An Intro to Bayesian Inference & Poker (aka Hand Reading)

[bennymacca]

That has to be one of the most terrible applications of maths to poker i have ever seen trishan. What complete rubbish. somebody raising 4 out of the first 5 hands still doesnt mean they are a maniac, and working out that they are a <whatevers> chance of being one is completely stupid.

Who would actually do these calculations at a live table, and how would they be any better than ones initial feelings on a player? Completely stupid. Apologies to AGJ, who might actually provide a decent example of this theories practicality, but if thats all Bayes theorem has got...

Take a more extreme case such as 2/3 or 4/5 at the start rather than just one hand, and you can see that he is a huge favorite to be maniac. As trishan says, it's all about adding prior information in, an is extremely powerful

[AJG]

yes, but my question is, is an answer of 8% THAT much better than an answer of "about 10% or probably a little less".

You are missing the point, which was how easily our intuition can mislead us when dealing with matters of probability... If you actually answered "about 10%", well done, but also consider it then goes on to solve it mathematically which will automatically provide more accuracy than an estimate. Just like you dont question Pokerstove outputting u have 35.13576% equity, instead saying "35 is good enough".
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All I have presented so far in this thread (and Trishan's examples to), is akin to the preface chapter in a book, outlining the raw math and concepts that will be discussed, and a primitive example to whet the appetite. Yet there is voiced opinion that "The book overall has nothing to offer over (my own) current thinking"....

Please remember that only the most fundamental aspects of this subject have been presented thus far, and that the examples given so far have been, of neccessity, simplified.... to the point of being almost useless on their own. And I thought I had made that clear in the OP and since.
All the things maccatak mentions (stack size, position, tilt etc) can all be incorporated into this, indeed they must be for it to be of real benefit.
A quote from a well respected source may serve well here:

"In poker, Bayes' Theorem allows us to refine our judgments about probabilities based on new information that we observe. In fact, strong players use Bayes' Theorem constantly as new information appears to continually refine their probability assesments; the process of Bayesian inference is at the heart of hand reading and exploitive play, as we shall see in Part II"

And this is true, that it is a Bayesian process, whether they realize it or not...

Maccatak: Trishan almost directly quoted MOP (pp 38-39) in the example you said was "the most ridiculous application of math to poker you had even seen" and "complete rubbish"... Take that one up with Chen and Ankenman mate.... And it is more designed (imo) to show how much influence a single event can have on refining prior assumptions, than to imply we should all label someone who open raises the first hand they play a maniac...
And please dont be so quick to dismiss new ideas cos they dont make sense or you cant see their utility given your current understanding of them. Personally I invariably find that, when I do this, it is my own understanding that is lacking, not the utility of the idea(s).
I'm guessing you havent read the article under the first link in the OP? (Bayesian reasoning at work here to )
I have tried to emphasise how important a general understanding of Bayes' Theorem is:

And again I will repeat something (key) from the OP:
Bayes' Theorem captures mathematically (and extremely accurately and elegantly), the natural process that any (thinking) player tries to apply at the poker table in using any and all information at hand to narrow his opponent's possible range. So, given this, understanding the theorem in and of itself, will enable us to execute this process more accurately.

Nowhere is it proposed that this is to be as quantified as pot odds or outs actually at the tables either...

Baby steps...
ANY presentation of new ideas (to the audience) needs to start at the absolute beginning and therefore basics/fundamentals. And like many other subjects, the fundamentals soon give way to large areas of practical application, but only once the fundamentals are grasped (and the more the better), hence why they are presented first. Don't get me wrong, I don't claim to fully understand them myself => part of the reason I started this topic.

Side Note:

I am reminded af something a university professor once said to me. He asked "What makes someone an expert Aaron?"... and i (like most i think) answered "Someone who knows a lot about something" to which he replied "NO!... An expert is someone who knows the fundamentals of their chosen field inside out, like 2nd nature. Then they can apply those fundamentals to any (new) situation presented to them, and solve the problem (or whatever depending on the field). Sure... time and experience also lead to the expert knowing alot, but that is merely a side-effect".
This conversation completely changed the way i look at learning (and imho to its great benefit)

Also, for the sake of future discussion on this, lets assume we are talking about a cash game or atleast, just ignore tournament effects like ICM and other tournament equity considerations, as it really has nothing to do with hand reading (the decision we make to be sure, but that is not the topic here)

[AJG]

Pop Quiz:

$1/$2 cash game, Effective Stacks are 80BB, Villain is a solid 21/17/5, and the following action occurs:

We open raise to $7 from the CO with AT, the btn and sb folds, and the BB 3bets us to $20, we call....

The flop falls: KT4
Villain cbets $28

1) Using combinatorics, how less likely is it he holds a K?

2) If we know he will only continue with a pair of Kings or better, is shoving a profitable play?

3) If it is, how big must the effective stack be before this is no longer profitable, or if not how small before it becomes profitable?

Let us assign a range of {TT+,AQs+,AKo} for an opponent (slightly tighter than 5%), as he 3bet us OOP... (and so all who try to solve use the same range)

And our EV = X(pot size) + (1 - X)(-L x Villain's Equity + W x Hero Equity)
L is our max loss, W max win.
X is villains fold %

Edited: minor error in 1st solution (but basic result still the same) - and added Q3 above.

HR - quiz 1.zip

My Solution

(1.32 KiB) Downloaded 91 times

I think posting answers as attached zipped text files will help in not giving hints to future readers - without them actively seeking them ?? (and only a few extra mouse clicks) - while allowing solutions to be submitted... - Forum doesnt allow text files as attachments ?? Hence the zip...

===============================================

User beware: garbage in is garbage out as with most poker formulas. How accurate your results will be will be entirely dependent on your 'prior' probabilities.

FYP

Only because terminology matters to have a coherent discussion with as little amgiguity as possible

[maccatak11]

OK, decent question, ill have a crack at it at some stage in the next day or two.

But in reality, a situation like the one above (where we are shove/folding the vast majority of the time) doesn't come up very often, especially in a cash game scenario like you mentioned.

Lets say, instead of 80BB effective stacks, that we had 200BB effective stacks. We aren't going to shove any more on this flop, so then what do we do?

Assuming our villain will c-bet almost all of his 3-betting range OOP pre flop, lots of us here might flat call and re-evaluate on the turn, or raise it up to find out where we are.

Lets say we call, because we are so deep, and any of the following happen:
- the board pairs kings
- the board flushes
- a straight card completes
- They check turn
- They fire a second barrell

now what? There are so many differences in board texture, a lot of the assumptions need to be made for this formula to be applied in the examples you have given.


What considerations can we make regarding villains perception about how we might have played previously on the table? Has he/she 3-bet us OOP before? What does villain think we are calling a 3-bet in position with? Have we shown down light from LP before? etc etc etc

[maccatak11]

I do like Trishans point however, that learning the Maths here gives credence to what we might do intuitively anyway.

I guess its like when we first starting thinking about the game, and start wondering whether to call a bet with our flush draw. Most regular pub players could rationalise "well, its not much more for me to call, i have an ok chance of hitting my flush, and i could win all dose moneyz"

After later learning about outs and pot odds, they might reach the same conclusion that they should either call or fold, but now they have a better understanding of why they should.

Im still not sold on Bayes theorem for an intermediate/advanced-beginner like myself. I guess im thinking about work versus profit. Will the work needed to study and implement something reasonably complex (compared to other simple maths concepts in poker) be worth it?

[bennymacca]

to be completely honest, i rarely explicitly calculate pot odds in my head anymore.

firstly, because it is pretty much the easiest tell in the world to see someone counting their outs, and secondly, because it is pretty much second nature to me, i just do it intuitively nowadays.

but there WAS a time where i learned it, and found that some instinctive things i do were right, some were wrong.

this is a similar thing. it may already confirm 90% of what you are doing right, but it may illuminate the 10% that you need work on.

that 10% makes the difference between a winning micro stakes player and a winning mid stakes player, which is pretty much were all of us are on the road to at the present time

[John Miller]

All too deep for me ....back to patty cake..

[AJG]

For macatak to read, no need to take up space for all:

But in reality, a situation like the one above (where we are shove/folding the vast majority of the time) doesn't come up very often, especially in a cash game scenario like you mentioned.

Who said anything about folding? im never folding to villain's bet there! are you? Perhaps the point of the example is to show how often it should come up, and perhaps if you think it shouldn't very often, then it is your reasoning or assumptions here that are flawed?


Lets say, instead of 80BB effective stacks, that we had 200BB effective stacks. We aren't going to shove any more on this flop, so then what do we do?
Work out the questions above, I think you will find number 3 rather insightful...

Assuming our villain will c-bet almost all of his 3-betting range OOP pre flop, lots of us here might flat call and re-evaluate on the turn, or raise it up to find out where we are.
This doesnt mean shoving isnt a viable play?

Lets say we call, because we are so deep, and any of the following happen:
- the board pairs kings Baye's T says they even less likely have K!
- the board flushes Awesome, hopefullly he does have KK or AK and cant fold it
- a straight card completes Again, awsome, we now have more outs on the river (and better implied odds)
- They check turn Cool, we freeroll to our flush outs
- They fire a second barrell Yeah ok, this one depends...
But this is all irrelevant if we shove the flop. Also you are changing the parameters of the scenario by saying 'so deep we just flat'

now what? There are so many differences in board texture, a lot of the assumptions need to be made for this formula to be applied in the examples you have given.
No there aren't, that is why I have given them... They are overly simplified to concentrate on the concepts...

You have mentioned board texture a few times now - what would you say if I told you that all reasoning based on board texture was purely Bayesian?


What considerations can we make regarding villains perception about how we might have played previously on the table? Has he/she 3-bet us OOP before? What does villain think we are calling a 3-bet in position with? Have we shown down light from LP before? etc etc etc

Seriously?
Here you are implying level 3 thought. Maybe you do so, I dont know, but I DO know that at < 50NL, doing so is largely pointless.
The only level we need be on is (Opponent thinking level) + 1...
And our opponents are rarely above Level 1...
Anything over and above 'opponent level' + 1, and we start to run into trouble by out-levelling ourselves at the table(s).

And of course, so far these are (as I have already said umpteen times) overly simplified examples.
You mention the work v reward ratio, yet seem to want to jump straight in the deep end!
And a large point here is that Bayes' Theorem can show us situations where our intuitive thought processes are actually wrong.
Why dont you come along for this ride with us, instead of trying to jump forward so far? After all, the journey is often reward in itself...

Im still not sold on Bayes theorem for an intermediate/advanced-beginner like myself. I guess im thinking about work versus profit. Will the work needed to study and implement something reasonably complex (compared to other simple maths concepts in poker) be worth it?
What exactly is an "intermediate/advanced-beginner"?
As I have quoted, it is the very heart of Hand Reading... So, if you want to be good at this, it follows that understanding the underlying principles (ie Bayes' Theorem) is going to be useful no?
And the effort is a 'once off' (although continuous to improve) which you could potentially use for the remainder of you 50+ years of future poker?
Or think of it this way, even if it only has limited benefit a micros (which it will), how much better at hand reading will you be when you hit 100NL if you start now? And then at >=200NL? (where u will get crushed if you do not read hands well)
And TBH, if you were so concerened about work v profit, you probably wouldnt even bother playing an under $20 tourney online... Or a 50NL cash table...

TBH, this thread is not about "selling Bayes' Theorem and its application to poker", and I am not going to try to do so (anymore than I already have - which I hadnt intended at all). It is for the discussion of its application... And hopefully to get better at hand reading theory. But hopefully, after some development, it will also convince any skeptics that "Yes, there is something here that will up my profits proportional to the effort" but tbh if it doesnt, too bad (for the skeptics)

Another even simpler example (in terms of information content):

Villain opens for 3x preflop from UTG, with a frequency of 10%, equating to an approximate range of {88+,ATs+,KTs+,QJs,AQo+}.
He is a known tight player who will only call a 3bet OOP with a premium (top <3%) hand in the range {JJ+,AKs}
(as you will see this is all the apriori information we actually need)

Villain's original range contains 98 combos.
Villain's call 3bet range contains 28 combos.
(You should already see where this is headed...)

Lets say we have AQ in the CO.
Now his 3bet calling range can only have 6xJJ,KK 3xAA,QQ, and 3xAKs = 21 combos. (or ~21% of the time he is opening with a premium hand)

Therefore he will fold to our 3bet (98 - 21) / 98 = 77 out of 98 = 78% of the time.
Thats a hell of a lot of folding.
Lets say we have 72o (or any random hand), which doesnt touch his 3bet calling range, he will still fold to a 3bet (98 - 28) / 98 = 70 of 98 times he open raises UTG, which is just over 70% and thats a hell of a lot of folding.

If we 3bet to 10BB we need a fold ~67% of the time to break even (at 75% we have EV of 1.2BB, almost that of blind stealing), and this isn't counting those times when we can actually continue after the flop connects with our random hand (and position!)...

Conclusion:
We show an immediate profit by 3betting here to 10BB with ATC...

Would you have thought this before? (I certainly didnt)

Consider when we have KJ which effects the 3bet calling range (in terms of combos) exactly as AQ did, so he is still folding 78% of the time.
Taking this a step further:
All our having KJ really does is strengthen his range, when he does call...
Likewise our holding AQ weakens his range, when he does call...
(can you see why this is so?)

The basic question here is "How often does villain have a premium hand when he raises utg @ 10%?", and as you can see the answer is : not often!

Now, I am not suggesting you 3bet bluff this type of player every time they raise UTG. Far from it...
But hopefully it will increase your 3bet bluffing frequency preflop enough that it increases your profit....
I would guess that alot of players likely have a preflop 3bet bluffing range of exactly NULL, so this simply means "introduce one!"
And if you consider at approx his average folding % we have almost the EV of a blind steal, perhaps making this play as a bluff as often as we would attempt a blind steal is a good starting point? Or maybe choose hands outside your '3bet for value' range (just your 3bet range if u never do this as a bluff) that have good equity against his calling range, like SCs down to 54s and small pocket pairs (even 64s has >22% here) so that you feel 'more secure' in trying this... Start with only a few hands then add more as you get a feel for how well this bluff works.
And just think, if you only do this (bluff pre for 4.5BB) successfully once every 200 hands (per table - if that is relevant) you will increase your winrate by over 2BB/100! not bad given most long term winning winrates are (significantly) <10BB/100
And there are more benefits besides... (for example: balancing your 3bet range)

Note: The article Garth quote-posted in this sub-forum is very much analogous to this, in that Hero rebluff shoved the flop precisely because villain doesnt have a hand often enough to call, making it a +EV play.
(and how exactly did Hero know this?)
(and what was Garth's comment at the bottom?)

[AJG]

A new player sits at your table, and folds every hand for the first 2 orbits.
Then he plays a hand, open raising from MP. The hand goes to showdown and he has KK.

1) What can you infer about their preflop play?
2) What impact does them folding approx the 1st 15 hands have?
3) What impact does them showing KK the first hand they do play have?

[AJG]

Please think on prev question before peeking:

Clearly we infer he is likely somewhat tight preflop...

Impact of folding 15 hands in a row? -> 100% toward the conclusion
Impact of showing down KK? -> 0% toward the conclusion.

Now I am guessing atleast some people will say, "but showing down KK first hand is also evidence of preflop tightness?", ie has > 0% impact on our conclusion.

No, its not.

Consider that ANY (reasonable) strategy will play KK preflop... So we can infer nothing from him showing this hand...

Put in terms of Bayes' Theorem (using multiple observations/events):
H = Hypothesis that player is tight preflop
O1 = Folded first X hands
O2 = Open raised KK

P(H|O1&O2) = P(O1&O2|H) x P(H) / P(O1&O2)

From this (and one of the points about Bayes' Theorem itself in OP) we see that the LHS is proportional to P(O1&O2|H) which is 'the probability that he folds the 1st X hands (15 here) AND open raises KK GIVEN he is a tight preflop player (our hypothesis)'
P(O1&O2) = P(O1) x P(O2)
Clearly P(O2) = 1.0 (ie its a certainty - who is NOT open raising KK??)

So this just reduces to P(H|O1) = P(O1|H)xP(H)/P(O1)
ie, we ignore the fact he showed KK.

Now, what if he shows down T9s instead of KK?
Well, we reconsider our assumption of "Tight preflop" (which we likely made before his cards were revealed anyway), and conclude he probably just got dealt garbage 15 hands in a row... And we can only do this, because open raising T9s from MP does actually give us some information regarding his preflop strat (well more than KK does). Why? because P(Open raises KK) = 1.0, while P(Open raises T9s) < 1. And likely quite abit less than 1 (certain).
Yes, ok, some players might limp their Kings, or sometimes limp them hoping for a raise to 3bet. So then compare P(Doesnt OPEN FOLD KK from MP) with P(Doesnt OPEN FOLD T9s from MP)... Now its easier to see as 1.0 : < 1.0

So the point of this, is we also need to consider how much a certiain observation can actually tell us. Alot of people would think that showing KK further emphasises that the player is tight preflop, but infact it tells us nothing!

As I am the only one posting in this thread anymore, I didnt want to leave it on a question, but won't be posting anymore...