# Kalman Filter: continuous state space, discrete observations

This is just an idea. How can we model the kalman filter to get the state representation in continuous space when the observations to the system are actually from the discrete space. The discrete observation, for example, can be sequence of digits (0-9) where the digits themselves do not have numerical meaning but the sequence of observations are correlated. I know we can try Hidden Markov Model (HMM) for this, but I would like to get the continuous (real-valued vector) space state representation rather than the discrete space as in HMM.

• You can get a "real valued" vector out of the HMM approach. The HMM approach will give a probability $p_i$ for each state $i = 0\ldots 9$. Then you just form $\sum_{i=0}^9 ip_i$ and you have the "mean state". Whether this makes any sense for your application is another thing; it may, but if the application is digit recognition, I'd rather get "1" for recognition than "1.23952756".
– Peter K.
Apr 24 '13 at 11:45