prior probability is a blog inspired by Bayesian methods, computer science, and Oliver Wendell Holmes’s prediction theory of law. We also like to blog about the philosophy of probability, spontaneous order, and prediction markets. In addition, we feature our love of maps — especially unorthodox and unusual maps — as well as various odds and ends of special interest to us, including all things Cuban, Caribbean, and Latin American. (Image below courtesy of Giovanni Parmigiani.)

Hey, where did you get your priors?

14 Responses to About

  1. Hello: I finally hear from you. Love aunt Julie

  2. Mayo says:

    The “washout theorems” lead to converge in very special cases with non-extreme priors, assuming iid data, and supposing you are concerned with assessing an event where there are long-run repetitions. None of these assumptions hold in appraising scientific hypotheses. Even then it can be shown that two agents will have posteriors that differ by as much as one wants. (Kyburg). But the real problem with appealing to wash-out theorems is that this has nothing to do with our critical appraisal of scientific inferences and the data before us right now.We scrutinize how well or poorly tested claims are, and do not settle for “if you keep going on with iid long-run repetitions of the same study, you should eventually converge”. We want approximate truth, not mere coherence. Finally, the possibility of eliciting subjective priors has been deemed such a waste of time in science that it is rarely done in practice, to my knowledge. The most popular Bayesian approaches appeal to nonsubjective or “default” priors that do not influence the data too much. But even there, it’s unclear what they mean (they are not beliefs, being improper) and there are rival default prior systems that recommend very different nonsubjective priors (e.g., Berger vs Bernardo priors). You can find discussion on my blog and related publications.

    • enrique says:

      Thanks for your thoughtful comment … Also, please “stay tuned” as I’m in the process of reading chapter 3 of your excellent book on “Error and Growth of Experimental Knowledge” to better appraise your critique of Bayesian subjectivism in the domain of science …

      • Mayo says:

        Enrique: Thank you. That chapter is focused more on subjective Bayesianism in philosophy of science. Obviously, they are connected, but I think Bayesian statistics, in practice, deserves its own discussion. I think the situation regarding the foundations of Bayesian statistical practice has changed a lot in recent times: with exceptions, it seems that even those who still believe in subjective philosophical foundations, deep down, do not apply it. Or, do you think they do? That’s practically the only topic that requires updating in a new edition of EGEK (Mayo 1996). Here’s a relevant post: http://errorstatistics.com/2012/11/21/irony-and-bad-faith-deconstructing-bayesians-reblog/

    • enrique says:

      I took the liberty of reblogging your amazing slides on “The science wars and the statistics wars,” which I found very helpful. I am enjoying your blog very much, and you are rapidly becoming my favorite philosopher of statistics … I only wish I had discovered your work (and your blog) earlier!

    • Charlie says:

      Guys, I need to read up on this to understand whats going on. what are prior probabilities. what is a “prior probability,” who are Berger and Bernardo?

      • Hi Charlie. I’m embarrassed to say that I don’t know off the top of my head who Berger and Bernardo are … But I will find out.

        As for prior probabilities, I would be happy to say a word or two. Briefly, a “prior” (or “prior probability”) is a degree of belief one has that a proposition is true or false. The main thing is that our beliefs are not all or nothing (i.e. either TRUE or FALSE) but instead we have degrees of belief ranging from 0 to 1. For example, if you strongly believe it won’t rain today (but are not 100% sure that it won’t rain), you might assign this belief a value of .9 or .8, depending on how strong your belief is. Notice too that degrees of belief are subjective: I might assign a low value, while you might assign a high value.

  3. Katelyn Scafidi says:

    Hi Prof. Pujol! I am still following your blogs! I would like to catch up and see how you are and pick your brain about some topics. Let me know if you are ever free.

  4. Pingback: Was Holmes a Bayesian? | prior probability

  5. Paul says:


    I am reaching out to you (a shot in the dark)…as I found you on twitter and I have a probability question that I am trying to figure out. I’ve tried on my own without much success. 😦

    I am a financial advisor in California and the article below was published in the Wall Street Journal a couple of weeks back and it really got me curious about the process of figuring out how the answer was determined. I’ve highlighted the sentences I am trying to figure out how they came up with the answer of 143 flips. If you have any interest in having a little fun and letting me know the process (and calculations) in solving the problem I would great appreciate it! If you are not interested do know anyone that might be? Thank you again in advance.

    WSJ Article below:

    Is it luck or skill? Picking a fund manager who can beat the index is tough, but picking one who beats it through actual ability is far more difficult.

    Victor Haghani, a co-founder of one of the best-known investment firms in history, says the most surprising thing is that people have great confidence that they can pick these super-talented fund managers. Currently the chief executive of Elm Partners, which espouses index investing for wealthy clients, Mr. Haghani will try to prove that to you with a simple test.

    He and two colleagues told several hundred acquaintances who worked in finance that they would flip two coins, one that was normal and the other that was weighted so it came up heads 60% of the time. They asked the people how many flips it would take them to figure out, with a 95% confidence level, which one was the 60% coin. Told to give a “quick guess,” nearly a third said fewer than 10 flips, while the median response was 40. The correct answer is 143.

    Mr. Haghani’s belief in indexing means he has a vested interest in the outcome. His earlier experience as an active investor gives him perspective on how hard it is to beat the market. Mr. Haghani was a co-founder of Long Term Capital Management, the hedge fund that had spectacular results from exploiting real market anomalies before its failure nearly took down the global financial system in 1998.

    The research applies directly to picking fund managers. We already know that most active managers fail to beat an index fund in any given year, yet many people pay up for managers they believe have the skill to do so.

    For example, performance of the Fairholme Fund won its manager, Bruce Berkowitz, the distinction of domestic equity fund manager of the decade in 2010 from Morningstar. The fund had beaten its benchmark most years, and its compound annual return was an impressive 13 percentage points better than peers on an annualized basis.

    The fund slumped spectacularly the very next year, lagging behind the S&P 500 by nearly 35 percentage points. The inevitable schadenfreude in the financial media elicited a rebuke from Mr. Berkowitz in his 2011 letter to shareholders. “One circling of the sun is too short a time to differentiate between good and lucky,” he wrote.

    He was right, but perhaps not in the way that he meant. Writing in 2014, after two more years—one bad and one good—for Fairholme, Mark Hebner of Index Fund Advisors used statistics to determine if skill was responsible for Fairholme’s past success. Using a measure of how much Mr. Berkowitz’s fund tended to deviate from its benchmark and the same 95% confidence interval used in the coin-flipping paper, he concluded that it was too soon to tell. It would take 18 years to get an answer—longer than the fund had then been in business and far longer than most investors’ patience with a manager. For managers with more volatile results than Mr. Berkowitz’s, Mr. Hebner determined that it could take several hundred years of performance to discern the manager’s true ability.

    There are other explanations for investing success that aren’t sustainable. Strategies such as value investing can fall in and out of favor. And, unlike loaded coins, an investment strategy that is a legitimate winner may not stay that way because others can try to mimic it. Long Term Capital Management produced spectacular results, but then its strategy was copied, and Mr. Haghani saw the erosion of their edge firsthand.

    The confidence investors place in their ability to pick skilled managers is ultimately costly. Investors buy top-performing and top-rated funds without any ability to determine if skill or luck produced the gains. The result is that the typical investor in active funds lags behind even those funds’ return by quite a bit.

    For example, when Mr. Berkowitz began his triumphant decade, his fund was tiny. By the end, it was large and then probably got a further boost from the award. During the year that it lagged behind the market by 35 percentage points, it took a lot more people’s savings down with it. For some successful funds, the effect is so stark that a fund manager retires with vast personal wealth and a wonderful reputation while never generating a net dollar of value for investors.

    Coin flipping can get expensive.

    Thank you again and I appreciate any direction you can provide.

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