I am facing a posterior distribution in a [MCMC][1] 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?


  [1]: https://en.wikipedia.org/wiki/Markov_chain_Monte_Carlo