Algorithm 2 Posterior-Weighted RUL Estimation.

Require: Estimated parameters $\widehat{\mathrm{\Theta }}$; absorbing state K; observed degradation trajectory ${\left\{y_{it}\right\}}_{t=1}^{T_i}$ for a new unit i. 1:Emission Evaluation:   Compute Gaussian emission densities $b_k\left(y_{it}\right)$ for all $t=1,\dots ,T_i$ and $k=1,\dots ,K$ using the linear emission model in (1). 2:Forward–Backward Recursions:   Run the scaled forward and backward recursions ((6)–(8)) to obtain ${\alpha }_{it}\left(k\right)$ and ${\beta }_{it}\left(k\right)$.   Compute smoothed posteriors ${\gamma }_{it}\left(k\right)$ via (9). 3:Fundamental Matrix:   Extract the transient submatrix $Q={\widehat{P}}_{1:(K-1),1:(K-1)}$.   Compute the fundamental matrix $F={(I_{K-1}-Q)}^{-1}$. 4:Compute RUL Path:   For each $t=1,\dots ,T_i$, evaluate $${\widehat{\mathrm{RUL}}}_i\left(t\right)={\gamma }_{it}^{\top }F{\mathbf{1}}_{K-1}$$ using (34). 5:return Full trajectory ${\{{\widehat{\mathrm{RUL}}}_i\left(t\right)\}}_{t=1}^{T_i}$ and the current estimate ${\widehat{\mathrm{RUL}}}_i\left(T_i\right)$.