Quantum-like Bayesian networks#

Idea. Keep an ordinary Bayesian network, but marginalise hidden variables over amplitudes instead of probabilities, so that unobserved causes can interfere (Moreira and Wichert, 2014, 2016).

Use it for: decision support where people (not the network) make the judgement under unresolved uncertainty: diagnosis, credit and fraud assessment, judicial reasoning, expert elicitation.

Models and functions#

Name	Description
`BayesNet(variables, parents, cpt)`	Discrete network; CPTs as nested dicts or functions
`classical_marginal(net, query, evidence)`	Exact classical inference by enumeration
`quantum_like_marginal(net, query, evidence, phases)`	Quantum-like inference, one phase per hidden configuration
`for_network(net, query, conditions)`	Returns a fit-able `QLBNModel` class with exactly the phases in use
`ClassicalBNModel`	Baseline with no free parameters

Mathematics#

P_q(Q = v | e) = α | Σ_h √P(v, e, h) · e^{iθ_h} |²,   α normalises over v

With two hidden configurations a phase difference of π/2 gives classical inference.

Data format#

conditions = {'TestA+': {'TestA': 'pos'}, 'TestA-': {'TestA': 'neg'}}   # evidence per condition
counts = {'TestA+': [n_yes, n_no], 'TestA-': [n_yes, n_no]}            # judgements of the query

Quick start#

from quantum_mind.core import compare
from quantum_mind.families.qlbn import BayesNet, for_network, ClassicalBNModel
QL = for_network(net, 'Disease', conditions)
compare([(QL, dict(net=net, query='Disease', conditions=conditions)),
         (ClassicalBNModel, dict(net=net, query='Disease', conditions=conditions))], counts)

Circuit#

quantum_mind.circuits.qlbn_circuit(net, query, evidence, phases): the amplitudes are loaded with StatePreparation, the hidden register goes through Hadamard gates and is post-selected on |0…0⟩, which sums the amplitudes. Exact against the analytic model (tests).

Limitations#

Up to 8 hidden configurations in the fit-able model; phases are free parameters unless a heuristic fixes them; enumeration is exponential in the number of hidden variables.

References#

Moreira and Wichert (2014) Applied Soft Computing 25. Moreira and Wichert (2016) Frontiers in Psychology 7:11.