Bayes Theorem

p(B/A) = p(A/B)*p(B)/p(A/B)*p(B)+p(A/B*)*p(B*)

p(B*) = probability of B not occurung.

Here is a good wiki about Bayes Theorem as it applys to Guassians

http://www.scholarpedia.org/article/Bayesian_statistics

The calculations look fine, i did something very similar to this in a presentation where we were trying to teach maths in context - as for taking all these other complicated mathematical approaches it really is just making things harder than needed, and I can't see how finding out how much other people at the table are winning or losing will help you in any applicable way. I think decent players in live soft games can easily have bigger than 10bb/100 and if you haven't done it for long you may just think variance is on your side.

Regardless of your spread of data, if your up, the calculations will show you are a winning player. It's nice to know this stuff and kind of satisfying if you've laboured over a calculator for hours, but it's more of a macro poker thing and not really gonna effect how you play so in some ways futile, maybe good if your trying to forcast long term profit, but the interval is always undoubtedly large.

Well that leaves me with an interesting question.

I have played about 250-260 hours live, and I am up about $5500 at 3-5NL and 5-10 NL. I definitely have had some swings. 3 of my worst losing nights were at 5-10 though ($1200, $1300, and $1700) though I've now won the last 2 times I played 5-10NL. I generally only have time to play once a week. I have learned a lot since I started playing live in march. Prior to that I was a small loser at 4-8 limit and small stakes NL online. After quitting for about 6-7 months, I started playing again earlier this year. In march I started playing
3-5 NL and have been playing at least once a week since then, occasionally several times a week. Do I sound like someone who hasthe potential to be a winning player?

and how would I roughly calculate my BB per 100 and my standard deviation?
Thanks in advance.

Knowing your confidence intervals is very important for bankroll strategy.

Here is an example of where guassian noise has made an incorrect assumption as to where the most likely position of the thing is using classical statistics.

Green Dot = Best Guess (same as your sample winrate in poker)

White Dot = Real (same as your 'real' winrate in poker)


[IMG] Uploaded with ImageShack.us[/IMG]


Does this make sense. The green dot is your sample winrate. The white dot is the 'real' winrate which we cannot know. However we can use bayesian techniques to get closer to it.

BTW -4bb/100 was wrong.

I did the stupidest thing like EVER. I included my winnings. Which completely ****s up the priori distribution lol

I'm not sure what you mean, but a lot of published player stats do show that at low stakes 6-max -4bb/100 is pretty typical for the rake loss. That gets less at higher stakes obviously.

My winnings account for a huge amount of the winnings in my relatively small database.

If I take my sample away then the game becomes a big loser.

I need a sample where my own impact on the sample itself is minimal.

-4bb/100 hands might still be right I suppose (jumped the gun assuming it was wrong)

What does your database consist of?

Care to explain this. It seems you are dealing with some kind of sampling problem, but I'm not sure what is is.

This doesn't change the mean result, unless your personal style somehow generated much more or less rake than average. I would guess, that also SD is not that significantly skewed because of your own results.

Yep, the mean stays the same.

But I think my presence changes a priori distribution (because I pwn lol)

Ohhhhh... I'm sorry you just wanna know if you're a winning player. OK LMAO

That takes an hour at most. If you can't recognize the fish at at 1/2 game - you're probably not a long term winner.

Jesus christ, why are you even posting. This really is the epitome if a waste of your, and everyone else's time. I'm serious here - I'm not angry or anything, but do you even understand what the question being posed is, or the proposed method? Or are you just shooting off your mouth?

Ok, I sounded like an ass in my post which had no merit or proper contribution to this thread.

What I do want to say is that if all you want to know is whether or not you are a winner in a game just sitting and playing for a few hours will give you a pretty good indication. If you have a feeling that your c-bets are bluff-raised or floated with air then you may not have such an easy time as say in a game where fish call down 3 streets with K2o on a 6KJxx board.


As for the proposed method, its just reworking stats on your past few cash game sessions to obtain an interval of BB/100 values.

If you're pretty decent, you might have an idea from playing in a game whether you're +ev in it or not. However, pretty much everyone who's ever played poker is pretty biased about their own game and skills.

Yes, exactly. To some people, it's important, useful, or interesting to figure out how much poker is going to be worth to them BEFORE they've play a hundred thousand hands and can look back over history. It's really a pretty good idea for a player to be constantly analyzing his skill, the quality of available games, etc.

I used to play in a pretty good live game - terrible players. I added up the costs, estimated my win rate given my experience, and decided there were tougher games that were more +ev and didn't play there any more. This game was full of total fish, but also the rake was very high and the game was very slow in hands/hour, so even though I'm sitting at the table able to easily evaluate that I'm better than almost anyone else there, I was able to mathematically work out that it wasn't a good use of my time. See what I'm getting at?

I see your point. The thing is that OP's end result was "I am 95% confident I make from -3.08 to 36.84 BB/100" which is such a wide range that you can pretty much get a similar "feel" for how you will do in the game just by sitting in it for a few hours.

For me its pretty easy to figure out whether or not I'm outclassed or will have trouble winning in a game. For example sitting down at 2/5 on a Teusday afternoon - not +EV, only regs there. Weekends - +EV, mad fish.

Provided his math is correct he's very nearly ascertained that he's not -ev in the game. This is more than most people can say. Well - a lot of people say it, but it's more than most people actually Know.

I don't mean to be rude but your "feeling" for how you'll do in a game isn't really worth as much as some real numbers.

This is wrong, and nobody has really clearly explained why.

The first thing you need to understand is the difference between a parameter and a statistic. A parameter is the "thing you're trying to figure out, or infer" (in this case the parameter is your true winrate) and a statistic is a "sample from the thing you're trying to figure out" (in this case the statistic is your observed winrate over 1250 hands).

Now, a confidence interval, or any other frequentist method, makes no inference about the probability of a parameter given a statistic. It only tells you the probability of a statistic given a parameter. Those two sentences might sound the same, but they are completely different mathematically. For example, the sentence "99 percent of terrorists are Muslim" means something completely different than "99 percent of Muslims are terrorists".

You cannot and will not ever be able to use confidence intervals or any frequentist method to say anything about the probability of you having any particular true winrate based on any observed sample of hands you play, REGARDLESS of how big that sample is. You can only use them to calculate the probability of getting a particular sample IF in theory you had a certain true winrate.

For example, in this case you have an observed winrate of 16.88 BB/100 and you found confidence intervals of -3.08 to 36.84 BB/100. All that means is that if your true winrate were anywhere from -3.08 to 36.84 BB/100, and you took an infinite number of 1250 hand samples, 95 percent of those samples would include your observed 16.88 BB/100. However, it says nothing about the probability of your true winrate. Your true winrate could be -37.5 BB/100 or 42.33 BB/100, and neither is more or less likely.

The only way you can calculate the probability of you having a particular true winrate given the observed sample is to use Bayesian Inference.

As someone who has studied both Bayesian Inference and frequentist methods, and not to mention plays poker for a living, take it from me you are wasting your time with confidence intervals and standard deviations and whatever other math calculations you are trying to make. No one can ascertain much of anything from your 50 hour sample.

If I were betting against someone on what your observed winrate will be over your next 50 hours of play, I could probably get myself a nice +EV bet by building a simple little model that would probably not even take into account your last 50 hours of play. I'd do better by just asking you a bunch of questions.

^^^ I think you just got pwned. jk

But I really do have to defend my "feeling" argument. OP only wanted to know if he was winning player and he clearly can't figure out whether he's 10BB/100 winner or 0.1BB/100 winner so the "real numbers" aren't really that convincing.

Also, since OP was talking about live poker I think my argument is even more valid since it is much easier to get a feel for your competition when playing live.

Care to explain this to someone, who doesn't understand statistics very deeply?

I don't understand, why 42.33 bb/100 and -37.5 bb/100 would be equally likely true winrates. I'll illustrate my point by tweaking the numbers a little more obvious to my cause: you play 1,000,000 hands with 6 bb/100 observed winrate (SD 50 bb/100). Are you now stating, that true winrate of 7 bb/100 is equally likely to for example -2 bb/100? Even without any knowledge about player pool's winrate distribution, I think it would be safe to say, that true winrate is VERY likely in the range 3 - 9 bb/100.

it depends on what % of the opulatioon have those winrates. More people have -2bb so that is more likely

Even if we have some distribution of winrates in the player pool, this doesn't make any sense. It's probably true, that there are more -2 bb/100 players than there are 7 bb/100 players, but -2 bb/100 player playing million hands with observed winrate of 6 bb/100 is astronomically unlikely, yet for 7 bb/100 winner it's somewhat expected. Even if you combine these two distributions (unless you set player pools winrate distrubtion's parameters to unrealistic values) there is no way -2 bb/100 is more likely true winrate for said player.

poiu,

I am not stating anything. I am talking about what frequentist methods (particularly confidence intervals) state. And to answer your question, after 1M observed hands, it's not that frequentist methods state one true winrate to be equally likely as another (perhaps that was unclear in my post), it's that they make no claim about the probability of your true winrate. They don't believe parameters to be random variables (i.e. they feel your true winrate is a fixed number, whatever that number is, therefore there is no probability of it being x or y.)

My opinion, if it's not clear, is that that's absurd. Talking about a true winrate is a matter of philosophy and semantics. It's irrelevant, as stated. The crux of what we're really talking about is predictive value, or "what's our best estimate at this person's future winrate", and that question is incredibly valuable.

And 1M observed hands would be quite valuable in answering that question, but even then one cannot give any kind of actual numerical range prediction without looking at the population and using some form of Bayesian Inference. After all, while 1 million sounds like a large number, it's still just a number. How do I know how meaningful a number it is in this context? I need more information.

Let's say a minor league baseball player gets called up to the majors tomorrow and has 2 hits in 4 at-bats. Would most people take the over or under on him hitting .500 the rest of the season? Of course they'd take the under, and their reasoning would be something along the lines of "lots of players hit .500 in one game, but nobody hits .500 for an entire season." So they are essentially using a prior, and they intuitively feel the amount of variance that comes with 4 at-bats.

But what about 40 at-bats? Or 400? What's the natural variance that comes with those numbers? And what does the population do over that timeframe? If I tell you a certain player's batting average over 400 at-bats, will you be able to guess what he'll do in the next 400? This is where our intuitive "feel" breaks down and we need a system of reasoning that answers these questions more accurately. That's all Bayesian Inference really is.

I might agree with your general sentiment, but not for the reasons you give. In other words, I too may NOT predict a player with an observed winrate of 6 BB/100 over 1M hands to post a winrate of -2 BB/100 in his next 1M hands, but the reason isn't because of his observed 1M hands.

I might have reasons (separate from his observed winrate) to believe he belongs to a certain sub-population. For example, let's say he plays 40 hours a week, posts on Two Plus Two, has been playing poker for 3 years, and has read 10 poker books. Well, what does the population of players who fit THAT criteria look like? What if their mean winrate is not -2 BB/100 but actually 12 BB/100? Well then I might actually predict this player to have a HIGHER than 6 BB/100 winrate for his next 1M hands, despite the fact that the entire population mean winrate is -2 BB/100. Make sense?

Thank you for explaining this. I see the point. Just out of curiosity, if we had no idea about the winrate distribution in player pool (even that mean is negative because of rake), and observed said result (6 bb/100 for 1M hands with 50 bb/100 SD), can't we say anything useful about the "true winrate" with Bayesian approach? For this question we can assume, that playing environment is constant, so said player plays consistently against a homogenous field. True winrate in this context means the winrate players results converge to after enough hands (it obviously converges, because of constant environment and level of play).

If we think "true winrate" as a random variable, doesn't the sample give us some probability distribution, even without any assumptions about player pool's winrate distribution?

Makes sense, but doesn't help very much, because at least I have no idea, what kind winrate distribution players chosen with that criteria have. Most likely such players have positive winrate on average, but at least I cannot estimate much more precisely without further knowledge about player pool.

If you know nothing about said player except the observed result, couldn't you really say nothing about his expected performance in next 1M hands against same field?