EcoQBNs: First Application of Ecological Modeling with Quantum Bayesian Networks

A recent advancement in modeling was the development of quantum Bayesian networks (QBNs). QBNs generally differ from BNs by substituting traditional Bayes calculus in probability tables with the quantum amplification wave functions. QBNs can solve a variety of problems which are unsolvable by, or are too complex for, traditional BNs. These include problems with feedback loops and temporal expansions; problems with non-commutative dependencies in which the order of the specification of priors affects the posterior outcomes; problems with intransitive dependencies constituting the circular dominance of the outcomes; problems in which the input variables can affect each other, even if they are not causally linked (entanglement); problems in which there may be >1 dominant probability outcome dependent on small variations in inputs (superpositioning); and problems in which the outcomes are nonintuitive and defy traditional probability calculus (Parrondo’s paradox and the violation of the Sure Thing Principle). I present simple examples of these situations illustrating problems in prediction and diagnosis, and I demonstrate how BN solutions are infeasible, or at best require overly-complex latent variable structures. I then argue that many problems in ecology and evolution can be better depicted with ecological QBN (EcoQBN) modeling. The situations that fit these kinds of problems include noncommutative and intransitive ecosystems responding to suites of disturbance regimes with no specific or single climax condition, or that respond differently depending on the specific sequence of the disturbances (priors). Case examples are presented on the evaluation of habitat conditions for a bat species, representing state-transition models of a boreal forest under disturbance, and the entrainment of auditory signals among organisms. I argue that many current ecological analysis structures—such as state-and-transition models, predator–prey dynamics, the evolution of symbiotic relationships, ecological disturbance models, and much more—could greatly benefit from a QBN approach. I conclude by presenting EcoQBNs as a nascent field needing the further development of the quantum mathematical structures and, eventually, adjuncts to existing BN modeling shells or entirely new software programs to facilitate model development and application.

The x and y axes of the Hilbert space representation are represented by two column vectors: and the superposition state S --representing each state simultaneously --is represented by: defines the probability amplitudes (wave amplitudes, described by complex numbers; see text) and = the phase of the amplitude (a "shift" of the waveform).
As noted in the text, the probability amplitude squared is the classical state probability value, which is calculated by multiplying the amplitude by its complex conjugate. In the Hilbert state space representation, for habitat state H2 (text Fig. 1), this is calculated as: As noted in the text, the solution of ( 2) = 0.5 denotes complete uncertainty in this two-state system because the value of the wave form shift parameter ϴ has not been specified, which would otherwise denote the degree to which the two state wave functions differ and would potentially result in non-uniform probability outcomes.

Calculations of conditional probabilities of conditions for the bat habitat example discussed in the text, based on a non-ecological example given in Trueblood et al. (2015).
With: r = riparian habitat, with condition i, w = woodland, scrub, or savanna habitat, with condition j, and H = bat habitat condition, with conditions good or poor, joint probabilities are determined from the conditional probability table (text Fig. X2) as: Similarly, in the quantum Bayesian network (QBN), let bat habitat, riparian, and woodland habitat conditions be associated with projectors P, Q, and S, respectively. Then their joint probabilities can be described by use of Born's Rule describing the absolute magnitudes of the vectors: where the conditional state is given by: If the projectors Q and S do not commute, then the conditional state will depend on the order in which these two variables are specified, so that , ≠ , which further denotes that This is an ecological noncommutative condition, for example, if the bat were to first seek the overall woodland condition and then within that seek the riparian condition, and if the probability (frequency) of using that site was different than if it first followed riparian areas to where it would embed within a woodland condition. By the ecological principle of hierarchical habitat selection, the former is more likely the case, where the bat would seek the woodland condition that is more generally available across the landscape, and then within that seek specific and more constrained riparian conditions. (However, for this species, such habitat selection behavior is speculative and intended here for demonstration of the statistical modeling approach.) From the above framework, in a QBN, the probability values in the traditional conditional probability table (CPT; text Fig. 2) are replaced by probability amplitudes, as detailed in the main text. Particularly if the system being described is ecologically noncommutative such as described above, then the resulting CPT can be referred to as conditional quantum probability table or CQPT, and different CQPT values would need to be calculated for different sequences with which the input variables (habitat selection sequences) are specified. Note that noncommutative conditions --the sequence in which prior conditions are specified --essentially violate local Markov conditions and are thus not solvable with traditional Bayesian calculus without introducing additional complexity with latent variables that specify the sequences and their unique outcomes, whereas a QBN can inherently deal with such conditions.

S3
Calculations of prior probabilities of conditions for the bat habitat example discussed in the text and for the conditional quantum probability table discussed in Supplement 2.
In the CQPT, the prior probabilities are derived as follows. First, presume that only 10% of a given landscape consists of riparian areas, which is the basis for the value 0.1 in the CPT of text