Comments on: A Frequentist And A Bayesian Walk Into Infinity… http://popsych.org/a-frequentist-and-a-bayesian-walk-into-infinity/ The Internet's Best Evolutionary Psycholo-guy Wed, 03 Jan 2018 01:05:13 +0000 hourly 1 http://wordpress.org/?v=3.4.2 By: Do People “Really” Have Priors? | Pop Psychology http://popsych.org/a-frequentist-and-a-bayesian-walk-into-infinity/#comment-535 Do People “Really” Have Priors? | Pop Psychology Tue, 05 Mar 2013 00:14:51 +0000 http://popsych.org/?p=916#comment-535 [...] as well, so this question applies well to any statistician). Like many concepts in statistics, this one seems to both be useful in certain situations and able to easily lead one astray in others. Today I’d like to consider a thought experiment [...] [...] as well, so this question applies well to any statistician). Like many concepts in statistics, this one seems to both be useful in certain situations and able to easily lead one astray in others. Today I’d like to consider a thought experiment [...]

]]> By: A Frequentist And A Bayesian Walk Into Infinity… | Pop Psychology | Statistics- Bayes or Frequentist | Scoop.it http://popsych.org/a-frequentist-and-a-bayesian-walk-into-infinity/#comment-388 A Frequentist And A Bayesian Walk Into Infinity… | Pop Psychology | Statistics- Bayes or Frequentist | Scoop.it Fri, 16 Nov 2012 14:39:43 +0000 http://popsych.org/?p=916#comment-388 [...] A Frequentist And A Bayesian Walk Into Infinity… Posted on November 10, 2012 by Jesse Marczyk. I'm going to preface this post by stating that statistics is not my primary area of expertise. Admittedly, this might not be the best way of ...  [...] [...] A Frequentist And A Bayesian Walk Into Infinity… Posted on November 10, 2012 by Jesse Marczyk. I'm going to preface this post by stating that statistics is not my primary area of expertise. Admittedly, this might not be the best way of …  [...]

]]> By: Min http://popsych.org/a-frequentist-and-a-bayesian-walk-into-infinity/#comment-387 Min Fri, 16 Nov 2012 03:06:13 +0000 http://popsych.org/?p=916#comment-387 "Further, the amount of time that the Bayesian would believe that the fair coin was heavily biased would, over time, be vastly overwhelming by the amount of time they’d believe it was fair." With a flat prior, which is what I think you have in mind, the Bayesian probability for the next toss at each point will be closer to 50:50 than the frequentist probability estimate. Work it out. “Further, the amount of time that the Bayesian would believe that the fair coin was heavily biased would, over time, be vastly overwhelming by the amount of time they’d believe it was fair.”

With a flat prior, which is what I think you have in mind, the Bayesian probability for the next toss at each point will be closer to 50:50 than the frequentist probability estimate. Work it out.

]]> By: Jesse Marczyk http://popsych.org/a-frequentist-and-a-bayesian-walk-into-infinity/#comment-386 Jesse Marczyk Fri, 16 Nov 2012 02:00:08 +0000 http://popsych.org/?p=916#comment-386 Ah. I was getting at the idea that if, after each toss, you asked for the posteriors, you'd see them representing an infinite number of values between -1 and 1, where -1 is totally biased towards tails and 1 totally towards heads. The specific degree of bias falls into 3 general categories (towards heads, tails, or neither), but the precise degree can be represented across an infinite number of possible values in that range. Ah. I was getting at the idea that if, after each toss, you asked for the posteriors, you’d see them representing an infinite number of values between -1 and 1, where -1 is totally biased towards tails and 1 totally towards heads. The specific degree of bias falls into 3 general categories (towards heads, tails, or neither), but the precise degree can be represented across an infinite number of possible values in that range.

]]> By: Jeff WItmer http://popsych.org/a-frequentist-and-a-bayesian-walk-into-infinity/#comment-385 Jeff WItmer Fri, 16 Nov 2012 01:57:14 +0000 http://popsych.org/?p=916#comment-385 Perhaps I don't understand what you are envisioning. Suppose the first 1000 tosses are HTH followed by 997 Tails; then we see some heads, about 50-50 for the next several thousand tosses. We agree that this is highly unlikely, but possible. OK, then for a stretch of time (say, from toss 12 through toss 3000 or so) the Bayesian would think the coin to be biased towards tails. I count that as coming to the conclusion that the coin is unfair in one way, not "a relatively infinite number of ways." Perhaps I don’t understand what you are envisioning. Suppose the first 1000 tosses are HTH followed by 997 Tails; then we see some heads, about 50-50 for the next several thousand tosses. We agree that this is highly unlikely, but possible. OK, then for a stretch of time (say, from toss 12 through toss 3000 or so) the Bayesian would think the coin to be biased towards tails. I count that as coming to the conclusion that the coin is unfair in one way, not “a relatively infinite number of ways.”

]]> By: Jesse Marczyk http://popsych.org/a-frequentist-and-a-bayesian-walk-into-infinity/#comment-384 Jesse Marczyk Thu, 15 Nov 2012 23:18:14 +0000 http://popsych.org/?p=916#comment-384 My point does not assume the Bayesian only cares about the last X number of observations. At some point, the Bayesian will have flipped the coin N times. Given an infinite amount of time, there will be strings of consecutive heads or tails that are 2N long, 5N, 0.5N or any constant times N, no matter how large N is. Yes, it's Incredibly improbable and exponentially more improbable as N increases. Further, the amount of time that the Bayesian would believe that the fair coin was heavily biased would, over time, be vastly overwhelming by the amount of time they'd believe it was fair. Accordingly, to the extent this would actually pose a problem in real-world data collection is debatable but, for the purposes of this example, the Bayesian would continually update their posteriors to, at various points, represent beliefs about the bias of the coin flips from almost completely biased towards tails or heads. My point does not assume the Bayesian only cares about the last X number of observations. At some point, the Bayesian will have flipped the coin N times. Given an infinite amount of time, there will be strings of consecutive heads or tails that are 2N long, 5N, 0.5N or any constant times N, no matter how large N is. Yes, it’s Incredibly improbable and exponentially more improbable as N increases. Further, the amount of time that the Bayesian would believe that the fair coin was heavily biased would, over time, be vastly overwhelming by the amount of time they’d believe it was fair. Accordingly, to the extent this would actually pose a problem in real-world data collection is debatable but, for the purposes of this example, the Bayesian would continually update their posteriors to, at various points, represent beliefs about the bias of the coin flips from almost completely biased towards tails or heads.

]]> By: Jeff Witmer http://popsych.org/a-frequentist-and-a-bayesian-walk-into-infinity/#comment-383 Jeff Witmer Thu, 15 Nov 2012 23:04:06 +0000 http://popsych.org/?p=916#comment-383 "They would also come to the conclusion that the coin is unfair in a relatively infinite number of ways, depending on what time in the flipping process they make conclusions." No, they would not. The quoted sentence seems to presume that the Bayesian would at any give time only consider the most recent observations (say, the last 1000). After a few hundred tosses the data collected will have overwhelmed the prior and from that point forward the posterior interval around .5 is going to have so much density that an occasional string of many heads (or tails) won't change the posterior distribution much at all. “They would also come to the conclusion that the coin is unfair in a relatively infinite number of ways, depending on what time in the flipping process they make conclusions.” No, they would not. The quoted sentence seems to presume that the Bayesian would at any give time only consider the most recent observations (say, the last 1000). After a few hundred tosses the data collected will have overwhelmed the prior and from that point forward the posterior interval around .5 is going to have so much density that an occasional string of many heads (or tails) won’t change the posterior distribution much at all.

]]> By: Min http://popsych.org/a-frequentist-and-a-bayesian-walk-into-infinity/#comment-382 Min Wed, 14 Nov 2012 17:35:01 +0000 http://popsych.org/?p=916#comment-382 A Bayesian would not necessarily come to the conclusion that the coin is fair. To do so would require what I. J. Good called a Type II probability distribution, i. e., a prior distribution of priors, and one that included a prior that the coin is fair. :) A Bayesian would not necessarily come to the conclusion that the coin is fair. To do so would require what I. J. Good called a Type II probability distribution, i. e., a prior distribution of priors, and one that included a prior that the coin is fair. :)

]]> By: Min http://popsych.org/a-frequentist-and-a-bayesian-walk-into-infinity/#comment-381 Min Wed, 14 Nov 2012 17:32:32 +0000 http://popsych.org/?p=916#comment-381 Thanks for the clarification. :) Unfortunately, I am very busy through the weekend, but I would like to come back to this topic next week. :) Thanks for the clarification. :)

Unfortunately, I am very busy through the weekend, but I would like to come back to this topic next week. :)

]]> By: Min http://popsych.org/a-frequentist-and-a-bayesian-walk-into-infinity/#comment-380 Min Wed, 14 Nov 2012 17:31:12 +0000 http://popsych.org/?p=916#comment-380 OC, that is true for the frequentist theory, as well. :) OC, that is true for the frequentist theory, as well. :)

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