Who Were the Sea Peoples?

When I first heard of the axiomatic formulation of probability, it was presented to me as "resolving" the dispute between the frequentist interpretation of probability and the subjective interpretation. (Interestingly, John Maynard Keynes was one of the leading proponents of the subjective interpretation, while Richard von Mises was a prominent champion of the frequentist interpretation. So Keynes had running disputes with both of the von Mises brothers.) And the Wikipedia page just linked to describes the axiomatic formulation of probability as a "rival" to the frequentist interpretation. But to see the axiomatic formulation as rival to those other theories is a serious mistake (as  Kolmogorov  himself seemed to recognize -- see below). The frequentist and subjective theories of probability are concerned with the relation of probabilistic statements to the real world. They are essentially asking, "If we say, for instance, that the odds of rain today are 30...

Here : "None of this means Mickelson is doomed; it only means that he'll have to defy the odds." This statement reflects the modern tendency to assign the mathematical formulas of probability the same role as "Fate" played in the classical worldview, or witchcraft does for the Zande . There are no "odds" working against Mickelson simply because the 36-hole leader at Royal Troon has only won The Open two out of eight times. Mickelson is only competing with the other golfers on the course with him: all he has to do is to keep scoring better than they do, and he will win. He does not, besides beating them, also have to defeat some divine being called "the odds." If eight out of the previous eight 36-hole leaders had gone on to win, that would not make Mickelson's task one bit easier, and if zero out of eight had won, it would not make his task one bit harder. (All of this, of course, is barring the possibility that Mickelson has adopt...

Nate Silver has been getting some flack for his declaration that Donald Trump had a 2% chance of winning the GOP nomination. One thing to note in Silver's defense: "2% chance" is not "no chance": 2% chance events happen! But the real problem is elsewhere: Silver thinks we can assign "objective" probabilities to one-off events. But to assign a probability to any potential happening in an "objective" way, we have to abstract from the particular circumstances of time and place absolutely everything that cannot be reduced to a number by which we can "objectively" place that potential happening into a class with other, past happenings taken to be "identical" to the potential happening in all relevant features except those differing numbers. (For instance, we will have to turn each presidential candidate into a point in a vector space, where the number of "factors" we choose to include in our analysis are the dimen...

"Thus, however 'mystical' the content of Zande witchcraft beliefs may or may not be... they are actually employed by the Zande in a way anything but mysterious -- as an elaboration and defense of the truth claims of colloquial reason. Behind all these reflections upon stubbed toes, botched pots, and sour stomachs lies a tissue of commonsense notions the Zande apparently regarded as being true on their face: that minor cuts normally heal rapidly; that stones render baked clay liable to cracking... that in walking about Zandeland is unwise to daydream, for the place is full of stumps. And it is as part of this tissue of common sense assumptions, not of some primitive metaphysics, that the concept of witchcraft takes on its meaning and has its force... It is when ordinary expectations fail to hold, when the Zande man-in-the-field is confronted with anomalies or contradictions, that the cry of witchcraft goes up. It is, in this respect at least, a kind of dummy variable in...

March Madness is upon us, so it's time for more "sportscasters misunderstanding probability" fun. Here is one I hear a lot: Let's say that every single year, one of the four 14-seeds beats one of the four 3-seeds. Pundits will say, "Well, because one of the 14-seeds always wins, you should pick one to upset a 3." No: unless you have some special knowledge as to which 14-seed will win, you should pick all four 3s. Then, you will get 3 of the 4 games right. But if you pick (at random) a 14-seed to win, one time out of four you will get four right, but three times out of four you will get two wrong . That's an expected 1 and 1/2 wrong, as opposed to a certain one wrong by picking all the 3s.

We often see a news piece on a poll showing that Frip is ahead of Frap, 48% to 46%, followed by a declaration that Frip and Frap are "statistically tied." This phrase is nonsense . If the poll above was conducted properly, the odds are pretty good that Frip really is ahead, just perhaps not with "95% confidence." And the 95% confidence level is itself arbitrary: there is no real reason to prefer it to 93% confidence, or 97% confidence. And the magical power granted to passing the 95% threshold is completely unwarranted: if a poll one week shows Frip ahead with 94.9% confidence, it will be reported that Frip and Frap are in a "statistical tie," but if, the next week, Frip is ahead with 95.1% confidence, it will be reported that he has a "decisive lead." But if we applied statistical reasoning to the difference between the two polls themselves, we would surely find that their predictions were "statistically tied," and far more ...

Major league baseball analysts were asked the above question after the Mets had won the first two games against the Cubs. Several of them respond, "No, the Cubs are too good a team to be swept." I think they were not being good Bayesian reasoners. They had a prior: "The Cubs are a good team, and the Mets cannot sweep them." In other words, the Mets cannot win four straight games against a team as good as the Cubs. But they were failing to update this prior given new data. First of all, having won the first two games, the odds of the Mets "winning four straight games" became irrelevant: the only relevant odds were of their now winning two straight games. Secondly, the Mets' decisive wins in games one and two should have altered their evaluation of the relative strength of the two teams: the Mets were playing particularly well, while the Cubs were not, and thus the odds of the Mets winning two (more) straight games ought to have increased in t...

Keshav and I are going to play a chess match. Let us assume that when we have played in the past, we have been very evenly matched. There has been heavy betting on the match, and the Mob has gotten involved. They have persuaded the two of us to agree to fix the match, so that Keshav wins, 6 games to 4, and it is agreed as to which games he will throw, and which I will throw. There will also be a couple of games we play all out, just to make things look good, and a final fixed game, but for which it is unknown who will throw the game until the result of the two "real" games is known. You happen to have gotten a hot tip about the fix, but your tipster only knows the final result, and not which games are which. You can place rational bets on the series or some number of games in the series, using the fact that there is a 60% chance Keshav will win any randomly chosen game in the series. In other words, you will accept any bet on Keshav that offers an expected return of over $...

See Nate Silver, here : "The Cleveland Cavaliers led for much of Game 1 of the finals against the Golden State Warriors and had a better than 70 percent shot at winning it when LeBron James put the Cavs up by four with 5:08 left to play." Having just finished reviewing a book on Shackle, who largely holds the same view as Mises on case probability *, I find it interesting to ask just what Silver means here. Clearly, Silver is drawing upon a database of results that contains information on how often teams up four in a game with 5:08 to play won that game. Perhaps he has 200 records that match that criterion, and the team up four won 140 of those games. "But so what?" ask Mises and Shackle. The Cavaliers aren't going to play the Warriors 200 times, starting from this identical situation, so they can win 70% of the time. This game won't result in a Schrödinger's cat situation, with the Cavaliers 70% victorious and 30% defeated; no, one team or the o...

"For example, consider a case where a person is presented with the following three options "(a) Receive $500,000 with certainty. (b) A 95 percent probability of receiving $1 million dollars and a 5 per cent probability of receiving just $100. (c) A 50 per cent probability of receiving $900,000 and a 50 per cent probability of receiving $100,000 "For both shackles model and Prospect Theory , option (a) becomes the referenced point and the rival strategies have to be reframed as follows for Prospect Theory: "(b) A 95 per cent probability of gaining $500,000 and a 5 per cent chance of losing $499,900 (c) A 50 per cent probability of gaining $400,000 and a 50 per cent probability of losing $400,000 "In terms of prospect theory, loss aversion guarantees that (a) will be preferred over (c); option (b) may also be rejected in favour of (a) due to the downside risk being assigned a much bigger weight than 5 per cent, whereas in terms of traditional subjecti...

Earl and Littleboy describe an empirical test of Shackle's (non-probabilistic) theory of choice under uncertainty done by Hey in the 1980s. The subjects were presented with three uncertain situations, asked to produce an exhaustive list of what might occur in each case, and then to rate each in terms of possibility, probability, and potential surprise. The authors sum up the result of the experiment as follows: "Though Hey's findings were problematic for both Shackleian and probabilistic analyses of expectations, he felt on balance they were more damaging to Shackle's approach" (p. 134). However, going by the description in the book, this does not seem like a very fair test of Shackle's theory. Shackle says that in typically situations, people do not try to make an exhaustive list of possibilities and then assign a probability weight to each: instead, they considered only certain attention-grabbing possibilities. To test this, Hey apparently told the subj...

When looking at a formal theory of choice under uncertainty, there are (at least) three questions one might ask about it: 1) Is it mathematically sound? 2) Should people be applying it in some or all situations? If some, which ones? 3) Do people actually reason that way? The authors of G. L. S. Shackle sometimes seem to be mixing these three questions together, to the detriment of their analysis. For instance, they contend, "Whether probability is relevant [to single cases, rather than to a sequence of repeated trials,] is testable even by simple thought experiments" (p. 71). Well, first of all, what the authors next describe are not thought experiments à la Einstein, but experiments they have thought about someone performing. (The difference being they are asking "Think about this: how will people really respond in this situation?" rather than claiming the thought experiment itself proves anything.) And what they claim (I think correctly) is that, in the ...

It is interesting how Shackle and Mises are partners in this matter. The authors of G. L. S. Shackle discuss Shackle's rejection of the notion that a singular event can meaningfully be said to have probability X. Shackle puts forward an example where England and Australia are to have a cricket match. But instead of the usual coin toss deciding who bats first, England has managed to get the matter on the toss of a die, where a one will mean Australia bats first, and any other number, England will do so. He asks, "Can we now give any meaningful answer whatever to the question 'Who will bat first?' except 'We do not know'?" (p. 63) The authors reject Shackle's agnosticism here, claiming: "Of course, the right answer is 'England will, most probably'" (p. 64). But this simply begs the question that Shackle was raising: In the case of a single event, what exactly do we mean when we assert its probability is X? Of course everyone agr...

I parodied the kind of thing they write here . But what is the error in their reasoning? This issue is not merely of academic interest: medical doctors and juries are both prone to commit statistical errors that render their practical judgments fallacious. Doctors for instance, have overestimated the likelihood that a patient has a disease, given a positive test result, by a factor of 33 . And the prosecutor's fallacy can convince juries despite it being a fallacy: "In another scenario, a crime-scene DNA sample is compared against a database of 20,000 men. A match is found, that man is accused and at his trial, it is testified that the probability that two DNA profiles match by chance is only 1 in 10,000. This does not mean the probability that the suspect is innocent is 1 in 10,000. Since 20,000 men were tested, there were 20,000 opportunities to find a match by chance." In fact, in this situation, there is an 86% chance an innocent man will match the DNA profile f...

We need to have a name for the pairing, in current sports reporting, of absolute worship of "statistics" combined with absolute ignorance about how to do probabilistic reasoning. "Stapidity"? For instance, the NBA draft lottery is a domain in which we can be sure pure probabilistic reasoning applies, since it is deliberately set up that way. If a team has a 42.3% chance of getting the top pick, that's that: there is no point looking at "recent history" to see how teams in that position did, since in a random sampling, we expect to see subsets with different distributions of results than we will get as our sample size approaches infinity. And we know with certainty (unless we suspect the NBA has a broken random number generator) that in the limit, 42.3% of such teams will wind up with the top pick. And yet : "And as the fine folks at ESPN Stats & Information pointed out, recent history says not to be too confident the Lakers will keep ...

Here is John Hollinger's description of how he gets his NBA playoff predictions : "As always, the output of a product is only as good as its input, so let's explain a little about how this is derived. The computer starts with the day's Hollinger Power Rankings. Then, in each of the 5,000 times it replays the season, it makes a random adjustment up or down to allow for the possibility that a team will play better or worse than it has done thus far." I've been trying to think this through: why run simulations at all? The power rankings must establish some relationship between teams, such that, say, when a 94.5 plays an 86.7 it will likely win by three points, or something like that. Now you can produce some random wiggles and determine what the likelihood 94.5 will win is. Then use similarly derived likelihoods for all remaining games to get all of the teams' final records. In other words, my first impression here is that the run of 5000 simulations ...

and what do you get? Another completely unknown probability, 100 billion times larger than the first. What you can't say without knowing the first probability is that the second one must be close to one, right? Not according to Jesus Diaz : "This means that the chances of life and habitable planets in our galaxy alone is overwhelmingly high. So high that it's impossible to deny that it's out there." Nope. Having no idea how life arose on earth, we have no idea how probable it is on any other earth-like planet.

Y'all know I'm a big fan of Nate Silver, as a prognosticator. But as a philosopher of probability, he's pretty darned bad. Consider this: "Some of the historical cases of teams that defied even longer odds are well-known. Pennsylvania, in 1979, overcame what we estimate were 500-to-1 odds against reaching the Final Four."  Pennsylvania did not "overcome" these odds to reach the Final Four. "The odds" were not out on the court, blocking Penn's shots or grabbing rebounds away from them. Silver should read his Keynes: "500-to-1" is a prognosticator's subjective judgment about what will happen in the future, not something real * that a team must "overcome." The best way to understand such odds is to see them as a statement that the person positing them would be willing to make a 501-to-1 bet that Penn would get in, and a 499-to-1 bet that they would not. * Yes, as Oakeshott said, everything is real if we do not t...

Create two classes into which we may place singular events: 1) Those events which will transpire; and 2) Those events which will not transpire. Next, assign probabilities: A) Events in class 1 have a probability of 1. B) Events in class 2 have a probability of 0. Now, decide whether the event in which you are interested belongs in class 1 or class 2. Read off the corresponding probability from A or B! That was easy, wasn't it?

Scott Sumner, in aside to his case for an GDP prediction market, writes the following concerning Intrade's market on the Supreme Court's Obamacare decision: The market didn’t ”fail” at all, the 80% forecast was probably the optimal forecast... Sure, there was always some uncertainty, that’s what 80% means. That’s why the market didn’t price in a 100% chance of the law being overturned.... Consider the following analogy: Two prediction markets are set up to predict the toss of the coin before the next Super Bowl. One says 50% odds of heads and the other says 58% odds of heads. Then the coin is tossed, and it’s heads. Which market “failed?” I’d say the market with the 58% forecast. They made a bad forecast and simply got lucky This raises an interesting issue in probability theory, related to the Mises brothers' concerns about case probability: For a unique event that will never be repeated, what, exactly, does it mean to have an "optimal" forecast? ...