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PL/NL Texas Hold'em >> Micro Stakes

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bozzer
Pooh-Bah


Reged: 05/29/06
Posts: 2140
Loc: in with the 2p2 lingo
Instareads: Bayes theorum and small sample sizes
      #10933141 - 06/25/07 03:01 PM

Warning: this post features some reasonably advanced/experimental ideas (which I am of course reluctant to disclose). if you're a beginning player, feel free to read by all means, but I'd recomend focusing on some more core concepts.

Warning 2: there is some maths in this post. I haven't hidden it, but it should be possible to get the point without going through the maths. i recommend you do though: it's fairly easy and probably quite useful.



Instareads: Bayes theorum and small sample sizes
by bozzer


A few weeks ago there was a hand history posted where KAT21 was wondering how much faith to put in his early information on the villain: that he'd played 5/6 hands, he'd raised 3 of them, and the two he'd shown down were pretty budget. KAT then proceeded to get involved in quite a marginal situation.

Several responders, including me, suggested that the OP should fold and get some better reads on villain before deciding to commit his stack in a dicey spot.

In a cash game though, we're looking for any edge we can get. Did we have enough information to conclude that the villain was likely to be a maniac and proceed accordingly? Or should we wait until we have 30 hands on a villain before adjusting?

I'm going to suggest that when we're faced with some unusual data we should actually be adjusting a lot more quickly than might currently be standard here at uNL.

To do this I'll introduce something called Bayes' Theorum, which is at the core of any mathematical hand analysis in poker.

Let's have a look at two situations: a high VPIP over a small sample, and getting 3bet a lot in a short space of time.


high vpip

Imagine you're sitting down to start a session in your normal $25NL game. During your first orbit you notice that one player has voluntarily put money in the pot in 5 out of the last 6 hands.

your hud hasn't fired up yet, but when you notice him call your raise with



you start to wonder how you're going to respond if you get a lot of action.

What do you think the chance is that if we sat with him for 50 hands he'd end up playing more than half his hands?

Make a mental note of your estimate, and we'll come back to that.

To get a better answer to the question than just guessing we need...


BAYES' THEORUM

here's an unrelated simple poker problem to get you thinking:

you're playing an opponent who only calls a preflop raise in the big blind with pocket pairs 22-JJ. When he hits a set he goes all in 70% of the time. To balance his play he bets all in with his unimproved pocket pairs 15% of the time. You have



and raise. He calls in the big blind and the flop comes Q 9 4 rainbow. Sure enough he goes all in for 10 big blinds more, offering us ~2:1 on the call.



After the flop comes down we can see he's got a set 6/54 = 11% of the time (if you can't see this, read this post). After he shoves, what percentage of the time does he have a set?

If you've never done this sort of problem before grab a calculator and see if you can work it out.

Make a note.

Got it?

Answer:
He has a set 36% of the time.


Large numbers of otherwise intelligent members of the population find this kind of calculation intuitively very difficult to grasp. I suspect poker players will be better, but if you didn't get it right it's important to realise you need all three pieces of information: the chance he has a set before he acts, the chance he takes this action with a set, and the chance he takes this action without a set.

If you like equations, here's one:



or

p(he has a set given that he shoved) = p(he shoves when he has a set) * p(he has a set) / [p(he shoves when he has a set) * p(he has a set) + p(he shoves without a set) * p(he is without a set) ]

And that's Bayes theorum.

I find it easiest to think about it this way:

how often does he get a set and shove it? 0.11 * 0.7 = 0.07.
how often does he miss a set but still shove? 0.89 * 0.15 = 0.13.
Then it's just 7:13 = 7/20 = 35%.

If you're not keen for the maths, the main message is that probabilities are conditional - they depend on other probabilities. The probability he has a set when he shoves is dramatically affected by how often he shoves the large proportion of times when he does not have a set. They also depend heavily on the prior probabilities - how often he has a certain hand to begin with.

When you're using Poker Stove to give a guy a preflop range and then narrowing it down, often you're saying 'he never plays aces like that, so we'll take them out of his range completely'. That's ok but it's very rough. Or you can say 'maybe he plays aces like that a third of the time so I'll leave 2 out of 6 of his AA combos in his range. That's pretty much what we're doing with Bayes' theorum, just a little more precisely.


back to the high vpip stuff

What does this have to do with working out whether to trust your early reads? Surely it's a straight probabilities question to work out how unlikely it is for a sane person to be playing 5 out of 6 hands? Don't we just go 'hmm I VPIP about 20% of my hands so 0.2*0.2*0.2*0.2*0.2*0.8 = some tiny number so I call!'

That is part of the problem, but we also need to know the prior probabilities - if there weren't any maniacs playing at 25nl, then we would just assume he's a tight player who is getting cards. Happily there are plenty of maniacs at $25nl, but how many exactly?

I got these prior probabilities by definining the question a little more precisely: 'how often does this player turn out to be in the top 10% of loose players at the level?'. I looked at my Poker Tracker database for 25nl and filtered it to players I had over 50 hands on, to give a total of 385 players. 38 players had a VPIP over 52%, for an average VPIP of 62%. The other 90% of players had a VPIP of 52% or less, for an average of 26%.

We already know intuitively that very loose players will have much larger chance of playing 5/6 hands, but we also know they are uncommon. We need use the new information we have on the way he played his first 6 hands to ‘skew’ the prior probability of there being only a 10% chance that he is a very loose player.

Here's the Bayes maths:

p(plays 5/6 with a 62% VPIP) = (0.62)^5 * (1 - 0.62)^1 = 3.4%
p(plays 5/6 with a 26% VPIP) = (0.26)^5 * (1 - 0.26)^1 = 0.088%

3.4 * 10% = 0.34

0.088 * 90% = 0.079

p(in top 10% of loose players given that he played 5/6 hands) = 34 / (34 + 7.9) = 82%

So it seems that if next time you see someone play 5 out of their first 6 hands at your favourite $25nl table, there’s a very good chance they're a nutter. You'd be a fool if you didn't start adjusting before your HUD has even loaded up.


am I getting 3bet light?

In this second case, we're often dealing with very small sample sizes at uNL, and current HUDs don't display anything at all. But if you're paying attention you can quickly get an idea of whether you're getting reraised light without seeing any showdown hands.

Here's the scenario. It's $50nl. You've openraised in the cutoff the last 5 times in a row. 2 out of those 5 times the button has 3bet you. Predictably your holdings have been crap and you've had to fold each time. This hand, you raise first in with



in the CO. The button again raises. How likely is he to be 3betting you light?

Let's assume a standard 3betting range is {JJ+,AQ+}. That's 4.2% of hands. Lets give a wide 3betting range as { 66+, ATs+, KJs+, QTs+, suited conectors 65s+, suited one gappers 75s+, AQo+, KQo }. That's 12.5% of hands.

Unfortunately I don't have any data on how common 3betting light is, so I'll just have to take a guess. We'll say that 10% of unknown players at $50nl have a wide 3betting range in this situation (CO vs BTN). That might be a bit high or even a bit low depending on where you play, but it doesn't really matter.

Time for the maths:

p(tight 3b range RR 3/6 times) = 0.042^3 * (1-0.042)^3 = 0.000057

p(wide 3b range RR 3/6 times) = 0.125^3 * (1-0.125)^3 = 0.00087

these are very small numbers but it doesn't matter because this unlikely event (getting RR 3 times out of 6) has already happened. we need to know how likely that it's been happening without the goods:

5.7 * 90% = 5.13

87 * 10% = 8.7

8.7 / 8.7+5.13 = 62%

So he's more likely than not to be three betting you light according to my rough assumptions. This isn't massive, but considering how rare I have assumed light 3bettors are this is a significant result. How you choose to play AQ is another question, but if you don't use this information you are giving up an edge.

Of course villain isn't unknown, because we've sat with him for nearly 6 orbits. If he happens to have decent PFR numbers during the 30 or so other hands we've played with him, and we guess that a third of people raising 17%+ of hands are gonna have the wide three betting range, we're suddenly looking at an 88% chance of this 3bet being light given his past actions.

And if he's read this post, and is using Bayesian reasoning to work out that you are raising maybe 25%+ of your hands from the cutoff there's an even greater chance he's using a light 3betting range!

Maybe this is an obvious example, or maybe you disagree completely with my assumptions. To be honest it doesn't really matter. I could have used several different situations or tweaked different variables, but still arrived at the same general point: lots of aggression in a small sample size can be surprisingly significant data, and we can't afford to ignore it.

I hope this post has convinced you of that point. If it hasn't, I suggest you open up a spreadsheet program and start playing around with some numbers you think are more pertinent or realistic. Hopefully if I have convinced you that this is important you've already opened a spreadsheet to start getting a feel for some different situations.

I also hope that I've introduced Bayes theorum in the context of hand reading reasonably simply for those who don't know it, and used it in an interesting new context for those who already did. If you're still unclear on Bayes theorum see http://yudkowsky.net/bayes/bayes.html for an introduction.

I got the idea for this post from Mathematics of Poker by Chen and Ankemann; there might be more about this topic in there.


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moneybreaker
newbie


Reged: 05/15/07
Posts: 45
Re: Instareads: Bayes theorum and small sample sizes [Re: bozzer]
      #10933274 - 06/25/07 03:11 PM

wow. skimmed trough this post, looks interesting. i'll read it later when i'm done playing.
thanks for the effort put into your post.


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C4LL4W4Y
veteran


Reged: 02/15/07
Posts: 1415
Loc: NJ
Re: Instareads: Bayes theorum and small sample sizes [Re: bozzer]
      #10933488 - 06/25/07 03:26 PM

Great post, I'm interested in working out some of these math problems myself.

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AJGibson
old hand


Reged: 06/27/06
Posts: 748
Loc: Not thinking
Re: Instareads: Bayes theorum and small sample sizes [Re: moneybreaker]
      #10933580 - 06/25/07 03:33 PM

Though this does change your final answer, there is a slight flaw in your working. Your equation should your equation read :-

p(plays 5/6 with a 62% VPIP) = 6*[ (0.62)^5 * (1 - 0.62)^1 ]= 6 * 3.4% = 20.4%
p(plays 5/6 with a 26% VPIP) = 6*[ [(0.26)^5 * (1 - 0.26)^1 ] =6 *0.088% = 0.528 %

As you calculated the probability only folded the last of six hands, not one out of six.

You made a similar error on the 3-bet section, that's why you probabilities are so tiny. You should multiply by 20 to get the correct probablity. Again this doesn't effect your result as these factors cancel.

Good work.


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Fiksdal
banned


Reged: 08/22/06
Posts: 4444
Re: Instareads: Bayes theorum and small sample sizes [Re: AJGibson]
      #10943710 - 06/26/07 08:19 AM

Bozzer

Clearly what you are presenting here are key concepts when it comes to information in poker. This post is a lot greater than most of the people who read uNL will realize. My guess is most of them just sees the math and starts skimming further down.

You are right though. Many times have I posted hands with like a 20 hand sample size on villain, and people have responded saying that information is totally useless. Obviously they don't realize that while the information is of less quality than, say, a 200 hand sample, it is of course not completely worthless in any way. As you explain very well in your post, it is very obvious that is says something about how likely villain is to be looose or tight, etc. Meaning a villain with a 100% VPIP over 10 hands, could be a total nit who was dealt aces 10 hands in a row, it is theoretically possible. (chance of a dude being dealt aces 10 times in a row is (52*52)^10, and when this number is > 0, it is possible to happen. However it is quite obvious that it is way more likely that villain is just a total monkey.

Understandably, few people are gonna be able to actually apply this to their game. However, if those people read this post a couple of times and try to understand it, it will make them better players, simply because a mind that has thought through these things will automatically have better intuition at the tables.

Great post Bozzer.


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relativity_x
old hand


Reged: 09/19/06
Posts: 947
Loc: 3 bet min-raising
Re: Instareads: Bayes theorum and small sample sizes [Re: Fiksdal]
      #10943896 - 06/26/07 08:50 AM

I've sat down at several tables and ran 60/60 or 40/20 over the first 6/10 hands. After 100, I was back down to 25/12

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ama0330Moderator
more whining, less poker


Reged: 04/20/06
Posts: 5704
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Re: Instareads: Bayes theorum and small sample sizes [Re: Fiksdal]
      #10943915 - 06/26/07 08:54 AM

This is a really good post.

I've never been much for math and I'm still not, and to be honest, though I get the point of this post and realise its utility, I wont be using the concepts involved in it. Basically my standard response when someone shows me huge math equations then comes to whatever conclusion is "you're overthinking".

Im a huge believer in Occams Razor. For example, in your first example, you go through the theorem and through a quite sound mathematical analysis arrive at the conclusion that:

Quote:

So it seems that if next time you see someone play 5 out of their first 6 hands at your favourite $25nl table, there’s a very good chance they're a nutter. You'd be a fool if you didn't start adjusting before your HUD has even loaded up.




I would generally have arrived at the same conclusion without really considering the probability that this were true, based upon what I know about the dynamics of 25nl and how people play. It just seems obvious that someone behaving that way would be raising a very wide range.

I'm happy to admit that In a way I am just stubborn and don't really want to learn the math behind concepts such as this, but I really believe that there is so much human psychology involved in poker that it is extremely difficult to place a mathematical formula over it to arrive at precise conclusions. I've 3bet 53o from the CO before just because I felt like it - is there an equation which will encompass such an anomaly?

The other thing that I think is often neglected is adustment... for example in your second hand, you say that you have been 3bet twice from 5 opening raises. If I was to be 3bet just once from the BB I would immediately consider my CO opening range and whether or not to tighten up. If I was to be 3bet twice by the same player in the space of 5 orbits I would definitely consider that player to be capable 3betting somewhat light, no question. Accordingly I would most likely tighten up my range and play back more aggressively with more marginal holdings. Note how I don't even consider math when I make that decision. Maybe thats a flaw of mine, but I think that to 3bet someone 3 times in 5 orbits after an open each time says more to me than a hot run of cards, especially OOP.

I think that if you were to play the same player for many thousand hands HU, as happens at the nosebleed stakes, this sort of thinking becomes invaluable. But I feel like so often in uNL, we have to make our decisions in a vacuum. How many players at your site would you truly confidently be able to say play in such a way as you could assign a probabiiity to him or her having a particular holding? To me, defining a player as:

Quote:

an opponent who only calls a preflop raise in the big blind with pocket pairs 22-JJ. When he hits a set he goes all in 70% of the time. To balance his play he bets all in with his unimproved pocket pairs 15% of the time.




is, imo, more mathematical convenience than fact. I feel like you could never be able to make such an assessment of any player. I know that even an approximation is helpful and I'm not saying it's useless, but I'm saying that its imprecise, and its utility, to me, does not extend that far past the laboratory.

To those who havent read any of my post and are just reading this para, I'm not pissing on Bozzers parade, I think his post is excellent. I'm merely stating that I believe that there is too much imprecision, mostly involved with human psychology, in poker to make math such as this crucial for ones development as a player.

Or maybe I'm just ignorant.


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ama0330Moderator
more whining, less poker


Reged: 04/20/06
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Re: Instareads: Bayes theorum and small sample sizes [Re: ama0330]
      #10943927 - 06/26/07 08:55 AM

also tl;dr



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Gelford
Regular Joe


Reged: 11/15/05
Posts: 6392
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Re: Instareads: Bayes theorum and small sample sizes [Re: ama0330]
      #10943953 - 06/26/07 09:01 AM

Quote:

I've 3bet 53o from the CO before just because I felt like it - is there an equation which will encompass such an anomaly?




Actually there is believe me or not


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maccamack
addict


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Re: Instareads: Bayes theorum and small sample sizes [Re: Fiksdal]
      #10944003 - 06/26/07 09:08 AM

Thanks - great post. Definite food for thought.

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