Evidence or Marginal? What Moment-Vector Propagation Computes in Polynomial Density Neurons

원문 초록

When a neuron's input is lost, or known through a belief $q$, what should it output? For a neuron storing a joint polynomial density $N$ with input density $D$, two formulas compete. $p_\mathrm{CD}=E_q[N]/E_q[D]$ costs one matrix-vector product; Hierarchical Correlation Reconstruction introduces it as a constant-denominator approximation to $p_\mathrm{EU}=E_q[N/D]$. For a density model both are exact answers to different questions: $p_\mathrm{CD}$ is Bayesian updating with the belief as virtual evidence; $p_\mathrm{EU}$ installs the belief as a new marginal (Jeffrey's rule). With a missing input $p_\mathrm{CD}$ returns the model's conditional; $p_\mathrm{EU}$ generally does not. This post hoc reading is proved, and illustrated outside the pair family up to $k=8$ inputs; it has no pre-registered test, only one exploratory real-data pilot. Post hoc, closed forms fall within a factor 2 of all 112 measured gaps, and negative density begins near input correlation $1/k$. In a pre-registered synthetic study (up to 32 inputs), 5 of 15 hypotheses are supported, 2 inconclusive and 8 not supported, including all three on transfer beyond equicorrelated Gaussians. Post hoc, where $p_\mathrm{EU}$ is defined, truncation usually outweighs the gap for Beta beliefs in pair neurons; triple neurons under flat and data beliefs are exceptions.