BB/100, standard deviation, and recognizing the winning player

[StellarWind]

[ QUOTE ]
9 months is MORE than long enough to establish you as a winning player. The level of ignorance on this board on the issue of the relevance of runs of results is staggering. If you're a winning player over 20,000 hands, you're going to be a winning player if you keep doing what you're doing. Stop propagating this ridiculous notion that anything under a 100,000 hand sample size is irrelevant and people who have been playing winning poker for the better part of a year don't know whether they're winners. It is simply incorrect and foolish.

[/ QUOTE ]
Matador posted this recently in this forum and was not surprisingly incinerated for his trouble.

I don't agree with the sweeping generalizations implicit in his statement. It doesn't apply to all players and all situations. Nevertheless his basic point is correct.

Before I elaborate on that, I offer a problem for you to think about. I'm a pretty honest guy away from the table so I will tell you straight up that this is a trap. You've been warned.

A poker player joins Party and begins playing 2/4. After 10000 hands he has a measured win rate of 2 BB/100 and his standard deviation is 10 BB/100. What is the probability that he is a winning player in this game? By winning I mean that his intrinsic win rate is greater than zero.

You should make the usual mathematical assumption that his advantage (or lack thereof) over the game is constant. The game is equally good every day and he plays about the same every day.

That's it. I'll wait for some responses before I post again.

[Grisgra]

[ QUOTE ]
A poker player joins Party and begins playing 2/4. After 10000 hands he has a measured win rate of 2 BB/100 and his standard deviation is 10 BB/100. What is the probability that he is a winning player in this game?

[/ QUOTE ]

Without making further tricky assumptions, the probability is ~97.5% that he is a winning player. You could probably jack that up a bit -- he has a SD that is pretty low, implying that he doesn't play loose/wacky, implying that he has a style that is shared by most winners (i.e., I think that 99% of losing players have an SD>10), but putting all such trickiness aside . . .

About 97.5%. I agree with the Evil Matador/Cinnamonwind/Sophia/Lansing/Busterstacks here. Tired of people apologizing for "only" having 30,000 hands or whatnot. You need a shitload of hands to gain a lot of certainty about your BB/100 -- but you don't need more than 20k or 30k (or in this low-SD case, 10k) to simply know whether you're a winning player or not when you're pumping out 2BB/100 or more.

[set57hike]

To assess the question, we will conduct a hypothesis test.

Null hypothesis: The player is a break even gambler. (population mean = 0)
Alternative hypothesis: The player is a winning player.
(population mean > 0)

As he has 10000 hands, this means he has 100 samples of size 100. It is reasonable to assume that each sample is approximately normally distributed. The standard deviation for each sample is given to be 10BB per 100 hands. We will treat the standard deviation given as the population standard deviation to avoid using a student t distribution.

We form the test statistic by computing
(sample mean-pop mean under the null hypothesis)/(pop stan deviation/sqrt(sample size))
=(2-0)/(10/sqrt(100))
=2/1=2

With this value for the test statistic, we get a p-value of
P(Z>2)=0.02275

So, at any significance level of 2.275% or above, we can conclude that this player is a winner (although we cannot say what level of a winner he is.) This is fairly strong evidence, but depending on your own personal requirements, you may hold out for stronger evidence. To explain the p-value: It means that, if we repeated this experiment many, many times, and if the player really is simply a break even player (like we said under the null hypothesis), then 2.2% of the time he would get similar or even more successful results than he had in this sample. You wouldn't want to send someone to the electric chair on this evidence, but it is fairly convincing.

However, many people use 1BB/100 hands as the measure of a good player. Repeating similar steps above we come up with a p value of P(Z>1)=.158655.
This is significantly weaker evidence, as we would only believe this player is "good" with a significance level of 15.6% or higher. Most people would not believe you based on this statistical evidence and this definition of a good player, as the highest value for significance level we are willing to live with is usuallly 10%. (Remember, a higher value for significance level means we are making more type 1 errors).

So, it depends what you define as a winning player and how stringent your standardard of evidence are. Be careful... it doesn't make sense to say what the probability is that he is a winning player. He IS or he ISN'T a winning player... that part isn't random under our assumptions. It is a subtle difference, but you can only quantify how likely you are to make a mistake in deciding whether he is a winning player. That decision is random as it is based on a random sample of his play.

[lehighguy]

The frustration is that around the time you have 20,000 hands or so you want to know if this can be your job or not. After 20,000 hands the allure of online poker has worn of and your mainly playing for the money. So people want to know, can I win more if I keep playing. Is it worth my time. Should I quict my job/buy that car/make financial decision X because I can rely on my poker money.

Consider myself. I'm a college senoir. I'm either gonna get a real world job or play poker in a few months. I don't have 100,000 hands of data, but I need to have a reliable BB/HR in order to make financial decisions. This is what leads people to try and fudge the numbers a bit and hope for the best.

My friend won 1,000s on his first 10,000+ hands, now he works 30 hours a week to pay of credit card debts. So I don't believe that you have a large enoguh sample size at all. In fact, the odds are clear from the information you gave.

EV: 2bb/100
SD: 10BB/100
Probability of 0bb/100 or greater is something like 55% according to statistics.

[Grisgra]

[ QUOTE ]
In fact, the odds are clear from the information you gave.

EV: 2bb/100
SD: 10BB/100
Probability of 0bb/100 or greater is something like 55% according to statistics.

[/ QUOTE ]

Completely wrong! The question is very basic -- set57hike broke the answer down very nicely in his post.

[steamboatin]

Mason Malmuth covers this very well in Gambling Theory and Other Topics.

The chapter titled "What about the losers?" pg 56 dicusses how far ahead a loser can be in the short run.

The example used is a 30-60 lowball game and it has a standard deviation of $650 per hour. A player that is actually losing $30 per hour could statistically be ahead $5,866 after ten hours of play. $16,500 after 100 hours of play and after 1,000 hours of play it is theorectically possible for a $30 an hour loser to be ahead $31,664.

After 10,000 hours of play the best this player can expect is to be down, $105,000.

Now this in no way proves that your boy is a loser and it in no way proves that he is a winner. What it proves is, we don't if he is a winner or a loser.

I realize that you are a borderline if not full fledged troll and I probably shouldn't waste my time with this post but i am not doing it for you.

I believe we have a responsibility to the lurkers. The problem with a post like yours is a lurker may use your incorrect information to make a very important decision. Many people read and study these posts and some accept the information at face value.

<font color="red">ATTN: LURKERS!!!!! </font>

DO NOT QUIT YOUR JOB OR MAKE IMPORTANT DECISIONS BASED UPON POSTS.

If you are even remotely considering relying upon poker for any portion of your income, you must read and study, Gambling Theory and Other Topics by Mason Malmuth. It is available right from this site, the shipping is quick and you get a discount so the shipping is for all practical purposes, free.

[Piz0wn0reD!!!!!!]

[ QUOTE ]
I agree with the Evil Matador/Cinnamonwind/Sophia/Lansing/Busterstacks here.

[/ QUOTE ]

what does busterstacks have to do with this?

[Michael Davis]

StellarWind used to give some of the best advice ever read on the small stakes forum. I don't know what defines a troll, but certainly he doesn't fit.

-Michael

[Piz0wn0reD!!!!!!]

Im not sure why this is still open to debate. There are mathmatical formulas you can use to detremine your RoR, WR confidence, varience etc. Math doesnt lie.

[fnord_too]

[ QUOTE ]
Math doesnt lie.

[/ QUOTE ]

But it is often done incorrectly or misinterpretted....

[Piz0wn0reD!!!!!!]

[ QUOTE ]
[ QUOTE ]
Math doesnt lie.

[/ QUOTE ]

But it is often done incorrectly or misinterpretted....

[/ QUOTE ]

While this is true, people who think 10k hands is absolute 100% hard evidence of winning play prolly arn't good at math (or even know of these equations).

[Grisgra]

We all know that Busterstacks is yet another pseudonym for Cinnamonwind/Sophia/etc/etc/etc. Duh.

[Grisgra]

And yet with an SD of 10, 100% is a lot closer to the truth than 55%. There *are* a lot of people here who seem to think that you can't tell whether you're a winning player after 30-40k hands at 2BB/100.

[Piz0wn0reD!!!!!!]

[ QUOTE ]
We all know that Busterstacks is yet another pseudonym for Cinnamonwind/Sophia/etc/etc/etc. Duh.

[/ QUOTE ]
HAHAHAHAHAHAHAHA youre kidding right?
i personally know buster in real life, and i assure you he is not even similar to that troll.

Well, he does flame a lot. But thats it.

[Grisgra]

Hmmm . . . maybe YOU are really Busterstacks . . . which means . . .

!!!

/images/graemlins/grin.gif

[Piz0wn0reD!!!!!!]

[ QUOTE ]
And yet with an SD of 10, 100% is a lot closer to the truth than 55%. There *are* a lot of people here who seem to think that you can't tell whether you're a winning player after 30-40k hands at 2BB/100.

[/ QUOTE ]

where did you get 55% from? 97.5% is NOT 100%. Not even CLOSE.
I think you are misunderstanding me. If i win 2BB/100 for 10k hands, im am highly LIKELY to be a winning player. 2.5% is a LARGE margin of error IMHO. Also, after such a small sample, i really have no idea what my WR is. I can calculate its accuracy, but i assure you it will be a WIDE range (including -BB/100).

It is very possible (and happens all the time) for a losing player to go on a 10k hands winning streak. esp in limit and esp in SH play.

[Piz0wn0reD!!!!!!]

[ QUOTE ]
Hmmm . . . maybe YOU are really Busterstacks . . . which means . . .

!!!

/images/graemlins/grin.gif

[/ QUOTE ]

lol!

we totaly have 2 seperate conversations going on here between the two of us. I guess now this is a thrid /images/graemlins/smirk.gif

[Grisgra]

[ QUOTE ]
[ QUOTE ]
And yet with an SD of 10, 100% is a lot closer to the truth than 55%. There *are* a lot of people here who seem to think that you can't tell whether you're a winning player after 30-40k hands at 2BB/100.

[/ QUOTE ]

where did you get 55% from? 97.5% is NOT 100%. Not even CLOSE.
I think you are misunderstanding me. If i win 2BB/100 for 10k hands, im am highly LIKELY to be a winning player. 2.5% is a LARGE margin of error IMHO. Also, after such a small sample, i really have no idea what my WR is. I can calculate its accuracy, but i assure you it will be a WIDE range (including -BB/100).

It is very possible (and happens all the time) for a losing player to go on a 10k hands winning streak. esp in limit and esp in SH play.

[/ QUOTE ]

A few points:

1) I'm talking about answering the question "Am I A Winning Player?", not "What is my BB/100 within 1.0BB/100?"

2) I got the 55% from lehighguy in this thread -- I've seen others have that exact same sense, that after 10-20k hands you only have a vague sense of whether you're a winning player.

Obviously, 97.5% is very close to 100% . . . c'mon! We make decisions based on less information every day. If I play at a given limit, and decide that I'll only move up once I'm at least 90% sure that I'm at least a 1BB/100 winner at that limit -- well, that kind of information (depending on your sample win rate) can easily be gained within 10-20k hands (e.g., 20k hands at 3BB/100 with a SD of 16).

Now, it's true that people go through 10k stretches of break-even play all the time. But their SDs usually aren't 10. And it's true that people go through 20k stretches of break-even play . . . some of the time. But these same people are folks that have 100k to 200k hands in, probably -- it'd be rare for their first 20k hands to be break-even while actually being decent winning players. And obviously, if a player is looking at 20k hands separately from his other 180k without a very good reason, there's some big-time selection bias going on.

[Piz0wn0reD!!!!!!]

[ QUOTE ]

Obviously, 97.5% is very close to 100% . . . c'mon! We make decisions based on less information every day.

[/ QUOTE ]

In poker and math the difference between 97.5% and 100% is CRUCIAL.

[Piz0wn0reD!!!!!!]

[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
And yet with an SD of 10, 100% is a lot closer to the truth than 55%. There *are* a lot of people here who seem to think that you can't tell whether you're a winning player after 30-40k hands at 2BB/100.

[/ QUOTE ]

where did you get 55% from? 97.5% is NOT 100%. Not even CLOSE.
I think you are misunderstanding me. If i win 2BB/100 for 10k hands, im am highly LIKELY to be a winning player. 2.5% is a LARGE margin of error IMHO. Also, after such a small sample, i really have no idea what my WR is. I can calculate its accuracy, but i assure you it will be a WIDE range (including -BB/100).

It is very possible (and happens all the time) for a losing player to go on a 10k hands winning streak. esp in limit and esp in SH play.

[/ QUOTE ]

A few points:

1) I'm talking about answering the question "Am I A Winning Player?", not "What is my BB/100 within 1.0BB/100?"

2) I got the 55% from lehighguy in this thread -- I've seen others have that exact same sense, that after 10-20k hands you only have a vague sense of whether you're a winning player.

Obviously, 97.5% is very close to 100% . . . c'mon! We make decisions based on less information every day. If I play at a given limit, and decide that I'll only move up once I'm at least 90% sure that I'm at least a 1BB/100 winner at that limit -- well, that kind of information (depending on your sample win rate) can easily be gained within 10-20k hands (e.g., 20k hands at 3BB/100 with a SD of 16).

Now, it's true that people go through 10k stretches of break-even play all the time. But their SDs usually aren't 10. And it's true that people go through 20k stretches of break-even play . . . some of the time. But these same people are folks that have 100k to 200k hands in, probably -- it'd be rare for their first 20k hands to be break-even while actually being decent winning players. And obviously, if a player is looking at 20k hands separately from his other 180k without a very good reason, there's some big-time selection bias going on.

[/ QUOTE ]

im not sure what you point is. Please clarify.

[Grisgra]

[ QUOTE ]
In poker and math the difference between 97.5% and 100% is CRUCIAL.

[/ QUOTE ]

It's only crucial about 5 times in 200, and that's not very often /images/graemlins/smile.gif.

[Grisgra]

[ QUOTE ]

im not sure what you point is. Please clarify.

[/ QUOTE ]

??

My point is that often, you can have a very large degree of confidence in whether you are a winning player after a mere 20k to 30k hands. In the original example, 10k hands is sufficient to gain confidence. I'm not one who needs to hit p-value &lt;0.001 before I make judgements about whether my play is sufficiently good or not.

[Piz0wn0reD!!!!!!]

[ QUOTE ]
[ QUOTE ]
In poker and math the difference between 97.5% and 100% is CRUCIAL.

[/ QUOTE ]

It's only crucial about 5 times in 200, and that's not very often /images/graemlins/smile.gif.

[/ QUOTE ]

its enough not to bet the farm.

[Piz0wn0reD!!!!!!]

[ QUOTE ]
[ QUOTE ]

im not sure what you point is. Please clarify.

[/ QUOTE ]

??

My point is that often, you can have a very large degree of confidence in whether you are a winning player after a mere 20k to 30k hands. In the original example, 10k hands is sufficient to gain confidence. I'm not one who needs to hit p-value &lt;0.001 before I make judgements about whether my play is sufficiently good or not.

[/ QUOTE ]

I think we are basicaly arguing the same point.

[Grisgra]

[ QUOTE ]
[ QUOTE ]
[ QUOTE ]

im not sure what you point is. Please clarify.

[/ QUOTE ]

??

My point is that often, you can have a very large degree of confidence in whether you are a winning player after a mere 20k to 30k hands. In the original example, 10k hands is sufficient to gain confidence. I'm not one who needs to hit p-value &lt;0.001 before I make judgements about whether my play is sufficiently good or not.

[/ QUOTE ]

I think we are basicaly arguing the same point.

[/ QUOTE ]

Well, I don't know that we were, but if so, that's fine. The original poster's point was that there seem to be a lot of people on here who think that you can't tell whether you're a winner after 20-30k hands, simply because there are long-term winners out there who have 20-30k stretches where they break even or win hardly anything at all. They don't understand the statistics behind the whole deal, and it's causing them grief.

Again, to everyone out there: Just because pros will have 10k or 20k hand stretches of breakeven play does NOT mean that after your FIRST 20k hands, you can't get a very very good idea of whether you're a winning player or not.

Admittedly, not something to bet the farm on, or to make claims about specific BB/100 on /images/graemlins/smile.gif.

[Piz0wn0reD!!!!!!]

[ QUOTE ]
...does NOT mean that after your FIRST 20k hands, you can't get a very very good idea of whether you're a winning player or not.

[/ QUOTE ]

This is exactly my point. But i stress that it is A VERY GOOD IDEA OF and not solid 100% accurate proof.

Baisicaly dont quit your day job till you get some more hands in.

[Grisgra]

I agree re the whole job thing -- I'd want 90% certainty that I was making at least my desired X/hr and 99% certainty I was making my minimal X/hr before I did any quit-my-day-job nonsense. I think that if you've got good alternate options that quitting your day job is generally a bad idea in any case, unless you're in the $200/hr+ range, but that's for another thread . . .

[Piz0wn0reD!!!!!!]

[ QUOTE ]
I agree re the whole job thing -- I'd want 90% certainty that I was making at least my desired X/hr and 99% certainty I was making my minimal X/hr before I did any quit-my-day-job nonsense. I think that if you've got good alternate options that quitting your day job is generally a bad idea in any case, unless you're in the $200/hr+ range, but that's for another thread . . .

[/ QUOTE ]

not sure what you do, but the highest paying job ive ever had was at dominos.

I didnt follow my own advice to some extent.

I went pro with a small BR and no cusion. In fact, my job was wagered over 1 game of backgamon. I win, i go pro. I lose, i work for another month. I won the game by 1 pip!!! if i ever write a book, this will be in it for sure.

[ZZZ]

That 97.5% is all well and good. It means that a break even player has only a 2.5% of winning 2BB/100 or better over his first 10K hands.

However this does not mean that a random player who makes 2BB/100 over his first 10K hands has a 97.5% chance of being a winning player. That chance depends on both the win rate and the distibution of winners vs. losers coming into Party Poker.

Let's say you knew that 99.9% of players coming into the system were losing players. Then the majority of players of who make 2BB/100 over their first 10K hands would be losers, and you would justified in thinking that a random player with those numbers is less than 50% likely to be a winning player. Obviously, that 99.9% number is too high, but it shows you must consider the fact that more losers than winners come into the system when trying to find the probability that a random player is a winning player.

It's a bit like taking a random coin from your pocket and flipping heads 10 times in a row, and trying to come up with the probability that the coin is biased. You wouldn't say that an unbiased coin would only have a 0.1% chance of coming up heads 10 times in a row, so there is a 99.9% chance the coin is biased.

ZZZ

[SomethingClever]

[ QUOTE ]
[ QUOTE ]
We all know that Busterstacks is yet another pseudonym for Cinnamonwind/Sophia/etc/etc/etc. Duh.

[/ QUOTE ]
HAHAHAHAHAHAHAHA youre kidding right?
i personally know buster in real life, and i assure you he is not even similar to that troll.

Well, he does flame a lot. But thats it.

[/ QUOTE ]

/images/graemlins/grin.gif /images/graemlins/grin.gif /images/graemlins/grin.gif /images/graemlins/grin.gif

This gave me a good laugh too.

[sammy_g]

About quitting your day job after 10K hands:

The problem is not so much that you're only 97.5% sure you're a winning player, although it's still something to consider when making a decision like this. It's that you aren't at all sure what your true earn is. There's a significant chance that your win rate really falls in the 0-1BB/100 range, which isn't exactly a comfortable living unless you play pretty high.

[Grisgra]

Well, this is the kind of thing I was referring to re all the "trickiness" you can look at. e.g., "prior" information.

The fact that your SD is only 10 is something in the "more likely to be a winner" column.
The fact that the guy is a 2+2'er is something in the "more likely to be a winner" column.

If 99.9% of folks in general are losers, well, pretty odd if the very first person you checked turned out to be a significant winner . . . but you could work that into your calculations, sure.

[Grisgra]

[ QUOTE ]
not sure what you do, but the highest paying job ive ever had was at dominos.

[/ QUOTE ]

Fair enough, I make a little more than that /images/graemlins/smile.gif. But if you're young and can go to college to make a guaranteed 60-100k/yr (or more) + benefits, but instead choose to grind out 80-100k/yr without benefits and without any guarantee that online poker is going to stay as wonderful as it is right now -- well, I think it's wider to go the college route and just make some poker money on the side. Obviously depends on the person, but you've got to think long-term here.

[Piz0wn0reD!!!!!!]

[ QUOTE ]
[ QUOTE ]
not sure what you do, but the highest paying job ive ever had was at dominos.

[/ QUOTE ]

Fair enough, I make a little more than that /images/graemlins/smile.gif. But if you're young and can go to college to make a guaranteed 60-100k/yr (or more) + benefits, but instead choose to grind out 80-100k/yr without benefits and without any guarantee that online poker is going to stay as wonderful as it is right now -- well, I think it's wider to go the college route and just make some poker money on the side. Obviously depends on the person, but you've got to think long-term here.

[/ QUOTE ]

i went to college.

[Piz0wn0reD!!!!!!]

[ QUOTE ]
About quitting your day job after 10K hands:

The problem is not so much that you're only 97.5% sure you're a winning player, although it's still something to consider when making a decision like this. It's that you aren't at all sure what your true earn is. There's a significant chance that your win rate really falls in the 0-1BB/100 range, which isn't exactly a comfortable living unless you play pretty high.

[/ QUOTE ]

theres a lot more than just that wrong with quiting your job after 10k hands...

[Grisgra]

Silly liberal-arts major /images/graemlins/smile.gif.

[Piz0wn0reD!!!!!!]

[ QUOTE ]
Silly liberal-arts major /images/graemlins/smile.gif.

[/ QUOTE ]

comprehensive sound arts.

[chabibi]

the way the cards are dealt have a huge degree of variance. i was wondering if winrate calculation could take in to account this variance. what i mean is if you were to look at your miscellaneous stats to see what % of your hands were pairs, two pairs, 3 of a kinds etc.. could you not get a better idea of winrate. for example after 10 000 hands someone with a winrate of 5BB/100h and a sd of 16 BB/100h might set up a confidence interval and find out that he is 95% certain that his winrate is between 1.9BB/100 and 8.1BB/100 (forgive me if my math is wrong) but what if this player saw that after 7 cards dealt he held a full house 5% of the time and a straight 5% of the time and just a high card 13% of the time obviously this player received more than his fare share of premium hands.. would his winrate not be skewed in the upwards direction despite a 95% CI that he is a solid winning player

[steamboatin]

that is one of the reasons that 10,000 hand isn't enough to tell if someone is a winner. At only 60 hands per hour that 167 hours of play. At 100 hands per hour that's 100 hours of play. A losing player could theorectically still be ahead after 1,000 hours of play and that is straight out of Gambling Theory and Other Topics by Mason Malmuth. All the loser has to do is run good and get an unusual amount of good cards exactly as you state.

[passion]

Not to be nitpicky, but I don't think you can pinpoint the probability that the player in question is a winner. What you you can assertain is the probability that in a sample of 10,000 hands a break-even player with a sample standard deviation of 10BB/100 would show a win rate of 2BB/100 or higher. That probability is about 0.025.

Passion

[ThePimpulator]

[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
In poker and math the difference between 97.5% and 100% is CRUCIAL.

[/ QUOTE ]

It's only crucial about 5 times in 200, and that's not very often /images/graemlins/smile.gif.

[/ QUOTE ]

its enough not to bet the farm.

[/ QUOTE ]

Speak for yourself...

[Piz0wn0reD!!!!!!]

[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
In poker and math the difference between 97.5% and 100% is CRUCIAL.

[/ QUOTE ]

It's only crucial about 5 times in 200, and that's not very often /images/graemlins/smile.gif.

[/ QUOTE ]

its enough not to bet the farm.

[/ QUOTE ]

Speak for yourself...

[/ QUOTE ]

i recomend you read about bankroll requirements and related topics for your sake.

[StellarWind]

Congrats to ZZZ and passion for seeing the point. Many of the rest of you fell into the trap.

[ QUOTE ]
What is the probability that he is a winning player in this game? By winning I mean that his intrinsic win rate is greater than zero.

[/ QUOTE ]
The correct response is that I did not provide sufficient information to answer the question.

97.5% is the probability that a breakeven player with SD = 10 BB/100 will not win 2 BB/100 or more over 10000 hands. 97.5% is also the probability that a 2 BB/100 win rate player will not be a loser over 10000 hands.

Note that each of those two formulations started with a player's true win rate and made a statement about the likelihood of a particular result.

So given that a player won 2 BB/100 over 10000 hands, what is the probability that he is a winning player? This question starts with a result and attempts to determine the likelihood of a particular win rate. You cannot do that with standard deviation. Inverting the conditions and saying "97.5%" is completely invalid.

Let me say that again. 97.5% is completely wrong in theory and practice.

Consider the following change to the problem. The player has won 8 BB/100 over 10000 hands. How likely is it that his true win rate is at least 6 BB/100? Do the arithmetic and the same 97.5% comes out. This methodology also will tell you that he has a 50% chance of having a higher win rate than 8 BB/100.

These 97.5% and 50% numbers are absurd and fail to consider the fact that true win rates of 8 BB/100 for Party 2/4 are effectively unheard of. It is much more likely that one of the many merely very good players has had an exceptional run several standard deviations above his average. As unlikely as such runs are, with so many players playing so many hands, someone is bound to have one fairly often.

Returning to the original question, the 97.5% answer is just as flawed for the 2 BB/100 win rate. The mistake is just less conspicuous.

[ QUOTE ]
What is the probability that he is a winning player in this game?

[/ QUOTE ]
You cannot answer this question based on playing results unless I give you the a priori probabilities of different true win rates. For example if I had said this was a randomly selected Party 2/4 player then a diligent poster could assemble stats for all Party 2/4 players and compute a correct answer. BTW, that answer would be a lot less than 97.5% because 2 BB/100 is far above the average Party player. It's a lot more likely that one of the innumerable -0.5 BB/100 player got lucky than a rare 4.5 BB/100 player had a bad run.

I could also have said this player was randomly selected from Party 2/4 players with more than 500 posts at 2+2. The chance of this player being a winning player is much higher.

If you play sensible TAG poker, don't tilt or have other major flaws, and play against obviously weak opposition, then it is very likely that you are a winning player. 20000 hands of comfortably winning results is very reassuring because it confirms the things we already knew. Paradoxically, while a losing result is worrisome, it is very likely to be a bad run. This assumes the bad results aren't tilting you as they so often do.

OTOH, if your VP$IP is 40% and you run wild bluffs against any loose/passive who sucks out on you, then I really don't care about your results and I refuse to believe you are a winning player. Your winnings are variance.

[Grisgra]

[ QUOTE ]
Congrats to ZZZ and passion for seeing the point. Many of the rest of you fell into the trap.


[/ QUOTE ]

If you're going to get all whiny "ha ha you don't have enough information" on us then you probably should have posted to the Probability forum instead. As I said in my initial post, yes, prior information would certainly be handy, BUT in the absence of prior information other than these basic statistics, there's nothing wrong with saying there's ~97% chance that this guy is a winning player.

If you want to say that "BUT if I gave you prior information that completely messed up the statistics, the 97.5% figure would be incorrect!" . . . well, duh. My understanding of the underlying Bayesian philosophy is pretty rusty, but I'm pretty damn sure that in the absence of prior information, one is allowed to flip the frequentist assumptions in hypothesis testing and move from "Assuming the null, there is a X% chance of seeing data like this" to "Given data like this, there is an X% chance that the null is correct".

Frankly, though, from what we know (not unreasonable win rate, the guy's SD is only 10, he's a 2+2'er playing Party 2/4), I think the chance he's a winning player is &gt;97.5%. This is a case where the prior information bumps up the percentage, not knocks it down.

EDIT: I'm assuming the guy in the example is a 2+2'er, as every post on here about people wondering about whether they are winning players are, by definition, posts by 2+2'ers.

[StellarWind]

I told you it was a trap. It's dangerous to use mathematics without considering whether it applies to the real world situation.

[ QUOTE ]
BUT in the absence of prior information other than these basic statistics, there's nothing wrong with saying there's ~97% chance that this guy is a winning player.

[/ QUOTE ]
Yes there is something wrong. That is one of the primary points of my post. 97.5% is an irrelevant number. It has nothing to do with the probability that the player in question is a winning player.

[ QUOTE ]
Frankly, though, from what we know (not unreasonable win rate, the guy's SD is only 10, he's a 2+2'er playing Party 2/4), I think the chance he's a winning player is &gt;97.5%. This is a case where the prior information bumps up the percentage, not knocks it down.

[/ QUOTE ]
Quite possibly. The answer could be higher or lower than 97.5%. Once again the point is that we shouldn't even be talking about 97.5%.

It's important to understand this because many players actually do this when they move up levels. They look at their win/loss result, number of hands, and standard deviation and compute a bogus probability that they are beating the game. Good players with bad results despair because the computation says they are 98.5% (or whatever) certain they are not beating the game. They move back down without a valid reason. Players with serious leaks and good results are ecstatic for no legitimate reason and don't subject their game to necessary scrutiny. Or maybe they move up again.

It is crucial to put the numbers aside for a moment and answer the real questions. How well am I playing? How well are my opponents playing? How favorable is the structure of the game (rake, etc.)? If you are a good player and you put the necessary effort into answering these questions you will know whether you are beating the game. Now take another look at the numbers. If they seem to match your conclusion you know where you stand with reasonable certainty. You don't need 100K hands to say you are a winning player because you are not relying primarily on the results.

[steamboatin]

Three winners no matter how bad it hurts you to admit, that I was right.

[ QUOTE ]

Now this in no way proves that your boy is a loser and it in no way proves that he is a winner. What it proves is, we don't if he is a winner or a loser.

I realize that you are a borderline if not full fledged troll and I probably shouldn't waste my time with this post but i am not doing it for you.


[/ QUOTE ]

Steamboatin maybe weak on math, but he can damn sure read!!

Props to Mason and GTOT.

[jason1990]

[ QUOTE ]
I told you it was a trap. It's dangerous to use mathematics without considering whether it applies to the real world situation.

[ QUOTE ]
BUT in the absence of prior information other than these basic statistics, there's nothing wrong with saying there's ~97% chance that this guy is a winning player.

[/ QUOTE ]
Yes there is something wrong. That is one of the primary points of my post. 97.5% is an irrelevant number. It has nothing to do with the probability that the player in question is a winning player.

[/ QUOTE ]
As a mathematician, I would like to state, for the record, why some people consider this number to be relevant. Take any player, 2+2er or not, winning player or not -- any player whatsoever. Have that player repeatedly play blocks of 10000 hands. For each block, record the measured win rate, W, and the measured standard deviation, S. In the long run, about 97.5% of those blocks will generate numbers W and S such that the "intrinsic" win rate, as you call it, is greater than W-S/5. This will be true of any player with any win rate and any standard deviation. (Assuming some degree of consistency and independence, of course.)

Now, what you or others choose to infer or not infer from this is a different story.

[Grisgra]

[ QUOTE ]
97.5% is an irrelevant number. It has nothing to do with the probability that the player in question is a winning player.

[/ QUOTE ]

This is a horribly egregious overstatement given the situation you described.

[StellarWind]

[ QUOTE ]
Take any player, 2+2er or not, winning player or not -- any player whatsoever. Have that player repeatedly play blocks of 10000 hands. For each block, record the measured win rate, W, and the measured standard deviation, S. In the long run, about 97.5% of those blocks will generate numbers W and S such that the "intrinsic" win rate, as you call it, is greater than W-S/5. This will be true of any player with any win rate and any standard deviation. (Assuming some degree of consistency and independence, of course.)

[/ QUOTE ]
This is absolutely correct. 97.5% of those 10000-hand blocks will be less than 2 standard deviations above the player's true win rate.

Rephrased, there is only a 2.5% chance that a randomly selected 10000-hand block will be more than 2 standard deviations above the player's true win rate.

But if you randomly select a Party 2/4 10000-hand block and discover that the the measured win rate W is 2 BB/100, the probability that the block will be more than 2 standard deviations above the player's true win rate is much greater than 2.5%.

This is true because we have general knowledge about the underlying distribution of true win rates for Party 2/4 players. The average player has a win rate around -2.5 BB/100 (rake). Players with win rates above 2 BB/100 ("good players") are much rarer than players below 2 BB/100 ("bad players"). A randomly selected 2 BB/100 result is much more likely to be a bad player running well instead of a good player running badly.

Now some posters want to ignore our knowledge of the underlying distribution of true win rates. More accurately, they are implicitly assuming a uniform distribution of win rates, so that for example 6 BB/100 players, 2 BB/100, and -2 BB/100 players are equally common. That's why I said it's dangerous to use mathematics without considering whether it applies to the real world situation. Garbage in, garbage out. The mathematics work out nicely but the answers don't describe reality and are useless.

The same effect happens in reverse when an established winning player moves up a level to a game he should be able to beat and gets clobbered. The first consideration is tilt and that has to be carefully ruled out. But assuming you do rule out tilt, the next possibility is that the player is running badly. Because we have all of this a priori information indicating the player should be a winning player, the probability that losing results are a bad run is much higher than the formula indicates.

[Grisgra]

[ QUOTE ]
A randomly selected 2 BB/100 result is much more likely to be a bad player running well instead of a good player running badly.

[/ QUOTE ]

And a randomly selected 2+2'er?

[ QUOTE ]
Now some posters want to ignore our knowledge of the underlying distribution of true win rates.
More accurately, they are implicitly assuming a uniform distribution of win rates, so that for example 6 BB/100 players, 2 BB/100, and -2 BB/100 players are equally common.

[/ QUOTE ]

Now you're just talking out of your butt. Nobody has made such a claim.



[/ QUOTE ]

[jason1990]

[ QUOTE ]
Now some posters want to ignore our knowledge of the underlying distribution of true win rates. More accurately, they are implicitly assuming a uniform distribution of win rates

[/ QUOTE ]
Actually, they are assuming nothing. Or, more accurately, they are using the frequency interpretation of probability theory. If you want to regard these non-random parameters (like intrinsic win rate) as random variables and impose some prior distribution based on experience or intuition, then this is the Bayesian interpretation of probability theory. Proponents of these perspectives have been at odds for centuries. Which is why I said, "what you or others choose to infer or not infer from this is a different story."

And, by the way, there is no "uniform" distribution that makes every real number equally likely.

[StellarWind]

[ QUOTE ]
And, by the way, there is no "uniform" distribution that makes every real number equally likely.

[/ QUOTE ]
I know that. But once you get a few standard deviations away from the number you are looking at it doesn't matter much anymore. It is sufficient that the distribution be approximately uniform over a reasonable interval surrounding 2 BB/100.

This is the answer to an objection that has not been raised yet. Once you have truly large numbers of hands the standard deviation drops to a small fraction of a BB/100. The distribution does become approximately uniform within a few standard deviations of the measured win rate because the interval in question becomes very small. Now the computation using standard deviation does closely approximate the actual probabilities of different underlying win rates.

[ QUOTE ]
Actually, they are assuming nothing. Or, more accurately, they are using the frequency interpretation of probability theory. If you want to regard these non-random parameters (like intrinsic win rate) as random variables and impose some prior distribution based on experience or intuition, then this is the Bayesian interpretation of probability theory. Proponents of these perspectives have been at odds for centuries. Which is why I said, "what you or others choose to infer or not infer from this is a different story."

[/ QUOTE ]
Mathematical philosophy should not enter into this because the topic of discussion is a real world system: Party Poker 2/4. You could actually do an experiment to see which theory is more accurate.

Assemble a giant database of all Party 2/4 results. Randomly select a block of 20000 hands. Divide the block into 10000 hands taken from the first, third, fifth, etc. orbits and 10000 hands taken from the second, fourth, sixth, etc. orbits. Conditions for the two halves should be approximately the same and the player should have an equally good chance to do well in either half.

Now examine the results of the first half and assume that the result happens to be between 1.8 and 2.2 SD over breakeven. Otherwise you discard this block and continue to randomly select blocks until you find a result within the range.

The approach to computing true win rates that you are calling the "frequency interpretation" says that the result for the second half of the block is equally likely to be better or worse than the result for the first half. You could make even odds bets on repeated trials and neither "over" nor "under" would have the best of it.

Actually "under" has a huge edge here. The selected first half results are far above the approximately -2.5 BB/100 average win rate. Most of the blocks chosen will be from players who won a lot in the first half because they ran well. The second half results won't be nearly as good on average.

All serious poker players apply this concept. Everytime Joe Newbie pops up and happily announces his 5 BB/100 win rate we all tell him the same thing. Enjoy your run but it probably won't last. We know that such a high win rate in online small stakes is very rare. It is much more likely that Joe is just a decent player who is running well. In doing this we are completely ignoring the measured win rate and standard deviation and 97.5% and everything else that goes with it. And we are absolutely justified in doing so.

[StellarWind]

[ QUOTE ]
[ QUOTE ]
Now some posters want to ignore our knowledge of the underlying distribution of true win rates.
More accurately, they are implicitly assuming a uniform distribution of win rates, so that for example 6 BB/100 players, 2 BB/100, and -2 BB/100 players are equally common.

[/ QUOTE ]

Now you're just talking out of your butt. Nobody has made such a claim.


[/ QUOTE ]
That's true. No one made such a claim. But I said "implicitly assuming" and you are even if you don't realize it. You can't do what you did unless the underlying distribution of win rates is uniform over the interval you are working over. But the actual distribution of win rates within a few standard deviations of 2 BB/100 is not remotely flat. It's a very steep curve. As a result 97.5% isn't even a decent approximation of the correct answer.

Many players don't realize this and draw completely wrong conclusions from their playing results.

[scrub]

Great thread.

scrub

[jason1990]

[ QUOTE ]
Mathematical philosophy should not enter into this because the topic of discussion is a real world system: Party Poker 2/4. You could actually do an experiment to see which theory is more accurate.

[/ QUOTE ]
Oh, but it does enter into it. Suppose I want to know the probability that I am a winning player. I am not a random Party Poker player. I am me. I am a particular, fixed player.

So what does it mean, "the probability that I am a winning player"? Either I am or I am not. So the probability must be either 0 or 1. For it to be something else, we must attach an interpretation to the statement.

If you want to pretend that I am a Party Poker player who was randomly selected by someone or something, and thereafter attach some probability distribution to my intrinsic win rate and standard deviation, then this is the Bayesian interpretation. But in reality, I was not randomly selected by anyone, other than maybe myself, since I was the one who initially wanted to know if I am a winning player. I am not a random player, I am a fixed player. Hence, my intrinsic win rate is not a random variable, it is a constant. If I take this attitude, that is the frequentists interpretation (as it relates to confidence intervals).

The experiment you described has the Bayesian interpretation built into it. It has nothing to do with me and says nothing about whether or not I am a winning player. For example, suppose that tomorrow I switch from Party to Absolute Poker. And suppose that, on average, the players on AP are better than those on Party. According to the Bayesian interpretation, the probability I am a winning player has just increased, since now I am a random AP player. But, of course, since we're talking about a fixed player (me), that makes no sense. I, in fact, have less chance of being a winning player, since the competition will be tougher.

At any rate, the experiment you describe does not address my hypothetical question at the beginning of this post, which has to do specifically with whether or not I am a winning player. It only addresses the question of whether or not a randomly selected PP player is a winner. This is why it already has the Bayesian interpretation built into it. As such, the experiment cannot say that the Bayesian interpretation is the correct one. It can only say that if it is correct, then it is accurate.

I agree with the spirit of what you are saying. It is my opinion (and I think it is also yours) that a player should not use statistics as the sole (or even primary) guage of whether or not he is a winning player. It is much more important for him to analyze his play, discuss hands, and try to determine if he is making the correct decisions at the table, rather than to focus on the statistical results which can often be very misleading. But to support this idea by attacking the frequentists' method of statistical analysis doesn't make sense. The frequentists' interpretation is a valid one and is applied by scientists worldwide. It is powerful and effective in a wide array of real world systems, including this one. It just so happens that in poker, we have a better way of determining if someone is a winner: look at how he plays.

Finally, to be fair, the Bayesian interpretation is powerful, effective, and applied by scientists worldwide. So what you're suggesting about incorporating prior information is a reasonable suggestion, from a statistical point of view. But it's not the only valid method. And it's still no substitute for analyzing one's play.

[LinusKS]

Not to beat a dead horse - but this looks like the right track to me.

If you thought that - hypothetically - 90% of players were long-term losers, and 10% were long-term winners, and you assumed - for the sake of argument - that half the players who do win win at a rate of less than 2BB/100, the math would look something like this:

Out of 10,000 random people, you'd have 9000 losers, and 500 2/BB winners. You'd expect 2.5% of those losers to show a 2/BB win-rate after 10,000 hands, or 225 of them.

You'd also expect 97.5% of the 500 2/BB players would show a profit after 10,000 hands - but not necessarily a full 2BB/100. If you assume half the 2BB/100 players were behind their true win-rate, and half were ahead, 50% of them would be at or above 2/BB, meaning roughly 250 of them.

That would mean that out of those 10,000 people (leaving aside for the moment the 5% winners who win less than 2/BB), you'd expect 475 of them to show a 2/BB result after 10,000 hands.

Out of those 475 people, 250 of them - or a little over half - would be true 2/BB players. The other 225 would be part of the 2.5% of losing players who happened to have gotten very lucky.

Of course, that all depends on the assumption that 90% of players are losers, and that half of winners win at a long-term rate of less than 2BB/100.

[ QUOTE ]
That 97.5% is all well and good. It means that a break even player has only a 2.5% of winning 2BB/100 or better over his first 10K hands.

However this does not mean that a random player who makes 2BB/100 over his first 10K hands has a 97.5% chance of being a winning player. That chance depends on both the win rate and the distibution of winners vs. losers coming into Party Poker.

Let's say you knew that 99.9% of players coming into the system were losing players. Then the majority of players of who make 2BB/100 over their first 10K hands would be losers, and you would justified in thinking that a random player with those numbers is less than 50% likely to be a winning player. Obviously, that 99.9% number is too high, but it shows you must consider the fact that more losers than winners come into the system when trying to find the probability that a random player is a winning player.

It's a bit like taking a random coin from your pocket and flipping heads 10 times in a row, and trying to come up with the probability that the coin is biased. You wouldn't say that an unbiased coin would only have a 0.1% chance of coming up heads 10 times in a row, so there is a 99.9% chance the coin is biased.

ZZZ

[/ QUOTE ]

[StellarWind]

I think we are mostly in agreement. Let me say you have added a lot to this thread. Thanks.

[ QUOTE ]
But to support this idea by attacking the frequentists' method of statistical analysis doesn't make sense.

[/ QUOTE ]
I'm not. The frequentist method and the Bayesian method are both powerful tools for solving real world problems. The difficulty is deciding which method is applicable in a specific situation.

[ QUOTE ]
The experiment you described has the Bayesian interpretation built into it.

[/ QUOTE ]
True. I introduced the random Party 2/4 player who wins 2 BB/100 for two reasons. First it is really interesting in its own right. Second I wanted to open minds to the idea that the 97.5% approach has limitations before heading into deeper waters.

[ QUOTE ]
It has nothing to do with me and says nothing about whether or not I am a winning player.

[/ QUOTE ]
This is the hard topic that I wanted to address.

A winning player in the larger sense of the term has the ability to evaluate his play, his opponents' play, and the overall nature of the game and decide whether he can beat it. This is one of the many skills that a successful poker player has to master. It comes from study, analysis, reviewing one's own play, getting advice from others, reading 2+2, etc.

I'm going to assume that we are discussing such a player. Most of us fall into this category. My advice to those who don't do these things is that you are playing for entertainment and should not count on winning.

So even before you compute your results, you have a large amount of qualitative information about whether you are a winning player in this game and by a little or a lot. It isn't easy to reduce to numbers but it exists. You may for example believe that: you are probably winning about 2.5 BB/100, 4 BB/100 or better is unrealistic, and 1 BB/100 or less is also not likely.

Then you compute your results over a fairly small sample and get an actual win rate and standard deviation. How do you interpret this result? The frequency approach effectively discards everything you know about your play except the actual results and draws conclusions like '97.5% likely to be a winning player'.

This happens in many disciplines when numeric and nonnumeric information need to be combined to solve a problem. It usually isn't obvious how to do this and too often the nonnumeric information is completely ignored. Numbers rule.

The results of ignoring most of the available information are often bad and the winning poker player issue is not an exception. Even though you cannot precisely combine your results with your understanding of whether you are beating the game, you need to try.

If you think you are a 2.5 BB/100 player and your results are 2.38 BB/100, the likelihood that you are actually a losing player is much lower than the frequency approach would suggest. How much lower depends on how much trust you can place in your game evaluation skills. A seasoned professional may be virtually certain that he is a winning player in which case the chance of being a loser drops to almost zero.

On the opposite side, results that conflict with our assessment of the game have a much higher probability of being outliers than the frequency approach would suggest. Once again, the more certain you are of your evaluation the more you should discount the results. To take an extreme example, if a top online limit professional sits down in a soft 0.5/1 game, conscientiously plays his best game, and happens to lose, then I attach almost no credit to that result. The chance that he is a good player who achieved a -4 SD result through bad luck is far greater than the chance that he is a losing player who achieved a normal result.

Typical applications of these concepts:

1. You think you are overmatched at this level, but your results in BB/100 and standard deviation indicate that you are probably beating the game.

Conclusion: There is a much higher chance that you have been lucky than the numbers are telling you. Get out unless you are willing to lose (e.g. for the experience).

2. You think you are a winning player and lo and behold, you are doing very nicely. But you only have 20000 hands and "The Fear" keeps following you around.

Conclusion: There are no guarentees but the odds are very high that you are a winning player. Remain observant and open to new information, but stop listening to the gloom and doom of the 100000-hand club.

3. You are a very successful player trying a slightly tougher game. You know of no reason why you shouldn't be able to kill the game, but the blood that is flowing is yours.

Conclusion A: You've tilted even though you don't realize it. You are out of your comfort zone and you have stopped playing your normal game. Suspend play and take a really hard look at your hands.

Conclusion B: The odds that this is a bad run are much higher than the numbers indicate. Don't panic and don't bail. Keep doing what you do and it will turn around.

Sorry about having two conclusions. Life isn't always easy. Figure out for yourself what is going on.

4. You have several hundred thousand hands and you are barely breaking even. You know you should be beating the game comfortably but you've heard bad runs can last a very long time.

Conclusion: Face reality. You are extremely unlikely to be running 10 standard deviations below your actual win rate. As your sample size increased the weight of information contained in it also increased. Your subjective self knowledge is not reliable enough to offset the data you have collected.

The Bayesian interpreation has not stopped working. It just stopped mattering. The probability that your disappointing results are due to chance may be five or ten times higher than the frequency method says, but that is still an extremely small number.

[Grisgra]

[ QUOTE ]
3. You are a very successful player trying a slightly tougher game. You know of no reason why you shouldn't be able to kill the game, but the blood that is flowing is yours.

Conclusion A: You've tilted even though you don't realize it. You are out of your comfort zone and you have stopped playing your normal game. Suspend play and take a really hard look at your hands.

Conclusion B: The odds that this is a bad run are much higher than the numbers indicate. Don't panic and don't bail. Keep doing what you do and it will turn around.

Sorry about having two conclusions. Life isn't always easy. Figure out for yourself what is going on.


[/ QUOTE ]

Look, everybody, it's Grisgra trying to play 10/20 6-max . . .

[jason1990]

That was a very nice post. Let me just say that I am not a militant anti-Bayesian. I think the method should probably be applied more often than it is. But I wanted to comment on one issue.

[ QUOTE ]
4. You have several hundred thousand hands and you are barely breaking even. You know you should be beating the game comfortably but you've heard bad runs can last a very long time.

Conclusion: Face reality. You are extremely unlikely to be running 10 standard deviations below your actual win rate. As your sample size increased the weight of information contained in it also increased. Your subjective self knowledge is not reliable enough to offset the data you have collected.

The Bayesian interpreation has not stopped working. It just stopped mattering. The probability that your disappointing results are due to chance may be five or ten times higher than the frequency method says, but that is still an extremely small number.

[/ QUOTE ]
Let me take the Bayesian approach and suppose I am able to quantify my self-analysis and construct a prior distribution which represents my opinion of my win rate. I collect data for N hands and then compute my posterior distribution. Of course, the prior influences the posterior. If I thought I was winner before I played, then my conclusion after I play will lean toward being a winner.

In this example #4, it seems that "reality kicked in". The number of hands, N, is so large that the effect of the prior on the posterior is negligible and the player must face the fact that he is not a 2BB/100 player. But how large must N be before "reality kicks in"? Well, this will of course be a function of the variance of the prior. The smaller the variance of the prior (i.e. the more confident I am from the beginning that I am a winner) the longer it will take for reality to kick in.

My conclusion is that, applied carelessly, the Bayesian approach might only serve to influence my conclusions based on my own opinions and my own confidence in those opinions, removing a sense of objectivity that I might want to achieve.

To temper this conclusion, let me acknowledge that it's an entirely different story for an experienced player who is moving to a new level. In that case, the player will have actual data from past experience to help him formulate a prior which is based on facts and is "objective", at least as far as that is possible. Also, such a player would probably be cautious and not introduce unnecessary levels of confidence (i.e. too small a variance) in his prior distribution.

But I can imagine even a "humble" player with no experience being misled by the Bayesian method. He might think, "I've read SSH, I read the forums, I'm a smart guy...I should be able to beat this game." But he may be misunderstanding key concepts without even knowing it. When he sees a low win rate, he leans towards thinking it is due to chance and may not work as diligently at correcting his game as he might have if he had just took the win rate at face value.

[MarkD]

I haven't read this entire thread yet but it hit's home right now as I'm in the middle of a 15k losing streak which makes me doubt my own abilities to beat the game. This self-doubt always creeps up when I am losing but this time the feeling is greater than ever beforse since this is my longest losing streak in terms of # of hands, and also, at 200BB's, it is my biggest losing streak to date.

[StellarWind]

[ QUOTE ]
My conclusion is that, applied carelessly, the Bayesian approach might only serve to influence my conclusions based on my own opinions and my own confidence in those opinions, removing a sense of objectivity that I might want to achieve.

[/ QUOTE ]
This is a very real danger. I agree with everything you wrote.