Comments on Statistics as inverse probabilityTypePad2013-04-22T00:26:23Zjunkchartshttp://junkcharts.typepad.com/numbersruleyourworld/tag:typepad.com,2003:http://junkcharts.typepad.com/numbersruleyourworld/2013/04/statistics-as-inverse-probability/comments/atom.xml/Max commented on 'Statistics as inverse probability'tag:typepad.com,2003:6a00d8341e992c53ef01901c5b6431970b2013-05-19T21:37:30Z2013-05-21T03:48:44ZMaxThe example seems related to Gambler's fallacy http://en.wikipedia.org/wiki/Gambler%27s_fallacy<p>The example seems related to Gambler's fallacy <a href="http://en.wikipedia.org/wiki/Gambler%27s_fallacy" rel="nofollow">http://en.wikipedia.org/wiki/Gambler%27s_fallacy</a></p>zbicyclist commented on 'Statistics as inverse probability'tag:typepad.com,2003:6a00d8341e992c53ef01901bc48247970b2013-05-02T14:00:52Z2013-05-03T05:07:04ZzbicyclistWell, it depends on how many coins you started off with. If we started off with some very, very, very...<p>Well, it depends on how many coins you started off with. If we started off with some very, very, very large number of coins (to Gelman's point, an impossible number), it would not be unusual that we'd get one fair coin that landed heads 1,000,000 times in a row.</p>
<p>But we can easily move this problem into the space of the possible. If we look at how advisory (not index) mutual funds are marketed, a strategy is to start a large number of funds with small seed money. Some strategies will perform well, possibly by accident. These get heavily promoted and the rest get quietly folded.</p>
<p>We're told that the fund has a glowing track record (for instance, it has always gone up -- the equivalent in this space of always coming up heads) but not given full information about the sampling space. </p>
<p>So, the best conclusion in these situations isn't that those who run the fund are geniuses for having a run of X excellent years, but people who've flipped a whole bunch of coins and will return to coin-flip accuracy as soon as you invest in the fund. </p>
<p>(I note the sum of coin flips is a random walk, and there's a whole literature on stock prices as random walks)</p>Jordan commented on 'Statistics as inverse probability'tag:typepad.com,2003:6a00d8341e992c53ef017eeabc6da4970d2013-05-01T19:00:51Z2013-05-02T03:12:29ZJordan...and yes, that's a reference to a Dominican beer.<p>...and yes, that's a reference to a Dominican beer.</p>Jordan commented on 'Statistics as inverse probability'tag:typepad.com,2003:6a00d8341e992c53ef019101b4cec5970c2013-05-01T18:54:54Z2013-05-02T03:12:29ZJordanIn my daily experiences with data (albeit not a large sample size) I don't know if the coin is fair....<p>In my daily experiences with data (albeit not a large sample size) I don't know if the coin is fair. To extend the analogy, I'm usually flipping some foregin bottle cap with little prior knowledge as to whether the cap comes up on the "Presidente" side more often than the other side.</p>Kaiser commented on 'Statistics as inverse probability'tag:typepad.com,2003:6a00d8341e992c53ef019101b09be7970c2013-05-01T04:20:50Z2013-05-02T03:12:29ZKaiserhttp://junkcharts.typepad.com/numbersruleyourworldAndrew: I like your formulation.<p>Andrew: I like your formulation.</p>Andrew Gelman commented on 'Statistics as inverse probability'tag:typepad.com,2003:6a00d8341e992c53ef01901baa841b970b2013-04-29T00:14:27Z2013-04-29T00:57:36ZAndrew Gelmanhttp://andrewgelman.comThe comment by Dimitri above is interesting because it reveals a form of innumeracy. (1/2) to the millionth power is...<p>The comment by Dimitri above is interesting because it reveals a form of innumeracy. (1/2) to the millionth power is not "extremely unlikely," it's in any real physical sense impossible.</p>
<p>I will throw in one more twist. We are given three incompatible statements:<br />
1. The coin is "fair." This concept is not clearly defined, but I will take it to mean that, when flipped, there is a 1/2 chance it lands heads.<br />
2. It was flipped 1 million times.<br />
3. All flips turned out heads.<br />
Any two of these three statements can be true, but not all three.</p>
<p>One possibility is Kaiser's, that 2 and 3 are true, hence 1 is false. Another possibility is that 1 and 2 are true, but 3 are false. After all, how would we know that a coin came up heads a million times. We're taking someone's word for it. See Section 3 of <a href="http://www.stat.columbia.edu/~gelman/research/published/feller8.pdf" rel="nofollow">this paper</a> for a similar example.</p>Jordan commented on 'Statistics as inverse probability'tag:typepad.com,2003:6a00d8341e992c53ef017d4313c08c970c2013-04-24T15:14:34Z2013-04-27T15:56:29ZJordanGood explanation of the difference between probability and statistics. I wish I had this many years ago. I am only...<p>Good explanation of the difference between probability and statistics. I wish I had this many years ago. I am only now getting a grasp on the difference between probability and null hypothesis significance testing (NHST) versus exploratory data analysis and Bayesian thinking. I was taught the former, when what I often want is the latter.</p>Dimitri commented on 'Statistics as inverse probability'tag:typepad.com,2003:6a00d8341e992c53ef017eea83ecdf970d2013-04-24T01:00:32Z2013-04-27T15:56:29ZDimitriWhat if the fairness of the coin was properly established with some robust scientific method and then it came up...<p>What if the fairness of the coin was properly established with some robust scientific method and then it came up heads 1,000,000 times? Extremely unlikely, but we are talking probability concepts here, so it is conceptually possible. I guess the book's author was trying to demonstrate that even if a guaranteed fair coin had a long streak, that does not influence the next single toss outcome.<br />
In regards to betting on that single outcome, I'd say people who rush to make bets in such cases are betting on the low probability of the streak continuing, rather than on the constant 0.5 of the next toss.</p>Kaiser commented on 'Statistics as inverse probability'tag:typepad.com,2003:6a00d8341e992c53ef017d430ca73b970c2013-04-23T15:17:18Z2013-04-23T15:17:18ZKaiserhttp://junkcharts.typepad.com/numbersruleyourworldTom: Exactly. That's the origin of the term "inverse probability".<p>Tom: Exactly. That's the origin of the term "inverse probability".</p>Tom commented on 'Statistics as inverse probability'tag:typepad.com,2003:6a00d8341e992c53ef017eea80f7a5970d2013-04-23T15:14:06Z2013-04-27T15:56:29ZTomIt seems to me that in your explanation above, the statistician and the probabilist are working the problem from opposite...<p>It seems to me that in your explanation above, the statistician and the probabilist are working the problem from opposite ends. The statistician is starting with the sampled data and inferring something about the population and sampling frame (in this case, the coin and the sampling procedure). The probabilist is starting with known characteristics of the population (i.e. the coin is fair) and the sampling procedure (each sample comes from an independent event), then estimates the probability of seeing a particular sample distribution. Is this fair, or have I misunderstood something?</p>