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Embarrassing update: as pointed out by Vladimir Nesov in the comments, all of my quantitative points below are incorrect. To maximize expected winnings, you should bet on whichever alternative you judge to be most likely. If you have a so-called logarithmic utility function — which already has the property of growing faster for small amounts than large — you should bet proportional to your odds on each answer. In fact, it’s exactly arguments like these that lead many to conclude that the logarithmic utility function is in some sense “correct”. So, in order to be led to betting more on the low-probability choices, one needs a utiltity that changes even faster for small amounts and slower for large amounts. But I disagree that this is “implausible” — if I think that is the best strategy to use, I should adjust my utility function, not change my strategy to match one that has been externally imposed. Just like probabilities, utility functions encode our preferences. Of course, I should endeavor to be consistent, to always use the same utility function, at least in the same circumstances, taking into account what economists call “externalities”.

The original post follows, mistakes included.

An even more unlikely place to find Bayesian inspiration was Channel 4’s otherwise insipid game show, “The Million Pound Drop”. In the version I saw, B-list celebs start out with a million pounds (sterling), and are asked a series of multiple-choice questions. For each one, they can bet any fraction of their remaining money on any set of answers; any money bet on wrong answers is lost (we’ll ignore the one caveat, that the contestants must wager no money on at least one answer, which means there’s always the chance that they will lose the entire stake).

Is there a best strategy for this game? Obviously, the overall goal is to maximize the actual winnings at the end of the series of questions. In the simplest example, let’s say a question is “What year did England last win the football world cup?” with possible answers “1912”, “1949”, “1966”, and “never”. In this case (assuming you know the answer), the only sensible course is to bet everything on “1966”.

Now, let’s say that the question is “When did the Chicago Bulls last win an NBA title?” with possible answers, “1953”, “1997”, “1998”, “2009”. The contestants, being fans of Michael Jordan, know that it’s either 1997 or 1998, but aren’t sure which — it’s a complete toss-up between the two. Again in this case, the strategy is clear: bet the same amount on each of the two — the expected winning is half of your stake no matter what. (The answer is 1998.)

But now let’s make it a bit more complicated: the question is “Who was the last American to win a gold medal in Olympic Decathlon?” with answers “Bruce Jenner”, “Brian Clay”, “Jim Thorpe”, and “Jess Owens”. Well, I remember that Jenner won in the 70s, and that Thorpe and Owens predate that by decades, so the only possibilities are Jenner and Clay, whom I’ve never heard of. So I’m pretty sure the answer is Jenner, but I’m by no means certain: let’s say that I’m 99:1 in favor of Jenner over Clay.

In order to maximize my expected winnings, I should bet 99 times as much on Jenner as Clay. But there’s a problem here: if it’s Clay, I end up with only one percent of my initial stake, and that one percent — which I have to go on and play more rounds with — is almost too small to be useful. This means that I don’t really want to maximize my expected winnings, but rather something that economists and statisticians call the “utility function”, or conversely, to minimize the loss function, functions which describes how useful some amount of winnings are to me: a thousand dollars is more than a thousand times useful than one dollar, but a million dollars is less than twice as useful as half a million dollars, at least in this context.

So in this case, a small amount of winnings is less useful than one might naively expect, and the utility function should reflect that by growing faster for small amounts and slower for larger amounts — I should perhaps bet ten percent on Clay. If it’s Jenner, I still get 90% of my stake, but if it’s Clay, I end up with a more-useful 10%. (The answer is Clay, by the way.)

This is the branch of statistics and mathematics called decision theory: how we go from probabilities to actions. It comes into play when we don’t want to just report probabilities, but actually act on them: whether to actually prescribe a drug, perform a surgical procedure, or build a sea-wall against a possible flood. In each of these cases, in addition to knowing the efficacy of the action, we need to understand its utility: if a flood is 1% likely over the next century and would cost one million pounds, but would save one billion in property damage and 100 lives if the flood occurred, we need to compare spending a million now versus saving a billion later (taking the “nonlinear” effects above into account) and complicate that with the loss from even more tragic possibilities. One hundred fewer deaths has the same utility as some amount of money saved, but I am glad I’m not on the panel that has to make that assignment. It is important to point out, however, that whatever decision is made, by whatever means, it is equivalent to some particularly set of utilities, so we may as well be explicit about it.

Happily, these sorts of questions tend to arise less in the physical sciences where probabilistic results are allowed, although the same considerations arise at a higher level: when making funding decisions…

I've come across a couple bits of popular/political culture that give me the opportunity to discuss one of my favorite topics: the uses and abuses of probability theory.

The first is piece by Nate Silver of the New York Times' FiveThirtyEight blog, dedicated to trying to crunch the political numbers of polls and other data in as transparent a manner as possible. Usually, Silver relies on a relentlessly frequentist take on probability: he runs lots of simulations letting the inputs vary according to the poll results (correctly taking into account the "margin of error" and more than occasionally using other information to re-weight the results of different polls. Nonetheless, these techniques give a good summary of the results at any given time -- and have been far and away the best discussion of the numerical minutiae of electioneering for both the 2008 and 2010 US elections.

But yesterday, Silver wrote a column: A Bayesian Take on Julian Assange which tackles the question of Assange's guilt in the sexual-assault offense with which he has been charged. Bayes' theorem, you will probably recall if you've been reading this blog, states that the probability of some statement ("Assange is innocent of sexual assault, despite the charges against him") is the product of the probability that he would be charged if he were innocent (the "likelihood") times the probability of his innnocence in the absence of knowledge about the charge (the "prior"):

P(innocent|charged, context) ∝ P(innocent | context) × P(charged|innocent, context)

To figure out these probabilities, there are no simulations we can perform -- we can't run a big social-science model of Swedish law-enforcement, possibly in contact with, say, American diplomats, and make small changes and see what happens. We just need to assign probabilities to these statements.

But even to do that requires considerable thought, and important decisions about the context in which we want to make these assignments. For Silver, the important context is that there is evidence that other governments, particularly the US, may have an ulterior motive for wanting to not just prosecute, but persecute Assange. Hence, the probability of his being unjustly accused [P(charged|innocent, context)] is larger than it would be for, say, an arbitrary Australian citizen traveling in Britain. Usually, Bayesian probability is accused of needing a subjective prior, but in this case the context affects and adds a subjective aspect to the likelihood.

Some of the commenters on the site make a different point: given that Assange is, at least in some sense, a known criminal (he has leaked secret documents, which is likely against the law), he is more likely to commit other criminal acts. This time, the likelihood is not affected, but the prior: the commenter believes that Assange is less likely to be innocent irrespective of the information about the charge.

Next: game shows.

Today is Ada Lovelace Day, “an international day of blogging to draw attention to women excelling in technology.” I — along with more than a thousand other people — have pledged to write about a female role model in technology.

Ada Lovelace was Byron’s daughter and worked with computer pioneer Charles Babbage on his “Computing Engines” — and is widely thought of as the first computer programmer. A reconstruction of the “Difference Engine” is on view at the Science Museum around the corner from here, and if you’re reading this on 24 March, you can go and talk to Ada herself!

But I want to talk not about a programmer, but a computer. That is, a computer named Henrietta Swan Leavitt. In the early 20th Century, some (always male) astronomers had batteries of (almost always female) “computers” working for them, doing their calculations and other supposedly menial scientific work.

Leavitt — who had graduated from Radcliffe College — was employed by Harvard astronomer Charles Pickering to analyze photographic plates: she counted stars and measured their brightness. Pickering was particularly interested in “variable stars”, which changed their brightness over time. The most interesting variable stars changed in a regular pattern and Leavitt noticed that, for a certain class of these stars known as Cepheids, the brighter ones had longer periods. Eventually, in 1912, she made this more precise, and to this day the “Cepheid Period-Luminosity Relationship” remains one of the most important tools in the astronomers box.

It’s easy enough to measure the period of a Cepheid variable star: just keep taking data, make a graph, and see how long it takes to repeat itself. Then, from the Period-Luminosity relationship, we can determine its intrinsic luminosity. But we can also easily measure how bright it appears to us, and use this, along with the inverse-square relationship between intrinsic luminosity and apparent brightness, to get the distance to the star. That is, if we put the same star twice as far away, it’s four times dimmer; three times as far is nine times dimmer, etc.

This was just the technique that astronomy needed, and within a couple of decades it had led to a revolution in our understanding of the scale of the cosmos. First, it enabled astronomers to map out the Milky Way. But at this time, it wasn’t even clear whether the Milky Way was the only agglomeration of stars in the Universe, or one amongst many. Indeed, this was the subject of the so-called “great debate” in 1921 between American astronomers Harlow Shapley and Heber Curtis. Shapley argued that all of the nebuale (fuzzy patches) on the sky were just local collections of stars, or extended clouds of gas, while Curtis argued that some of them (in particular, Andromeda) were galaxies — “Island Universes” as they were called — like our own. By at least some accounts, Shapley won the debate at the time.

But very soon after, due to Leavitt’s work, Edwin Hubble determined that Curtis was correct: he saw the signature of Cepheid stars in (what turned out to be) the Andromeda galaxy and used them to measure the distance, which turned out to be much further away than the stars in the galaxy. A few years later, Hubble used Leavitt’s Period-Luminosity relationship to make an even more startling discovery: more distant galaxies were receding from us at a speed (measured using the galaxy’s redshift) proportional to their distance from us. This is the observational basis for the Big Bang theory of the Universe, tested and proven time and again in the eighty or so years since then.

Leavitt’s relationship remains crucial to astronomy and cosmology. The Hubble Space Telescope’s “Key Project” was to measure the brightness and period of Cepheid stars in galaxies as far away as possible, determining Hubble’s proportionality constant and set an overall scale for distances in the Universe.

The social situation of academic astronomy of her day strongly limited Leavitt’s options — women weren’t allowed to operate telescopes, and it was yet more difficult for her as she was deaf, as well. Although Leavitt was “only” employed as a computer, she was eventually nominated for a Nobel prize for her work — but she had already died. We can only hope that the continued use of her results and insight to this day is a small recompense and recognition of her life and work.

By far the best article I’ve read about the British healthcare system, appeared this morning… in the New York Times. It discussed the NHS’s National Institute for Health and Clinical Excellence (NICE), the organization that rations pharmaceuticals in the UK (although you’ll rarely hear the word “ration” used).

When NICE’s decisions are discussed in the UK, it is inevitably in the context of some sad, sick patient denied access to some treatment that could extend his life. But this article took the logical, if unpleasant, stand that, given finite resources, some sort of rationing is inevitable. In the US, this is done by the “free” market — the scare-quotes are to remind us that healthcare spending per patient in the US is several times anywhere else in the world. What that means is that it’s great to be rich in the US — quick access to any drug, any test, any procedure, but only if you (or your insurer) will pay. But in the rest of the civilized world health-care is largely provided directly or indirectly by the government, irrespective of the patient’s wealth or employment. This is fair in an egalitarian sense, of course, but not necessarily in the libertarian sense Americans often prefer: why shouldn’t I be able to spend my money if I have it?

Conversely, this correctly pushes some of the criticism back onto the drug companies. Drug pricing is a particularly contrived manifestation of the invisible hand of the market: true costs are muddied by extensive R&D budgets, and demand is confused by governments and insurance companies willingness — or otherwise — to pay. (Elsewhere in the Times, my Imperial compatriot Olivia Judson touches on the interaction of scientists and drug companies as part of a larger piece on science, politics, Bush and Obama.)

This is not to say that the NHS system is perfect. It suffers from an infamous “postcode lottery”, as different geographical parts of the NHS system make different decisions about the way their resources will be used — that is, rationed. And despite the fact that the NHS is one of the largest single employers in the world, it is still too small for its task: initial doctor’s appointments are usually restricted to about ten minutes, and the waiting times for surgery and complicated tests can be months long. But it truly is egalitarian: I once came across Elvis Costello and Diana Krall in an NHS Hospital ER (A&E, as it’s known in the UK).

I would like to think that the passing of one Scooter — Phil Rizzuto, Yankees Shortstop and broadcaster — was of greater cultural significance than the pardonning of the other one (since they didn’t manage to indict the likely mastermind behind the plot before he could resign. Probably not.

Flooding cripples New York subway system:

Flooding from torrential overnight rains crippled the New York City subway system this morning. Delays of at least 30 minutes were reported on all subway lines, and customers were urged to forgo the subways entirely and take buses if possible. The thunderstorm caused havoc across the region, forcing thousands of people, like the pedestrians who crowded the Manhattan Bridge in both directions, to walk to work or work from home.

Of course, Bangladesh, India and Nepal have been struggling under far worse flood conditions caused by the monsoon, as they do almost every year.

My thoughts and sympathy go out to the residents of my old neighborhood, Osney Island, in Oxford, where it’s just started to flood before the waters peak (we hope) later today. Good luck to all the residents — stay strong, stay dry!

Once again, my Union, the University and College Union, has sort-of voted to boycott Israeli academics. It’s only “sort of”, because, like last time, the decision comes about from a vote of activists present at the UCU annual conference, not of the membership at large. Indeed, the vote has been opposed by the General Secretary of the Union, not to mention the British and Israeli governments, the not particularly pro-Israel Observer newspaper, and even the New York Times. No matter what your feelings about the present Israeli government and its actions with respect to the Palestinians, such a boycott is, at best, an empty gesture. At worst, it actively works against progressive causes espoused by the many Israeli academics who are among the vocal critics of their own government. And, of course, it is bad for scholarship, which, we often say, should at least endeavor to rise above politics.

A few weeks ago, the UK’s National Union of Journalists made a similar gesture, one that will likely have even more repercussions for me and other physicists. Nobel Laureate Steven Weinberg was slated to give a plenary address at the coming PASCOS meeting at Imperial next month. Unfortunately, and rather bizarrely, Weinberg has decided to use the NUJ’s decision (this was before the UCU’s meetings) as a reason to back out of his engagement, citing this as an example of “a widespread anti-Israel and anti-Semitic current of British opinion, especially in the intellectual establishment.” The crucial word in that sentence is, of course, that meek connective “and”. An anti-Israel bias is pretty evident here, but whether or not this translates into actual anti-semitism remains unclear. (The US, for example, has plenty of pro-Israel anti-semitism in the form of fundamentalist Christians hastening the coming of their rapture/apocalypse, for which a strong Israel seems to be required in a perverse reading of Revelations.) Alas, Weinberg’s not-quite-empty gesture is certainly bad for scholarship at best, and at worst deprives him of an actual pulpit from which he could have propounded his views.

A beautiful but frightening picture of the fire burning near the Griffith Park Observatory in Los Angeles. Evidence that we were probably never meant to live in that part of the world? (Photo courtesy Monica Almeida/New York Times)

…they are part of an an ancient Jewish conspiracy, and so it pisses off the anti-semites…