CSMC with a backward pass from the ancestors

Given the reference trajectory x_{1:T}, a resampling scheme, and the kernels K_T and K_{t,j,j} of AT, FM-AT, BS, FM-BS, or M-BS, with the probabilities (S3), Algorithm S2 of the supplementary material consists of the following steps.

(a) Run steps (a) and (b) of Algorithm S1, with c = b_t in Algorithm S1.b. (b) Draw l_T ∼ K_T(b_T, ·). (c) For t = T - 1, …, 1, draw l_t from the kernel K_{t,l_{t+1},l_{t+1}} started from the ancestor a_t^{l_{t+1}} of the particle l_{t+1}. (d) Return (x_1^{l_1}, …, x_T^{l_T}).

The joint distribution of its variables and its Markov kernel are defined here for a selection rule r.

noncomputable def CSMC.PreviousState.Model.algLawAnc {X : Type u_1} [MeasurableSpace X] {n N : ℕ} {μ : MeasureTheory.Measure X} (M : Model μ n) (resdist : Fin n → (Fin N → X) → (Fin N → ℝ) → (Fin N → Fin N) → ℝ) (r : SelectionRule) : ProbabilityTheory.Kernel (Traj X n) ((Pos X n N × Anc n N × Idx n N) × Idx n N)

The joint distribution, given the reference trajectory, of the particle positions, the ancestor indices, and the indices of the reference, (y, a, b), and of the indices l of the output, generated by Algorithm S2 with the resampling scheme resdist and the kernels of the selection rule r.

Equations

• One or more equations did not get rendered due to their size.

noncomputable def CSMC.PreviousState.Model.csmcAnc {X : Type u_1} [MeasurableSpace X] {n N : ℕ} {μ : MeasureTheory.Measure X} (M : Model μ n) (resdist : Fin n → (Fin N → X) → (Fin N → ℝ) → (Fin N → Fin N) → ℝ) (r : SelectionRule) : ProbabilityTheory.Kernel (Traj X n) (Traj X n)

The Markov kernel defined by Algorithm S2 with the resampling scheme resdist and the kernels of the selection rule r.

Equations

• M.csmcAnc resdist r = (M.algLawAnc resdist r).map fun (z : (CSMC.Pos X n N × CSMC.Anc n N × CSMC.Idx n N) × CSMC.Idx n N) => CSMC.trajOf z.1.1 z.2