Probabilities, Odds, and Relativistic Speeds


Your grace hath laid the odds o' the weaker side.



In another note we discussed combining probabilities. Here we consider the same subject from a slightly more abstract and general standpoint, and note a correspondence with the composition of relativistic velocities.


Consider a set of N+1 logical variables C0, C1, ..., CN, each of which has either the value T (true) or F (false). We stipulate that each of the 2N+1 possible configurations has a fixed probability, and furthermore that the probability of any configuration is the same as that of the complementary configuration. In other words, the system is invariant under exchange of True and False for all the variables. Next we define



where over-bars signify logical negation. Thus cj is true if and only if Cj agrees with C0.  There are 2N possible configurations of the variables c1 to cN, and we can assign a fixed probability to each configuration. For example, with N = 3 we have the following 8 possible configurations (using 1 to denote True and 0 to denote False), each with some definite probability.



Now we stipulate that c1, c2,..., cN are independent logical variables, meaning that the probability of the intersection of any subset of these events equals the product of the probabilities of the individual variables. (Note that pairwise independence is not sufficient to ensure complete independence.) These requirements fully determine the probabilities for each of the eight possible system configurations in terms of the probabilities of the individual variables. For example, the value of p5 in the table above is given by



If all three of the variables C1, C2, and C3 agree with each other, what is the probability that they agree with C0? The answer is simply p7/(p0 + p7), since p0 and p7 are the probabilities of the two configurations in which all three of the variables have the same value, and p7 is the configuration in which they agree with C0.  Thus, letting P7 denote the probability in question, we have



Taking the reciprocal and subtracting 1 from both sides, it follows that (1−P7)/P7 = p0/p7, and inserting the values of p0 and p7 we get



On the other hand, suppose C1 and C3 agree with each other, but C2 has the opposite value. In this case, what is the probability that C1 and C3 agree with C0? We will denote this probability by P5. The answer is p5/(p2 + p5), because p2 and p5 are the probabilities of the two configurations in which C1 and C3 agree and C2 differs. Thus we get



In general, we can partition the N values of Cj (j=1 to N) into two sets A and B according to their values, and ask what is the probability that C0 agrees with the value of the variables in A. For convenience we create a vector s, and put sj equal to +1 if Cj is in A, and put sj = −1 if Cj is in B. Then the probability in question is



Incidentally, if P(X) is the probability of an event X, then the “odds” O(X) of that event are defined as P(X)/(1−P(X)). Suppose the odds of a given hypothesis H are O(H), and we want to know how the odds of H would be affected given some new evidence E. Thus we want O(H|E), so the effect of evidence E is to multiply the original odds by the factor O(H|E)/O(H). This is sometimes called a Bayes factor. We have



Noting the identities



we find that the Bayes factor satisfies the relation



In our previous example, with N = 3, the hypothesis H was that C0 is True, and we can consider the effect of the “evidence” E corresponding to (say) C3 being True. The probability of C3 being True given that H is True is simply P(c3) = p1 + p3 + p5 + p7, because this is the sum of the probabilities for the configurations in which C3 agrees with C0. On the other hand, the probability of C3 being True given that H is False is the complement of this, i.e., it is 1–P(c3) = p0 + p2 + p4 + p6, because this is the sum of the probabilities for the configurations in which C3 does not agree with C0. Thus the Bayes factor for the evidence C3 is



For independent variables, each piece of evidence contributes a factor of this form, consistent with our previous result.


As an aside, we note some interesting aspects of the conceptual transition from the system of N+1 logical variables Cj to the system of N logical variables cj. Recall that we defined cj for j = 1 to N as the condition that Cj agrees with C0, and we stipulated that the probability of agreement with C0 is independent of the value of C0.  Thus given the 2N+1 possible configurations {C0,C1,...,CN} we assign the same probability to complementary configurations. In effect, we treat each configuration and its complement as “the same configuration”. This is reminiscent of how we model elliptical geometry as the points on the surface of an ordinary sphere but with the stipulation that opposite (anti-podal) points on the sphere are treated as “the same point”. Having imposed this complementary symmetry, we can consider just the 2N configurations {c1,c2,...cN}, and we then stipulate that these N variables are completely independent (not just pairwise independent). We’ve seen that these stipulations, together with specified values of the N probabilities P(cj), are sufficient to completely determine the probabilities of each of the 2N possible configurations of the cj, and hence each of the 2N+1 configurations of the Cj. Each configuration of the cj represents two complementary configurations of the Cj, and we assign half the probability of the former to each of the latter. The system resulting from these stipulations is formally symmetrical in the Cj for j = 1 to N, but obviously not symmetrical with C0, unless all the probabilities P(cj) equal 1/2. This corresponds to an asymmetry involving relativistic velocities discussed below.


There’s an interesting formal correspondence between relation (1) and the relativistic speed composition formula. Suppose that for each probability P we define a new variable V by the relation V = 2P−1. The above relation is



In this form the reciprocation of the factors in the product can be equivalently given by simply negating the respective V parameter. Therefore, if we re-define Vj by the relation Vj = ±(2Pj−1) where the sign is positive if Cj is in A and negative if Cj is in B, we can omit the exponent sj and write the above relation in the form



Now consider a set of N particles moving at constant speeds along a single line. Let v1 denote the signed speed of one particle relative to some given system K0 of standard inertial coordinates, and let v2 denote the signed speed of a second particle in terms of the standard inertial rest frame coordinates K1 of the first particle. Similarly let v3 denote the signed speed of a third particle in terms of the standard inertial rest frame coordinates K2 of the second particle, and so on. Then, according to the special theory of relativity, the composition of all these speeds (i.e., the speed of the Nth particle in terms of K0) is the speed v given by



This is formally identical to (2), showing that the composition of (co-linear) speeds in special relativity corresponds to the composition of (re-scaled) probabilities for independent conditionals in probability theory. In another note we discussed a different mapping, P = v2,  between probabilities and velocities, whereby the square of a velocity (in units with c = 1) is identified with a probability, and we showed there how the relativistic combination of perpendicular velocities corresponds to the basic law of probability for independent events. Here we’ve described another correspondence, this one based on the linear mapping P = (v+1)/2, applied to co-linear velocities.


These two mappings are not as incommensurate as they might seem, because the combination formula for three mutually perpendicular speeds (presented in the other note) is



which can be factored as



where k = 1 for perpendicular speeds, and k = −1 for co-linear speeds.


As mentioned above, there is an asymmetry between the logical variables Cj for j = 1 to N and the “reference” logical variable C0. This mirrors the asymmetry, under reciprocation, between the velocities in the relativistic velocity composition formula discussed in the note “More Symmetry”.


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