A Hyperbolic Sum Rule for Probability: Solving Recursive (“Chicken and Egg”) Problems

We prove that the probability of "<i>A</i> or <i>B</i>", denoted as <i>p</i>(<i>A</i> or <i>B</i>), where <i>A</i> and <i>B</i> are events or hypotheses that may be recursively dependent, is given by a "Hyperbolic Sum Rule" (<b>HSR</b>), which is relationally isomorphic to the hyperbolic tangent double-angle formula. We also prove that this HSR is Maximum Entropy (<b>MaxEnt</b>). Since this recursive dependency is commutative, it maintains the symmetry between the two events, while the recursiveness also represents temporal symmetry within the logical structure of the HSR. The possibility of recursive probabilities is excluded by the "Conventional Sum Rule" (<b>CSR</b>), which we have also proved to be MaxEnt (with lower entropy than the HSR due to its narrower domain of applicability). The concatenation property of the HSR is exploited to enable analytical, consistent, and scalable calculations for multiple hypotheses. Although they are intrinsic to current artificial intelligence and machine learning applications, such calculations are not conveniently available for the CSR, moreover they are presently considered intractable for analytical study and methodological validation. Where, for two hypotheses, we have <i>p</i>(<i>A</i>|<i>B</i>) > 0 and <i>p</i>(<i>B</i>|<i>A</i>) > 0 together (where "<i>A</i>|<i>B</i>" means "<i>A</i> given <i>B</i>"), we show that <i>either</i> {<i>A</i>,<i>B</i>} is independent <i>or</i> {<i>A</i>,<i>B</i>} is recursively dependent. In general, recursive relations cannot be ruled out: the HSR should be used by default. Because the HSR is isomorphic to other physical quantities, including those of certain components that are important for digital signal processing, we also show that it is as reasonable to state that "<i>probability is physical</i>" as it is to state that "<i>information is physical</i>" (which is now recognised as a truism of communications network engineering); probability is <i>not</i> merely a mathematical construct. We relate this treatment to the physics of Quantitative Geometrical Thermodynamics, which is defined in complex hyperbolic (Minkowski) spacetime.