I am facing a posterior distribution in a MCMC application that aims to sample an unobservable variable $x=\{x_t\}_{t=0}^{T}$ given an observed series $y=\{y_t\}^T_{t=0}$.

However, the conditional posteriors reads as $$p(x_t | y_{t+1}, y_t, y_{t-1} ,x_{t-1}, x_{t+1}, \Theta),$$ with $\Theta$ being a vector of additional structural parameters. According to my understanding, this would be a smoothing problem, since knowledge of $y_{t+1}$ is required to infer the value of $x_t$.

However, the articles dealing with the same problem refer to the series $x$ as filtered series.

Am I missing something here?