Charles Sanders Peirce: "Suppose that we have two rules of inference, such that, of all the questions to the solutions of which both can be applied, the first yields correct answers 81/100, and incorrect answers the remaining 19/100; while the second yields correct answers 93/100, and incorrect answers to the remaining 7/100. Suppose, further, that the two rules are entirely independent as to their truth, so that the second answers correctly 93/100 of the questions which the first answers correctly and also 93/100 of the questions which the first answers incorrectly, and answers incorrectly the remaining 7/100 of the questions which the first answers correctly, also 7/100 of the questions which the first answers incorrectly [i.e., local independence]."
"We may here conveniently make use of another mode of expression. Probability is the ratio of the favorable cases to all the cases. Instead of expressing our results in terms of this ratio we make use of another - the ratio of favorable to unfavorable cases. This last ratio may be called the chance [odds] of an event. Then the chance of a true answer by the first mode of inference is 81/19 and by the second is 93/7 and the chance of a correct answer from both when they agree is (81*93)/(19*7) or (81/19)*(93/7), or the product of the chances of each singly yielding a true answer."
"It will be seen that a chance is a quantity which may have any magnitude however great. An event in whose favor there is an even chance or 1/1 has a probability of 1/2. An argument having an even chance can do nothing toward reinforcing others. Since according to the rule its combination with another would only multiply the chance of the latter by 1."
"Probability and chance undoubtedly belong primarily to consequences and are relative to premises; but we may, nevertheless, speak of the chance of an event absolutely, meaning by that the chance of the combination of all arguments in reference to it which exist for us in the given state of our knowledge. Taken in this sense it is incontestable that the chance of an event has an intimate connection with the degree of our belief in it. Belief is certainly something more than a mere feeling; yet there is a feeling of believing, and this feeling does and ought to vary with the chance of the thing believed, as deduced from all the arguments. Any quantity which varies with the chance might, therefore, it would seem, serve as a thermometer [!] for the propensity of belief. Among all such quantities there is one which is peculiarly appropriate [i.e., has the correct properties]. When there is a very great chance, the feeling of belief ought to be very intense. Absolute certainty, or an infinite chance, can never be attained by mortals, and this may be represented appropriately by infinite belief. As the chance diminishes the feeling of believing should diminish, until an even chance is reached, or it should completely vanish and not incline either toward or away from the proposition. When a chance becomes less, then a contrary belief should spring up and should increase in intensity as the chance diminishes, and as a chance almost vanishes (which it can never quite do) the contrary belief should tend toward an infinite intensity. Now there is one quantity, which more simply than any other, fulfills these conditions; it is the logarithm of the chance [log-odds]. But there is further consideration which must, if permitted, fix us to this choice for our thermometer. It is that our belief ought to be proportioned to the weight of evidence, in this sense, that two arguments which are entirely independent, neither weakening nor strengthening each other, ought, when they concur, to produce a belief equal to the sum of the intensities of belief which either would produce separately. Now we have seen that the chances of independent concurrent arguments are to be multiplied together to get the chance of their combination, and therefore the quantities which best express the intensities of belief should be such that they are to be added [!] when the chances are multiplied in order to produce a quantity which corresponds to the combined chance. Now, the logarithm is the only quantity which fulfills this condition. There is a general law of sensibility, call Fechner's psycho-physical law. It is that the intensity of any sensation is proportional to the logarithm of the external force which produces it. It is entirely in harmony with this law that the feeling of belief should be as the logarithm of the chance. This latter being the expression of the state of fact which produces the belief."
"The combination of independent concurrent arguments takes a very simple form when expressed in terms of the intensity of belief, measured [!] in the proposed way. It is this. Take the sum of all the feelings [i.e., log-odds] of belief which would be produced separately by all the arguments pro and subtract from that the similar sum for arguments con, and the remainder is the feeling of belief we ought to have on the whole. This is a procedure which men often resort to, under the name of balancing reasons."
"These considerations constitute an argument in favor of the conceptualist view. The kernel of it is that the conjoint probability of all the arguments in our possession, with reference to any fact, must be intimately connected with the just degree of our belief in that fact; and this point is supplemented by various others showing the consistency of the theory with itself and with the rest of our knowledge." [Italics are Peirce's]
C. S. Peirce (1878) Illustration of the Logic of Science by C.S. Peirce, Assistant in the United States Coast Survey. Fourth Paper: The Probability of Induction. Popular Science Monthly, pp. 705-718. Here from pp. 707-709.
In current notation, B_{1} is the solving-ability of rule 1 and B_{2} of rule 2, D is the difficulty of the questions. Then: log_{e}(81/19) = B_{1} - D
log_{e}(93/7) = B_{2} - D
and, concatenating the two rules,
log_{e}(81/19)+log_{e}(93/7) = log_{e} ((81/19)*(93/7))
= (B_{1}-D) + (B_{2}-D)
Thus Peirce solved the problem of obtaining additive, i.e., linear, measures from probabilities (75 years before Rasch). Peirce also solved the problem of concatenating non-physical quantities (50 years before it was proclaimed insoluble by Norman Campbell, 86 years before the possibility of its solution was demonstrated by Luce and Tukey, and 117 years before its re-invention by Ben Wright as Team concatenation.)
Pierce, C.S. Was the Rasch Model Almost the Peirce Model? … Rasch Measurement Transactions, 2000, 14:3 p.756-7
Please help with Standard Dataset 4: Andrich Rating Scale Model
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