Timeline for What is the intuition behind beta distribution?
Current License: CC BY-SA 4.0
49 events
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| Oct 4 at 20:10 | comment | added | FountainTree | Great intuition! The beta distribution is our belief in a binomial! | |
| Oct 16, 2022 at 1:40 | comment | added | Galen | Here is a Python gist for David Robinson's plots of the beta distribution. | |
| May 26, 2022 at 15:26 | comment | added | Alex Ramalho | Amazing explanation. Even for someone who doesn't follow/know the rules of baseball ⚾️😁 thanks! | |
| May 21, 2020 at 21:29 | comment | added | Sau001 | You made it so easy. Thank you. | |
| May 19, 2020 at 7:13 | comment | added | Jazz | Statistics is fun when people like @DavidRobinson exists. Thanks, David. It made my life easier :) | |
| Apr 12, 2020 at 18:53 | history | edited | Alexis | CC BY-SA 4.0 |
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| Jan 29, 2020 at 18:28 | comment | added | max | IIUC, this answer says "The intuition for the beta distribution is that it makes math easy when we do Bayesian updates". I feel this is not an intuition, but a technical convenience. Intuition would be "If we have this setup, the outcome has a beta distribution". For example, I just saw such an intuition in Stéphane Laurent's answer below. | |
| Jan 9, 2018 at 7:20 | comment | added | ego_ | Isn't this sorely lacking the point that beta distribution is CONTINUOUS...this example is basically a binomial approximation to a Bernoulli process (which one often approximates beta-distributios with)? Wouldn't a better example be more along the lines of how large a proportion of the day a player spend on training? That can take any value between 0-1, i.e. being CONTINUOUS, whereas the number of hits he make will be DISCRETE and can't take any value between 0-1. | |
| Dec 5, 2017 at 19:45 | comment | added | TimeString | Hi @DavidRobinson, how did you decide $\alpha_0$ and $\beta_0$ to be 81 and 239, but not something else but with the same ratio (like 40, 115)? Is it because you're assuming a player plays (hits or misses) 300 times in a season? | |
| Sep 22, 2017 at 18:04 | comment | added | James Bowery | I see. So the narrower the distribution as specified by a and b, the more difficult it is to shift the mean of the probabilities. | |
| Sep 22, 2017 at 13:33 | comment | added | David Robinson |
@JamesBowery If the combined magnitude were much higher or lower it wouldn't be the same distribution, because it would have a much lower or higher, respectively, standard deviation. That is, the magnitude is covered by the second reason: "this distribution lies almost entirely within (.2, .35)".
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| Sep 22, 2017 at 13:05 | comment | added | James Bowery | @DavidRobinson you said "I came up with these parameters for two reasons", but you left out why you chose the combined magnitude which is, basically, how much weight you place on the prior. You can get about the same distribution by choosing much larger or smaller numbers resulting, respectively, in much less or more weight on new samples. | |
| Apr 23, 2017 at 15:08 | comment | added | David Robinson | @Wboy Thanks! I answer that here, by using the method of moments to get from a desired mean and standard deviation to the beta parameters. | |
| Apr 22, 2017 at 13:13 | comment | added | Wboy | Great explanation! can I ask, how did you get 81 and 219? I understand it fits after the graph plot, but how do I even go about obtaining the params? | |
| Apr 8, 2017 at 0:11 | history | edited | rolando2 | CC BY-SA 3.0 |
Corrected a tiny flaw; walks don't affect batting average.
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| Mar 21, 2017 at 22:38 | comment | added | David Robinson | @MichaelChirico true- in fact, that's what I did in the e-book that expands on my answer here varianceexplained.org/r/empirical-bayes-book | |
| Mar 21, 2017 at 22:34 | comment | added | MichaelChirico | It may be useful to, instead of using a heuristic to fit the Beta prior, fit it (e.g. with MLE) to historical batting averages, which are not too hard to find. | |
| Nov 9, 2016 at 14:55 | comment | added | Pig | Thanks for your answer - while what you said makes sense, roughly fitting mean and variance can't be the only way of modeling prior, ya? Should I consider this as a all-purpose generic model, and if my situation is more specific, I should specifically engineer a model for the prior and update the posterior as data comes in? Are there more general-purpose parametrized models like beta distribution? | |
| Apr 1, 2016 at 18:34 | history | bounty ended | Glen_b | ||
| S Sep 27, 2015 at 11:17 | history | suggested | KerrBer | CC BY-SA 3.0 |
The base alpha parameter should be 81 throughout (was sometimes written as 82).
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| Sep 27, 2015 at 9:47 | review | Suggested edits | |||
| S Sep 27, 2015 at 11:17 | |||||
| Jul 30, 2015 at 4:52 | comment | added | wsw | Very nice answer, but you glossed over the concept of Bayesian estimation towards the end, which is why some readers did not understand how you obtained the posterior distribution. | |
| Jul 4, 2015 at 2:00 | comment | added | David Robinson | @bernie2436 That's right! | |
| Jul 3, 2015 at 22:31 | comment | added | bernie2436 | @Davidrobinson awesome. To clarify in your last comment "at that point" means on the y axis when x is just so slightly over .300? | |
| Jul 3, 2015 at 22:15 | comment | added | David Robinson | @bernie2436 For example: suppose there is a probability of 2% that the player's batting average is between .300 and .301. To get the probability density at that point, we would do .02 / (.301 - .300), which computes a density of 20. That will be the value at that point | |
| Jul 3, 2015 at 22:13 | comment | added | David Robinson | @bernie2436 It's a probability density. Technically a probability density has an intuitive meaning only when you integrate it; i.e. find the area under the curve. For example, the probability that a player's batting average is between .300 and .350 is equal to the area under the curve from .3 to .35. (If you try this with smaller and smaller intervals, you can see how the y-axis at each point is chosen!) | |
| Jul 3, 2015 at 22:05 | comment | added | bernie2436 | @DavidRobinson great answer. How should I interpret the y-axis of that distribution? I see how to generate it from the beta function -- but what does it mean? | |
| Nov 15, 2014 at 4:53 | history | edited | David Robinson | CC BY-SA 3.0 |
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| S Dec 30, 2013 at 20:25 | history | suggested | gkcn | CC BY-SA 3.0 |
Correct beta parameter is 219, not 221
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| Dec 30, 2013 at 20:18 | review | Suggested edits | |||
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| Oct 23, 2013 at 21:50 | comment | added | David Robinson | @wrongusername: it's because the beta is the conjugate prior of the binomial. I link to the math that proves this result in the post. | |
| Oct 23, 2013 at 21:29 | comment | added | wrongusername | Why the beta distribution in particular? Is it just because it works? It seems quite similar to the binomial distribution, except I don't understand the -1 in the exponent. | |
| S Oct 2, 2013 at 9:49 | history | suggested | Zhubarb | CC BY-SA 3.0 |
Changed the wording to emphasise that the Beta distribution is ideal for use as a prior for the Binomial distribution, not for all distributions.
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| Oct 2, 2013 at 9:44 | review | Suggested edits | |||
| S Oct 2, 2013 at 9:49 | |||||
| Jul 13, 2013 at 23:38 | comment | added | David Robinson | @user27997 Those gave the desired mean of .27, and a standard deviation that is very roughly realistic for batting averages (about .025). Incidentally, I give an explanation of how to calculate α and β from a desired mean and variance here. | |
| Jul 13, 2013 at 20:25 | comment | added | user27997 | @DavidRobinson - Nice explanation! Could you clarify where the initial values of α=81 and β=219 are coming from? or they are just examples? | |
| Jan 22, 2013 at 21:49 | comment | added | dimitriy | @DavidRobinson: Suppose that we are dealing with restaurant reviews. You have a restaurant with 1000 transactions, but you observe binary ratings on only 500 of them. The "silent" ones are a mixture of negatives and positives. One area where ignoring this is problematic is popular restaurants, where people are reluctant to be the millionth to rate them. | |
| Jan 22, 2013 at 20:20 | comment | added | David Robinson | @DimitriyV.Masterov: Could you explain what you mean? While missing data is an issue in many classification problems, in this case (predicting a binomial or multinomial using a prior) there's nothing you can do but ignore it. (That is to say, what good does it do you to know that there were 30 other hits if you don't know anything about their result?) | |
| Jan 22, 2013 at 18:49 | comment | added | dimitriy | @DavidRobinson: Any idea what to do if you have some unobserved outcomes? For baseball this does not make much sense since games are public, but for something like product or restaurants ratings this is rather important. | |
| Jan 16, 2013 at 21:44 | comment | added | Mike Dunlavey | + I like your explanation of how you update the distribution when you have more data. | |
| Jan 16, 2013 at 18:08 | comment | added | Neil G | You should point out that the prior need not be beta-distributed (unless you go with the Jeffreys' prior, $\alpha_0=\beta_0=1/2$ — only the likelihood must be beta distributed. | |
| Jan 15, 2013 at 20:22 | history | edited | David Robinson | CC BY-SA 3.0 |
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| Jan 15, 2013 at 19:58 | comment | added | dimitriy | Here's a similar example from John Cook using binary Amazon seller rankings with different number of reviews. The discussion of choosing a prior in the comments is particularly illuminating: johndcook.com/blog/2011/09/27/bayesian-amazon/#comments | |
| Jan 15, 2013 at 18:52 | vote | accept | ffriend | ||
| Jan 15, 2013 at 18:08 | history | edited | David Robinson | CC BY-SA 3.0 |
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| Jan 15, 2013 at 17:05 | history | edited | David Robinson | CC BY-SA 3.0 |
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| Jan 15, 2013 at 17:00 | history | edited | David Robinson | CC BY-SA 3.0 |
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| Jan 15, 2013 at 16:52 | history | edited | David Robinson | CC BY-SA 3.0 |
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| Jan 15, 2013 at 16:41 | history | answered | David Robinson | CC BY-SA 3.0 |