  In this paper, we revisit the Kalman filter theory. After giving the
intuition on a simplified financial markets example, we revisit the maths
underlying it. We then show that Kalman filter can be presented in a very
different fashion using graphical models. This enables us to establish the
connection between Kalman filter and Hidden Markov Models. We then look at
their application in financial markets and provide various intuitions in terms
of their applicability for complex systems such as financial markets. Although
this paper has been written more like a self contained work connecting Kalman
filter to Hidden Markov Models and hence revisiting well known and establish
results, it contains new results and brings additional contributions to the
field. First, leveraging on the link between Kalman filter and HMM, it gives
new algorithms for inference for extended Kalman filters. Second, it presents
an alternative to the traditional estimation of parameters using EM algorithm
thanks to the usage of CMA-ES optimization. Third, it examines the application
of Kalman filter and its Hidden Markov models version to financial markets,
providing various dynamics assumptions and tests. We conclude by connecting
Kalman filter approach to trend following technical analysis system and showing
their superior performances for trend following detection.
