Bayesian causal inference in perception

Inference about whether sensory signals have a common source or separate sources. Bayes' theorem compares hypotheses through

P (C ∣ x) \propto P (x ∣ C) P (C)

. A common-source hypothesis can permit cue combination; separate-source hypotheses can preserve distinct estimates. Reliability weighting conditional on a common source is therefore not the entire causal-inference problem.

Table of contents
- Gaussian temporal-discrepancy shrinkage Bayesian causal inference in perception

Gaussian temporal-discrepancy shrinkage ( $E [δ ∣ y] = y τ^{2} / (s^{2} + τ^{2})$ )

 0  0

For a common-event model, let

y ∣ δ \sim N (δ, s^{2})

and

δ \sim N (0, τ^{2})

with positive variances. The posterior exponent is

- [(y - δ)^{2} / s^{2} + δ^{2} / τ^{2}] /2

. Completing the square yields posterior precision

s^{- 2} + τ^{- 2}

and mean

y τ^{2} / (s^{2} + τ^{2})

. The estimate is pulled toward zero, more strongly when the measurement is uncertain. This conditional model does not decide whether a common event should be assumed.