TITLE: What are "Filtering" and "Smoothing" with regards to Hidden Markov Models?
QUESTION [8 upvotes]: The Wikipedia article about Hidden Markov Models mentions "filtering" and "smoothing" tasks, see here: http://en.wikipedia.org/wiki/Hidden_Markov_model#Filtering. 
It gives a brief explanation but no motivating examples and no references to the equivalent algorithm or technique, if any exists. 
For example, does filtering imply that for every hidden variable there is "error" state and we want to compute likelihood of our sensor being in that state in order to drop the measurement? Or is there something more to it?
In terms of missing links, e.g. I notice the http://en.wikipedia.org/wiki/Kalman_filter article that seems to mention HMM. Is "Kalman" proper way to refer to HMM based filtering mentioned above? Or are there lots of different types of filtering all based on HMM?

REPLY [4 votes]: In the dynamic linear models literature, a distinction is often made between filtering, smoothing and predicting. At a given time $t$, filtering refers to the estimation of an unobserved state $x_t$ given an observed measurement $y_t$. Smoothing is the retroactive estimation of states $x_{t-k}, k \gt 0$ at past times, and predicting is the estimation of future states $x_{t+k}, k \gt 0$.