evaluating our hypothesis after the fact.
To update a hypothesis after an observation, you should consider how likely the observation is under the hypothesis compared to if the hypothesis is false.
That's the key to building up knowledge at the poker table - and in life.