Updating Probability Estimates
When we gain information about a particular player's cards
(due to their betting action, for example), I don't know the mathematically correct way to update the probabilities for the other players and the deck. For concreteness, I give a toy problem here. Suppose we have two players and a five card deck, with cards A, B, C, D, and E. Each player now has two cards, with one card remaining in the deck. We don't know either players' cards -- we are just watching the game. Somehow we have estimated the probability that each player has each two card hand. Below, the two columns give the probability of each hand for Player 1 and Player 2. The matrix below shows the probability that they have each card, and the conditional probability of each second card, given the first. (The columns and the matrix are just two representations of the same information.) Player 1 Prior -------------- AB .20 AC .05 AD .05 AE .05 BC .20 BD .10 BE .10 CD .10 CE .10 DE .05 ___ _ A__ B__ C__ D__ E__ .35 A ___ 4/7 1/7 1/7 1/7 .60 B 1/3 ___ 1/3 1/6 1/6 .45 C 1/9 4/9 ___ 2/9 2/9 .30 D 1/6 1/3 1/3 ___ 1/6 .30 E 1/6 1/3 1/3 1/6 ___ Player 2 Prior -------------- AB .05 AC .05 AD .15 AE .10 BC .05 BD .10 BE .10 CD .10 CE .15 DE .15 ___ _ A__ B__ C__ D__ E__ .35 A ___ 1/7 1/7 3/7 2/7 .30 B 1/6 ___ 1/6 1/3 1/3 .35 C 1/7 1/7 ___ 2/7 3/7 .50 D .30 .20 .20 ___ .30 .50 E .20 .20 .30 .30 ___ And from the above we can infer a probability distribution on the deck, given below. Deck Prior ---------- A .30 B .10 C .20 D .20 E .20 Now, we also somehow have an estimate of the probability distribution of the actions Player 1 will take, depending on which hand he has, shown below on the left. Player 1 Action Probablities ---------------------------- ___ Fold Call Raise AB .00 .10 .90 AC .05 .25 .70 AD .05 .30 .65 AE .10 .60 .30 BC .20 .05 .80 BD .40 .10 .50 BE .50 .10 .40 CD .70 .25 .05 CE .80 .20 .00 DE .75 .25 .00 Suppose that Player 1 calls. Then we can update our estimates of the likelihood that Player 1 has each hand. This results in the distribution below. Player 1 After Calling ---------------------- AB .121 AC .076 AD .091 AE .182 BC .061 BD .061 BE .061 CD .152 CE .121 DE .076 ____ _ A___ B__ C___ D__ E___ .470 A ____ .257 .162 .194 .387 .304 B .398 ____ .201 .201 .201 .410 C .185 .149 ____ .371 .295 .380 D .239 .161 .400 ____ .200 .440 E .414 .139 .275 .173 ____ Now, Player 1's action should also lead us to update our estimates for the deck and Player 2. A naive first try is to adjust the deck's and Player 2's single card probabilities so that each card's location probabilities sum to 1. For example, Player 1's A-probability went from .35 to .47, so we multiply Player 2's and Deck's A-probability by .53/.65. Similarly, we multiply the probabilities for B, C, D, and E by .696/.40, .59/.55, .62/.70, .56/.70, respectively. Naive Deck Estimate After Player 1's Call ----------------------------------------- A .245 B .174 C .215 D .177 E .160 ----- 0.971 = sum Naive Player 2 Estimate After Player 1's Call --------------------------------------------- A .285 B .522 C .375 D .443 E .400 ----- 2.025 = sum But these don't even sum to 1 and 2, as they must. So this approach is too simple-minded. Another approach would be to solve a system of linear equations. We want Deck(A) + Player2(A) = .530 Deck(B) + Player2(B) = .696 Deck(C) + Player2(C) = .590 Deck(D) + Player2(D) = .620 Deck(E) + Player2(E) = .560 Deck(A) + Deck(B) + Deck(C) + Deck(D) + Deck(E) = 1.0 Player2(A) + Player2(B) + Player2(C) + Player2(D) + Player2(E) = 1.0 But these equations in no way account for the prior distributions estimated for the deck and Player 2. So, what is the mathematically correct way to update our estimated distributions for the deck and Player 2 in this situation? The answer should satisfy all of the equations above, but also in some way reflect the prior estimated distributions. Thanks in advance! |
Re: Updating Probability Estimates
In your table titled "Player 1 Action Probabilities," the values in the BC row do not sum to 1.
heihojin |
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