Since second-order probability distributions assign probabilities to probabilities there is uncertainty on two levels. Although different types of uncertainty have been distinguished before and corresponding measures suggested, the distinction made here between first- and second-order levels of uncertainty has not been considered before. In this paper previously existing measures are considered from the perspective of first- and second-order uncertainty and new measures are introduced. We conclude that the concepts of uncertainty and informativeness needs to be qualified if used in a second-order probability context and suggest that from a certain point of view information can not be minimized, just shifted from one level to another.
Key words: Uncertainty, entropy, second-order probability. Nau, "Uncertainty aversion with second-order utilities and probabilities," Management Science, vol. Utkin and T. Ekenberg and J. Gardenfors and N. Sahlin, "Decision, probability and utility: Selected readings. Cambridge University Press, , ch. Cooman and P. Walley, "A possibilistic hierarchical model for behaviour under uncertainty," Theory and Decision 52 4 , pp.
At present, we have no actual reconstructions based on this idea, and so do not know whether there are unrecognized difficulties with it. Doubtless, there are other conceivable circumstances i. Any new information which could make our old estimate seem absurd would be, to put it mildly, highly cogent; and it would seem important that we state explicitly what this information is so we can take full advantage of it.
Maximum Entropy Bayesian Methods
This situation of unspecified information — intuition feels it but does not define it — is not anomalous, but the usual situation in exploring this neglected part of probability theory. It is not an occasion for dispute, but for harder thinking on a technical problem that is qualitatively different from the ones scientists are used to thinking about. One more step of that harder thinking, in a case very similar to this, appears in our discussion of the kangaroo problem below. In any event, as was stressed at the Laramie Workshop and needs to be stressed again, the question of the choice of N cannot be separated from the choices of m and n, the number of pixels into which we resolve the blurred image and the reconstruction, and u, v, the quantizing increments that we use to represent the data d k and the reconstruction p i for calculational purposes.
In most problems the real and blurred scenes are continuous, and the binning and digitization are done by us. Presumably, our choices of N, m, n, u, v all express something about the fineness of texture that the data are capable of supporting; and also some compromises with computation cost.
Although computer programmers must necessarily have made decisions on this, we are not aware of any discussion of the problem in the literature, and the writer's thinking about it thus far has been very informal and sketchy. More work on these questions seems much needed. In this connection, we found it amusing to contemplate going to the "Fermi statistics" limit where n is very large and we decree that each pixel can hold only one dot or none, as in the halftone method for printing photographs.
Also one may wonder whether there would be advantages in working in a different space, expanding the scene in an orthogonal basis and estimating the expansion coefficients instead of the pixel intensities. But the data give no evidence at all about the last n-m coordinates There might be advantages in a computational scheme that, by working in these coordinates, is able to deal differently with those aj that are well determined by the data, and those that are undetermined.
Perhaps we might decree that for the former "the data come first". But for the latter, the data never come at all. In any event, whatever our philosophy of image reconstruction, the coordinates a. Computational algorithms for carrying out the decomposition 7 are of course readily available Chambers, As we see from this list of unfinished projects, there is room for much more theoretical effort, which might be quite pretty analytically and worthy of a Ph.
It is interesting to compare the solutions of this problem given by various algorithms that have been proposed. Gull and Skilling , applying the work of Shore and Johnson, find the remarkable result that if the solution is to be found by maximizing some quantity, entropy is uniquely determined as the only choice that will not introduce spurious correlations in the matrix 11 , for which there is no evidence in the data. The maximum entropy solution is then advocated on grounds of logical consistency rather than multiplicity.
I want to give an analysis of the kangaroo problem, with an apology in advance to Steve Gull for taking his little scenario far more literally and seriously than he ever expected or wanted anybody to do. My only excuse is that it is a conceivable problem, so it provides a specific example of constructing priors for real problems, exemplifying some of our Tutorial remarks about deeper hypothesis spaces and measures.
And, of course, the principles are relevant to more serious real problems — else the kangaroo problem would never have been invented. What bits of prior information do we all have about kangaroos, that are relevant to Gull's question? Our intuition does not tell us this immediately, but a little pump priming analysis will make us aware of it.
Jaynes: 35 Monkeys, Kangaroos, and N In the first place, it is clear from 11 that the solution must be of the form:. So for any finite N there are a finite number of integer solutions N i,j.
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Any particular solution will have a multiplicity N! W 13 This seems rather different from the image reconstruction problem; for there it was at least arguable whether N makes any sense at all. The maximum entropy scene was undeniably the one the monkeys would make; but the monkeys were themselves only figments of our imagination. Now, it is given to us in the statement of the problem that we are counting and estimating attributes of kangaroos, which are not figments of our imagination; their number N is a determinate quantity.
Therefore the multiplicities W are now quite real, concrete things; they are exactly equal to the number of possibilities in the real world, compatible with the data. It appears that, far from abandoning monkeys, if there is any place where the monkey combinatorial rationale seems clearly called for, it is in the kangaroo problem! These are the predictions made by uniform weighting on our first monkey hypothesis space HI.
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Here we can start to discover our own hidden prior information by introspection; at what value of N do you begin feeling unhappy at this result? Now that argument unavailable; for N is a real, determinate quantity. So what has wrong this time? I feel another Sermon coming on. But at some point someone says: "This conclusion is absurd; I don't believe it!
It is well established by many different arguments that Bayesian inference yields the unique consistent conclusions that follow from the model, the data, and the prior information that was actually used in the calculation. Many times, the writer has been disconcerted by a Bayesian result on first finding it, but realized on deeper thought that it was correct after all; his intuition thereby became a little more educated. The same policy — entertain the possibility that your intuition may need educating, and think hard before rejecting a Bayesian result — is recommended most earnestly to others.
As noted in our Tutorial, intuition is good at perceiving the relevance of information, but bad at judging the relative cogency of different pieces of information. If our intuition was always trustworthy, we would have no need for probability theory. Over the past 15 years many psychological tests have shown that in various problems of plausible inference with two different pieces of evidence to consider, intuition can err — sometimes violently and in opposite directions — depending on how the information is received.
Some examples are noted in Appendix A. This unreliability of intuition is particularly to be stressed in our present case, for it is not limited to the untrained subjects of psychological tests. Throughout the history of probability theory, the intuition of those familiar with the mathematics has remained notoriously bad at perceiving the cogency of multiplicity factors.
This observed property of frequencies, to become increasingly stable with increasing number of observations, is seen as a kind of Miracle of Nature — the empirical fact underlying probability theory — showing that probabilities are physically real things. Yet as Laplace noted, those frequencies are only staying within the interval of high multiplicity; far from being a Miracle of Nature, the great majority of all things that could have happened correspond to frequencies remaining in that interval.
If one fails to recognize the cogency of multiplicity factors, then virtually every "random experiment" does indeed appear to be a Miracle of Nature, even more miraculous than In most of the useful applications of direct probability calculations — the standard queueing, random walk, and stochastic relaxation problems — the real function of probability theory is to correct our faulty intuition about multiplicities, and restore them to their proper strength in our predictions.
In particular, the Central Limit Theorem expresses how multiplicities tend to pile up into a Gaussian under repeated convolution. This can lead to very bad estimates of a parameter whose multiplicity varies greatly within the region of high likelihood. It behooves us to be sure that we are not committing a similar error here. Bear in mind, therefore, that in this problem the entire population of kangaroos is being sampled; as N increases, so does the amount of data that is generating that estimate Estimates which improve as the square root of the number of observations are ubiquitous in all statistical theory.
But if, taking note of all this, you still cannot reconcile 18 to your intuition, then realize the implications. Anyone who adamantly refuses to accept 18 is really saing: "I have extra prior information about kangaroos that was not taken into account in the calculation leading to TW7 More generally, having done any Bayesian calculation, if you can look at the result and know it is "wrong"; i.
You should have used it. Indeed, unless you can put your finger on the specific piece of information that was left out of the calculation, and show that the revised calculation corrects the difficulty, how can you be sure that the fault is in the calculation and not in your intuition? The monkey calculation on HI has only primed the mental pump; at this point, the deep thought leading us down to H2 is just ready to begin: What do we know about kangaroos, that our common sense suddenly warns us was relevant, but we didn't think to use at first?
There are various possibilities; again, intuition feels them but does not define them. Indeed, any prior information that establishes a logical link between these two attributes of kangaroos will make that argument inapplicable in our problem. Had our data or prior information been different, in almost any way, they would have given evidence for correlations and MAXENT would exhibit it. The "no correlations" pheraonenon emphasized by the kangaroo rationale is a good illustration of the "honesty" of MAXENT i. Of course, if we agree in advance that our probabilities are always to be found by maximizing the same quantity whatever the data, then a single compelling case like this is sufficient to determine that quantity, and the kangaroo argument does pick out entropy in preference to any proposed alternative.
This seems to have been Steve Gull's purpose, and it served that purpose well. The H2a case is rather unrealistic, but as we shall see it is nevertheless a kind of caricature of the image reconstruction problem; it has, in grossly exaggerated form, a feature that was missing from the pure monkey picture. H2b ; More realistically, although there are several species of kangaroos with size varying from man to mouse, we assume that Gull intended his problem to refer to the man-sized species who else could stand up at a bar and drink Foster's?
The species has a common genetic pool and environment; one is much like another. In this state of prior knowledge, learning that one kangaroo is left-handed makes it more likely that the next one is also left-handed.
Principle of maximum entropy
This positive correlation not between attributes, but between kangaroos was left out of the monkey picture. The same problem arises in survey sampling. But as we demonstrate below, this would not follow from Bayes' theorem with the monkey prior 13 , proportional only to multiplicities. In that state of prior knowledge call it I o , every kangaroo is a separate, independent thing; whatever we learn about one specified individual is irrelevant to inference about any other. Following Harold Jeffreys instead, we elect to think more deeply. Our state of knowledge anticipates some positive correlation between kangaroos, but for purpose of defining H2, suppose that we have no information distinguishing one kangaroo from another.
Then whatever prior we assign over the 4 N possibilities, it will be invariant under permutations of kangaroos. For it is a well-known theorem that a discrete distribution over exchangeable kangaroos or exchangeable anything else is a de Finetti mixture of multinomial distributions, and the problem reduces to finding the weighting function of that mixture. For easier notation and generality, let us now label the four mutually exclusive attributes of kangaroos by 1, 2, 3, 4 instead of 11, 12, 21, 22 , and consider instead of just 4 of them, any number n of mutually exclusive attributes, one of which kangaroos must have.
As it stands, 19 expresses simply a mathematical fact, which holds independently of whatever meaning you or I choose to attach to it. Xfl as a set of "real" parameters which define a class of hypotheses about what is generating our data. Then the factor N Nn p N This suggests that we interpret the generating function as G x We could easily restate everything so that the misconception could not arise; it would only be rather clumsy notationally and tedious verbally.
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However, this is a slightly dangerous step for a different reason; the interpretation 21 , 22 has a mass of inevitable consequences that we might or might not like. So before taking this road, let us note that we are here choosing, voluntarily, one particular interpretation of the theorem But the choice we are making is not forced on us, and after seeing its consequences we are free to return to this point and make a different choice.
Nn I I through 19 , then they are, so to speak, not real at all, only figments of our imagination. They are, moreover, not necessary to solve the problem, but created on the spot for mathematical convenience; it would not make sense to speak of having prior knowledge about them. They would be rather like normalization constants or MAXENT Lagrange multipliers, which are also created on the spot only for mathematical convenience, so one would not think of assigning prior probabilities to them.
Njx N n given some data D. But let us see the Bayesian solution. Suppose our data consist of sampling M kangaroos, M X][ k-1 k-1 k-1 x However, they gave only the choices, not the circumstances; intuitively, just what prior information is being expressed by 27? Johnson ; he showed, generalizing an argument that was in the original work of Bayes, that if in 19 all choices of Ni In a posthumously published work Johnson, he gave a much more cogent circumstance, which in effect asked just John Skilling's question: "Where would the next photon come from?
Johnson showed that if 28 is to hold for all N,Ni , this requires that the prior must have the Dirichlet-Hardy form 27 for some value of k. For recent discussions of this result, with extensions and more rigorous proofs, see Good , Zabell This intuitive insight of Johnson still does not reveal the meaning of the parameter k. This is the reciprocal of the familiar Bose-Einstein multiplicity factor number of linearly independent quantum states that can be made by putting N bosons into n single-particle states. Indeed, the number of different scenes that can be made by putting N dots into n pixels or N kangaroos into n categories, is combinatorially the same problem; one should not jump to the conclusion that we are invoking "quantum statistics" for photons.
Note that the monkey multiplicity factor W N is the solution to a very different combinatorial problem, namely the number of ways in which a given scene can be made by putting N dots into n pixels. However, the relevant quantity 31 is a ratio of such integrals, which does not become singular. In the limit it remains a proper i. But these results seem even more disconcerting to intuition than the one 18 which led us to question the pure monkey rationale. There we felt intuitively that the parameter q should not be determined by the data to an accuracy of 1 part in Does it seem reasonable that merely admitting the possibility of a positive correlation between kangaroos, should totally wipe out multiplicity ratios of 10 1 0 0 :l, as it appears to be doing in 32 , and even more strongly in 36?
In the inference called for, relative multiplicities are cogent factors. We expect them to be moderated somewhat by the knowledge that kangaroos are a homogeneous species; but surely multiplicities must still retain much of their cogency. Common sense tells us that there should be a smooth, continuous change in our results starting from the pure monkey case to a more realistic one as we allow the possibility of stronger and stronger correlations. Instead, 32 represents a discontinuous jump to the opposite extreme, which denies entropy any role at all in the prior probability.
In what sense, then, can we consider small values of k to be "uninformative"? A major thing to be learned in developing this neglected half of probability theory is that the mere unqualified epithet "uninformative" is meaningless. A distribution which is uninformative about variables in one space need not be in any sense uninformative about related variables in some other space.
As we learn in quantum theory, the more sharply peaked one function, the broader is its Fourier transform; yet both are held to be probability amplitudes for related variables. Our present problem exhibits a similar "uncertainty relation". The monkey multiplicity prior is completely uninformative on the sample space S of n N possibilities. It is for us to say which, if either, of these limits represents our state of knowledge. This depends, among other things, on the meaning we attach to the variables. In the present problem the Xj are only provisionally "real" quantities, introduced for mathematical convenience, the integral representation 19 being easy to calculate with.
But we have avoided saying anything about what they really mean. We now see one of those inevitable consequences of assigning priors to the xj[, that the reader was warned he might or might not like. This observation opens up another interpretive question about the meaning of a de Finetti mixture, that we hope to consider elsewhere. Now let us examine the opposite limit of If the kj[ are all equal, this reverts to a constant times the pure monkey multiplicity from whence we started.
So it is the region of large k, not small, that provides the smooth, continuous transition from the "too good" prediction One way to define an intuitive meaning for the parameters kj[ is to calculate Johnson's predictive function f N,Nj[ in 28 or its generalization f i N,N i. Although the incident happened a long time ago, some comments about it are still needed because the thinking of Venn persists in much of the recent statistical literature.
With today's hindsight we can see that Venn suffered from a massive confusion over "What is the Problem? Venn , not a mathematician, ignored his derivation — which might have provided a clue as to what the problem is — and tried to interpret the result as the solution to a variety of very different problems. Of course, he chose his problems so that Laplace's solution was indeed an absurd answer to every one of them. Apparently, it never occurred to Venn that he himself might have misunderstood the circumstances in which the solution applies. Fisher , pointed this out and expressed doubt as to whether Venn was even aware that Laplace's Rule had a mathematical basis and like other mathematical theorems had "stipulations specific for its validity".
Fisher's testimony is particularly cogent here, for he was an undergraduate in Caius College when Venn was still alive Venn eventually became the President of Caius College , and they must have known each other. Furthermore, Fisher was himself an opponent of Laplace's methods; yet he is here driven to defending Laplace against Venn. Yet we still find Venn's arguments repeated uncritically in some recent "orthodox" textbooks; so let the reader beware.
Now in the 's and 's Laplace's result became better understood by many: C D. Broad, H. Jeffreys, D. Wrinch, and W. Johnson all here in Cambridge also. Their work being ignored, it was rediscovered again in the 's by de Finetti, who added the important observation that the results apply to all exchangeable sequences, de Finetti's work being in turn ignored, it was partly rediscovered still another time by Carnap and Kemeny, whose work was in turn ignored by almost everybody in statistics, still under the influence of Venn. It was only through the evangelistic efforts of I. Good and L.
Savage in the 's and 's and D. Lindley in the 's and 's, that this exchangeability analysis finally became recognized as a respectable and necessary working part of statistics. Today, exchangeability is a large and active area of research in probability theory, much as Markov chains were thirty years ago.
We think, however, that the autoregressive models, in a sense intermediate between exchangeable and markoffian ones, that were introduced in the 's by G. Udny Yule also here in Cambridge, and living in the same room that John Skilling now occupies , offer even greater promise for future applications. In the 's, more than years after Laplace started it, great mathematical generalizations are known but we are still far from understanding the useful range of application of exchangeability theory, because the problem of relating the choices to the circumstances is only now being taken seriously and studied as a technical problem of statistics, rather than a debating point for philosophers.
Indeed, our present problem calls for better technical understanding than we really have at the moment. But at least the mathematics flows on easily for some distance more. Then 40 can be written Mp. We may interpret the k's also in terms of the survey sampling problem. Starting from the prior information I D and considering the data Ml..
Comparing with 40 we see that the Rule of Succession has two different meanings; this estimated fraction p is numerically equal to the probability that the next kangaroo sampled will have attribute 1. As we have stressed repeatedly, such connections between probability and frequency always appear automatically, as a consequence of Bayesian theory, whenever they are justified.
Generally, the results of survey samplings are reported as estimated fractions of the total population N, rather than of the unsampled part L - N-M. Examining the dependence of 48 on each of its factors, we see what Bayes1 theorem tells us about the interplay of the size of the population, the prior information, and the amount of data. Note, however, that 48 is not directly comparable to 18 because in 18 we used Steve Gull's data on kangaroos to restrict the sample space before introducing probabilities.
Now suppose we have sampled an appreciable fraction of the entire population. How does this affect the answer to the original kangaroo problem, particularly in the region of large N where we were unhappy before? But now, admitting the possibility of a positive correlation between kangaroos must, from the theorem, induce some correlation between their attributes. At this point our intuition can again pass judgment; we might or might not be happy to see such correlations. Our first analysis of the monkey rationale on HI was a mental pump-priming that made us aware of relevant information correlations between kangaroos that the monkey rationale did not recognize, and led us down into H2.
Now the analysis on H2 has become a second mental pump-priming that suddenly makes us aware of still further pertinent prior information that we had not thought to use, and leads us down into H3. When we see the consequences just noted, we may feel that we have overcorrected by ignoring a nearness effect; it is relevant that correlations between kangaroos living close together must be stronger than between those at opposite ends of the Austral continent.
In the U. But an exchangeable model insists on placing the same correlations between all individuals. Jaynes: Monkeys, Kangaroos, and N 53 In image reconstruction, we feel intuitively that this nearness effect must be more important than it is for kangaroos; in most cases we surely know in advance that correlations are to be expected between nearby pixels, but not between pixels far apart.
But in this survey we have only managed to convey some idea of the size of the problem. To find the explicit hypothesis space H3 on which we can express this prior information, add the features that the data are noisy and N is unknown; and work out the quantitative consequences, are tasks for the future. Therefore, however you go at it, when you finally arrive a satisfactory prior, you are going to find that monkey multiplicity factor sitting there, waiting for you.
This is more than a mere philosophical observation, for the following reason. In image reconstruction or spectrum analysis, if entropy were not a factor at all in the prior probability of a scene, then we would expect that MAXENT reconstructions from sparse data, although they might be "preferred" on other grounds, would seldom resemble the true scene or the true spectrum.
This would not be an argument against MAXENT in favor of any alternative method, for it is a theorem that no alternative using the same information could have done better. Resemblance to the truth is only a reward for having used good and sufficient information, whether it comes from the data or the prior. More important, the moral of our Sermons on this in the Tutorial was that if such a discrepancy should occur, far from being a calamity, it might enable us to repeat the Gibbs scenario and find a better hypothesis space.
In many cases, empirical evidence on this resemblance to the truth or lack of it for image reconstruction can be obtained. It might be thought that there is no way to do this with astronomical sources, since there is no other independent evidence. For an object of a previously uncharted kind, this is of course true, but we already know pretty well what galaxies look like.
If Roy Frieden's MAXENT reconstruction of a galaxy was no more likely to be true than any other, then would we not expect it to display any one of a variety of weird structures different from spiral arms? If they did not, nobody would have any interest in them. The clear message is this: if we hold that entropy has no role in the prior probability of a scene, but find that nevertheless the MAXENT reconstructions consistently resemble the true scene, does it not follow that MAXENT was unnecessary? It seems to us that there is only one way this could happen. If so, how much data would we need to approach this condition?
In March the writer found, in a computer study of a one-dimensional image reconstruction problem, that when the number of constraints was half the number of pixels the feasible set had not contracted very much; it still contained a variety of wildly different scenes, having almost no resemblance to the true one. So this amount of data still seems "sparse" and in need of MAXENT; any old algorithm would have given any old result, seldom resembling the truth. After months of puzzlement over this statement, I finally learned what John Skilling meant by it, through some close interrogation just before leaving Cambridge.
Indeed, it requires only a slight rephrasing to convert it into a technically correct statement: "The MAXENT reconstruction has no more likelihood than any other with equal or smaller Chisquared. Jaynes: Monkeys, Kangaroos, and N 55 The point is that "likelihood" is a well-defined technical term of statistics. What is being said can be rendered, colloquially, as "The MAXENT reconstruction is not indicated by the data alone any more strongly than any other with equal or smaller Chisquared. In any such problem, a specific choice within the feasible set must be made on other considerations than the data; prior information or value judgments.
Procedurally, it is possible to put the entropy factor in either. The difference is that is we consider entropy only a value judgment, it is still "preferred" on logical consistency grounds, but we have less reason to expect that our reconstruction resembles the true scene because we have invoked only our wishes, not any actual information, beyond the data.
In my view, the MAXENT reconstruction is far more "likely" in the colloquial sense of that word to be true than any other consistent with the data, precisely because it does take into account some highly cogent prior information in addition to the data. MAXENT images and spectrum estimates should become still better in the future, as we learn how to take into account other prior information not now being used. Indeed, John Skilling's noting that bare MAXENT is surprised to find isolated stars, but astronomers are not; and choosing "prior prejudice" weighting factors accordingly, has already demonstrated this improvement.
Pragmatically, all views about the role of entropy seem to lead to the same actual class of algorithms for the current problems; different views have different implications for the future. For diagnostic purposes in judging future possibilities it would be a useful research project to explore the full feasible set very carefully to see just how great a variety of different scenes it holds, how it contracts with increasing data, and whether it ever contracts enough to make MAXENT unnecessary as far as resemblance to the truth is concerned.
Yet the subjects in the test corresponding to odds of information. For them, "the though the prior information single datum. The field is reviewed by Donmell and Du Charme It is perhaps not surprising that the intuitive force of prior opinions depends on how long we have held them. Persons untrained in inference are observed to commit wild irrationalities of judgment in other respects. Slovic et al report experiments in which subjects, given certain personality profile information, judged the probability that a person is a Republican lawyer to be greater than the probability that he is a lawyer.
Hacking surveys the history of the judicial problem and notes that the Bayesian probability models of jury behavior given by Laplace and long ignored, account very well for the performance of modern English juries. Cohen reports on controversy in the medical profession over whether one should, in defiance of Bayesian principles, test first for rare diseases before common ones.
Such findings not only confirm our worst fears about the soundness of jury decisions, but engender new ones about medical decisions. These studies have led to proposals — doubtless years overdue — to modify current jury systems. The services of some trained Bayesians are much needed wherever important decisions are being made.
Computational Methods for Data Analysis. Organizational Behavior and Human Performance, 14, Statistical Methods and Scientific Inference. The Estimation of Probabilities. Daniell Image Reconstruction with Incomplete and Noisy Data. Nature, , Skilling The Maximum Entropy Method. In Indirect Imaging, J.
Roberts, ed. Research 10 Hacking, Ian Historical Models for Justice. Letter in Insurance Record, p. Cambridge University Press. Reprinted by Dover Publishing Co. Probability, the Deduction and Induction Problems. Mind, 15 Kahneman, D. Tversky On the Psychology of Prediction. Reprint- 16 Kyburg, H. Smokier Studies in Subjective Probability. Paris, ; Oeuvres Completes, 9, Fischhoff and S. Lichtenstein Behavioral Decision Theory. Johnson's Sufficientness Postulate1. Annals of Statistics, 10, Our purpose is to dispel the three more common objections raised against the rationale and results of the approach.
To do so we restrict the scope of the formalism: We consider only such experiments that can be repeated N, N not necessarily large , times. Their value is independent of the number, N, of repetitions of the experiment. What very much does depend on N is the variance of the frequency. Either by signing into your account or linking your membership details before your order is placed. Your points will be added to your account once your order is shipped.
Click on the cover image above to read some pages of this book! Throughout applied science, Bayesian inference is giving high quality results augmented with reliabilities in the form of probability values and probabilistic error bars. Maximum Entropy, with its emphasis on optimally selected results, is an important part of this. Across wide areas of spectroscopy and imagery, it is now realistic to generate clear results with quantified reliability. This power is underpinned with a foundation of solid mathematics. The annual Maximum Entropy Workshops have become the principal focus of developments in the field, and which capture the imaginative research that defines the state of the art in the subject.
The breadth of application is seen in the thirty-three papers reproduced here, which are classified into subsections on Basics, Applications, Physics and Neural Networks. Audience: This volume will be of interest to graduate students and researchers whose work involves probability theory, neural networks, spectroscopic methods, statistical thermodynamics and image processing.