Source
Knowles and Ghahramani (2007), Infinite sparse factor analysis and infinite independent components analysis, LIT-1626, read in full in NOTE-1209: §3, eqs. (9) and (12)–(14).
What was actually shown
The ratio. With the other sources fixed, the residual ε = y_t − Σ_{j≠k} g_j z_jt x_jt satisfies ε = g_k x + noise when z_kt = 1 and ε = noise when z_kt = 0. With x ~ N(0, 1), the marginal likelihood ratio is r_l = σ̂ exp{μ̂²/(2σ̂²)}, where σ̂² = σ_ε²/(σ_ε² + g_k′g_k) and μ̂ = g_k′ε/(σ_ε² + g_k′g_k) are the posterior variance and mean of x, (9). The posterior odds of z_kt = 1 multiply r_l by the IBP’s prior odds, m_{k,−t}/(β + N − 1 − m_{k,−t}).
At zero signal. If ε has no component along g_k, μ̂ = 0 and r_l = σ̂ <
- The data then favour z_kt = 0, by the factor σ̂, which is how much the prior’s spread on x exceeds the posterior’s. A stronger loading (larger g′g) or a wider prior on x makes the factor smaller.
The check. At D = 5, σ_ε² = 0.7, random g and ε: quadrature 0.310857, formula 0.310857; with ε projected off g: quadrature 0.259695, σ̂ = 0.259695. The Laplacian version, (14), gave 0.322171 both ways (record script, 2026-10-27).
What this does not say
- Not that the mask is identified in general. With the loading and other sources unknown, splits, rescalings and absorption remain (CASE-165, items 3–4).
- Not that the data decide without the prior. The factor is the prior’s: with x fixed at a value near zero instead of integrated out, “on” and “off” fit equally.
- Not about a sequence. The paper’s model has no dynamics; the infinite factorial HMM adds them (LIT-1577).
Connections
- CASE-165, item 3. Qualifies “availability is told from activation only by the prior and the transition, not by the data”: true with the activation fixed; with the activation integrated out, the data carry an Occam factor against holding an unused feature, whose size the activation’s prior sets.
- THEORY-1043. The mask’s prior is the IBP; its prior odds term is the buffet’s m/(N − m) at β = 1.