快猫短视频

WE ARE in a bar, and agree to toss a coin for the next round. Heads, I pay; tails, the drinks are on you. What are your chances of a free pint?

Most people 鈥� sober ones, at least 鈥� would agree: evens.

Then I flip the coin and catch it, but hide in it the palm of my hand. What鈥檚 your probability of free beer now?

Broadly speaking, there are two answers: (1) it is still 50 per cent, until you have reason to think otherwise; (2) assigning a probability to an event that has already happened is nonsense.

Which answer you incline towards reveals where you stand in a 250-year-old, sometimes strangely vicious debate on the nature of probability and statistics. It is the spat between frequentist and Bayesian statistics, and it is more than an esoteric problem. 鈥淭he frequentist-Bayesian debate is the only scientific controversy that actually does affect everybody鈥檚 life,鈥� says of Carnegie Mellon University in Pittsburgh, Pennsylvania. A drugs company testing a new drug can come to apparently very different conclusions according to which method it uses to analyse its results. A jury might reach a different decision after hearing evidence presented in frequentist and Bayesian terms. 鈥淚t鈥檚 not just philosophy, and it鈥檚 not just mathematics. It really is concrete,鈥� says Wasserman.

The two approaches have often seemed at loggerheads. But statisticians are slowly coming to a new appreciation: in a world of messy, incomplete information, the best way might be to combine the two very different worlds of probability 鈥� or at least mix them up a little.

Can you get your head around uncertainty? (Image: Eugenia Loli)

To fully appreciate the profundity of our bar bet, let鈥檚 start with an old T-shirt slogan: 鈥淪tatistics means never having to say you鈥檙e certain.鈥� Drawing conclusions without all the facts is the bread-and-butter of statistics. How many people in a country support legalising cannabis? You can鈥檛 ask all of them. Is a run of hotter summers consistent with natural variability, or a trend? There鈥檚 no way to look into the future to say definitively.

鈥淟ifetime chance of being killed by a dog in the US: 1 in 103,798; Source: 鈥�

Answers to such questions generally come with a probability attached. But that single number often masks a crucial distinction between two different sorts of uncertainty: stuff we don鈥檛 know, and stuff we can鈥檛 know.

Can鈥檛-know uncertainty results from real-world processes whose outcomes appear random to all who look at them: how a die rolls, where a roulette wheel stops, when exactly an atom in a radioactive sample will decay. This is the world of frequentist probability, because if you roll enough dice or observe enough atoms decaying, you can get a reasonable handle on the relative frequency of different outcomes, and can construct a measure of their probabilities.

Ignorance is Bayesian

Don鈥檛-know uncertainty is more slippery. Here individual ignorance, not universal randomness, is at play. What鈥檚 the sex of William and Kate鈥檚 new baby? We don鈥檛 yet know 鈥� although it is already a given. Who will win the cricket world cup? That is not a given 鈥� the tournament is still ongoing but the preliminary rounds will at least have given you a sense of who is in with a chance (assuming you care).

How to approach these different types of uncertainty divides frequentists and Bayesians. A strict frequentist has no truck with don鈥檛-know uncertainty, or any probability measure that can鈥檛 be derived from repeatable experiments, random number generators, surveys of a random population sample and the like. A Bayesian, meanwhile, doesn鈥檛 bat an eyelid at using other 鈥減riors鈥� 鈥� knowledge gleaned from past voting patterns in the cricket example, for example 鈥� to fill in the gaps. 鈥淏ayesians are happy to put probabilities on statements about the world,鈥� says Tony O鈥橦agan, a statistician at the University of Sheffield, UK, who researches Bayesian methods. 鈥淔requentists aren鈥檛.鈥�

The coin-in-the-pub example shows where these two views diverge. Before I flip the coin, frequentist and Bayesian probabilities line up: 50 per cent. Then the source of uncertainty changes from intrinsic randomness to personal ignorance. Only if you were inclined to Bayesian ways of working would you be happy to quote a probability figure. That figure might be 50 per cent 鈥� or perhaps a telltale flicker of a victorious smile on my face might persuade you to downgrade your chances of a free drink to just 20 per cent, say. 鈥淚n the Bayesian approach we try to answer questions by bringing all the relevant evidence to bear on it, even when the contribution of some of that evidence to the question depends on subjective judgements,鈥� says O鈥橦agan.

Bayesianism takes its name from the English mathematician and Presbyterian minister Thomas Bayes. In an essay published in 1763, two years after his death, he set out a new approach to a fundamental puzzle: how to work backwards from observations to hidden causes when your information is incomplete. Imagine you have a box of a dozen doughnuts, half cream, half jelly-filled. It鈥檚 relatively straightforward to calculate the probability of pulling out five jelly doughnuts in a row. But the backwards problem, working out the probable contents of an unknown box when you鈥檝e just pulled out five jellies, is trickier. Bayes鈥檚 innovation was to provide the seed of a mathematical framework that allowed you to start with a guess (perhaps you鈥檝e bought boxes of doughnuts from that store before), and refine it as further data came to light.

In the late 18th and early 19th centuries, Bayesian-style methods helped tame a range of inscrutable problems, from estimating the mass of Jupiter to calculating the number of boys born worldwide for every girl. But it gradually fell out of favour, victim of a dawning era of big data. Everything from improved astronomical observations to newly published statistical tables of mortality, disease and crime conveyed a reassuring air of objectivity. Bayes鈥檚 methods of educated guesswork seem hopelessly old-fashioned, and rather unscientific by contrast. Frequentism, with its emphasis on dispassionate number crunching of the results of randomised experiments, came increasingly into vogue.

The advent of quantum theory in the early 20th century, which re-expressed even reality in the language of frequentist probability (see 鈥�Random reality鈥�), provided a further spur to that development. The two strands of thought in statistics gradually drifted further apart. Adherents ended up submitting work to their own journals, attending their own conferences and even forming their own university departments. Emotions often ran high. The author Sharon Bertsch McGrayne recalls that when she started researching her book on the history of Bayesian ideas, The Theory That Would Not Die, one frequentist-leaning statistician berated her down the phone for attempting to legitimise Bayesianism. In return, Bayesians developed a sort of persecution complex, says at Carnegie Mellon. 鈥淪ome Bayesians got very self-righteous, with a kind of religious zealotry.鈥�

Flexible friend

In truth, though, both methods have their strengths and weaknesses. Where data points are scant and there is little chance of repeating an experiment, Bayesian methods can excel in squeezing out information. Take astrophysics as an example. A supernova explosion in a nearby galaxy, the Large Magellanic Cloud, seen in 1987, provided a chance to test long-held theories about the flux of neutrinos from such an event 鈥� but detectors picked up only 24 of these slippery particles. Without data, frequentist methods fell down 鈥� but the flexible, information-borrowing Bayesian approach .

It helped that well-grounded theories provided good priors to start that analysis. Where these don鈥檛 exist, a Bayesian analysis can easily be a case of garbage in, garbage out. It鈥檚 one reason why courts of law have been wary to adopt Bayesian methods, even though on the face of it they are an ideal way to synthesise messy evidence from many sources. In a 1993 New Jersey paternity case that used Bayesian statistics, the court decided jurors should use their own individual priors for the likelihood of the defendant having fathered the child, even though this would give each juror a different final statistical estimate of guilt. 鈥淭here鈥檚 no such thing as a right or wrong Bayesian answer,鈥� says Wasserman. 鈥淚t鈥檚 very postmodern.鈥�

Finding good priors can also demand an impossible depth of knowledge. Researchers searching for a cause for Alzheimer鈥檚 disease, for instance, might test 5000 genes. Bayesian methods would mean providing 5000 priors for the likely contribution of each gene, plus another 25 million if they wanted to look for pairs of genes working together. 鈥淣o one could construct a reasonable prior for such a high dimensional problem,鈥� says Wasserman. 鈥淎nd even if they did, no one else would believe it.鈥�

To be fair, without any background information, standard frequentist methods of sifting through many tiny genetic effects would have a hard time letting the truly important genes and combinations of genes rise to the top of the pile. But this is perhaps a problem more easily dealt with than conjuring up 25 million coherent Bayesian guesses.

Frequentism in general works well where plentiful data should speak in the most objective way possible. One high-profile example is the search for the Higgs boson, completed in 2012 at the CERN particle physics laboratory near Geneva, Switzerland. The analysis teams concluded that if in fact there were no Higgs boson, then a pattern of data as surprising as, or more surprising than, what was observed would be expected in only one in 3.5 million hypothetical repeated trials. That is so unlikely that the team felt comfortable rejecting the idea of a universe without a Higgs boson.

That wording may seem convoluted, and highlights frequentism鈥檚 main weakness: the way it ties itself in knots through its disdain for all don鈥檛-know uncertainties. The Higgs boson either exists or it doesn鈥檛, and any inability to say one way or the other is purely down to lack of information. A strict frequentist can鈥檛 actually make a direct statement of the probability of its existing or not 鈥� as indeed the CERN researchers were careful not to (although certain sections of the media and others were freer).

Head-to-head comparisons can point to the confusions that can arise, as was the case with a of two heart-attack drugs, streptokinase and tissue plasminogen activator, in the 1990s. The first, frequentist analysis gave a 鈥減 value鈥� of 0.001 to a study seeming to show that more patients survived after the newer, more expensive tissue plasminogen activator therapy. This equates to saying that if the two drugs had the same mortality rate, then data at least as extreme as the observed rates would occur only once in every 1000 repeated trials.

鈥淧robability of two children in a class of 25 sharing a birthday: 56.9%鈥�

That doesn鈥檛 mean the researchers were 99.9 per cent certain the new drug was superior 鈥� although again it is often interpreted that way. When other researchers conducted a using the results of previous clinical trials as a prior, they found a direct probability of the new drug being superior of only about 17 per cent. 鈥淚n Bayesianism we鈥檙e directly addressing the question of interest, talking about how likely it is to be true,鈥� says of the University of Cambridge. 鈥淲ho wouldn鈥檛 want to talk about that?鈥�

Perhaps it鈥檚 just a case of horses for courses, but don鈥檛 the strengths and weaknesses of both approaches suggest we might be better off combining elements of both? Kass is one of a new breed of statisticians doing just that. 鈥淭o me statistics is like a language,鈥� he says. 鈥淵ou can be conversant in both French and English and switch back and forth comfortably.鈥�

Stephen Senn, a drugs statistician at the Luxembourg Institute of Health agrees. 鈥淚 use what I call 鈥榤ongrel statistics鈥�, a little bit of everything,鈥� he says. 鈥淚 often work in a frequentist mode, but I reserve right to do Bayesian analyses and think in a Bayesian way.鈥�

鈥淕lobal probability of being born a girl: 48.3%; Source: CIA factbook鈥�

Kass points to an analysis he and his colleagues did on the firing rates of a couple of hundred neurons in the visual-motor region of the brain in monkeys. Prior work in basic neurobiology provided them with information on how fast these neurons should be firing, and how quickly the rate might change over time. They incorporated this into a Bayesian approach, then switched gears to evaluate their results under a standard frequentist framework. The Bayesian prior gave the methods enough of a kick-start to allow frequentist methods to detect even tiny differences in a sea of noisy data. The two approaches together .

Sometimes, Bayesian and frequentist ideas can be blended so much they create something new. In large genomics studies, a Bayesian analysis might exploit the fact that a study testing the effects of 2000 genes is almost like 2000 parallel experiments, and cross-fertilise the analyses, using the results from some to establish priors for others and using that to hone the conclusions of a frequentist analysis. 鈥淭his approach gives quite a bit better results,鈥� says Jeff Leek of Johns Hopkins University in Baltimore, Maryland. 鈥淚t鈥檚 really transformed the way we analyse genomic data.鈥�

It breaks down barriers, too. 鈥淚s this approach frequentist? Bayesian?鈥� asked Harvard University statistician Rafael Irizarry . 鈥淭o this applied statistician, it doesn鈥檛 really matter.鈥�

Not that the arguments have entirely gone away. 鈥淪tatistics is essentially the abstract language that science uses on top of data to tell stories about how nature works, and there is not one unique way to tell stories,鈥� says Kass. 鈥淭wo hundred years from now there might be some breakthrough connecting Bayesianism and frequentism into a grand synthesis, but my guess is that there will always be at least one versus the other.鈥�

So chances are in 2215, two people will still be sitting on pub stools arguing about their chances of free beer.

Read more: 鈥�Chance: How randomness rules our world鈥�

This article appeared in print under the headline 鈥淧robability wars鈥�