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Questions about PT4 and variance

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  1. #1

    Default Questions about PT4 and variance

    Just wondering if anyone knows how PT4 creates the "Luck Bell Curve" in leaktracker? is it for flopped straights/flushes/sets or for overall made?

    For the last 140,000 hands my luck bell curve shows straights being right at the very end of the unlucky section (literally bottom 0.5% of the curve) with my flushes section sitting in the 'unlucky' zone and my sets section sitting just above the midline in the normal zone.

    In the last 10k hands it feels like I'm making a lot more of my straight/flush draws and I'm wondering if this is now just variance starting to even out that curve. How many hands would it take to even out variance like that if my straights made % is really that much lower than average? Is there a way to exploit it in the future when it feels like variance is starting to even out?
  2. #2
    MadMojoMonkey's Avatar
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    You can not exploit variance. Variance is there because there is randomness - E.g. the shuffling of the cards.

    Variance - as in - the results vary.

    If you COULD exploit variance, then it would mean that the randomness was not random.
  3. #3
    Variance is one of my favourite subjects.

    Every single card dealt in every hand is subject to variance, because it is an random event. Using software like PT4, we can look at Cwon all in adj and our luck bell curve and say we are lucky or unlucky but these are really only tiny aspects of your total variance which cannot be measured. However, we can calculate the likelihood of how far we are away from our actual/real winrate, which over large sample sizes is likely to be pretty close to our winrate we see in PT4.

    We can still be really unlucky or lucky over 150K hands but the vast majority of the population (approx 95%) will be between slightly unlucky and slightly lucky.

    As the number of hands we play tends towards infinity then variance tends towards zero.

    In conclusion, in Poker, we shouldn't dwell on how lucky or unlucky we are as we cannot control it and we cannot measure it.
  4. #4
    Ok. I guess what I meant is that part of my brain really wants me to call draws in marginal situations as its expecting my 'luck bell' curve to even out from such an extreme end. I know it eventually will and I should just keep playing solid poker, but it is a bit frustrating when you keep missing monster draws in huge pots!
  5. #5
    MadMojoMonkey's Avatar
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    Quote Originally Posted by DJAbacus View Post
    Variance is one of my favourite subjects.
    You should study it, then. I'm happy to answer questions on the subjects of math and physics. Fire away.

    Quote Originally Posted by DJAbacus View Post
    As the number of hands we play tends towards infinity then variance tends towards zero.
    Umm.... What? That statement is incredibly misleading - and flat out false for non-convergent statistics.

    Props for correct use of "tends toward infinity" there, though.

    Quote Originally Posted by DJAbacus View Post
    In conclusion, in Poker, we shouldn't dwell on how lucky or unlucky we are as we cannot control it and we cannot measure it.
    You just described how it is measured.

    ***
    I'm guessing you've had some college level calculus, but you haven't taken either a Probability or a Statistics class, yet.
    ?

    You have the passion, so buy a book. Study this stuff for yourself, even if you're not in school. If you are in school, then you'll be miles ahead of your own education. It's win-win.
  6. #6
    Watch out everyone it's the Variance police. I'm guessing you don't get out much.

    If you play an infinite number of hands then variance will be equal to zero and therefore our winrate is therefore our 'true winrate' or 'skill level' if you like.

    In the following video, over an infinite sample size the 'normal distribution' curve will be effectively an infinitely thin line.

    https://www.youtube.com/watch?v=5416oRqykJw

    So all we need to realise is that the more hands we play the more likely our winrate reflects our skill level.

    So if we play 250K hands and our winrate is 5BBs/100 then it is likely that our skill level is closer to this than if we have a 5BBs/100 winrate over 50K hands.


    Quote Originally Posted by MadMojoMonkey View Post

    You just described how it is measured.

    Ok, maybe I should have said it cannot be measured exactly



    ***
  7. #7
    MadMojoMonkey's Avatar
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    Quote Originally Posted by DJAbacus View Post
    Watch out everyone it's the Variance police. I'm guessing you don't get out much.
    I could explain the finer points between convergent and non-convergent stats.

    I could explain why that video has nothing to do with poker or any stat you'd like to estimate that would help guide your decisions in a poker hand.

    I could explain why the normal distribution is NOT an unbiased estimator for statistics based on Bernoulli trials.

    I could, but I see that you've got it all figured out.

    Sorry to have wasted your time, DJ. Where can I buy your book on probability and statistics?


    @renegaderob: If there's anything you'd like clarification on, please keep asking questions until you fully understand.
    Last edited by MadMojoMonkey; 03-11-2015 at 04:04 PM.
  8. #8
    Why would you start having a go at someone who is

    1) Very helpful
    2) Very knowledgable about lots of maths topics
    3) Very nice
  9. #9
    Quote Originally Posted by MadMojoMonkey View Post
    I could explain why that video has nothing to do with poker or any stat you'd like to estimate that would help guide your decisions in a poker hand.
    It's simple really.

    You bet on tails.

    You flip a coin 30 times and it comes up tails 20 times. You have been lucky.

    It comes up heads 20 times. You have been unlucky.

    You flip the coin an infinite number of times and you will win 50% of the time, neither lucky or unlucky.

    Poker is a more complicated scenario, with many variables but over an infinite number of hands we win or lose based on our 'Poker Skills' as we are neither lucky nor unlucky over an infinite sample.
  10. #10
    Quote Originally Posted by ImSavy View Post
    Why would you start having a go at someone who is

    1) Very helpful
    2) Very knowledgable about lots of maths topics
    3) Very nice
    4) Extremely patronising.
  11. #11
    Maths Question for Mojo.

    2 players play infinite HU poker hands.

    They both shove PF with KK and AA

    Over an infinite sample who has won the most money if we only look at the AA vs KK hands. You must show your working out.
  12. #12
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    DJA im not a probability fiend but it seems like you are falling for the gamblers fallacy if you really think it all evens out. If you flip coins for 1:1 odds approaching infinity times, there will be approaching infinity different outcomes that fall along a gaussian curve. It is extremely unlikely that you would break even.
  13. #13
    Quote Originally Posted by Renton View Post
    DJA im not a probability fiend but it seems like you are falling for the gamblers fallacy if you really think it all evens out. If you flip coins for 1:1 odds approaching infinity times, there will be approaching infinity different outcomes that fall along a gaussian curve. It is extremely unlikely that you would break even.
    Yes, it is extremely unlikely that you will get exactly 5000 heads if you flip a coin 10000 times but it is still the most likely outcome. It is highly likely that you will be close to 5000, in other words, it is highly likely that you will be between slightly lucky and slightly unlucky.
    Last edited by DJAbacus; 03-12-2015 at 11:35 AM.
  14. #14
    I'm a bit confused now as to what should happen with variance. Using my pt example, if I'm sitting at 150k hands on the bottom 0.5% for straights made, should that eventually even out so it sits in the 'normal' part of the bell curve or can it continue to stay like that as one of the infinitely different results Renton mentioned?
  15. #15
    MadMojoMonkey's Avatar
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    Quote Originally Posted by DJAbacus View Post
    Maths Question for Mojo.

    2 players play infinite HU poker hands.

    They both shove PF with KK and AA

    Over an infinite sample who has won the most money if we only look at the AA vs KK hands. You must show your working out.
    wikipedia - Binomial Distribution

    Accounting for the +1/-1 value of win/loss means the expected value is
    EV(X) = np - n/2

    but p = 50% = 1/2
    so
    EV(X) = n/2 - n/2 = 0

    check

    The variance of X is
    var(X) = np(1 - p) = n/4

    So no matter how many times you play at even odds, the expected outcome (read the mode of the data set) is 0 or no change.
    The variance is ever increasing with n. Meaning that the bell curve is ever widening with n. Meaning that the peak of the bell curve is getting smaller and smaller as the curve becomes more and more flat with increasing n.

    Or to put it another way:
    Quote Originally Posted by Renton View Post
    If you flip coins for 1:1 odds approaching infinity times, there will be approaching infinity different outcomes that fall along a gaussian curve. It is extremely unlikely that you would break even.
    ... even though the most likely result would be to break even, the probability of having any specific amount becomes vanishingly slight.
  16. #16
    MadMojoMonkey's Avatar
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    Quote Originally Posted by DJAbacus View Post
    Yes, it is extremely unlikely that you will get exactly 5000 heads if you flip a coin 10000 times but it is still the most likely outcome. It is highly likely that you will be close to 5000, in other words, it is highly likely that you will be between slightly lucky and slightly unlucky.
    For n = 10,000 trials, the variance is n/4 = 2,500.
    The standard deviation is sqrt(var) = sqrt(n)/2 = 100/2 = 50

    This means that ~76% of the data will fall between -50 and +50.


    For n = 1,000,000 trials, the standard deviation is sqrt(n)/2 = 1,000/2 = 500

    This means that ~76% of the data will fall between -500 and +500.

    So you see that while the data is "clumped" near zero... the spread of the clump is ever widening.
  17. #17
    Quote Originally Posted by MadMojoMonkey View Post
    For n = 10,000 trials, the variance is n/4 = 2,500.
    The standard deviation is sqrt(var) = sqrt(n)/2 = 100/2 = 50

    This means that ~76% of the data will fall between -50 and +50.


    For n = 1,000,000 trials, the standard deviation is sqrt(n)/2 = 1,000/2 = 500

    This means that ~76% of the data will fall between -500 and +500.

    So you see that while the data is "clumped" near zero... the spread of the clump is ever widening.
    Yes, but we are not taking the amount of hands into consideration here, only the amount of trials.

    So if we did 100,000 trials of 50K hands

    and then 100,000 trials of 250K hands

    and then 100,000 trials of 1M hands

    The shape of these 3 normal distributions would look very different.
  18. #18
    MadMojoMonkey's Avatar
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    trials = hands

    Even if not... all you did was say, OK, I want to generate thousands of distributions, then overlap them all.

    That result, taking the trials in separate sets of experiments, will have higher variance than if you took all of the trials as a single experiment.

    And none of them would yield a normal distribution.
    The normal distribution is for continuous statistics, and this is discreet statistics.

    The distribution here is the Binomial distribution, for which I linked to the wiki above.

    ***
    You're seriously digging in your heels, when you should be taking notes.
  19. #19
    MadMojoMonkey's Avatar
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    Quote Originally Posted by DJAbacus View Post
    Yes, but we are not taking the amount of hands into consideration here, only the amount of trials.

    So if we did 100,000 trials of 50K hands

    and then 100,000 trials of 250K hands

    and then 100,000 trials of 1M hands

    The shape of these 3 normal distributions would look very different.
    This is a different set of distributions entirely.

    Now you're talking about the average of the averages... or the mean of the means.
    E.g. You calculate the mean from one sample of 50k hands.
    Then you do that 100,000 times.

    Now you want to know the distribution of the means. And you want to know the mean of that.
    You are asking about the mean of means.
    (just to clarify, 'cause it's an odd thing to understand the first time through.)

    The statistics which apply to statistics is a different set of math again. We're basically invoking the law of large numbers and regression to the mean.

    Which yields a convergent distribution, but not necessarily convergent with 0 variance - in this case, it would be convergent with var(X) = sqrt(n)/2.
    So here's a new topic we haven't touched on yet. Sometimes perfect information yields a probability distribution. Meaning perfect information is still embedded with uncertainty.
    In this case, it's because we're getting meta about our stats, and we're being very clever and careful about quantifying what we "know" and what we "expect."
  20. #20
    You seem to like to use a lot of mathematical terminology. You're explanations aren't clear and I not even sure if they are relevant.

    Have a look at this Poker Variance Calculator.

    http://pokerdope.com/poker-variance-calculator/

    It shows that your winrate is more reliable over larger number of hands played because variance has less effect on your winrate the more hands you play.

    These are my results. Have a play yourself.

    If your winrate is 5BB/100 over 100 hands the probability that you are a losing player (below 0BB/100) is 48%.

    If your winrate is 5BB/100 over 1000 hands the probability that you are a losing player (below 0BB/100) is 44%

    If your winrate is 5BB/100 over 10,000 hands the probability that you are a losing player (below 0BB/100) is 31%.

    (Currently my winrate is 6BB/100 over 19,000 hands so there is still a 20% chance that due to variance, I am a losing player (below 0BBs/100) at 2nl atm)

    If your winrate is 5BB/100 over 100,000 hands the probability that you are a losing player (below 0BB/100) is 6%.

    If your winrate is 5BB/100 over 1,000,000 hands the probability that you are a losing player (below 0BB/100) is 0% (negligible).

    In conclusion. the more hands you play, the more reliable your winrate becomes as variance tends towards zero as the number of hands you play tends towards infinity.

    This was the main point of my first post in this thread.
    Last edited by DJAbacus; 03-12-2015 at 06:08 PM.
  21. #21
    MadMojoMonkey's Avatar
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    Now you're asking a different question again.

    Like I explained above, this notion of your winrate after n hands being more reliable (when asking the question, how likely is it that the actual value is greater than 0)... this does NOT mean the variance is less.

    This is because the variance increases proportionally to the square root of n, whereas winrate is based on BR, which increases at a constant rate.

    The variance is always increasing, but it increases at an ever-slower rate as n increases.
    BUT the winrate represents a constant (unchanging) slope - a straight line.

    The winrate is constant, but the variance ... whatever the slope ... increases at a slower rate.

    So the winrate is climbing up past the 0 axis steaily. At first, the variance increases at a very fast rate... but then it slows down.

    Here:

    The bottom graph is the short term, and the top graph is the long term.

    These are graphs of a 4.3bb/100hands winrate @ 80bb/1hand variance
    These are typical values for an mediocre to good TAG player, beating the micros.
    The red and blue lines show the upper and lower bounds for a 95% Confidence Interval.
    The dotted green line is EV(bankroll), and the slope of that line is the winrate.

    The green line looks curved in the upper (long term) graph because the x-axis is distorted to a logarithmic scale to better show the data.



    The variance dominates in the short term. However, at ~30,000 hands, you can see that the constant slope of the winrate has equaled out with the ever-decreasing slope of the variance. (That's where there's a valley in the blue line in the top graph.) After that, the variance stops expanding in the negative, and since the slope is always less than the winrate's slope, the winrate dominates from that point on. At ~120,000 hands, you can see that the lower bound of the 95% CI crosses the x-axis, indicating a <5% chance of being a losing player at that point.

    At no time does the variance shrink. Just the portion which falls below the 0 mark on the x-axis (Profit).

    ***
    Also note that the green line represents the center of a distribution that is symmetrical above and below that point for that value. Since the green line is always further from the x-axis as n increases, the portion of the distribution which falls below the x-axis is always less of a %-age of the distribution for that value of n.

    So, even though the variance is increasing at all times, even in the short term, the fact that the winrate is positive means that the probability of being a losing player is always less with increasing n.
    Last edited by MadMojoMonkey; 03-12-2015 at 09:20 PM.
  22. #22
    I look at it like this.

    I win $100 a day playing poker.

    On the first day, variance is calculated as - $50 so this has a huge affect on my winnings. 50% of my winnings.

    On day 50, variance is calculated for all 50 days and is still -$50, so hasn't changed but now this is now only 1% of my winnings.

    So over larger sample sizes variance has less affect.
    Last edited by DJAbacus; 03-13-2015 at 03:25 AM.
  23. #23
    MadMojoMonkey's Avatar
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    Quote Originally Posted by DJAbacus View Post
    I look at it like this.

    I win $100 a day playing poker.

    On the first day, variance is calculated as - $50 so this has a huge affect on my winnings. 50% of my winnings.

    On day 50, variance is calculated for all 50 days and is still -$50, so hasn't changed but now this is now only 1% of my winnings.

    So over larger sample sizes variance has less affect.
    Let's step back a bit. I accept that I was a bit direct a few posts back - you said patronizing. I'm not sure exactly what you thought was patronizing, but I do apologize for whatever tone I used that has made this conversation nearly impossible.


    I'm being perfectly straightforward with you about a subject that can be readily verified with a 10-second web search. I understand that it's a rather advanced mathematical concept, which is why I'm willing to take the time to explain it to you.

    Let's make sure we're on the same page and talking about one topic at a time.

    What's your definition of variance?
  24. #24
    Quote Originally Posted by MadMojoMonkey View Post
    Let's step back a bit. I accept that I was a bit direct a few posts back - you said patronizing. I'm not sure exactly what you thought was patronizing, but I do apologize for whatever tone I used that has made this conversation nearly impossible.


    I'm being perfectly straightforward with you about a subject that can be readily verified with a 10-second web search. I understand that it's a rather advanced mathematical concept, which is why I'm willing to take the time to explain it to you.

    Let's make sure we're on the same page and talking about one topic at a time.

    What's your definition of variance?
    Mojo, I appreciate your enthusiasm and think you are a great poster.

    I don't want an advanced Maths lesson.

    Put it this way.

    When it starts raining (and it rains a lot in England) I put up my umbrella so I don't get wet. I don't need to know the average size of the raindrops and the speed they are falling. Nor do I need to know what the water is made of. I just need to know that if I don't put up my umbrella I'm going to get wet. I may want to estimate the length of time it will rain for but I'm not digging out past records and using advanced mathematics here. I'm just looking up at the sky and thinking. It's pretty dark, it's probably going to rain for at least an hour.

    I appreciate that you have a better understanding of variance but I feel I have enough of an understanding of variance in poker, not to get wet.

    Cheers
    DJ
  25. #25
    just to throw a curve ball into the topic. its silly to say that you winrate stays the same as the number of hands increases because your skill level will also change as you get more experienced and do more study etc just as the populations skill rate will change over the same period of time
    .
  26. #26
    MadMojoMonkey's Avatar
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    Quote Originally Posted by DJAbacus View Post
    I don't want an advanced Maths lesson.
    The only reason this turned into a math lesson is because you have not believed a word I've said. Fine. You don't know me or my credentials. We disagree. Now. I can only convince you that my method is proven and based on sound logic with the math. So you've tied my hands here.

    As for your analogy: I bet you'd pay more attention to the drops if each one could potentially cause money to appear or disappear in your wallet.

    You started the conversation with this:

    Quote Originally Posted by DJAbacus View Post
    Variance is one of my favourite subjects.
    and now you're telling me this:

    Quote Originally Posted by DJAbacus View Post
    I appreciate that you have a better understanding of variance but I feel I have enough of an understanding of variance in poker, not to get wet.
    You have spent days in a conversation with an expert in the field (me) - who has disagreed with everything you've said on the subject and tried to set you straight. You have refused to change your opinion and you still "feel" like you understand variance?
    Why? SMH

    [pretentious]
    It's like you're just standing there with your eyes closed and you swear you know what you're looking at.
    [/pretentious]

    "To stand under and understand is all the same"
    -Shakespeare - Two Gentlemen of Verona

    You have this thing built up in your head, but if you don't humble yourself to the concept, you will never get it.
  27. #27
    Hey Mojo, what does the variance actually tell us? I just read this: http://www.mathsisfun.com/data/standard-deviation.html and arriving at the answer seems fairly straightforward, but I don't really know how to interpret the answer.

    What does variance = 21,704 actually mean in this example? I know what it is, the average of the squared differences from the mean, but it doesn't really mean anything to me.

    Also, what can a standundering of variance teach us about poker?
    Erín Go Bragh
  28. #28
    Quote Originally Posted by Keith View Post
    just to throw a curve ball into the topic. its silly to say that you winrate stays the same as the number of hands increases because your skill level will also change as you get more experienced and do more study etc just as the populations skill rate will change over the same period of time
    .
    Agreed. Mathematically, we have to assume that our skill level relative to the skill level of the other players in our 'pool' remains constant, as the number of hands we play increases, which it may or may not do.
  29. #29
    Quote Originally Posted by MadMojoMonkey View Post
    As for your analogy: I bet you'd pay more attention to the drops if each one could potentially cause money to appear or disappear in your wallet.
    I agree my analogy is flawed. I mean, an umbrella is infinitely more useful to someone standing in the rain than a Maths degree is to someone who wants to be a winning poker player.
  30. #30
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    Quote Originally Posted by seven-deuce View Post
    Hey Mojo, what does the variance actually tell us? I just read this: http://www.mathsisfun.com/data/standard-deviation.html and arriving at the answer seems fairly straightforward, but I don't really know how to interpret the answer.
    The variance quantifies the amount of spread in a sample of data.
    Quantifies - gives a specific number for / measurement of

    The square root of the variance is the standard deviation, which I find a more intuitive stat.
    In most texts, the default variable used to denote the variance is {sigma^2}, where {sigma} is the stdev.

    The standard deviation is nice because it tells us something important about the results.
    Specifically:

    ~68% of the data in the set is within +/-1 stdev of the mean.*
    ~95% of the data in the set is within +/-2 stdev of the mean.
    ~99.7% of the data in the set is within +/-3 stdev of the mean.
    ~99.994% of the data in the set is within +/-4 stdev of the mean.
    ~99.99994% of the data in the set is within +/-5 stdev of the mean.

    For most of us 2 sigma, or 2 standard deviations from the mean, is a very practical guide that compromises between poor results from small samples and getting any meaningful result. For scientific discoveries (like the announcement of the discovery of the Higgs Boson), 5 sigma is used.

    This picture is a normal distribution, and while it is not appropriate for poker statistics, it illustrates the point with a well-recognized picture.

    *I believe I have mis-stated this as ~76% in recent posts... maybe in another thread, but 68.3% is the correct number.

    Quote Originally Posted by seven-deuce View Post
    What does variance = 21,704 actually mean in this example?
    That link explains what I was going to say, and what I started above.

    The variance is not really too helpful as itself, but the square root of the variance gives the standard deviation, which is the more intuitive number. First of all, the [units] on the stdev are the same as the mean - which means that the value of the stdev can be compared to the mean apples-to-apples.

    In that example, (and I can't stress this enough) the variance is NOT 21,704. The variance is 21,704 mm^2.
    Without the units, the statement is false. (This is 99% a personal gripe against mathematicians from an engineer's perspective, but no mathematician will argue that I'm wrong, just that it's obvious and why don't I STFU.)
    *ahem*
    So 21,704 mm^2 is hard to make sense of. We're comparing lengths, but the variance is an area. Not helpful.
    When we take the square root of the variance, we get 147 mm.
    So now we have a variable which describes the spread of the data in the same units as the data. Very helpful.

    Mathematically, the variance and the standard deviation are exclusively dependent upon each other - meaning that both contain the exact same information. Since neither is any different than the other, we are free to choose whichever we find more intuitive to use.

    While we use the word "variance" in common communication to describe the spread of a set of data, it's mostly 'cause "standard deviation" doesn't roll off the tongue as well. (AFAIK)

    Quote Originally Posted by seven-deuce View Post
    I know what it is, the average of the squared differences from the mean, but it doesn't really mean anything to me.
    (I wrote this before I clicked your link, and it's nice to know that link covers it.)

    It's not, necessarily, an average. As pertains to poker stats The denominator in the summation is (n - 1), and not n.

    If the denominator is n, then it is the variance for a closed population.
    Since we're always talking about open populations with poker stats, the denominator needs to be (n - 1).

    Closed population = all data that can or ever will be acquired has been acquired.
    Open population = more data will eventually be added to the current data set.

    Quote Originally Posted by seven-deuce View Post
    Also, what can a standundering of variance teach us about poker?
    Understanding variance, and higher concepts like Confidence Intervals, allows us determine in advance how often we expect to be wrong when assigning Villains ranges based on stats. This is the most important thing. We can put a verifiable number on to how wrong we allow our guesses to be. We might be wrong for other reasons (especially in poker), but this much of our wrongness is accounted for.

    Why choose to be wrong at all? Why not choose to be 100% certain?
    Because the only way we can be 100% certain is if we say it's a number between 0% and 100%. That's not useful.

    OK, so why not just choose to say the stat is what the stat is, and that's enough?
    Because mathematically, that result by itself represents a very low Confidence Interval, even though it is the mode (most frequent outcome) of the distribution.

    By choosing a 95% CI, I choose to be wrong 1 time out of 20. If I want to use thinner error bars, I may choose to be wrong more often. If I choose a 75% CI, then I choose to be wrong 1 time out of 4, but my error bars are much thinner on my guess. If I try to use 99% CI, then for almost all poker applications, the predictions are too wide to be useful.
  31. #31
    MadMojoMonkey's Avatar
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    Quote Originally Posted by DJAbacus View Post
    I agree my analogy is flawed. I mean, an umbrella is infinitely more useful to someone standing in the rain than a Maths degree is to someone who wants to be a winning poker player.


    At least you have a sense of humor.

  32. #32
    Thanks for responding, feel free to wave the white flag any time.

    So the variance is always a squared number due to the calculation you perform to obtain it. And cm^2 implies area, like the area of a square is x cm^2. Is the area the variance is referring to the area of the bell curve? So the larger the area the wider the dispersion of the data around its mean? The higher the variance figure is the more dispersed the data is about its mean, and the lower the variance figure is the tighter data is clumped around the mean?

    Only when you square root the variance in order to get the standard deviation you're removing the square and hence converting it back into its original units mm, years or whatever it happened to be?

    And are they "the same" because they're the same information expressed in a different "format" for lack of a better word. If my assumption about the area was correct, then variance is the area the data points cover around the mean, and the std deviation is, well I dunno what it is yet.

    A good portion of that could be pure garbage, but I know how to calculate the variance and std deviation now, and basically understand what they indicate. The larger the variance is, the larger the std deviation will be. The smaller the variance, the smaller the std deviation. And large variance = a flatter curve, while small variance produces a steeper curve?
    Erín Go Bragh
  33. #33
    MadMojoMonkey's Avatar
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    Quote Originally Posted by seven-deuce View Post
    Thanks for responding, feel free to wave the white flag any time.
    Nevar!

    Quote Originally Posted by seven-deuce View Post
    So the variance is always a squared number due to the calculation you perform to obtain it. And cm^2 implies area, like the area of a square is x cm^2.
    Yes.

    Quote Originally Posted by seven-deuce View Post
    Is the area the variance is referring to the area of the bell curve?
    No. Every probability distribution has an associated Cumulative Distribution function, which is the integral of the prob. dist.
    An integral gives the area under the curve (of whatever function it integrates).

    Quote Originally Posted by seven-deuce View Post
    So the larger the area the wider the dispersion of the data around its mean? The higher the variance figure is the more dispersed the data is about its mean, and the lower the variance figure is the tighter data is clumped around the mean?
    Yes and yes (and yes).

    Nit picking: Variance is a value, or scalar. A figure is a 2D object (a form is a 3D object).

    Quote Originally Posted by seven-deuce View Post
    Only when you square root the variance in order to get the standard deviation you're removing the square and hence converting it back into its original units mm, years or whatever it happened to be?

    And are they "the same" because they're the same information expressed in a different "format" for lack of a better word.
    Yes.

    Quote Originally Posted by seven-deuce View Post
    If my assumption about the area was correct, then variance is the area the data points cover around the mean, and the std deviation is, well I dunno what it is yet.
    In your example, the only reason it's an "area" is because the data was in length units, and length units, squared, is an area.
    Don't put too much weight on my use of the word area as pertains to variance.

    If we were measuring the surface area of the dogs (why? STFU and measure dem bitches!), then the variance would be in mm^4... and we'd want to take the square root so that it was in mm^2, like the areas we measured in the data set.

    It's about the apples-to-apples comparison.

    Not only do the units match, but the value is directly cooperative with our mean.
    So, if our mean value is 20 with a stdev of 5, then that's easy to immediately read off:
    That's 20 +/- 5 @ ~68% CI and 20 +/- 10 @ ~95% CI

    Quote Originally Posted by seven-deuce View Post
    A good portion of that could be pure garbage, but I know how to calculate the variance and std deviation now, and basically understand what they indicate. The larger the variance is, the larger the std deviation will be. The smaller the variance, the smaller the std deviation. And large variance = a flatter curve, while small variance produces a steeper curve?
    Not much garbage, and it was my use of the word "area" that started it. Other than that, yes, yes, yes -
    *ugh*
    - since we're using cooperative distributions.

    There are distributions with no well-defined mean or variance, but thankfully they do not concern us in poker.
  34. #34
    MadMojoMonkey's Avatar
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    Quote Originally Posted by MadMojoMonkey View Post
    Not only do the units match, but the value is directly cooperative with our mean.
    So, if our mean value is 20 with a stdev of 5, then that's easy to immediately read off:
    That's 20 +/- 5 @ ~68% CI and 20 +/- 10 @ ~95% CI
    Sorry. This part is pretty bad on my part. These numbers assume we're talking about a random variable that follows the normal distribution, which I've repeatedly pointed out is of little-to-no use for us.

    However, the fact that +/- 1 stdev is ~68% CI is the constant thing here. The distribution may not be symmetrical with the plusses and minuses. For stats like VPIP and PFR, we would distrust them if they were symmetrical.

    If we have 1.01% PFR after 99 hands, and we fold the next hand, then we have 1.00% PFR. However, if we had raised the hand, then we'd have 2.00% PFR.
    So a fold causes a change of -0.01%, but a bet yields a change of +0.99%.
    If our error bars around the 1.01% were symmetrical, we'd be very suspicious about them. Well, some mathematicians were a while ago, and they figured out some workarounds... like the Wilson Score interval... which gives stable results and is not symmetrical.

    So the plusses will be bigger than the minuses for a stat with frequency less than 50%, and vice versa for a stat with freq. greater than 50%.
    Only a stat that is exactly 50% will have a symmetrical distribution.
  35. #35
    Okay, thanks for the help Mojo. I hoked out an old textbook that covers this stuff. I'll bump this thread with questions if I hit any snags.
    Erín Go Bragh
  36. #36
    MadMojoMonkey's Avatar
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    I also think it's worth noting that if we start talking about winrate, then we're in weird waters.

    The problem I have is that it seems to me that simply defining your winrate as your BR/hands makes for a poor estimation. I mean... when we're rolled for 20 - 50 BI, a swing of 4 BI is a significant portion of our total BR. Our estimated winrate would swing all over the place, based on randomness playing out as wins and losses. It ignores the path to the final result.

    A more stable estimation of winrate would be to look at a plot of BR vs hands and perform a linear regression on the data. The linear regression yields the straight line which minimizes the variance from the line. Note that it minimizes the variance in the vertical direction, and not perpendicular to the line, which I always have to think real hard to see. This is called a "best fit line" or a "trend line."

    The slope of that line would be the best mathematical prediction of winrate.

    In performing the linear regression, we have access to more stats - this time more directly about the winrate.

    We can calculate the variance in the slope, which would be the variance in our estimation of our winrate.

    I think that the variance in the slope could only converge on 0 if our "true" winrate was constant over all time and the overall skill of our opponents was the same. Otherwise... the variance in the slope will reflect those subtle, long-term changes.
  37. #37
    I'm confused, we don't define our win rate as br/hands.

    Slightly off topic do you know any good books about stats, never something I've been a huge fan of but I want to take a look at it, first year undergrad type stuff.
  38. #38
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    Quote Originally Posted by ImSavy View Post
    I'm confused, we don't define our win rate as br/hands.

    Slightly off topic do you know any good books about stats, never something I've been a huge fan of but I want to take a look at it, first year undergrad type stuff.
    Our win-rate is the rate of change of our BR/hands.
  39. #39
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    Quote Originally Posted by ImSavy View Post
    I'm confused, we don't define our win rate as br/hands.
    Spoony has it right.

    winrate is the change of BR with respect to (w.r.t.) change in hands.
    As a differential equation:
    {winrate} = d{BR}/d{hands}


    My point is that if we look at a sample at the end of n-hands, we can apply math and make an estimate of that rate of change... but it's a poor estimate based on 1 data point. (Well, 2... we call the starting BR 0 and the starting hand number 0, making a data point)

    If we look at the change over each individual hand, then we have a (n-1) more data points to work with. All that extra data allows us to use more refined and robust methods of estimating the average slope... which is the rate of change in BR w.r.t. hands.

    Now, my proposition is to assume that the winrate is a constant throughout the data set and to use a linear regression to model the slope of BR w.r.t. hands.

    Here's a pretty good argument for why a linear regression is a reasonable choice:
    TL;DR : it is
    Spoiler:
    We could do a quadratic regression (probably a bad idea) or a cubic regression (probably better than quad.). If we are looking at our BR over a very long period during which we've been moving up stakes, then we'd probably want to use an exponential regression. Of course, we could definitely do a Fourier analysis on the data to look for periodic (repeating) patterns.

    I'm saying, there are lots of ways to show a trendline through data, but the one that yields the lowest variance is not necessarily the best model of the phenomenon. For regressions like x^2, x^3, x^n, ... the bigger the n, the lower the variance until you get the n high enough that the line passes through every data point. Sure, great line... but does it model the situation? No.

    I think we're really interested in modeling our winrate as a constant over a sample size... so a linear regression works well, and gives us a good basis for extrapolation. A cubic fit would show a bit of wave in our winrate (which may or may not indicate anything other than card variance), but it would make a poor extrapolation (trending either to + or - very big very fast, just outside our data set, usually).


    Quote Originally Posted by ImSavy View Post
    Slightly off topic do you know any good books about stats, never something I've been a huge fan of but I want to take a look at it, first year undergrad type stuff.
    Sorry, but I don't. I didn't like my text book on the subject, although it was slightly less annoying than almost all other math text books.
    My favorite math book is umm...

    Here's the 7th edition (huge 1000+ page .pdf), but I have the 6th in hard back... apparently they took out the section on prob/stats.

    This came up, among dozens of other, which I didn't look at. It looks like a prob/stats textbook to me.

    Just google "probability and statistics textbook pdf" and see the 636,000 results
  40. #40
    spoonitnow's Avatar
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    Related math nerd note:

    If our "true" win-rate stayed constant, then our winnings wouldn't converge to their true value over time, but our win-rate would.
  41. #41
    Quote Originally Posted by MadMojoMonkey View Post
    Nevar!


    Yes.


    No. Every probability distribution has an associated Cumulative Distribution function, which is the integral of the prob. dist.
    An integral gives the area under the curve (of whatever function it integrates).


    Yes and yes (and yes).

    Nit picking: Variance is a value, or scalar. A figure is a 2D object (a form is a 3D object).


    Yes.


    In your example, the only reason it's an "area" is because the data was in length units, and length units, squared, is an area.
    Don't put too much weight on my use of the word area as pertains to variance.

    If we were measuring the surface area of the dogs (why? STFU and measure dem bitches!), then the variance would be in mm^4... and we'd want to take the square root so that it was in mm^2, like the areas we measured in the data set.

    It's about the apples-to-apples comparison.

    Not only do the units match, but the value is directly cooperative with our mean.
    So, if our mean value is 20 with a stdev of 5, then that's easy to immediately read off:
    That's 20 +/- 5 @ ~68% CI and 20 +/- 10 @ ~95% CI


    Not much garbage, and it was my use of the word "area" that started it. Other than that, yes, yes, yes -
    *ugh*
    - since we're using cooperative distributions.

    There are distributions with no well-defined mean or variance, but thankfully they do not concern us in poker.
    Quote Originally Posted by ImSavy View Post
    I'm confused, we don't define our win rate as br/hands.

    Slightly off topic do you know any good books about stats, never something I've been a huge fan of but I want to take a look at it, first year undergrad type stuff.

    http://www.amazon.co.uk/Quantitative.../dp/0077109023 - This book was recommended to us for a "Quantitative Analysis" class we had to take during year 1 of an accounting degree. I can't vouch for it because I never read it. You might be able to find a PDF online.
    Erín Go Bragh
  42. #42
    Quote Originally Posted by ImSavy View Post
    Slightly off topic do you know any good books about stats, never something I've been a huge fan of but I want to take a look at it, first year undergrad type stuff.
    Andy Field sells a ton of stats books, so fair to assume they're good. Generally speaking though, first year UG and A-Level Stats are no different in my experience, but the A-Level textbooks are much better written. EdExcel or any other exam board will have a decent text for sale.

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