1. Like all sciences, actuarial science has been developed piecemeal. From time to time theory has been constructed attempting to systematize and unify practices which had already been adopted by the working actuary. In the case of the general treatment of the derivation of rates of decrement from data, little has been published in the Journal to unify the various methods and techniques since W. J. H. Whittall's well-known paper written in 1893 just prior to the commencement of the investigation into the British Offices experience.

Ever since as a student I noted the remarks of a reviewer in the Journal tracing the effects of the Boer War in one of the English Life Tables, I have regarded the population mortality tables of a country as reflecting its political and social life. It is from this viewpoint that the mortality tables of England and the United States of America are examined. The subject of international comparisons is one fraught with difficulty, and in the following paper I have ever held before me William Farr's dictum at the head of this paper.

The paper deals with the mortality and sickness of merchant seamen and with certain other matters that were connected in one way or another with the main subject.

The work underlying the paper was undertaken for the Ministry of War Transport (now the Ministry of Transport) and a report differing little from the present paper was completed in March 1946. I am much indebted to the Minister of Transport for leave to publish the information and for his interest in the work. I should also like to put on record the encouragement I received at an earlier stage from his predecessor and throughout from Sir Cyril Hurcomb, the Secretary (Director General) of the Ministry. Though I am, of course, responsible for the statistical analysis, I have had much help and criticism from the Registrar-General of Shipping and Seamen, Mr W. J. Killingback, M.B.E. Both he and the office of the Registrar-General of Shipping and Seamen at Cardiff, especially Mr C. F. Knight, have given me valuable information and advice without which I should have been in still greater difficulty in understanding and interpreting the statistics. Mr E. A. J. Heath, F.I.A., who was an Assistant Statistical Adviser at the Ministry, made the analyses of the discharges on grounds of ill health which are given in paragraphs 15–18 and in Tables 6 and 7. His experience of permanent sickness insurance was a valuable asset; he also helped in other ways in connexion with the work.

In the paper which I submitted to the Institute on 28 April 1944 I discussed the most equitable method of treating appreciation or depreciation in the assets of an office distributing its surplus by means of a uniform reversionary bonus. I believe that the conclusions reached were correct within the limits set, but consideration of the discussion convinced me that these limits were too narrow, since they involved the treatment of the problem of appreciation or depreciation without reference to the other factors affecting equity in bonus distribution and excluded any solution based on the use of a system of distribution of profits other than a uniform reversionary bonus. I felt that the methods used in the paper might produce results of interest if applied to the general problem of securing equity when the conditions experienced differ from those assumed in calculating the premiums, and I was encouraged to proceed on these lines by the suggestion made by Mr H. E. Melville, speaking as Presidenti that the time was suitable for such an inquiry in view of the exceptionally wide fluctuations in the experience rates of interest over the past thirty years.

1. The methods of analysis leading to the usual formulae for the value of the benefits provided by a pension fund are contained in the historic papers of H. W. Manly, J.I.A. Vol. XXXVII, p. 193 and G. King, J.I.A. Vol. XXXIX, p. 129. The purpose of this paper is to set forth certain developments of the formulae of G. King from which the balance of forces within the fund may be investigated. It is emphasized that the formulae themselves are but algebraic shorthand for processes of calculation, and the deductions do not arise from the formulae but from the numerical results obtained. These results will vary from fund to fund according to the relative financial importance of the various portions of the benefits granted. Within the limits of this paper it is not possible to deal either with the financial situation of an existing fund or with the varying contributions and benefits met in current practice, and in order to produce results for discussion, calculations have been made on the basis of the benefits and contributions of a standard type of fund.

The main object of this paper is to propound and discuss a new indifference rule for the prior probabilities in the theory of inverse probability. Being invariant in form on transformation, this new rule avoids the mathematical inconsistencies associated with the classical rule of ‘uniform distribution of ignorance’ and yields results which, particularly in certain critical extreme cases, do not appear to be unreasonable. Such a rule is, of course, a postulate and is not susceptible of proof; its object is to enable inverse probability to operate as a unified principle upon which methods may be devised of allowing a set of statistics to tell their complete and unbiased story about the parameters of the distribution law of the population from which they have been drawn, without the introduction of any knowledge beyond and extraneous to the statistics themselves. The forms appropriate for the prior probabilities in certain other circumstances are also discussed, including the important case where the unknown parameter is a probability, or proportion, for which it is desired to allow for prior bias. Before proceeding to the main purpose of the paper, however, it is convenient to provide some background to the subject. Reference is first made to certain modern writers to indicate how the problem with which inverse probability is concerned occupies a central place in the foundations of scientific method and in modern philosophy. In quoting from these writers I am not to be taken as suggesting that they necessarily support the inverse probability approach to the problem. The next section of the paper contains some brief comments on the direct statistical methods which have been devised in recent times to side-step induction and inverse probability, and this is followed by a few remarks on the various definitions of probability.

Organization is a problem of analysis and its converse, synthesis. It involves first the exact determination of the separate elements to be coordinated, whether these be sources of supply, processes of manufacture or divisions of administration. Secondly, these elements must be translated into the appropriate functions which they necessitate; for example, specialized branches of labour or steps in accounting procedure. Finally, the separate functions must be integrated into a coherent pattern, sufficiently stable to be economical of effort while remaining sensitive to changes in any of the basic circumstances with which the plan has to deal.