## One More for the Bad Statistics in Astronomy File…

Posted in Bad Statistics, The Universe and Stuff with tags , , , , , on May 20, 2015 by telescoper

It’s been a while since I last posted anything in the file marked Bad Statistics, but I can remedy that this morning with a comment or two on the following paper by Robertson et al. which I found on the arXiv via the Astrostatistics Facebook page. It’s called Stellar activity mimics a habitable-zone planet around Kapteyn’s star and it the abstract is as follows:

Kapteyn’s star is an old M subdwarf believed to be a member of the Galactic halo population of stars. A recent study has claimed the existence of two super-Earth planets around the star based on radial velocity (RV) observations. The innermost of these candidate planets–Kapteyn b (P = 48 days)–resides within the circumstellar habitable zone. Given recent progress in understanding the impact of stellar activity in detecting planetary signals, we have analyzed the observed HARPS data for signatures of stellar activity. We find that while Kapteyn’s star is photometrically very stable, a suite of spectral activity indices reveals a large-amplitude rotation signal, and we determine the stellar rotation period to be 143 days. The spectral activity tracers are strongly correlated with the purported RV signal of “planet b,” and the 48-day period is an integer fraction (1/3) of the stellar rotation period. We conclude that Kapteyn b is not a planet in the Habitable Zone, but an artifact of stellar activity.

It’s not really my area of specialism but it seemed an interesting conclusions so I had a skim through the rest of the paper. Here’s the pertinent figure, Figure 3,

It looks like difficult data to do a correlation analysis on and there are lots of questions to be asked  about  the form of the errors and how the bunching of the data is handled, to give just two examples.I’d like to have seen a much more comprehensive discussion of this in the paper. In particular the statistic chosen to measure the correlation between variates is the Pearson product-moment correlation coefficient, which is intended to measure linear association between variables. There may indeed be correlations in the plots shown above, but it doesn’t look to me that a straight line fit characterizes it very well. It looks to me in some of the  cases that there are simply two groups of data points…

However, that’s not the real reason for flagging this one up. The real reason is the following statement in the text:

Aargh!

No matter how the p-value is arrived at (see comments above), it says nothing about the “probability of no correlation”. This is an error which is sadly commonplace throughout the scientific literature, not just astronomy.  The point is that the p-value relates to the probability that the given value of the test statistic (in this case the Pearson product-moment correlation coefficient, r) would arise by chace in the sample if the null hypothesis H (in this case that the two variates are uncorrelated) were true. In other words it relates to P(r|H). It does not tells us anything directly about the probability of H. That would require the use of Bayes’ Theorem. If you want to say anything at all about the probability of a hypothesis being true or not you should use a Bayesian approach. And if you don’t want to say anything about the probability of a hypothesis being true or not then what are you trying to do anyway?

If I had my way I would ban p-values altogether, but it people are going to use them I do wish they would be more careful about the statements make about them.

## The Law of Averages

Posted in Bad Statistics, Crosswords with tags , , on March 4, 2015 by telescoper

Just a couple of weeks ago I found myself bemoaning my bad luck in the following terms

A few months have passed since I last won a dictionary as a prize in the Independent Crossword competition. That’s nothing remarkable in itself, but since my average rate of dictionary accumulation has been about one a month over the last few years, it seems a bit of a lull.  Have I forgotten how to do crosswords and keep sending in wrong solutions? Is the Royal Mail intercepting my post? Has the number of correct entries per week suddenly increased, reducing my odds of winning? Have the competition organizers turned against me?

In fact, statistically speaking, there’s nothing significant in this gap. Even if my grids are all correct, the number of correct grids has remained constant, and the winner is pulled at random  from those submitted (i.e. in such a way that all correct entries are equally likely to be drawn) , then a relatively long unsuccessful period such as I am experiencing at the moment is not at all improbable. The point is that such runs are far more likely in a truly random process than most people imagine, as indeed are runs of successes. Chance coincidence happen more often than you think.

Well, as I suspected would happen soon my run of ill fortune came to an end today with the arrival of this splendid item in the mail:

It’s the prize for winning Beelzebub 1303, the rather devilish prize cryptic in the Independent on Sunday Magazine. It’s nice to get back to winning ways. Now what’s the betting I’ll now get a run of successes?

P.S. I used the title “Law of Averages” just so I could point out in a footnote that there’s actually no such thing.

## Uncertainty, Risk and Probability

Posted in Bad Statistics, Science Politics with tags , , , , , , , , on March 2, 2015 by telescoper

Last week I attended a very interesting event on the Sussex University campus, the Annual Marie Jahoda Lecture which was given this year by Prof. Helga Nowotny a distinguished social scientist. The title of the talk was A social scientist in the land of scientific promise and the abstract was as follows:

Promises are a means of bringing the future into the present. Nowhere is this insight by Hannah Arendt more applicable than in science. Research is a long and inherently uncertain process. The question is open which of the multiple possible, probable or preferred futures will be actualized. Yet, scientific promises, vague as they may be, constitute a crucial link in the relationship between science and society. They form the core of the metaphorical ‘contract’ in which support for science is stipulated in exchange for the benefits that science will bring to the well-being and wealth of society. At present, the trend is to formalize scientific promises through impact assessment and measurement. Against this background, I will present three case studies from the life sciences: assisted reproductive technologies, stem cell research and the pending promise of personalized medicine. I will explore the uncertainty of promises as well as the cunning of uncertainty at work.

It was a fascinating and wide-ranging lecture that touched on many themes. I won’t try to comment on all of them, but just pick up on a couple that struck me from my own perspective as a physicist. One was the increasing aversion to risk demonstrated by research funding agencies, such as the European Research Council which she helped set up but described in the lecture as “a clash between a culture of trust and a culture of control”. This will ring true to any scientist applying for grants even in “blue skies” disciplines such as astronomy: we tend to trust our peers, who have some control over funding decisions, but the machinery of control from above gets stronger every day. Milestones and deliverables are everything. Sometimes I think in order to get funding you have to be so confident of the outcomes of your research to that you have to have already done it, in which case funding isn’t even necessary. The importance of extremely speculative research is rarely recognized, although that is where there is the greatest potential for truly revolutionary breakthroughs.

Another theme that struck me was the role of uncertainty and risk. This grabbed my attention because I’ve actually written a book about uncertainty in the physical sciences. In her lecture, Prof. Nowotny referred to the definition (which was quite new to me) of these two terms by Frank Hyneman Knight in a book on economics called Risk, Uncertainty and Profit. The distinction made there is that “risk” is “randomness” with “knowable probabilities”, whereas “uncertainty” involves “randomness” with “unknowable probabilities”. I don’t like these definitions at all. For one thing they both involve a reference to “randomness”, a word which I don’t know how to define anyway; I’d be much happier to use “unpredictability”. Even more importantly, perhaps, I find the distinction between “knowable” and “unknowable” probabilities very problematic. One always knows something about a probability distribution, even if that something means that the distribution has to be very broad. And in any case these definitions imply that the probabilities concerned are “out there”, rather being statements about a state of knowledge (or lack thereof). Sometimes we know what we know and sometimes we don’t, but there are more than two possibilities. As the great American philosopher and social scientist Donald Rumsfeld (Shurely Shome Mishtake? Ed) put it:

“…as we know, there are known knowns; there are things we know we know. We also know there are known unknowns; that is to say we know there are some things we do not know. But there are also unknown unknowns – the ones we don’t know we don’t know.”

There may be a proper Bayesian formulation of the distinction between “risk” and “uncertainty” that involves a transition between prior-dominated (uncertain) and posterior-dominated (risky), but basically I don’t see any qualititative difference between the two from such a perspective.

Anyway, it was a very interesting lecture that differed from many talks I’ve attended about the sociology of science in that the speaker clearly understood a lot about how science actually works. The Director of the Science Policy Research Unit invited the Heads of the Science Schools (including myself) to dinner with the speaker afterwards, and that led to the generation of many interesting ideas about how we (I mean scientists and social scientists) might work better together in the future, something we really need to do.

## Digit Ratio Survey

Posted in Bad Statistics, Biographical with tags , , , on February 9, 2015 by telescoper

I was intrigued by an article I found at the weekend which reports on a (no doubt rigorous) scientific study that claims a connection between the relative lengths of index and ring fingers and the propensity to be promiscuous. The assertion is that people whose ring finger is longer than their index finger like to play around, while those whose index finger is longer than their ring finger are inclined to fidelity. Obviously, since the study involves the University of Oxford’s Department of Experimental Psychology, there can be do doubt whatsoever about its reliablity or scientific credibility, just like the dozens of other things supposed to be correlated with digit ratio. Ahem.

I do remember a similar study some time ago that claimed that men with with a longer index finger (2D) than ring finger (4D) (i.e. with a 2D:4D digit ratio greater than one) were much more likely to be gay than those with a digit ratio lower than one. Taken with this new finding it proves what we all knew all along: that heterosexuals are far more likely to be promiscuous than homosexuals.

For the record, here is a photograph of my left hand (which, on reflection, is similar to my right, and which clearly shows a 2D:4D ratio greater than unity):

Inspired by the stunning application of the scientific method described in the report, I have decided to carry out a rigorous study of my own. I have heard that, at least among males, it is much more common to have digit ratio less than one than greater than one but I can’t say I’ve noticed it myself. Furthermore previously unanswered question in the literature is whether there is a connection between digit ratio and the propensity to read blogs. I will know subject this to rigorous scientific scrutiny by inviting readers of this blog to complete the following simply survey. I look forward to publishing my findings in due course in the Journal of Irreproducible Results.

PS. The actual paper on which the report was based is by Rafael Wlodarski, John Manning, and R. I. M. Dunbar,

## Doomsday is Cancelled…

Posted in Bad Statistics, The Universe and Stuff with tags , on November 25, 2014 by telescoper

Last week I posted an item that included a discussion of the Doomsday Argument. A subsequent comment on that post mentioned a paper by Ken Olum, which I finally got around to reading over the weekend, so I thought I’d post a link here for those of you worrying that the world might come to an end before the Christmas holiday.

You can find Olum’s paper on the arXiv here. The abstract reads (my emphasis):

If the human race comes to an end relatively shortly, then we have been born at a fairly typical time in history of humanity. On the other hand, if humanity lasts for much longer and trillions of people eventually exist, then we have been born in the first surprisingly tiny fraction of all people. According to the Doomsday Argument of Carter, Leslie, Gott, and Nielsen, this means that the chance of a disaster which would obliterate humanity is much larger than usually thought. Here I argue that treating possible observers in the same way as those who actually exist avoids this conclusion. Under this treatment, it is more likely to exist at all in a race which is long-lived, as originally discussed by Dieks, and this cancels the Doomsday Argument, so that the chance of a disaster is only what one would ordinarily estimate. Treating possible and actual observers alike also allows sensible anthropic predictions from quantum cosmology, which would otherwise depend on one’s interpretation of quantum mechanics.

I think Olum does identify a logical flaw in the argument, but it’s by no means the only one. I wouldn’t find it at all surprising to be among the first “tiny fraction of all people”, as my genetic characteristics are such that I could not be otherwise. But even if you’re not all that interested in the Doomsday Argument I recommend you read this paper as it says some quite interesting things about the application of probabilistic reasoning elsewhere in cosmology, an area in which quite a lot is written that makes no sense to me whatsoever!

## German Tanks, Traffic Wardens, and the End of the World

Posted in Bad Statistics, The Universe and Stuff with tags , , , , , , , on November 18, 2014 by telescoper

The other day I was looking through some documents relating to the portfolio of courses and modules offered by the Department of Mathematics here at the University of Sussex when I came across a reference to the German Tank Problem. Not knowing what this was I did a google search and  a quite comprehensive wikipedia page on the subject which explains the background rather well.

It seems that during the latter stages of World War 2 the Western Allies made sustained efforts to determine the extent of German tank production, and approached this in two major ways, namely  conventional intelligence gathering and statistical estimation with the latter approach often providing the more accurate and reliable, as was the case in estimation of the production of Panther tanks  just prior to D-Day. The allied command structure had thought the heavy Panzer V (Panther) tanks, with their high velocity, long barreled 75 mm/L70 guns, were uncommon, and would only be encountered in northern France in small numbers.  The US Army was confident that the Sherman tank would perform well against the Panzer III and IV tanks that they expected to meet but would struggle against the Panzer V. Shortly before D-Day, rumoursbegan to circulate that large numbers of Panzer V tanks had been deployed in Normandy.

To ascertain if this were true the Allies attempted to estimate the number of Panzer V  tanks being produced. To do this they used the serial numbers on captured or destroyed tanks. The principal numbers used were gearbox numbers, as these fell in two unbroken sequences; chassis, engine numbers and various other components were also used. The question to be asked is how accurately can one infer the total number of tanks based on a sample of a few serial numbers. So accurate did this analysis prove to be that, in the statistical theory of estimation, the general problem of estimating the maximum of a discrete uniform distribution from sampling without replacement is now known as the German tank problem. I’ll leave the details to the wikipedia discussion, which in my opinion is yet another demonstration of the advantages of a Bayesian approach to this kind of problem.

This problem is a more general version of a problem that I first came across about 30 years ago. I think it was devised in the following form by Steve Gull, but can’t be sure of that.

Imagine you are a visitor in an unfamiliar, but very populous, city. For the sake of argument let’s assume that it is in China. You know that this city is patrolled by traffic wardens, each of whom carries a number on their uniform.  These numbers run consecutively from 1 (smallest) to T (largest) but you don’t know what T is, i.e. how many wardens there are in total. You step out of your hotel and discover traffic warden number 347 sticking a ticket on your car. What is your best estimate of T, the total number of wardens in the city? I hope the similarity to the German Tank Problem is obvious, except in this case it is much simplified by involving just one number rather than a sample.

I gave a short lunchtime talk about this many years ago when I was working at Queen Mary College, in the University of London. Every Friday, over beer and sandwiches, a member of staff or research student would give an informal presentation about their research, or something related to it. I decided to give a talk about bizarre applications of probability in cosmology, and this problem was intended to be my warm-up. I was amazed at the answers I got to this simple question. The majority of the audience denied that one could make any inference at all about T based on a single observation like this, other than that it  must be at least 347.

Actually, a single observation like this can lead to a useful inference about T, using Bayes’ theorem. Suppose we have really no idea at all about T before making our observation; we can then adopt a uniform prior probability. Of course there must be an upper limit on T. There can’t be more traffic wardens than there are people, for example. Although China has a large population, the prior probability of there being, say, a billion traffic wardens in a single city must surely be zero. But let us take the prior to be effectively constant. Suppose the actual number of the warden we observe is t. Now we have to assume that we have an equal chance of coming across any one of the T traffic wardens outside our hotel. Each value of t (from 1 to T) is therefore equally likely. I think this is the reason that my astronomers’ lunch audience thought there was no information to be gleaned from an observation of any particular value, i.e. t=347.

Let us simplify this argument further by allowing two alternative “models” for the frequency of Chinese traffic wardens. One has T=1000, and the other (just to be silly) has T=1,000,000. If I find number 347, which of these two alternatives do you think is more likely? Think about the kind of numbers that occupy the range from 1 to T. In the first case, most of the numbers have 3 digits. In the second, most of them have 6. If there were a million traffic wardens in the city, it is quite unlikely you would find a random individual with a number as small as 347. If there were only 1000, then 347 is just a typical number. There are strong grounds for favouring the first model over the second, simply based on the number actually observed. To put it another way, we would be surprised to encounter number 347 if T were actually a million. We would not be surprised if T were 1000.

One can extend this argument to the entire range of possible values of T, and ask a more general question: if I observe traffic warden number t what is the probability I assign to each value of T? The answer is found using Bayes’ theorem. The prior, as I assumed above, is uniform. The likelihood is the probability of the observation given the model. If I assume a value of T, the probability P(t|T) of each value of t (up to and including T) is just 1/T (since each of the wardens is equally likely to be encountered). Bayes’ theorem can then be used to construct a posterior probability of P(T|t). Without going through all the nuts and bolts, I hope you can see that this probability will tail off for large T. Our observation of a (relatively) small value for t should lead us to suspect that T is itself (relatively) small. Indeed it’s a reasonable “best guess” that T=2t. This makes intuitive sense because the observed value of t then lies right in the middle of its range of possibilities.

Before going on, it is worth mentioning one other point about this kind of inference: that it is not at all powerful. Note that the likelihood just varies as 1/T. That of course means that small values are favoured over large ones. But note that this probability is uniform in logarithmic terms. So although T=1000 is more probable than T=1,000,000,  the range between 1000 and 10,000 is roughly as likely as the range between 1,000,000 and 10,000,0000, assuming there is no prior information. So although it tells us something, it doesn’t actually tell us very much. Just like any probabilistic inference, there’s a chance that it is wrong, perhaps very wrong.

Which brings me to an extrapolation of this argument to an argument about the end of the World. Now I don’t mind admitting that as I get older I get more and  more pessimistic about the prospects for humankind’s survival into the distant future. Unless there are major changes in the way this planet is governed, our Earth may indeed become barren and uninhabitable through war or environmental catastrophe. But I do think the future is in our hands, and disaster is, at least in principle, avoidable. In this respect I have to distance myself from a very strange argument that has been circulating among philosophers and physicists for a number of years. It is called Doomsday argument, and it even has a sizeable wikipedia entry, to which I refer you for more details and variations on the basic theme. As far as I am aware, it was first introduced by the mathematical physicist Brandon Carter and subsequently developed and expanded by the philosopher John Leslie (not to be confused with the TV presenter of the same name). It also re-appeared in slightly different guise through a paper in the serious scientific journal Nature by the eminent physicist Richard Gott. Evidently, for some reason, some serious people take it very seriously indeed.

So what can Doomsday possibly have to do with Panzer tanks or traffic wardens? Instead of traffic wardens, we want to estimate N, the number of humans that will ever be born, Following the same logic as in the example above, I assume that I am a “randomly” chosen individual drawn from the sequence of all humans to be born, in past present and future. For the sake of argument, assume I number n in this sequence. The logic I explained above should lead me to conclude that the total number N is not much larger than my number, n. For the sake of argument, assume that I am the one-billionth human to be born, i.e. n=1,000,000,0000.  There should not be many more than a few billion humans ever to be born. At the rate of current population growth, this means that not many more generations of humans remain to be born. Doomsday is nigh.

Richard Gott’s version of this argument is logically similar, but is based on timescales rather than numbers. If whatever thing we are considering begins at some time tbegin and ends at a time tend and if we observe it at a “random” time between these two limits, then our best estimate for its future duration is of order how long it has lasted up until now. Gott gives the example of Stonehenge, which was built about 4,000 years ago: we should expect it to last a few thousand years into the future. Actually, Stonehenge is a highly dubious . It hasn’t really survived 4,000 years. It is a ruin, and nobody knows its original form or function. However, the argument goes that if we come across a building put up about twenty years ago, presumably we should think it will come down again (whether by accident or design) in about twenty years time. If I happen to walk past a building just as it is being finished, presumably I should hang around and watch its imminent collapse….

But I’m being facetious.

Following this chain of thought, we would argue that, since humanity has been around a few hundred thousand years, it is expected to last a few hundred thousand years more. Doomsday is not quite as imminent as previously, but in any case humankind is not expected to survive sufficiently long to, say, colonize the Galaxy.

You may reject this type of argument on the grounds that you do not accept my logic in the case of the traffic wardens. If so, I think you are wrong. I would say that if you accept all the assumptions entering into the Doomsday argument then it is an equally valid example of inductive inference. The real issue is whether it is reasonable to apply this argument at all in this particular case. There are a number of related examples that should lead one to suspect that something fishy is going on. Usually the problem can be traced back to the glib assumption that something is “random” when or it is not clearly stated what that is supposed to mean.

There are around sixty million British people on this planet, of whom I am one. In contrast there are 3 billion Chinese. If I follow the same kind of logic as in the examples I gave above, I should be very perplexed by the fact that I am not Chinese. After all, the odds are 50: 1 against me being British, aren’t they?

Of course, I am not at all surprised by the observation of my non-Chineseness. My upbringing gives me access to a great deal of information about my own ancestry, as well as the geographical and political structure of the planet. This data convinces me that I am not a “random” member of the human race. My self-knowledge is conditioning information and it leads to such a strong prior knowledge about my status that the weak inference I described above is irrelevant. Even if there were a million million Chinese and only a hundred British, I have no grounds to be surprised at my own nationality given what else I know about how I got to be here.

This kind of conditioning information can be applied to history, as well as geography. Each individual is generated by its parents. Its parents were generated by their parents, and so on. The genetic trail of these reproductive events connects us to our primitive ancestors in a continuous chain. A well-informed alien geneticist could look at my DNA and categorize me as an “early human”. I simply could not be born later in the story of humankind, even if it does turn out to continue for millennia. Everything about me – my genes, my physiognomy, my outlook, and even the fact that I bothering to spend time discussing this so-called paradox – is contingent on my specific place in human history. Future generations will know so much more about the universe and the risks to their survival that they won’t even discuss this simple argument. Perhaps we just happen to be living at the only epoch in human history in which we know enough about the Universe for the Doomsday argument to make some kind of sense, but too little to resolve it.

To see this in a slightly different light, think again about Gott’s timescale argument. The other day I met an old friend from school days. It was a chance encounter, and I hadn’t seen the person for over 25 years. In that time he had married, and when I met him he was accompanied by a baby daughter called Mary. If we were to take Gott’s argument seriously, this was a random encounter with an entity (Mary) that had existed for less than a year. Should I infer that this entity should probably only endure another year or so? I think not. Again, bare numerological inference is rendered completely irrelevant by the conditioning information I have. I know something about babies. When I see one I realise that it is an individual at the start of its life, and I assume that it has a good chance of surviving into adulthood. Human civilization is a baby civilization. Like any youngster, it has dangers facing it. But is not doomed by the mere fact that it is young,

John Leslie has developed many different variants of the basic Doomsday argument, and I don’t have the time to discuss them all here. There is one particularly bizarre version, however, that I think merits a final word or two because is raises an interesting red herring. It’s called the “Shooting Room”.

Consider the following model for human existence. Souls are called into existence in groups representing each generation. The first generation has ten souls. The next has a hundred, the next after that a thousand, and so on. Each generation is led into a room, at the front of which is a pair of dice. The dice are rolled. If the score is double-six then everyone in the room is shot and it’s the end of humanity. If any other score is shown, everyone survives and is led out of the Shooting Room to be replaced by the next generation, which is ten times larger. The dice are rolled again, with the same rules. You find yourself called into existence and are led into the room along with the rest of your generation. What should you think is going to happen?

Leslie’s argument is the following. Each generation not only has more members than the previous one, but also contains more souls than have ever existed to that point. For example, the third generation has 1000 souls; the previous two had 10 and 100 respectively, i.e. 110 altogether. Roughly 90% of all humanity lives in the last generation. Whenever the last generation happens, there bound to be more people in that generation than in all generations up to that point. When you are called into existence you should therefore expect to be in the last generation. You should consequently expect that the dice will show double six and the celestial firing squad will take aim. On the other hand, if you think the dice are fair then each throw is independent of the previous one and a throw of double-six should have a probability of just one in thirty-six. On this basis, you should expect to survive. The odds are against the fatal score.

This apparent paradox seems to suggest that it matters a great deal whether the future is predetermined (your presence in the last generation requires the double-six to fall) or “random” (in which case there is the usual probability of a double-six). Leslie argues that if everything is pre-determined then we’re doomed. If there’s some indeterminism then we might survive. This isn’t really a paradox at all, simply an illustration of the fact that assuming different models gives rise to different probability assignments.

While I am on the subject of the Shooting Room, it is worth drawing a parallel with another classic puzzle of probability theory, the St Petersburg Paradox. This is an old chestnut to do with a purported winning strategy for Roulette. It was first proposed by Nicolas Bernoulli but famously discussed at greatest length by Daniel Bernoulli in the pages of Transactions of the St Petersburg Academy, hence the name.  It works just as well for the case of a simple toss of a coin as for Roulette as in the latter game it involves betting only on red or black rather than on individual numbers.

Imagine you decide to bet such that you win by throwing heads. Your original stake is £1. If you win, the bank pays you at even money (i.e. you get your stake back plus another £1). If you lose, i.e. get tails, your strategy is to play again but bet double. If you win this time you get £4 back but have bet £2+£1=£3 up to that point. If you lose again you bet £8. If you win this time, you get £16 back but have paid in £8+£4+£2+£1=£15 to that point. Clearly, if you carry on the strategy of doubling your previous stake each time you lose, when you do eventually win you will be ahead by £1. It’s a guaranteed winner. Isn’t it?

The relationship of all this to the Shooting Room is that it is shows it is dangerous to pre-suppose a finite value for a number which in principle could be infinite. If the number of souls that could be called into existence is allowed to be infinite, then any individual as no chance at all of being called into existence in any generation!

Amusing as they are, the thing that makes me most uncomfortable about these Doomsday arguments is that they attempt to determine a probability of an event without any reference to underlying mechanism. For me, a valid argument about Doomsday would have to involve a particular physical cause for the extinction of humanity (e.g. asteroid impact, climate change, nuclear war, etc). Given this physical mechanism one should construct a model within which one can estimate probabilities for the model parameters (such as the rate of occurrence of catastrophic asteroid impacts). Only then can one make a valid inference based on relevant observations and their associated likelihoods. Such calculations may indeed lead to alarming or depressing results. I fear that the greatest risk to our future survival is not from asteroid impact or global warming, where the chances can be estimated with reasonable precision, but self-destructive violence carried out by humans themselves. Science has no way of being able to predict what atrocities people are capable of so we can’t make any reliable estimate of the probability we will self-destruct. But the absence of any specific mechanism in the versions of the Doomsday argument I have discussed robs them of any scientific credibility at all.

There are better grounds for worrying about the future than simple-minded numerology.

## Bayes, Laplace and Bayes’ Theorem

Posted in Bad Statistics with tags , , , , , , , , on October 1, 2014 by telescoper

A  couple of interesting pieces have appeared which discuss Bayesian reasoning in the popular media. One is by Jon Butterworth in his Grauniad science blog and the other is a feature article in the New York Times. I’m in early today because I have an all-day Teaching and Learning Strategy Meeting so before I disappear for that I thought I’d post a quick bit of background.

One way to get to Bayes’ Theorem is by starting with

$P(A|C)P(B|AC)=P(B|C)P(A|BC)=P(AB|C)$

where I refer to three logical propositions A, B and C and the vertical bar “|” denotes conditioning, i.e. $P(A|B)$ means the probability of A being true given the assumed truth of B; “AB” means “A and B”, etc. This basically follows from the fact that “A and B” must always be equivalent to “B and A”.  Bayes’ theorem  then follows straightforwardly as

$P(B|AC) = K^{-1}P(B|C)P(A|BC) = K^{-1} P(AB|C)$

where

$K=P(A|C).$

Many versions of this, including the one in Jon Butterworth’s blog, exclude the third proposition and refer to A and B only. I prefer to keep an extra one in there to remind us that every statement about probability depends on information either known or assumed to be known; any proper statement of probability requires this information to be stated clearly and used appropriately but sadly this requirement is frequently ignored.

Although this is called Bayes’ theorem, the general form of it as stated here was actually first written down not by Bayes, but by Laplace. What Bayes did was derive the special case of this formula for “inverting” the binomial distribution. This distribution gives the probability of x successes in n independent “trials” each having the same probability of success, p; each “trial” has only two possible outcomes (“success” or “failure”). Trials like this are usually called Bernoulli trials, after Daniel Bernoulli. If we ask the question “what is the probability of exactly x successes from the possible n?”, the answer is given by the binomial distribution:

$P_n(x|n,p)= C(n,x) p^x (1-p)^{n-x}$

where

$C(n,x)= \frac{n!}{x!(n-x)!}$

is the number of distinct combinations of x objects that can be drawn from a pool of n.

You can probably see immediately how this arises. The probability of x consecutive successes is p multiplied by itself x times, or px. The probability of (n-x) successive failures is similarly (1-p)n-x. The last two terms basically therefore tell us the probability that we have exactly x successes (since there must be n-x failures). The combinatorial factor in front takes account of the fact that the ordering of successes and failures doesn’t matter.

The binomial distribution applies, for example, to repeated tosses of a coin, in which case p is taken to be 0.5 for a fair coin. A biased coin might have a different value of p, but as long as the tosses are independent the formula still applies. The binomial distribution also applies to problems involving drawing balls from urns: it works exactly if the balls are replaced in the urn after each draw, but it also applies approximately without replacement, as long as the number of draws is much smaller than the number of balls in the urn. I leave it as an exercise to calculate the expectation value of the binomial distribution, but the result is not surprising: E(X)=np. If you toss a fair coin ten times the expectation value for the number of heads is 10 times 0.5, which is five. No surprise there. After another bit of maths, the variance of the distribution can also be found. It is np(1-p).

So this gives us the probability of x given a fixed value of p. Bayes was interested in the inverse of this result, the probability of p given x. In other words, Bayes was interested in the answer to the question “If I perform n independent trials and get x successes, what is the probability distribution of p?”. This is a classic example of inverse reasoning, in that it involved turning something like P(A|BC) into something like P(B|AC), which is what is achieved by the theorem stated at the start of this post.

Bayes got the correct answer for his problem, eventually, but by very convoluted reasoning. In my opinion it is quite difficult to justify the name Bayes’ theorem based on what he actually did, although Laplace did specifically acknowledge this contribution when he derived the general result later, which is no doubt why the theorem is always named in Bayes’ honour.

This is not the only example in science where the wrong person’s name is attached to a result or discovery. Stigler’s Law of Eponymy strikes again!

So who was the mysterious mathematician behind this result? Thomas Bayes was born in 1702, son of Joshua Bayes, who was a Fellow of the Royal Society (FRS) and one of the very first nonconformist ministers to be ordained in England. Thomas was himself ordained and for a while worked with his father in the Presbyterian Meeting House in Leather Lane, near Holborn in London. In 1720 he was a minister in Tunbridge Wells, in Kent. He retired from the church in 1752 and died in 1761. Thomas Bayes didn’t publish a single paper on mathematics in his own name during his lifetime but was elected a Fellow of the Royal Society (FRS) in 1742.

The paper containing the theorem that now bears his name was published posthumously in the Philosophical Transactions of the Royal Society of London in 1763. In his great Philosophical Essay on Probabilities Laplace wrote:

Bayes, in the Transactions Philosophiques of the Year 1763, sought directly the probability that the possibilities indicated by past experiences are comprised within given limits; and he has arrived at this in a refined and very ingenious manner, although a little perplexing.

The reasoning in the 1763 paper is indeed perplexing, and I remain convinced that the general form we now we refer to as Bayes’ Theorem should really be called Laplace’s Theorem. Nevertheless, Bayes did establish an extremely important principle that is reflected in the title of the New York Times piece I referred to at the start of this piece. In a nutshell this is that probabilities of future events can be updated on the basis of past measurements or, as I prefer to put it, “one person’s posterior is another’s prior”.