Source
Van Gael, Teh and Ghahramani (2008), The infinite factorial hidden Markov model, LIT-1577, read in full in NOTE-1199: §2.1 (the finite model, (3)–(6)), §2.4 (stick-breaking) and §2’s statement that “rows correspond to timesteps and the columns to features or Markov chains”.
What was actually shown
The definition. For M features over T time steps, column m has transition matrix [[1 − a_m, a_m], [1 − b_m, b_m]], with a_m ~ Beta(α/M, 1) and b_m ~ Beta(γ, δ) drawn independently, and every chain starts off. In the collapsed likelihood (6) the a-factor depends only on c⁰⁰ and c⁰¹ and the b-factor only on c¹⁰ and c¹¹. In the stick-breaking representation the b paired with the m-th largest a is “an independent draw from a Beta(γ, δ) distribution which is independent of M”.
The check. From (6), the probability that a feature on at t stays on at t + 1, given its counts, is (c¹¹ + γ)/(γ + δ + c¹⁰ + c¹¹), whatever c⁰¹, c⁰⁰ or α. The record computed it as a ratio of (6) for three count vectors. (The paper’s buffet prints it with δ for γ; NOTE-1199.) A priori, a feature on is kept with probability E[b] = γ/(γ + δ). That is 10/11 = 0.91 at the prior means the paper’s experiment uses (γ = 10, δ = 1), for a feature with a = 0.025 as for one with a = 0.5.
One sequence. The rows are time steps of a single object, and the iFHMM explains one multichannel time series. The paper has no axis of objects and derives no limit for several sequences sharing a feature’s (a_m, b_m).
What this does not say
- Not that prevalence is constant. A chain’s stationary on-fraction is a/(1 − b + a), which depends on b. The expected number of active features rises from α at the first step (THEORY-1019).
- Not that persistence is learned per feature from little data. The b’s are independent a priori. A posteriori a rare feature has few 1 → 1 counts and its b stays near the prior.
- Not a model of an organization’s members. Using it for N members needs a sharing structure the paper does not give.
Connections
- CASE-165, item 2. The case’s sticky term is, with its other inputs at zero, the same two-state chain with on-rate π_k and stay-rate σ(logit π_k + ρ). There the stay-rate is a function of the on-rate, which is why a rare feature is kept with probability 0.16 at ρ = 2. In the mIBP the two are independent, so the case’s “one ρ cannot do that for features of different prevalence” does not arise. NOTE-1199 gives the numbers.
- THEORY-1006. The dependent IBP keeps each feature’s prevalence fixed and sets persistence through its Gaussian process, and only per-feature length scales make persistence per feature. The mIBP has persistence per feature by construction and fixes no prevalence.
- NOTE-1172. The review describes this model in §7.4 in the same terms; this is the primary source.