Statisticians and electrical engineers are familiar with an analogous uncertainty between time and frequency in the analysis of time-series , and this obviously suggests the query :sx can a frequency 5n be associated with an energy E ?sx Physicists appeal to the relation E = h5n , where h is Planck's constant , but quite apart from the qualms expressed by " dinger ( 1958 ) about this relation , it is at least arguable that the frequency 5n is as fundamental in it as the energy E. I can therefore sympathize with ( though I am sceptical of ) the proposals by Bohm and de Broglie for a return to the interpretation of 5ps in terms of real ( deterministic ) waves ; I do not think these proposals will be rebutted until the statistical approach has been put on a more rational basis .sx Interesting attempts have been made by various writers , but none of these attempts so far has , to my knowledge , been wholly successful or very useful technically .sx For example , Lande@2 keeps to a particle formulation , whereas it is the particle , and its associated energy E , which seem to be becoming the nebulous concepts .sx Let me refer again to time-series theory , which tells us that the quantization of a frequency 5n arises automatically for circularly-defined series- for , if you will allow me to call it this , periodic 'time' ( more precisely in a physical context , for the angle variables which appear in the dynamics of bound systems) .sx A probabilistic approach via random fields thus has the more promising start of including naturally two of the features of quantum phenomena which were once regarded as most paradoxical and empirical- the Uncertainty Principle and quantization .sx This switch to fields is of course not new ; the real professionals in this subject have been immersed in fields for quite a while .sx However , I am not sure that what probabilists and what physicists mean here by fields are quite synonymous , and in any case it is the old probabilistic interpretation in terms of particles that we lay public still get fobbed off with .sx It would seem to me useful at this stage to make quite clear to us where , if anywhere , the particle aspect is unequivocal- certainly discreteness and discontinuity are not very relevant .sx Here I must leave this fascinating problem of probability in quantum mechanics , as I would like to turn to its function in the theory of information .sx ( 3 ) The concept of information .sx Information theory as technically defined nowadays refers to a theory first developed in detail in connection with electrical communication theory by C. Shannon and others , but recognized from the beginning as having wider implications as a conceptual tool .sx From its origin it was probably most familiar at first to electrical engineers , but its more general and its essentially statistical content made it a natural adjunct to the parts of probability theory hitherto studied by the statistician .sx This is recognized , for example , in an advertisement for a mathematical statistician from which I quote :sx Applicants should possess a degree in statistics or mathematics , and should if possible be able to show evidence of an interest in some specialized aspect of the subject such as , for example , decision theory , information theory or stochastic processes .sx It has not , I think , been recognized sufficiently in some of the recent conferences on information theory , to which mathematical statisticians 6per se have not always been invited .sx The close connection of the information concept with probability is emphasized by its technical definition in relation to an ensemble or population , and indeed , it may usefully be defined ( cf .sx Good ( 1950 ) , Barnard ( 1951) ) as - log p ( a simple and direct measure of uncertainty which is reduced when the event with probability p has occurred ) , although the more orthodox definition is the 'average information' - 15Sp log p , averaged over the various possibilities or states that may occur .sx It is also possible to extend this definition to partial or relative information , in relation to a change of ensembles or distributions from one to another .sx With this extended definition of - log p/p@7 , where p@7 relates to the new ensemble , the information can be positive or negative , and as the logarithm of a probability ratio will look familiar to statisticians , although it should be stressed that the probabilities refer to fully specified distributions , and the likelihood ratio of the statistician ( made use of so extensively by Neyman and E. S. Pearson ) only enters if the probabilities p and p@7 are interpreted as dependent on different hypotheses H and H@7 .sx For example , if p@7 is near p , differing only in regard to a single unknown parameter 5th , then where I(5th ) is R. A. Fisher's information function , under conditions for which this function exists .sx Formally , the concept of information in Shannon's sense can be employed more directly for inferring the value of 5th .sx To take the simplest case shorn of inessentials , if we make use Bayes's theorem to infer the value of a parameter 15th;r ; which can take one of only k discrete values , then our prior probability distribution about 15th;r ; will be modified by our data to a posterior probability distribution .sx If we measure the uncertainty in each such distribution by - 15Sp log p , we could in general expect the uncertainty to be reduced , but we can easily think of an example where the data would contradict our 6a priori notions and make us less certain than before .sx This seems to me to stress the subjective or personal element in prior probabilities used in this way , and my own view is that the only way to eliminate this element would be deliberately to employ a convention that prior distributions are to be maximized with respect to uncertainty .sx In the present example this would imply assuming a uniform prior distribution for 15th;r ; , and ensure that information was always gained from a sample of data ; it is somewhat reminiscent of arguments used by Jeffreys in recent years for standardizing prior distributions , but I think it important to realize that such conventions weaken any claim that these methods are the only rational ones possible .sx Whether or not the information concept in this sense finds any permanent place in statistical inference , there is no doubts [SIC] of its potential value in two very important scientific fields , biology and physics .sx This claim in respect to biology is exemplified by the Symposium on Information Theory in Biology held in Tennessee in 1956 ; and while we must be careful not to confuse the general function of new concepts in stimulating further research with the particular one of making a particular branch or aspect of a science more precise and unified , the use of the information concept in discussing capacities of nerve fibres transmitting messages to the brain , or coding genetic information for realization in the developed organism , should be sufficient demonstration of its quantitative value .sx As another illustration of the trend to more explicit and precise uses of the information concept in biology , we may consider the familiar saying that life has evolved to a high degree of organization , that in contrast to the ultimate degradation of dead matter , living organisms function by reducing uncertainty , that the significant feature of their relation with their environment is not their absorption of energy ( vital of course as this is ) , but their absorption of negative entropy .sx An attempt to measure the rate of accumulation of genetic information in evolution due to natural selection has recently been made by Kimura ( 1961 ) , who points out that a statement by R. A. Fisher that 'natural selection is a mechanism for generating an exceedingly high degree of improbability' indicates how the increase in genetic information may be quantitatively measured .sx While his estimate is still to be regarded as provisional in character , it is interesting that Kimura arrives at an amount , accumulated in the last 500 million years up to man , of the order of 10 :sx 8: 'bits' , compared with something of the order of 10 :sx 10: bits estimated as available in the diploid human chromosome set .sx He suggests that part of the difference , in so far as it is real , should be put down to some redundancy in the genetic coding mechanism .sx With regard to physics , I have already mentioned 'negative entropy' as a synonym for information , and this is in fact the link .sx Again we have the danger of imprecise analysis , and the occurrence of a similar probabilistic formula for information and physical entropy does not by itself justify any identification of these concepts .sx Nevertheless , physical entropy is a statistical measure of disorganization or uncertainty , and information in this context a reduction of uncertainty , so that the possibility of the link is evident enough .sx To my mind one of the most convincing demonstrations for the need of this link lies in the resolution of the paradox of Maxwell's demon , who circumvented the Second Law of Thermodynamics and the inevitable increase in entropy by letting only fast molecules move from one gas chamber to another through a trap-door .sx It has been pointed out by Rosenfeld ( 1955 ) that Clausius in 1879 went some way to explaining the paradox by realizing that the demon was hardly human in being able to discern individual atomic processes , but logically the paradox remains unless we grant that such discernment , while in principle feasible , at the same time creates further uncertainty or entropy at least equal ( on the average ) to the information gained .sx That this is so emerges from a detailed discussion of the problem by various writers such as Szilard , Gabor , and Brillouin ( as described in Brillouin's book) .sx ( 4 ) The ro@5le of time .sx I might have noted in my remarks on quantum theory that , whether or not time is sometimes cyclic , it appears in that theory in a geometrical ro@5le , reminiscent of time in special relativity , and not in any way synonymous with our idea of time as implying evolution and irreversible change .sx It is usually suggested that this latter ro@5le must be related to the increase of physical entropy , but when we remember that entropy is defined statistically in terms of uncertainty we realize not only that evolutionary time itself then becomes statistical , but that there are a host of further points to be sorted out .sx Let me try to list these :sx ( a ) In the early days of statistical mechanics , at the end of the last century , Maxwell's paradox was not the only one raised .sx Two others were Loschmidt's reversibility paradox , in which the reversibility of microscopic processes appeared to contradict the Second Law , and Zermelo's recurrence paradox , in which the cyclical behaviour of finite dynamic systems again contravened the Second Law .sx It should be emphasized that , while these paradoxes were formulated in terms of deterministic dynamics , they were not immediately dissipated by the advent either of quantum theory or of the idea of statistical processes .sx For I have just reminded you that time in quantum mechanics is geometrical and reversible ; and stationary statistical processes based on microscopic reversible processes are themselves still reversible and recurrent .sx The explanations of the paradoxes are based , in the first place , on the difference between absolute and conditional probabilities , and in the second , on the theory of recurrence times .sx The apparent irreversibility of a system is due to its being started from an initial state a long way removed from the more typical states in equilibrium and the apparent non-recurrence of such a state to the inordinately long recurrence time needed before such a state will return .sx ( b ) So far so good- but this conclusion applies to a system of reasonable size .sx We conclude that microscopic phenomena have no intrinsic time-direction , at least if this can only be defined in relation to internal entropy increase ( cf .sx Bartlett , 1956) .sx This is consistent with theoretical formulations in recent years of sub-atomic phenomena involving time-reversals .sx ( c ) We have also to notice that while the entropy of our given system will increase with external or given time , this relation is not reciprocal , for , if we first choose our time , a rare state in our stationary process will just as likely be being approached as being departed from .sx