Probabilities, Odds, and Relativistic Speeds 

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

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 C_{0}, C_{1}, ..., C_{N}, each of which has either the value T (true) or F (false). We stipulate that each of the 2^{N+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 overbars signify logical negation. Thus c_{j} is true if and only if C_{j} agrees with C_{0}. There are 2^{N} possible configurations of the variables c_{1} to c_{N}, 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 c_{1}, c_{2},..., c_{N} 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 p_{5} in the table above is given by 



If all three of the variables C_{1}, C_{2}, and C_{3} agree with each other, what is the probability that they agree with C_{0}? The answer is simply p_{7}/(p_{0} + p_{7}), since p_{0} and p_{7} are the probabilities of the two configurations in which all three of the variables have the same value, and p_{7} is the configuration in which they agree with C_{0}. Thus, letting P_{7} denote the probability in question, we have 



Taking the reciprocal and subtracting 1 from both sides, it follows that (1−P_{7})/P_{7} = p_{0}/p_{7}, and inserting the values of p_{0} and p_{7} we get 



On the other hand, suppose C_{1} and C_{3} agree with each other, but C_{2} has the opposite value. In this case, what is the probability that C_{1} and C_{3} agree with C_{0}? We will denote this probability by P_{5}. The answer is p5/(p2 + p5), because p_{2} and p_{5} are the probabilities of the two configurations in which C_{1} and C_{3} agree and C_{2} differs. Thus we get 



In general, we can partition the N values of C_{j} (j=1 to N) into two sets A and B according to their values, and ask what is the probability that C_{0} agrees with the value of the variables in A. For convenience we create a vector s, and put s_{j} equal to +1 if C_{j} is in A, and put s_{j} = −1 if C_{j} 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(HE), so the effect of evidence E is to multiply the original odds by the factor O(HE)/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 C_{0} is True, and we can consider the effect of the “evidence” E corresponding to (say) C_{3} being True. The probability of C_{3} being True given that H is True is simply P(c_{3}) = p_{1 }+ p_{3 }+ p_{5 }+ p_{7}, because this is the sum of the probabilities for the configurations in which C_{3} agrees with C_{0}. 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(c_{3}) = p_{0 }+ p_{2 }+ p_{4 }+ p_{6}, because this is the sum of the probabilities for the configurations in which C_{3} does not agree with C_{0}. 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 C_{j} to the system of N logical variables c_{j}. Recall that we defined c_{j} for j = 1 to N as the condition that C_{j} agrees with C_{0}, and we stipulated that the probability of agreement with C_{0} is independent of the value of C_{0}. Thus given the 2^{N+1} possible configurations {C_{0},C_{1},...,C_{N}} 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 (antipodal) points on the sphere are treated as “the same point”. Having imposed this complementary symmetry, we can consider just the 2^{N} configurations {c_{1},c_{2},...c_{N}}, 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(c_{j}), are sufficient to completely determine the probabilities of each of the 2^{N} possible configurations of the c_{j}, and hence each of the 2^{N+1} configurations of the C_{j}. Each configuration of the c_{j} represents two complementary configurations of the C_{j}, 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 C_{j} for j = 1 to N, but obviously not symmetrical with C_{0}, unless all the probabilities P(c_{j}) 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 redefine V_{j} by the relation V_{j} = ±(2P_{j}−1) where the sign is positive if C_{j} is in A and negative if C_{j} is in B, we can omit the exponent s_{j} and write the above relation in the form 



Now consider a set of N particles moving at constant speeds along a single line. Let v_{1} denote the signed speed of one particle relative to some given system K_{0} of standard inertial coordinates, and let v_{2} denote the signed speed of a second particle in terms of the standard inertial rest frame coordinates K_{1} of the first particle. Similarly let v_{3} denote the signed speed of a third particle in terms of the standard inertial rest frame coordinates K_{2} 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 K_{0}) is the speed v given by 



This is formally identical to (2), showing that the composition of (colinear) speeds in special relativity corresponds to the composition of (rescaled) probabilities for independent conditionals in probability theory. In another note we discussed a different mapping, P = v^{2}, 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 colinear 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 colinear speeds. 

As mentioned above, there is an asymmetry between the logical variables C_{j} for j = 1 to N and the “reference” logical variable C_{0}. This mirrors the asymmetry, under reciprocation, between the velocities in the relativistic velocity composition formula discussed in the note “More Symmetry”. 
