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

[bennymacca]

please stop putting ellipses everywhere its annoying...

[AJG]

Please stop making useless, non-constructive posts, it's annoying...


I am so very sorry to have annoyed you with my writing style benny the cunt.
Please accept my heartfelt and deepest apology.
Clearly my use of ellipses has completely eclipsed the actual content of the post(s).

But seriously...
You find it THAT annoying you need to post saying so? (or did you need to, to make your Friday post quota?)


Anyway, both out of context instances have been removed.
Happy now? Good...
If not, well... DILLIGAF?

[bennymacca]

just kind of wondering the point to this thread - it was constructive and a good thread, but then you started posting answers to your own replies in a pretty short space of time.

are you trying to teach us bayes theorem, or show us how much you know about bayes theorem?

just sayin

[AJG]

Well, only you and trishan actually posted anything that was not objection.
I posted a scenario to think on, and no-one replied (at all?). I didnt realize there was some time-frame involved and that 2 days should be plenty if people are actually interested in participating... Perhaps it was premature? How long would you suggest?

Also, that last one was the only answer I posted ?? unless you include the zipped text file attachment, which I explained the reason for in the post. And I always intended to post an 'answer' to that last, as the point was often information we think is relevant is actually completely irrelevant. I won't delete, but put in a spoiler if that helps the continuation of this thread

I think you/we often forget there are just as many lurkers as active posters - possibly more, like any forum. As I had decided not to continue posting further in this thread (which is a pity cos we havent even got to the meaty stuff yet), I thought that I should provide the answer for any and all who care to read it, now and in the future, then forget about it hence forth.

[AJG]

are you trying to teach us bayes theorem, or show us how much you know about bayes theorem?

All I have done is state Bayes' Theorem in its simplest possible form, and that it has application to Poker. How can that possibly be misconstrued as 'showing how much I know about it'?

And I cannot make assumptions regarding people's previous exposure to it (especially as it isn't even included in alot of undergraduate Math courses - even then only near the end), so needed to atleast state it and provide brief explanation for those who had never seen it. If that is misconstrued with 'teaching it to you', in the sense of 'I dont expect any of you fools to have heard of this before <snicker>', well I really dont know what to say to that... other than, no

[Sinitster]

interesting reading ...

it is like milk to me .......... pastuerised !

[trishan]

Come on boys play nice. I would really like to see more happening in this advanced discussion forum and it was nice to see a bit of activity, particularly on a subject like Bayes theorem. Aaron I think you might need to give these sorts of threads quite a bit more time before people respond with anything meaningful. The majority of people would have looked at the first post and gone too hard, or too long and moved on. Some would still be trying to digest the information. Others like myself who have some knowledge of this might want to brush up (I'm doing some revision).

We should really look at how we structure these threads. Perhaps an example at the start, few days to think about it then step by step go through explaining parts of theorem. Try to make it as engaging as possible. Some very basic probability concepts would be worthwhile too.

So what are the meaty examples of Bayes? I am still interested.

[AceLosesKing]

Please stop making useless, non-constructive posts, it's annoying...


I am so very sorry to have annoyed you with my writing style benny the cunt.
Please accept my heartfelt and deepest apology.
Clearly my use of ellipses has completely eclipsed the actual content of the post(s).

But seriously...
You find it THAT annoying you need to post saying so? (or did you need to, to make your Friday post quota?)


Anyway, both out of context instances have been removed.
Happy now? Good...
If not, well... DILLIGAF?

Ellipses annoy the hell out of me, I'm with benny the cunt (in fact it probably ticks me off even more).

[AJG]

Still, in a thread of this subject nature, you bother to post "Ohh i dont like your use of punctuation"
FFS

We should really look at how we structure these threads. Perhaps an example at the start, few days to think about it then step by step go through explaining parts of theorem. Try to make it as engaging as possible. Some very basic probability concepts would be worthwhile too..

That is what I tried to do. We have to start somewhere, and the beginning is usually a good place to do so. Yet the few examples (both of us gave) were met with some negativity (albeit mostly from 1 source, and yes positive responses also): seemingly because they were too basic or didnt indicate decent poker usage (to them) or for whatever reason. Of course there will always be those who for whatever reason don't see fit to engage in a particular discussion, or dont think it relevant for them, which is fine. Each to their own...

Trishan: I know there are 'meaty' uses for this stuff, but of course alot of the sources on it only hint at most of them. Part of the reason for starting this thread was to hopefully, collectively discover more of them. At this stage I have little of my own original thought to add and could mostly provide examples from other wiser heads (and even my own thoughts on it could probably be seen as extensions to those others, but you have to start somewhere right? And my own numbers etc havent been double checked like published results). Still, the hope is that those may inspire some further thought on the subject... Also, given that the entire subject has only been broached in a most elementary here way so far, there HAS to be more juicy uses for this stuff...
But the main thing to take away atm (which I have tried to emphasise a few times now) is that Bayesian thinking often indicates holes in our intuitive understanding of probability (which research into eduaction and human cognitive processes shows is typically pretty crap) - for example I asked quite a few players tonight if they thought an Ace on the flop made it less likely an opponent has one, to which EVERY SINGLE ONE OF THEM replied, "of course not - those cards are already dealt, how could it?" - or something similar. Hmmmm

It just pisses me a tad, to put work into developing new examples (so I am not just copying from a book or whatever), put effort into presenting it and making sure all the numbers add up etc, just to have people comment on the friggin grammar used. - You guys do know what ellipses are meant to indicate, yes?
And tbh, there are things that annoy all of us at times (pointless posts like the 2 ALK made in this thread alone shit me off no end), yet there is no reason to whinge about it. Deal with it. Its not like I am abusing anyone or anything, it just a few friggin dots! The fact that people choose to comment on THAT rather than the actual content is, at the very least, disheartening. Actually I think its also just plain rude! (A polite PM would be much better form?)

This forum gets so negative and petty at times its ridiculous. TBH, Im not at all sure I want to put any time into these posts anymore (and know others, who shall remain nameless, who feel very similar), atleast here on this forum. Because believe it or not, there ARE other forums where my input is actually appreciated and respected. And no-one gives a rats if I use a few ellipses or not.

[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.

The Problem
I still don't think you are convinced so I just wanted to show you another example based on the above comment where you suggest that someone who plays 4 out of 5 hands is not maniac. Yes we can never be certain that a player is a maniac in that situation but what Bayes theorem does is adjust our prior estimate of the probability the player is a maniac given further information, in this case raising 4 out of the first 5 hands.

Prior Estimate Based on HEM Stats
Now based on my HEM stats for people at 10NL against which I have played more than 50 hands we can make a few estimates at how people play. There are more than a few maniacs at this level but lets say we want to find the likelihood that this player is in the top 10% of loose players at this level. So using HEM I can see that the top 10% of loose players (based on VPIP) have an average VPIP of 67% (lowest VPIP of 56%). The other 90% of players have an average VPIP of 24%. In other words if I arranged the players with more than 50 hands history in my database from highest VPIP to lowest VPIP, the top 10% would have an average VPIP of 67% and can thus be classified as "maniacs".

Some preliminary calculations
Before using Bayes we can find out the probability that a player raises Y out of the the first X hands given a certain VPIP. It is calculated as follows:
P(player raises Y hands out first X hands given Z VPIP)
= 5 * [VPIP^(number of hands out of 5 raised)]*[(1-VPIP)^(number of hands folded)]
= 5 * [Z^Y]*[(1-Z)^(X-Y)]

The 5 at the start of the equation is the number of combinations the person could have played the hands in. The second bit in the square brackets calculates the probability of playing 4 out of 5 hands in a specific order.

Thus multiplying the probability of a player raising 4/5 hands in a specific order by the 5 specific ways it could be played gives the overall probability.

"If you are confused about this":

To find the probability that two or more events occur together (Probability Event A AND Event B AND Event C AND....Event N occur) you multiply the individual probabilities of each event.

The % of time a player will raise is equal to their VPIP and conversely the % they will fold is equal to (1-VPIP).

P(FRRRR) = P(F)*P(R)*P(R)*P(R)*P(R)

The probabilities of each individual combination occurring are equal:
P(FRRRR) =P(RFRRR) = P(RRFRR) = P(RRRFR) = P(RRRRF) = X%
As there are 5 individual combinations which could occur.
The probability of any of these 5 occurring is 5 * X%

Using the numbers above:
P(raises 4 out of 5 hands given a VPIP of 67%)
=5 * [.67^4*.33^1]
=.3325
P(raises 4 out of 5 hands given a VPIP of 24%)
=5 * [.24^4*.76^1]
=.01260

Straight away you can see that he is more likely to have a VPIP of 67% (33% likely) over one of 24% (1.25% likely) but we want to find the adjusted probability that this player is in the top 10% of loose players given he raised 4 out the first 5 hands..

Bayes Theorem
Bayes Theorem can help. The equation again:


Here A = player is in the top 10% of loose players at this level
B = raises 4 out of the first 5 hands

P(B|A) = P(raises 4 out first 5 hands given he is in top 10% of loose players at this level) = .3325
P(A) = .10
P(B) = (probability he is in the top 90% of players and raises the first 4/5 hands)*(probability he is in the top 10% of players and raises the first 4/5 hands) = [.9*.01260] + [.1*.3325] = .04459

P(A|B) = [.3325 * .1]/.04459
=.7457 or 74.57%

Conclusion
So this means he is 74.57% likely to be in the top 10% of the loosest players at this level given he has raised 4/5 hands. So when we sat down we knew he was 10% likely to fall into the top 10% of players who had an average VPIP of 67%. Now that he raise 4/5 hands he is 64.57% MORE likely to be in this category and thus to maximise EV you should start exploiting this immediately rather than waiting for more information.