Algorithm 3 HMM backward-filtering forward-sampling algorithm for block-sampling z1:T

1: let π0 and π = (π1|…|πK)T be the initial and transition distributions

2: let ${\mathbf{u}}_t\mathrm{\in }{ℝ}_{+}^K$ be the likelihoods p(yt | zt = k) for each k

3: let ${\mathbf{\zeta }}_t\in {\mathrm{ℝ}}_{+}^K$ be the backward messages p(yt+1:T | zt = k) at t for each k

4: normalize each ut to sum to 1, preventing underflow during computations

5: for t = 1,…,T do

6:   ũt ← ut/ (1Tut)

7: end for

8: calculate backward messages over all time points

9: ζ̃T = 1

10: for t = T − 1,…,1 do

11:  transmit messages backward : τt+1 ← ũt+1 ∘ ζ̃t+1, ζt ← πτt+1

12:  normalize : ζ̃t ← ζt/ (1Tζt)

13: end for

14: τ1 ← ũ1 ∘ ζ̃1

15: sample first time point

16: q1 ← π0 ∘ τ1, q̃1 ← q1/(1Tq)

17: z1 ~ q̃1

18: sample other time points

19: for t = 2,…,T do

20:  qt ← πzt−1 ∘ τt, q̃t ← qt/ (1Tqt)

21:  zt ~ q̃t

22: end for