Was Helmholtz a Bayesian? A review

Transcription

1 Perception, 2008, volume 37, pages 000 ^ 000 doi: /p5973 Was Helmholtz a Bayesian? A review Gerald Westheimer Division of Neurobiology, 144 Life Sciences Addition, University of California at Berkeley, Berkeley, CA , USA; gwestheimer@berkeley.edu Received 22 December 2007; in revised form 15 January 2008 Abstract. Modern developments in machine vision and object recognition have generated renewed interest in the proposal for drawing inferences put forward by the Rev. Thomas Bayes (1701 ^ 1759). In this connection the epistemological studies by Hermann Helmholtz (1821 ^ 1894) are often cited as laying the foundation of the currently popular move to regard perception as Bayesian inference. Helmholtz in his mature writings tried to reconcile the German idealist notions of reality-ashypothesis with scientists' quests for the laws of nature, and espoused the view that we ``attain knowledge of the lawful order in the realm of the real, but only in so far as it is represented in the tokens within the system of sensory impressions''. His propositions of inferring objects from internal sensory signals by what he called `unconscious inferences' have made Helmholtz be regarded as a proto-bayesian. But juxtaposing Bayes's original writings, the modern formulation of Bayesian inference, and Helmholtz's views of perception reveals only a tenuous relationship. ``Ich bin Bayesianer'' (Stegmu«ller 1973, page 117) 1 Introduction A large body of scholarship has grown around what is now called Bayesian statistics. The approach is widely used in, for example, machine vision and object recognition, and researchers in perception are being strongly urged to embrace it (Kersten et al 2004; Knill and Richards 1996). To enhance the argument, the name of Hermann Helmholtz (1821 ^ 1894), the founder and still most prominent contributor to visual science, is often invoked as having been a pioneer in thinking along the lines of modern Bayesian inference. To examine this claim it is necessary to study what both Bayes (1701 ^ 1759) and Helmholtz (figures 1 and 2) had actually written, as contrasted with the views now being imputed to them. Bayes's ideas, long in eclipse, had not yet resurfaced during Helmholtz's lifetime, nor had they assumed their modern form until well into the 20th century (Dale 1999). Hence the question can be raised whether Helmholtz, in a prescient way, had come to hold views that nowadays would be called Bayesian, even though, as we shall see, there is no consensus that Bayes actually held them himself in that form. 2 Modern formulation of Bayesian inference At present, one would call a Bayesian someone who subscribes to the standard formulation of Bayesian inference, as laid out in countless textbooks and articles. It runs something like the following (see, for example, MacKay 2003): Given an observed event E, and a set of hypotheses H 1 ; H 2 ; :::, then P(H 1 je ), the probability of the hypothesis H 1 given the event, is proportional to the product of two terms. The first is P(EjH 1, the probability of the event given that hypothesis H 1 ; it is called the likelihood. The second is P(H 1 ), the estimate that this hypothesis is in fact applicable, ie the prior probability. To reach an actual probability, a value between 0 and 1, a normalization factor is introduced to cover the full range of all hypotheses H 1, H 2, :::. Hence the usual formulation of the Bayes theorem nowadays is X P H 1 je ˆP E jh 1 6P H 1 P E jh i 6P H i Š. (1) i

2 2 G Westheimer Figure 1. The Rev. Thomas Bayes, F.R.S. (1701 ^ 1759) Figure 2. Exzellenz Hermann v. Helmholtz (1821 ^ 1894). Sometimes, a more particular formulation is used for cases in which there is a set of parameters w within the hypothesis H. For example, if H is the hypothesis that a Poisson distribution is involved, and w is the variance, the equation takes the following form X P w je, H ˆP E jw, H 6P w jh P E jw, H 6P wjh Š. The term P(E jw, H ) is unproblematic and involves the calculation of an inverse probability, viz the probability of the event E, given the hypothesis. Most of the discussion has centered on the origin and ultimate source of the `prior'. When applied to object recognition, the task is to decide whether a specific component in a captured picture is in fact the image of a real object. In this connection, the event is the presence in the image of a structure or a pattern that can be fully characterized, for example by the number and value of pixels. The hypothesis would be the existence of a specific object with a pre-defined structure in the real world. Enough would have to be known about the actual image and each of the hypothesized objects to calculate the probability that in the mode of transmission each of these objects would generate such an image. Also required are prior estimates of the probability of the objects being in fact out there in the real world. With this information it is possible to use the above equations to calculate the posterior probability that what has been received is in fact the image of a specific object. 3 What Bayes actually wrote It is of interest to compare the above formulation with what the Rev. Thomas Bayes had actually said. (1) In the posthumously published essay in Philosophical Transactions Bayes wrote that he was investigating how to solve the problem (Barnard 1958; Bayes 1763) (1) Bayes was not only accomplished as a mathematician but also had a deep understanding of the place of mathematics vis-a -vis the natural world: ``It is not the business of a Mathematician to show that a strait line or circle can be drawn, but he tells you what he means by these; and if you understand him, you may proceed further with him; and it would not be the purpose to object that there is no such thing in nature as a true strait line or perfect circle, for this is not his concern: he is not inquiring how things are in matter of fact, but supposing things to be in a certain way, what are the consequences to be deduced from them; and all that is to be demanded from him is, that his suppositions be intelligible, and his inferences just from the suppositions he makes.'' (cited in Barnard 1958, page 294)

3 Was Helmholtz a Bayesian? 3 ``Given the number of times in which an unknown event has happened and failed: required the chance that the probability of its happening in a single trial lies somewhere between any two degrees of probability that can be named.'' (page...) What follows has been described by one of its foremost interpreters as ``one of the most difficult works to read in the history of statistics'' (Stigler 1982). Bayes deals with the location along a distance AB of the point in which a thrown ball comes to rest, given that ``there shall be the same probability that it rests upon any one equal part of the plane as another, and that it must necessarily rest somewhere upon it''. First, a ball W is thrown once, landing in o (figure 3), but the position of o is not determined. What has been determined, though, is that a second similar ball, when thrown n times, landed p times to the right of o and q ˆ n p times to the left. Bayes is interested in the probability of o lying between any arbitrary points F and B along AB. In each ball throw ``there shall be the same probability that it rests upon any one equal part of the plane as another'' and ``the happening or failing of an event in different trials are so many independent events''. f b f b B o o A x Figure 3. Bayes considers the situation of a ball being thrown to land anywhere with equal probability along the distance AB, calling the event a success if it comes to rest to the right of a fixed point o whose position is unknown. The law governing the event is the binomial distribution with parameter x if one stipulates the event to have taken place on p occasions out of p q ˆ n trials. Bayes then goes on to calculate the chance of p successes in n trials for the interval between two points f and b within AB. It is the ratio of the area under the likelihood distribution of binomials between parameters f and b to that under the whole distribution between A and B. Bayes calls the situation in which a ball lands to the right of o as ``the happening of the event M'' and argues that its occurrence p times in p q ˆ n trials is governed by the binomial distribution (n!=p!q!)6x p 6(1 x) q where x is the ratio of the distances Ao and AB. In modern parlance, the ball coming to rest to the right of o is the event and the binomial distribution in the hypothesis. Bayes then presciently used what is now known as the inverse approach: For any other point along AB, say f, the chance that a ball tossed n times would on p occasions land to the right of f is also governed by the binomial, now (n!=p!q!)6f p 6 1 f ) q, where f is the distance Af. Hence the desired answer, viz the chance that o lies between f and B, is given by the sum of the binomials with all the parameters between f and B. In order to arrive at a probability, ie a value between zero and one, this would have to be divided by the sum for all points between A and B. According to Bayes, this is also the probability that a single throw from the same ensemble lands in that interval. Bayes approach is to identify the event E and the governing hypothesis w, H (that the chance of a tossed ball landing to the right of any dividing point of a line of unit length is given by the binomial distribution with the parameter w, the ratio of the P dividing point distance to the length of the whole line) and then to compute P E jw, H, the chance of the event occurring for the stipulated parameter range of the hypothesis. In other words, Bayes's operations take place in the realm of likelihood.

4 4 G Westheimer Just to illustrate what Bayes was after: If 5 out of 25 ball tosses gave a positive answer to the question whether the event has occurred, what is the probability that o lies in the middle 10% of the line AB, ie between f ˆ 0:45 and b ˆ 0:55? If B (x; n, p) ˆ n!= p!(n p)!š6x p 6(1 x) q, then the required probability is x ˆ 0:55 x ˆ 1:00 P P B x; 25, 5 B x; 25, 5 x ˆ 0:45 x ˆ 0 The argument proceeds along the following lines: For x ˆ 0:45, what is the probability that a single throw lands to the right of it? Here n ˆ p ˆ 1 and (0:45; 1, 1) ˆ 0:45. For one throw out of two it is B 0:45; 2, 1 ˆ0:495, because either both throws go into one or the other division of the whole distance, with probabilities of (0.45) 2 and (0.55) 2 respectively, or one throw goes into each, with probability 0:4560:55 ˆ 0:2475, which can occur in two ways. Finally, for exactly 5 throws out of 25, the binomial value B(0:45; 25, 5) is equal to ; P B x; 25, 5 for the range 0.45 to 0.55 computes to and for the full distance 1 4 x 4 0 to Hence, in this particular application of Bayes's problem, namely `If 5 out of 25 tosses had a positive outcome, what is the probability of 0:55 4 o 4 0:45?' the answer is =3.846 ˆ , or 1 in 154. On the other hand, for 0:3 4 x 4 0:2 itis0.453or1in2.2. But it is the prior that is the crucial component of all the Bayesian discussions. In Bayes's words, in arriving at the value of his expectation he wanted to consider whether there was anything that may give ``reason to think that, in a certain number of trials, [the event] should rather happen any one possible number of times than another''. Even here, unfortunately, Bayes's writing has led to debates. Having called the outcome of the ball-throw the ``event M'' in prop. 9 where he lays out the binomial proposition, he goes on to say: ``In what follows therefore I shall take for granted that the rule given concerning the event M in prop. 9 is also the rule to be used in relation to any event concerning the probability of which nothing at all is known antecedently to any trials made or observed concerning it.'' (cited in Barnard 1958, page 306). Such a phrasing might give the impression that Bayes rejected the idea of a prior, but this is not the case, for he states explicitly concerning both the fiduciary and test throws ``that if either... be thrown there shall be the same probability that it rests upon any one equal part of the plane as another'', ie the prior is flat. Stigler, who subjected the situation to the most thorough critical analysis, concluded that for Bayes the prior is associated with the locations of the subsequent throws and not those leading to the formulation of the hypothesis H, as is now embodied in the canon of Bayesian inference. As it happens, in the situation analyzed by Bayes, both have a flat prior and the distinction is not material. However, the substitution of P(E ) for P(H ) in equation (1) would reformulate Bayes's theorem into P H je ˆP E jh 6P E X P E jh 6P E Š and deprive it of its main attraction, namely the ability to assign a different probability to one hypothesis or model, or one range of parameters within a hypothesis, than to others. The starting point of modern Bayesian analyses is a particular set of observations and the assignment of higher probabilities to some than to others surely must relate to hypotheses and not observations.

5 Was Helmholtz a Bayesian? 5 Thus is seems that, if being a Bayesian means being a strict follower of Bayes, it does not necessarily imply unconditional acceptance of the current interpretation of all the terms of equation (1), even if there were no other challenges to their application. Bayes was ignored for at least a century and a half. When his work was resurrected, it was in the context of arguments between warring factions in statistics and probability theory who tried to come to terms with the extremely deep problem of defining probability. The situation is one in which Einstein's famous dictum (made in connection with geometry) might be rephrased: Insofar as probability definitions are mathematical, they do not refer to reality; insofar as they refer to reality they are not mathematical. In mathematics one looks for proofs of convergence of infinite series. But in situations where probability is invoked, even in as long a sequence of dice throws (or balls drawn from an urn with replacement) as might be contemplated, the outcome is ultimately uncertain and hence unsatisfactory in the strictest logicodeductive environment of mathematics. Again and again, in probability theory and statistics, recourse has to be sought in how the individual regards chance and would deal with it. Bayes writes about expectation, gain and loss, and it is not by accident that the one surviving original document of his was found in the archives of an insurance company. 4 Helmholtz's epistemology No single person, before or since, contributed more to the knowledge of the human sensory apparatus than Hermann Helmholtz, and throughout his career he kept concerning himself with questions of the origin of our visual experiences. He first broached the subject in an 1854 lecture, as a 34-year-old beginning professor of physiology in Ko«nigsberg, and returned to it in a variety of settings till almost the last essay he wrote during the year of his death in The introduction to part III of the first edition of the Handbuch der physiologischen Optik, which forms the basis of the English version, contains a sentence which best encapsulates the views most widely attributed to him. ``The general rule according to which visual representations determine themselves... is that we always find present in the visual field such objects as would have to exist in order for them to produce the same impression on the neural apparatus under the usual normal conditions of the use of our eyes.'' (Helmholtz 1867, page 428/1911, page 4) (2) In gauging the evolution of Helmholtz's views, however, it is important to realize that this passage was omitted in the second edition (published in 1895, but not available in an English translation). In its place Helmholtz wrote an extensive revision of the section which is more in line with the pivotal formulations to which we now turn. The roots of Helmholtz's epistemological writings are twofold. First, he was one of the ablest and most consequential practitioners of mid-nineteenth century natural scienceöand by far the most successful of those in sensory physiologyöwho erected a securely structured body of knowledge based on relatively uncomplicated empirical observations and deductive rules of manageable mathematical complexity. Second, he was firmly grounded in the academic and cultural (and later even the industrial) elite of that other solidly and successfully constructed element of nineteenth-century European history, the Prussian state. The two streams converged in 1878, when as Rector of Berlin University he took the occasion of the solemn Founder's-Day address to present the credo of his epistemological system. The speech represented a synthesis of (2) The wording of the passage differs somewhat from the one in Southall's English translation which, as many of the English versions of Helmholtz's writing, does not always capture the subtleties of Helmholtz's phrasing. All quotes in this paper have been rendered into English by the author.

6 6 G Westheimer the aims of a working scientist in the area of sensory perception and those of a loyal member of an intellectual community in which Kant and Goethe were revered. (3) Its very title affords an illustration of the difficulties encountered in viewing Helmholtz entirely via the English version of his writings. Helmholtz toyed with various other titles (Koenigsberger 1903), such as ``Prinzipien der Wahrnehmung'' (Principles of perception), ``Was ist wirklich?'' (What is real?), and a favorite citation from Goethe's Faust `Àlles Verga«ngliche ist nur ein Gleichniss'' (All that is transitory is only a metaphor). In the end he chose ``Tatsachen in der Wahrnehmung'' which surely should be rendered ``Facts in perception'' and not, as a prominent version would have it, ``The facts of perception''. He quickly cut to the chase: ``What is truth in our percepts and thoughts? In what way do our ideas correspond to reality? Philosophyöit tries to sift what in our knowledge and ideas is due to the influence of the material world in order to establish what belongs to the purely innate activity of the mind. Scienceöit, to the contrary, tries to sift what is definition, notation, manner of representation, hypothesis, in order to lay out, in a pure form, what belongs to the world of reality whose laws it seeks.'' (Helmholtz 1878/1903, page 218) In order to mediate between them, Helmholtz first of all recognizes and accepts the validity of the program of the two camps over which he as Rector was presiding. And no one was better suited for the mediating task than this iconic presence in natural science who at the same time could boast as family acquaintance the founding philosopher of German idealism, and predecessor in the rectorate a half-century earlier, Johann Gottlieb Fichte. Helmholtz was driven to his cogitations when, after sequentially analyzing the optical, anatomical, physiological, and psychophysical stages of vision, he confronted the nature and origin of perception. He realized that the physicalist/materialist approach that guided him through the physics and biology thus far had come to the end of its rope, here in the borderland between the material and mental worlds. Seeking guidance in Kant's writing, he could not refute that philosopher's proposition that a deep chasm divides the concept of an object's representation in the mind from that object's hypothesized existence in the outside world. Helmholtz was a medical graduate, a sometime Professor of Physiology at Heidelberg, now head of one of the great departments of Physics, and he owed it to his constituents in these professions and to his own scientific achievements to hold fast to the concept of a reality in which the laws articulated by natural science held sway. At the same time he was thoughtful and thoroughgoing and could not, therefore, ignore Kant's teaching. The reconciliation is elegant: ``The distinction between thought and reality is possible only when we know how to distinguish between what the `I' can change and what the `I' cannot change.... What we then attain is knowledge of the lawful order in the realm of the real, but only in so far as it is represented in the tokens within the system of sensory impressions.'' (Helmholtz 1878/ 1903, page 242) The phrasing becomes more impressive when it is realized that `I' and `not-i' (`Ich' and `Nicht-Ich') are essential formulations in Fichte's writing, who counterposed the incarnations of mind, will, faith, morals, to those of the material and of nature (Schmidt 1969) (3) Helmholtz's standing in the German Imperial establishment is attested to in a memo from the Minister of Education to Chancellor Bismarck relating to the appointment to the presidency of the new Imperial Bureau of Standards. Helmholtz should maintain his association with the University because it ``would retain a man who has been viewed for many years as its scientific head, who contributed more than anyone else to smooth out... the contrast between the natural sciences and the humanities, and who, in the arena of politics, fostered the moderately conservative tendency in which Berlin University is well in advance among German Universities'' (Koenigsberger 1903, page 353). In 1883 he was raised to the hereditary nobility and in 1891 he became Privy Councilor to be addressed `His Excellency'.

7 Was Helmholtz a Bayesian? 7 and who, as an idealist philosopher, even went so far as to posit that the `I' generates the `not-i'. All the ingredients of Helmholtz's argument are contained in the paragraph: there is a realm of the real with its lawful order; what we know about it is represented by tokens within the system of sensory impressions; and finally, our knowledge hinges on active explorationöreality is what remains invariant when the expected changes due to willed movements are factored out from the sensory impressions. 5 Helmholtz on inference The single most enveloping aspect of Helmholtz as a scientist and epistemologist was a belief in empiricism which he espoused throughout his career. Specifically, he continued to assert that our knowledge of the real world is derived by using our motor system as exploring organ to deduce invariances by trial and error. Generating a movement is the activity of the `I' which keeps track of the instructions. (A modern term is efference copy, meaning the record that is maintained of the outgoing or efferent signals from the central nervous system to the muscles.) The associated change in sensory signals is registered, and inferences can be drawn from a `before' and `after' comparison. Through knowledge of the actuated movement it can be determined what in the changes of the sensory impressions can be ascribed to the movements; what remains, by inference, is of the real world. Such inferences are the same as the deductions in the realm of ordinary logic, only here we are not aware of them and hence the appellation `unconscious inferences'. Helmholtz's clearest and most specific illustration of how he imagines this process to operate is in the arena of retinal local signs. He agrees with Lotze that each location of the retinal periphery has its own spatial value. In the fully developed organism a nexus has been established between this spatial value and the eye rotation necessary to foveate a target imaged on that peripheral retinal location. When a willed eye rotation has been executed, and both its extent and the shift in retinal location have been registered internally, the spatial location and extent of the object can be inferred. Helmholtz here has satisfied the imperatives both of the natural scientist of his day, to whom a real world was a given, and of Kantian thinkers, for whom the `Ding und sich' is in principle unreachable: the real world was a hypothesis. As a practicing scientist he is, of course, obliged to argue that there is nothing wrong with hypotheses, and in fact the whole enterprise of science restsöquite firmlyöon this premise: ``In its essence, each properly constituted hypothesis proposes a more general law of phenomena than we had obtained by immediate observations up to then, and is an attempt to rise to an ever more general and comprehensive set of laws. Any new facts asserted by such a hypothesis must be tested and affirmed by observations and experiments.'' (Helmholtz 1878/1903, page 242) He will not be pushed into a corner by absolutists and pure materialists: ``Any reduction of phenomena to underlying causes and forces asserts that we have found something permanent and final. Such an unconditional statement is, however, never justified; the incompleteness of our knowledge does not allow this, nor does the nature of conclusions from inferences on which, right from the beginning, our perceptions of the real are based.'' (page 243) Nor is he unaware of the problem of causality that had exercised Hume's mind: ``Every inductive inference is based on the trust that previously observed lawful behavior will be found valid in all cases that have yet to be observed. This is the trust in the lawfulness of all phenomena. But that lawfulness is the condition of comprehensibility.... The law of causality expresses a trust in the complete comprehensibility (vollkommende Begreifbarkeit) of the world. Comprehending, in the sense in which I have described it, is the method used by our thinking to subdue the world, to order facts, to predict the future.

8 8 G Westheimer It has the right and the duty to extend its method to occurrences, and it has indeed harvested great results in this way. For the utilization of the law of causality, however, we have no guarantee other than its success.'' (page 243) 6 Can Helmholtz's proposition be called Bayesian? Helmholtz routinely made the distinction between sensation (Empfindungen) and perception (Wahrnehmung). It is in connection with perception that he sponsored the proposition that it is based on sensory tokens and motor outflow signals, both internal to the organism, from which inferences are drawn about objects. These inferences are constantly updated and affirmed by further probing. Helmholtz's pronouncements were made in a general philosophical context; he was actually much more measured when offering specific instances. He argued all along that many of our visual abilitiesöspatial localization for exampleöare not inborn but acquired, and it was therefore incumbent on him to explain how they arose. When he employed the word `object' it was to refer to an elemental stimulus such as a specific target location or object point rather than to a visual feature such as a person or a tree. Still, in being explicitly on record that the real world and its laws are hypotheses inferred from internal sensory signals, Helmholtz's view of the process of perception does indeed have some similarity with the process utilized in machine vision in the attempts to identify objects from computer images. Does this make him a Bayesian? The Bayesian approach asks first of all: What is the probability of the event, given the hypothesis, P(E jh. Then, to arrive at the posterior probability of the hypothesis H, given the event E, ie P(H je, P(E jh is multiplied by a prior probability of the hypothesis, P(H ). If the object is the hypothesis, and the sensory tokens and the motor outflow signals are the event, then the act of perception is the inference of H given E, and the endpoint of the perceptual process is indeed equivalent to the endpoint of a Bayesian process. Thus far there is no dissonance. But what about the intervening steps? Does Helmholtz propose a stage in which the hypotheses (possible objects) are each tested against the event, ie the likelihoods are estimated so that each of an array of objects could have given rise to the particular set of sensory and motor signals experienced by the observer? In other words, does he believe that an internal record is available of each of the possible objects (ie hypotheses), for purposes of computation of its P(E jh )? And further: Can some analog be found in Helmholtz's proposition for priors, ie the availability of P(H ) to be associated with each P(E jh )? Although there are many signs of its evolution over his lifetime, Helmholtz's epistemological writings are remarkably consistent, and in none of them can one find substantiation that he was willing to frame the problem of perception in such a form. Present-day Bayesian statistics is, of course, not at all the same as it was at the time is was formulated in the midde of the 18th century. Helmholtz's work dates back before the modern era of Bayesian statistics and, as we have seen, Bayes himself gave much less emphasis to `priors' than his current disciples. Could Helmholtz perhaps be said to have been a Bayesian in the more original form? A reading of the original, cited above, makes the connection seem rather tenuous between Bayes's calculation of the probability for the location of a ball thrown on a table and Helmholtz's proposition for inferring the location of an object in the real world from sensory signals. Yet one can conjecture that, had he been shown the modern Bayesian equation and confronted with the challenge to update his writing, Helmholtz would not have abjured the proposition that hypotheses about real objects, based on the internal representations in our sensory signals, are developed by some mechanism involving inverse probabilities and priors. He agonized about these issues 150 years ago, being torn between

9 Was Helmholtz a Bayesian? 9 the opposing imperatives of the natural scientist trying for laws of reality, and the philosopher in quest for the ultimate source of our knowledge, He understood perhaps better than any of his contemporaries how this division of labor plays out: ``The natural sciences still rest on the same firm foundations that they had in Kant's days and whose productive utilization was best exemplified by Newton.... Kant's philosophy did not have the intention of augmenting the amount of our knowledge by means of pure thought because its prime tenet is that knowledge of reality must emerge from experience; its sole purpose was to examine the source of our knowledge and its justification, something that will always remain philosophy's task and that no generation can shirk with impunity.'' (Helmholtz 1855/1903, page 88) As Helmholtz understood, the ``firm foundations of natural sciences'', in their exemplary ``productive utilization... by Newton'', they involved observations and the testing of hypotheses. This holds now as it did then and, as everywhere else, in the arena of perception the enterprise rests on observables. Object recognition by machine vision tries to relate images, which are observable, with real-world objects which, though they are formulated in terms of hypotheses, are nevertheless in the realm of the tangible. Neurophysiological exploration utilizes known objects and tangible records of neural activity; the hypotheses about their relationship, if intended to have `productive utilization', have to be testable. Bayesian inference without a doubt has a defined place in these endeavors, which are often covered by the phrase `empirical Bayesian'. But the motto ``Perception as Bayesian Inference'' might be read as implying that human visual perception is a Bayesian act of inferring a hypothesized reality based on internal sensory impressions and on memory traces of a different kind of previous experiences and their frequency. Of course, as Rector of Berlin University, Helmholtz would not, and as a deeply consequent thinker could not, refute Kant's imperative of a transcendental reality. But he was alsoöand foremostöa natural scientist. In contrast with philosophical rumination about the immediacy of an Anschauung and the inferred nature of reality, research in human perception cannot but follow the procedures in the rest of natural science which lead from observables to hypothesized states, with the laws linking the two devised so as to be testable. If ``Perception as Bayesian Inference'' is taken to mean that the percept is the event and the object is the hypothesis, such a program to be viable needs the percept to be moved into the realm of the observable. The challenge to achieve this has been with the discipline at least since the advent of the Gestalt movement. If he lived now, Helmholtz would have congratulated the machine-vision community on what they are achieving, but as a student of the human visual apparatusöwilling as he might be to acknowledge the primacy of mental perceptual processes and to posit reality as hypothesisöwould have concentrated on the more traditional research methodology, wherein propositions are tested by presenting targets in the real world and inferences made about the internal state of the organism. References Barnard G A, 1958 ``Thomas Bayes's essay towards solving a problem in the doctrine of chances'' Biometrika ^ 315 Bayes T, 1763 `Àn essay towards solving a problem in the doctrine of chances'' Philosophical Transaction of the Royal Society of London ^ 418 Dale A I, 1999 A History of Inverse Probability from Thomas Bayes to Karl Pearson (New York: Springer) Helmholtz H von, 1855/1903 ``Ueber das Sehen des Menschen'', in Vortra«ge und Reden (Braunschweig: Vieweg) pp 85 ^ 117 Helmholtz H von, 1867 Handbuch der physiologischen Optik (Leipzig: Voss) Helmholtz H von, 1878/1903 ``Die Tathsachen in der Wahrnehmung'', in Vortra«ge und Reden (Braunschweig: Vieweg) pp 213 ^ 247