Inverse Probability: Its First Formulation

Principle of Induction

"Who will catalogue the innumerable cases of changes to which the air is submitted, so that from this one may conjecture what will be its future state a month from now, not to say a year from now? However, there lies another way before us, by which we may obtain our goal; and what cannot be derived a priori can at least be gathered a posteriori, that is from an event often observed in similar examples. For it should be presumed that each [event] will occur or not occur in the future as many times as it has been observed, in a similar state of affairs, to have occurred or not occurred in the past. If .... one has for many years past observed the aspect of the skies and noted how often it is serene or rainy, or if some one has very often observed two players and seen how often this one or that one came out victor in the game, by that very fact he has discerned the ratio which the numbers of cases probably have to each other, in which the same events, in circumstances similar to the previous ones, can henceforth occur or not occur."

Necessity of Replication

"Then, this too can escape no one, that to make a judgement in this manner on any event it is not sufficient to have taken one or two experiments, but one needs a large number of experiments. For the most stupid of men, I know not by what instinct of nature, by himself and without any instruction (a remarkable thing to be sure), is convinced that the more observations have been made, the less danger there is of wandering from one's aim."

Consistent Estimation

"There remains something further to be contemplated here, which perhaps has never before occurred to any one who thought on the matter. It remains to be investigated whether with the increase of the number of observations there is a corresponding continuous increase in the probability of obtaining the genuine ratio between the numbers of cases in which some event will occur and not occur to such extent that this probability will at last exceed any given degree of certitude; or does the problem, so to speak, have its own asymptote; that is: is there a certain degree of certitude which can never be exceeded no matter how often one multiplies the observations, for example, that we could never be certain beyond 1/2, or 2/3, or 3/4 parts of certitude that we had discovered the true ratio of cases. To make clear what I mean by an example, I suppose that without you knowing it there are hidden in an urn 3000 white pebbles and 2000 black pebbles, that in trying to determine the numbers of these pebbles you take out one pebble after another (each time replacing the pebble you have drawn before choosing the next, so as not to decrease the number of pebbles in the urn), and that you observe how often a white and how often a black pebble is withdrawn. The question is this: can you do this so often that it becomes ten times, one hundred times, one thousand times, etc., more probable (that is, that it be morally certain) that the numbers of whites and blacks chosen are in the same 3:2 ratio actually enjoyed by the pebbles, rather than any other different ratio. For unless this is so, I confess that our attempt at exploring the numbers of cases by experiments is done with."

Precision of Estimates

"Lest, however, these things be understood improperly, it is to be carefully noted that the ratio between the number of cases, which we are trying to determine by experiment, should not be taken as exact and indivisible (for then the contrary would result, and it would become less probable that the true ratio [3:2] would be found the more numerous were the observations). But it is a ratio taken in some latitude, that is, fenced by two limits which can be made as narrow as one might wish. That is to say, if in the example of the pebbles alluded to above we take two ratios 301/200 and 299/200 or 3001/2000 and 2999/2000, etc., of which one is immediately greater and the other immediately less than the ratio 3:2, it will be shown that it can be made more probable, that the ratio found by often repeated experiments will fall within these limits of the 3:2 ratio rather than outside them."

"This, therefore, is that problem which I propose to place in the open at this juncture, a problem which I pursued for twenty years, and one whose novelty and extreme utility, together with its comparable difficulty, can contribute both weight and value to all the other chapters of this treatise."

Jacob Bernoulli (1654-1705) Ars Conjectandi, Part 4. (Basel, 1713) Trans. Charles J. Lewis.

"Inductive inference is the only process known to us by which essentially new knowledge comes into the world."
Ronald A. Fisher (1935) The Design of Experiments

Bernoulli J. (1998) Inverse Probability: Its First Formulation. Rasch Measurement Transactions 12:1 p. 625.

Inverse Probability: Its First Formulation. Bernoulli J. … Rasch Measurement Transactions, 1998, 12:1 p. 625.

Rasch Publications
Rasch Measurement Transactions (free, online) Rasch Measurement research papers (free, online) Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch Applying the Rasch Model 3rd. Ed., Bond & Fox Best Test Design, Wright & Stone
Rating Scale Analysis, Wright & Masters Introduction to Rasch Measurement, E. Smith & R. Smith Introduction to Many-Facet Rasch Measurement, Thomas Eckes Invariant Measurement: Using Rasch Models in the Social, Behavioral, and Health Sciences, George Engelhard, Jr. Statistical Analyses for Language Testers, Rita Green
Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar Journal of Applied Measurement Rasch models for measurement, David Andrich Constructing Measures, Mark Wilson Rasch Analysis in the Human Sciences, Boone, Stave, Yale
in Spanish: Análisis de Rasch para todos, Agustín Tristán Mediciones, Posicionamientos y Diagnósticos Competitivos, Juan Ramón Oreja Rodríguez

To be emailed about new material on
please enter your email address here:

I want to Subscribe: & click below
I want to Unsubscribe: & click below

Please set your SPAM filter to accept emails from welcomes your comments:

Your email address (if you want us to reply):


ForumRasch Measurement Forum to discuss any Rasch-related topic

Go to Top of Page
Go to index of all Rasch Measurement Transactions
AERA members: Join the Rasch Measurement SIG and receive the printed version of RMT
Some back issues of RMT are available as bound volumes
Subscribe to Journal of Applied Measurement

Go to Institute for Objective Measurement Home Page. The Rasch Measurement SIG (AERA) thanks the Institute for Objective Measurement for inviting the publication of Rasch Measurement Transactions on the Institute's website,

Coming Rasch-related Events
Oct. 6 - Nov. 3, 2023, Fri.-Fri. On-line workshop: Rasch Measurement - Core Topics (E. Smith, Facets),
Oct. 12, 2023, Thursday 5 to 7 pm Colombian timeOn-line workshop: Deconstruyendo el concepto de validez y Discusiones sobre estimaciones de confiabilidad SICAPSI (J. Escobar, C.Pardo)
June 12 - 14, 2024, Wed.-Fri. 1st Scandinavian Applied Measurement Conference, Kristianstad University, Kristianstad, Sweden
Aug. 9 - Sept. 6, 2024, Fri.-Fri. On-line workshop: Many-Facet Rasch Measurement (E. Smith, Facets),


The URL of this page is