Archive for Bayesian probability

Bayes, Bridge and the Brain

Posted in Books, Talks and Reviews, The Universe and Stuff with tags , , , on April 12, 2012 by telescoper

I was having a chat over coffee yesterday with some members of the Mathematics Department here at the University of Cape Town, one of whom happens to be an expert at Bridge, actually representing South Africa in international competitions. That’s a much higher level than I could ever aspire to so I was a bit nervous about mentioning my interest in the game, but in the end I explained that I have in the past used Bridge (and other card games) to describe how Bayesian probability works; see this rather lengthy post for more details. The point is that as cards are played, one’s calculation of the probabilities of where the important cards lie changes in the light of information revealed. It makes much more sense to play Bridge according to a Bayesian interpretation, in which probability represents one’s state of knowledge, rather than what would happen over an ensemble of “random” realisations.

This particular topic – and Bayesian inference in general – is also discussed in my book From Cosmos to Chaos (which is, incidentally, now available in paperback). On my arrival in Cape Town I gave a copy of this book to my genial host, George Ellis, and our discussion of Bridge prompted him to say that he thought I had missed a trick in the book by not mentioning the connections between Bayesian probability and neuroscience. I hadn’t written about this because I didn’t know anything about it, so George happily enlightened me by sending a few review articles, such as this:

I can’t post it all, for fear of copyright infringement, but you get the idea. Here’s another one:

And another…

Nature Reviews Neuroscience 11, 605 (August 2010) | doi:10.1038/nrn2787-c1

A neurocentric approach to Bayesian inference    Christopher D. Fiorillo

Abstract A primary function of the brain is to infer the state of the world in order to determine which motor behaviours will best promote adaptive fitness. Bayesian probability theory formally describes how rational inferences ought to be made, and it has been used with great success in recent years to explain a range of perceptual and sensorimotor phenomena.

As a non-expert in neuroscience, I find these very interesting. I’ve long been convinced that from the point of view of formal reasoning, the Bayesian approach to probability is the only way that makes sense, but until reading these I’ve not been aware that there was serious work being done on the possibility that it also describes how the brain works in situations where there is insufficient information to be sure what is the correct approach. Except, of course, for players of Bridge who know it very well.

There’s just a chance that I may have readers out there who know more about this Bayes-Brain connection. If so, please enlighten me further through the comments box!

Guest Post – Bayesian Book Review

Posted in Bad Statistics, Books, Talks and Reviews with tags , , , on May 30, 2011 by telescoper

My regular commenter Anton circulated this book review by email yesterday and it stimulated quite a lot of reaction. I haven’t read the book myself, but I thought it would be fun to post his review on here to see whether it provokes similar responses. You can find the book on Amazon here (UK) or here ( USA). If you’re not completely au fait with Bayesian probability and the controversy around it, you might try reading one of my earlier posts about it, e.g. this one. I hope I can persuade some of the email commenters to upload their contributions through the box below!


The Theory That Would Not Die: How Bayes’ Rule Cracked the Enigma Code, Hunted Down Russian Submarines, and Emerged Triumphant from Two Centuries of Controversy

by Sharon Bertsch Mcgrayne

I found reading this book, which is a history of Bayes’ theorem written for the layman, to be deeply frustrating. The author does not really understand what probability IS – which is the key to all cogent writing on the subject. She never mentions the sum and product rules, or that Bayes’ theorem is an easy consequence of them. She notes, correctly, that Bayesian methods or something equivalent to them have been rediscovered advantageously again and again in an amazing variety of practical applications, and says that this is because they are pragmatically better than frequentist sampling theory – ie, she never asks the question: Why do they work better and what deeper rationale explains this? RT Cox is not mentioned. Ed Jaynes is mentioned only in passing as someone whose Bayesian fervour supposedly put people off.

The author is correct that computer applications have catalysed the Bayesian revolution, but in the pages on image processing and other general inverse problems (p218-21) she manages to miss the key work through the 1980s of Steve Gull and John Skilling, and you will not find “Maximum entropy” in the index. She does get the key role of Markov Chain Monte Carlo methods in computer implementation of Bayesian methods, however. But I can’t find Dave Mackay either, who deserves to be in the relevant section about modern applications.

On the other hand, as a historian of Bayesianism from Bayes himself to about 1960, she is full of superb anecdotes and information about
people who are to us merely names on the top of papers, or whose personalities are mentioned tantalisingly briefly in Jaynes’ writing.
For this material alone I recommend the book to Bayesians of our sort and am glad that I bought it.

Bayes’ Razor

Posted in Bad Statistics, The Universe and Stuff with tags , , , , , , , , , on February 19, 2011 by telescoper

It’s been quite while since I posted a little piece about Bayesian probability. That one and the others that followed it (here and here) proved to be surprisingly popular so I’ve been planning to add a few more posts whenever I could find the time. Today I find myself in the office after spending the morning helping out with a very busy UCAS visit day, and it’s raining, so I thought I’d take the opportunity to write something before going home. I think I’ll do a short introduction to a topic I want to do a more technical treatment of in due course.

A particularly important feature of Bayesian reasoning is that it gives precise motivation to things that we are generally taught as rules of thumb. The most important of these is Ockham’s Razor. This famous principle of intellectual economy is variously presented in Latin as Pluralites non est ponenda sine necessitate or Entia non sunt multiplicanda praetor necessitatem. Either way, it means basically the same thing: the simplest theory which fits the data should be preferred.

William of Ockham, to whom this dictum is attributed, was an English Scholastic philosopher (probably) born at Ockham in Surrey in 1280. He joined the Franciscan order around 1300 and ended up studying theology in Oxford. He seems to have been an outspoken character, and was in fact summoned to Avignon in 1323 to account for his alleged heresies in front of the Pope, and was subsequently confined to a monastery from 1324 to 1328. He died in 1349.

In the framework of Bayesian inductive inference, it is possible to give precise reasons for adopting Ockham’s razor. To take a simple example, suppose we want to fit a curve to some data. In the presence of noise (or experimental error) which is inevitable, there is bound to be some sort of trade-off between goodness-of-fit and simplicity. If there is a lot of noise then a simple model is better: there is no point in trying to reproduce every bump and wiggle in the data with a new parameter or physical law because such features are likely to be features of the noise rather than the signal. On the other hand if there is very little noise, every feature in the data is real and your theory fails if it can’t explain it.

To go a bit further it is helpful to consider what happens when we generalize one theory by adding to it some extra parameters. Suppose we begin with a very simple theory, just involving one parameter p, but we fear it may not fit the data. We therefore add a couple more parameters, say q and r. These might be the coefficients of a polynomial fit, for example: the first model might be straight line (with fixed intercept), the second a cubic. We don’t know the appropriate numerical values for the parameters at the outset, so we must infer them by comparison with the available data.

Quantities such as p, q and r are usually called “floating” parameters; there are as many as a dozen of these in the standard Big Bang model, for example.

Obviously, having three degrees of freedom with which to describe the data should enable one to get a closer fit than is possible with just one. The greater flexibility within the general theory can be exploited to match the measurements more closely than the original. In other words, such a model can improve the likelihood, i.e. the probability  of the obtained data  arising (given the noise statistics – presumed known) if the signal is described by whatever model we have in mind.

But Bayes’ theorem tells us that there is a price to be paid for this flexibility, in that each new parameter has to have a prior probability assigned to it. This probability will generally be smeared out over a range of values where the experimental results (contained in the likelihood) subsequently show that the parameters don’t lie. Even if the extra parameters allow a better fit to the data, this dilution of the prior probability may result in the posterior probability being lower for the generalized theory than the simple one. The more parameters are involved, the bigger the space of prior possibilities for their values, and the harder it is for the improved likelihood to win out. Arbitrarily complicated theories are simply improbable. The best theory is the most probable one, i.e. the one for which the product of likelihood and prior is largest.

To give a more quantitative illustration of this consider a given model M which has a set of N floating parameters represented as a vector \underline\lambda = (\lambda_1,\ldots \lambda_N)=\lambda_i; in a sense each choice of parameters represents a different model or, more precisely, a member of the family of models labelled M.

Now assume we have some data D and can consequently form a likelihood function P(D|\underline{\lambda},M). In Bayesian reasoning we have to assign a prior probability P(\underline{\lambda}|M) to the parameters of the model which, if we’re being honest, we should do in advance of making any measurements!

The interesting thing to look at now is not the best-fitting choice of model parameters \underline{\lambda} but the extent to which the data support the model in general.  This is encoded in a sort of average of likelihood over the prior probability space:

P(D|M) = \int P(D|\underline{\lambda},M) P(\underline{\lambda}|M) d^{N}\underline{\lambda}.

This is just the normalizing constant K usually found in statements of Bayes’ theorem which, in this context, takes the form

P(\underline{\lambda}|DM) = K^{-1}P(\underline{\lambda}|M)P(D|\underline{\lambda},M).

In statistical mechanics things like K are usually called partition functions, but in this setting K is called the evidence, and it is used to form the so-called Bayes Factor, used in a technique known as Bayesian model selection of which more anon….

The  usefulness of the Bayesian evidence emerges when we ask the question whether our N  parameters are sufficient to get a reasonable fit to the data. Should we add another one to improve things a bit further? And why not another one after that? When should we stop?

The answer is that although adding an extra degree of freedom can increase the first term in the integral defining K (the likelihood), it also imposes a penalty in the second factor, the prior, because the more parameters the more smeared out the prior probability must be. If the improvement in fit is marginal and/or the data are noisy, then the second factor wins and the evidence for a model with N+1 parameters lower than that for the N-parameter version. Ockham’s razor has done its job.

This is a satisfying result that is in nice accord with common sense. But I think it goes much further than that. Many modern-day physicists are obsessed with the idea of a “Theory of Everything” (or TOE). Such a theory would entail the unification of all physical theories – all laws of Nature, if you like – into a single principle. An equally accurate description would then be available, in a single formula, of phenomena that are currently described by distinct theories with separate sets of parameters. Instead of textbooks on mechanics, quantum theory, gravity, electromagnetism, and so on, physics students would need just one book.

The physicist Stephen Hawking has described the quest for a TOE as like trying to read the Mind of God. I think that is silly. If a TOE is every constructed it will be the most economical available description of the Universe. Not the Mind of God.  Just the best way we have of saving paper.


Deductivism and Irrationalism

Posted in Bad Statistics, The Universe and Stuff with tags , , , , , , , , , , , on December 11, 2010 by telescoper

Looking at my stats I find that my recent introductory post about Bayesian probability has proved surprisingly popular with readers, so I thought I’d follow it up with a brief discussion of some of the philosophical issues surrounding it.

It is ironic that the pioneers of probability theory, principally Laplace, unquestionably adopted a Bayesian rather than frequentist interpretation for his probabilities. Frequentism arose during the nineteenth century and held sway until recently. I recall giving a conference talk about Bayesian reasoning only to be heckled by the audience with comments about “new-fangled, trendy Bayesian methods”. Nothing could have been less apt. Probability theory pre-dates the rise of sampling theory and all the frequentist-inspired techniques that modern-day statisticians like to employ.

Most disturbing of all is the influence that frequentist and other non-Bayesian views of probability have had upon the development of a philosophy of science, which I believe has a strong element of inverse reasoning or inductivism in it. The argument about whether there is a role for this type of thought in science goes back at least as far as Roger Bacon who lived in the 13th Century. Much later the brilliant Scottish empiricist philosopher and enlightenment figure David Hume argued strongly against induction. Most modern anti-inductivists can be traced back to this source. Pierre Duhem has argued that theory and experiment never meet face-to-face because in reality there are hosts of auxiliary assumptions involved in making this comparison. This is nowadays called the Quine-Duhem thesis.

Actually, for a Bayesian this doesn’t pose a logical difficulty at all. All one has to do is set up prior probability distributions for the required parameters, calculate their posterior probabilities and then integrate over those that aren’t related to measurements. This is just an expanded version of the idea of marginalization, explained here.

Rudolf Carnap, a logical positivist, attempted to construct a complete theory of inductive reasoning which bears some relationship to Bayesian thought, but he failed to apply Bayes’ theorem in the correct way. Carnap distinguished between two types or probabilities – logical and factual. Bayesians don’t – and I don’t – think this is necessary. The Bayesian definition seems to me to be quite coherent on its own.

Other philosophers of science reject the notion that inductive reasoning has any epistemological value at all. This anti-inductivist stance, often somewhat misleadingly called deductivist (irrationalist would be a better description) is evident in the thinking of three of the most influential philosophers of science of the last century: Karl Popper, Thomas Kuhn and, most recently, Paul Feyerabend. Regardless of the ferocity of their arguments with each other, these have in common that at the core of their systems of thought likes the rejection of all forms of inductive reasoning. The line of thought that ended in this intellectual cul-de-sac began, as I stated above, with the work of the Scottish empiricist philosopher David Hume. For a thorough analysis of the anti-inductivists mentioned above and their obvious debt to Hume, see David Stove’s book Popper and After: Four Modern Irrationalists. I will just make a few inflammatory remarks here.

Karl Popper really began the modern era of science philosophy with his Logik der Forschung, which was published in 1934. There isn’t really much about (Bayesian) probability theory in this book, which is strange for a work which claims to be about the logic of science. Popper also managed to, on the one hand, accept probability theory (in its frequentist form), but on the other, to reject induction. I find it therefore very hard to make sense of his work at all. It is also clear that, at least outside Britain, Popper is not really taken seriously by many people as a philosopher. Inside Britain it is very different and I’m not at all sure I understand why. Nevertheless, in my experience, most working physicists seem to subscribe to some version of Popper’s basic philosophy.

Among the things Popper has claimed is that all observations are “theory-laden” and that “sense-data, untheoretical items of observation, simply do not exist”. I don’t think it is possible to defend this view, unless one asserts that numbers do not exist. Data are numbers. They can be incorporated in the form of propositions about parameters in any theoretical framework we like. It is of course true that the possibility space is theory-laden. It is a space of theories, after all. Theory does suggest what kinds of experiment should be done and what data is likely to be useful. But data can be used to update probabilities of anything.

Popper has also insisted that science is deductive rather than inductive. Part of this claim is just a semantic confusion. It is necessary at some point to deduce what the measurable consequences of a theory might be before one does any experiments, but that doesn’t mean the whole process of science is deductive. He does, however, reject the basic application of inductive reasoning in updating probabilities in the light of measured data; he asserts that no theory ever becomes more probable when evidence is found in its favour. Every scientific theory begins infinitely improbable, and is doomed to remain so.

Now there is a grain of truth in this, or can be if the space of possibilities is infinite. Standard methods for assigning priors often spread the unit total probability over an infinite space, leading to a prior probability which is formally zero. This is the problem of improper priors. But this is not a killer blow to Bayesianism. Even if the prior is not strictly normalizable, the posterior probability can be. In any case, given sufficient relevant data the cycle of experiment-measurement-update of probability assignment usually soon leaves the prior far behind. Data usually count in the end.

The idea by which Popper is best known is the dogma of falsification. According to this doctrine, a hypothesis is only said to be scientific if it is capable of being proved false. In real science certain “falsehood” and certain “truth” are almost never achieved. Theories are simply more probable or less probable than the alternatives on the market. The idea that experimental scientists struggle through their entire life simply to prove theorists wrong is a very strange one, although I definitely know some experimentalists who chase theories like lions chase gazelles. To a Bayesian, the right criterion is not falsifiability but testability, the ability of the theory to be rendered more or less probable using further data. Nevertheless, scientific theories generally do have untestable components. Any theory has its interpretation, which is the untestable baggage that we need to supply to make it comprehensible to us. But whatever can be tested can be scientific.

Popper’s work on the philosophical ideas that ultimately led to falsificationism began in Vienna, but the approach subsequently gained enormous popularity in western Europe. The American Thomas Kuhn later took up the anti-inductivist baton in his book The Structure of Scientific Revolutions. Kuhn is undoubtedly a first-rate historian of science and this book contains many perceptive analyses of episodes in the development of physics. His view of scientific progress is cyclic. It begins with a mass of confused observations and controversial theories, moves into a quiescent phase when one theory has triumphed over the others, and lapses into chaos again when the further testing exposes anomalies in the favoured theory. Kuhn adopted the word paradigm to describe the model that rules during the middle stage,

The history of science is littered with examples of this process, which is why so many scientists find Kuhn’s account in good accord with their experience. But there is a problem when attempts are made to fuse this historical observation into a philosophy based on anti-inductivism. Kuhn claims that we “have to relinquish the notion that changes of paradigm carry scientists ..closer and closer to the truth.” Einstein’s theory of relativity provides a closer fit to a wider range of observations than Newtonian mechanics, but in Kuhn’s view this success counts for nothing.

Paul Feyerabend has extended this anti-inductivist streak to its logical (though irrational) extreme. His approach has been dubbed “epistemological anarchism”, and it is clear that he believed that all theories are equally wrong. He is on record as stating that normal science is a fairytale, and that equal time and resources should be spent on “astrology, acupuncture and witchcraft”. He also categorised science alongside “religion, prostitution, and so on”. His thesis is basically that science is just one of many possible internally consistent views of the world, and that the choice between which of these views to adopt can only be made on socio-political grounds.

Feyerabend’s views could only have flourished in a society deeply disillusioned with science. Of course, many bad things have been done in science’s name, and many social institutions are deeply flawed. One can’t expect anything operated by people to run perfectly. It’s also quite reasonable to argue on ethical grounds which bits of science should be funded and which should not. But the bottom line is that science does have a firm methodological basis which distinguishes it from pseudo-science, the occult and new age silliness. Science is distinguished from other belief-systems by its rigorous application of inductive reasoning and its willingness to subject itself to experimental test. Not all science is done properly, of course, and bad science is as bad as anything.

The Bayesian interpretation of probability leads to a philosophy of science which is essentially epistemological rather than ontological. Probabilities are not “out there” in external reality, but in our minds, representing our imperfect knowledge and understanding. Scientific theories are not absolute truths. Our knowledge of reality is never certain, but we are able to reason consistently about which of our theories provides the best available description of what is known at any given time. If that description fails when more data are gathered, we move on, introducing new elements or abandoning the theory for an alternative. This process could go on forever. There may never be a final theory. But although the game might have no end, at least we know the rules….


A Little Bit of Bayes

Posted in Bad Statistics, The Universe and Stuff with tags , , , , , , on November 21, 2010 by telescoper

I thought I’d start a series of occasional posts about Bayesian probability. This is something I’ve touched on from time to time but its perhaps worth covering this relatively controversial topic in a slightly more systematic fashion especially with regard to how it works in cosmology.

I’ll start with Bayes’ theorem which for three logical propositions (such as statements about the values of parameters in theory) A, B and C can be written in the form

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



This is (or should be!)  uncontroversial as it is simply a result of the sum and product rules for combining probabilities. Notice, however, that I’ve not restricted it to two propositions A and B as is often done, but carried throughout an extra one (C). This is to emphasize the fact that, to a Bayesian, all probabilities are conditional on something; usually, in the context of data analysis this is a background theory that furnishes the framework within which measurements are interpreted. If you say this makes everything model-dependent, then I’d agree. But every interpretation of data in terms of parameters of a model is dependent on the model. It has to be. If you think it can be otherwise then I think you’re misguided.

In the equation,  P(B|C) is the probability of B being true, given that C is true . The information C need not be definitely known, but perhaps assumed for the sake of argument. The left-hand side of Bayes’ theorem denotes the probability of B given both A and C, and so on. The presence of C has not changed anything, but is just there as a reminder that it all depends on what is being assumed in the background. The equation states  a theorem that can be proved to be mathematically correct so it is – or should be – uncontroversial.

Now comes the controversy. In the “frequentist” interpretation of probability, the entities A, B and C would be interpreted as “events” (e.g. the coin is heads) or “random variables” (e.g. the score on a dice, a number from 1 to 6) attached to which is their probability, indicating their propensity to occur in an imagined ensemble. These things are quite complicated mathematical objects: they don’t have specific numerical values, but are represented by a measure over the space of possibilities. They are sort of “blurred-out” in some way, the fuzziness representing the uncertainty in the precise value.

To a Bayesian, the entities A, B and C have a completely different character to what they represent for a frequentist. They are not “events” but  logical propositions which can only be either true or false. The entities themselves are not blurred out, but we may have insufficient information to decide which of the two possibilities is correct. In this interpretation, P(A|C) represents the degree of belief that it is consistent to hold in the truth of A given the information C. Probability is therefore a generalization of the “normal” deductive logic expressed by Boolean algebra: the value “0” is associated with a proposition which is false and “1” denotes one that is true. Probability theory extends  this logic to the intermediate case where there is insufficient information to be certain about the status of the proposition.

A common objection to Bayesian probability is that it is somehow arbitrary or ill-defined. “Subjective” is the word that is often bandied about. This is only fair to the extent that different individuals may have access to different information and therefore assign different probabilities. Given different information C and C′ the probabilities P(A|C) and P(A|C′) will be different. On the other hand, the same precise rules for assigning and manipulating probabilities apply as before. Identical results should therefore be obtained whether these are applied by any person, or even a robot, so that part isn’t subjective at all.

In fact I’d go further. I think one of the great strengths of the Bayesian interpretation is precisely that it does depend on what information is assumed. This means that such information has to be stated explicitly. The essential assumptions behind a result can be – and, regrettably, often are – hidden in frequentist analyses. Being a Bayesian forces you to put all your cards on the table.

To a Bayesian, probabilities are always conditional on other assumed truths. There is no such thing as an absolute probability, hence my alteration of the form of Bayes’s theorem to represent this. A probability such as P(A) has no meaning to a Bayesian: there is always conditioning information. For example, if  I blithely assign a probability of 1/6 to each face of a dice, that assignment is actually conditional on me having no information to discriminate between the appearance of the faces, and no knowledge of the rolling trajectory that would allow me to make a prediction of its eventual resting position.

In tbe Bayesian framework, probability theory  becomes not a branch of experimental science but a branch of logic. Like any branch of mathematics it cannot be tested by experiment but only by the requirement that it be internally self-consistent. This brings me to what I think is one of the most important results of twentieth century mathematics, but which is unfortunately almost unknown in the scientific community. In 1946, Richard Cox derived the unique generalization of Boolean algebra under the assumption that such a logic must involve associated a single number with any logical proposition. The result he got is beautiful and anyone with any interest in science should make a point of reading his elegant argument. It turns out that the only way to construct a consistent logic of uncertainty incorporating this principle is by using the standard laws of probability. There is no other way to reason consistently in the face of uncertainty than probability theory. Accordingly, probability theory always applies when there is insufficient knowledge for deductive certainty. Probability is inductive logic.

This is not just a nice mathematical property. This kind of probability lies at the foundations of a consistent methodological framework that not only encapsulates many common-sense notions about how science works, but also puts at least some aspects of scientific reasoning on a rigorous quantitative footing. This is an important weapon that should be used more often in the battle against the creeping irrationalism one finds in society at large.

I posted some time ago about an alternative way of deriving the laws of probability from consistency arguments.

To see how the Bayesian approach works, let us consider a simple example. Suppose we have a hypothesis H (some theoretical idea that we think might explain some experiment or observation). We also have access to some data D, and we also adopt some prior information I (which might be the results of other experiments or simply working assumptions). What we want to know is how strongly the data D supports the hypothesis H given my background assumptions I. To keep it easy, we assume that the choice is between whether H is true or H is false. In the latter case, “not-H” or H′ (for short) is true. If our experiment is at all useful we can construct P(D|HI), the probability that the experiment would produce the data set D if both our hypothesis and the conditional information are true.

The probability P(D|HI) is called the likelihood; to construct it we need to have   some knowledge of the statistical errors produced by our measurement. Using Bayes’ theorem we can “invert” this likelihood to give P(H|DI), the probability that our hypothesis is true given the data and our assumptions. The result looks just like we had in the first two equations:

P(H|DI) = K^{-1}P(H|I)P(D|HI) .

Now we can expand the “normalising constant” K because we know that either H or H′ must be true. Thus

K=P(D|I)=P(H|I)P(D|HI)+P(H^{\prime}|I) P(D|H^{\prime}I)

The P(H|DI) on the left-hand side of the first expression is called the posterior probability; the right-hand side involves P(H|I), which is called the prior probability and the likelihood P(D|HI). The principal controversy surrounding Bayesian inductive reasoning involves the prior and how to define it, which is something I’ll comment on in a future post.

The Bayesian recipe for testing a hypothesis assigns a large posterior probability to a hypothesis for which the product of the prior probability and the likelihood is large. It can be generalized to the case where we want to pick the best of a set of competing hypothesis, say H1 …. Hn. Note that this need not be the set of all possible hypotheses, just those that we have thought about. We can only choose from what is available. The hypothesis may be relatively simple, such as that some particular parameter takes the value x, or they may be composite involving many parameters and/or assumptions. For instance, the Big Bang model of our universe is a very complicated hypothesis, or in fact a combination of hypotheses joined together,  involving at least a dozen parameters which can’t be predicted a priori but which have to be estimated from observations.

The required result for multiple hypotheses is pretty straightforward: the sum of the two alternatives involved in K above simply becomes a sum over all possible hypotheses, so that

P(H_i|DI) = K^{-1}P(H_i|I)P(D|H_iI),


K=P(D|I)=\sum P(H_j|I)P(D|H_jI)

If the hypothesis concerns the value of a parameter – in cosmology this might be, e.g., the mean density of the Universe expressed by the density parameter Ω0 – then the allowed space of possibilities is continuous. The sum in the denominator should then be replaced by an integral, but conceptually nothing changes. Our “best” hypothesis is the one that has the greatest posterior probability.

From a frequentist stance the procedure is often instead to just maximize the likelihood. According to this approach the best theory is the one that makes the data most probable. This can be the same as the most probable theory, but only if the prior probability is constant, but the probability of a model given the data is generally not the same as the probability of the data given the model. I’m amazed how many practising scientists make this error on a regular basis.

The following figure might serve to illustrate the difference between the frequentist and Bayesian approaches. In the former case, everything is done in “data space” using likelihoods, and in the other we work throughout with probabilities of hypotheses, i.e. we think in hypothesis space. I find it interesting to note that most theorists that I know who work in cosmology are Bayesians and most observers are frequentists!

As I mentioned above, it is the presence of the prior probability in the general formula that is the most controversial aspect of the Bayesian approach. The attitude of frequentists is often that this prior information is completely arbitrary or at least “model-dependent”. Being empirically-minded people, by and large, they prefer to think that measurements can be made and interpreted without reference to theory at all.

Assuming we can assign the prior probabilities in an appropriate way what emerges from the Bayesian framework is a consistent methodology for scientific progress. The scheme starts with the hardest part – theory creation. This requires human intervention, since we have no automatic procedure for dreaming up hypothesis from thin air. Once we have a set of hypotheses, we need data against which theories can be compared using their relative probabilities. The experimental testing of a theory can happen in many stages: the posterior probability obtained after one experiment can be fed in, as prior, into the next. The order of experiments does not matter. This all happens in an endless loop, as models are tested and refined by confrontation with experimental discoveries, and are forced to compete with new theoretical ideas. Often one particular theory emerges as most probable for a while, such as in particle physics where a “standard model” has been in existence for many years. But this does not make it absolutely right; it is just the best bet amongst the alternatives. Likewise, the Big Bang model does not represent the absolute truth, but is just the best available model in the face of the manifold relevant observations we now have concerning the Universe’s origin and evolution. The crucial point about this methodology is that it is inherently inductive: all the reasoning is carried out in “hypothesis space” rather than “observation space”.  The primary form of logic involved is not deduction but induction. Science is all about inverse reasoning.

For comments on induction versus deduction in another context, see here.

So what are the main differences between the Bayesian and frequentist views?

First, I think it is fair to say that the Bayesian framework is enormously more general than is allowed by the frequentist notion that probabilities must be regarded as relative frequencies in some ensemble, whether that is real or imaginary. In the latter interpretation, a proposition is at once true in some elements of the ensemble and false in others. It seems to me to be a source of great confusion to substitute a logical AND for what is really a logical OR. The Bayesian stance is also free from problems associated with the failure to incorporate in the analysis any information that can’t be expressed as a frequency. Would you really trust a doctor who said that 75% of the people she saw with your symptoms required an operation, but who did not bother to look at your own medical files?

As I mentioned above, frequentists tend to talk about “random variables”. This takes us into another semantic minefield. What does “random” mean? To a Bayesian there are no random variables, only variables whose values we do not know. A random process is simply one about which we only have sufficient information to specify probability distributions rather than definite values.

More fundamentally, it is clear from the fact that the combination rules for probabilities were derived by Cox uniquely from the requirement of logical consistency, that any departure from these rules will generally speaking involve logical inconsistency. Many of the standard statistical data analysis techniques – including the simple “unbiased estimator” mentioned briefly above – used when the data consist of repeated samples of a variable having a definite but unknown value, are not equivalent to Bayesian reasoning. These methods can, of course, give good answers, but they can all be made to look completely silly by suitable choice of dataset.

By contrast, I am not aware of any example of a paradox or contradiction that has ever been found using the correct application of Bayesian methods, although method can be applied incorrectly. Furthermore, in order to deal with unique events like the weather, frequentists are forced to introduce the notion of an ensemble, a perhaps infinite collection of imaginary possibilities, to allow them to retain the notion that probability is a proportion. Provided the calculations are done correctly, the results of these calculations should agree with the Bayesian answers. On the other hand, frequentists often talk about the ensemble as if it were real, and I think that is very dangerous…