Interest in the **filtering problem** dates back to the late 1930s–early 1940s. It was considered in Kolmogorov’s work on time series and Wiener’s on improving radar communication during WWII, which first appeared in 1942 as a classified memorandum nicknamed “The Yellow Peril”, so named after the colour of the paper on which it was printed.

Kalman extended this work to non-stationary processes. This work had military applications, notably the prediction of ballistic missile trajectories. Non-stationary processes were required to realistically model their launch and re-entry phases. Of course, non-stationary processes abound in other fields–even the standard Brownian motion of the basic Bachelier model in finance is non-stationary. Kalman’s fellow electrical engineers initially met his ideas with scepticism, so he ended up publishing in a mechanical engineering journal. In 1960, Kalman visited the NASA Ames Research Center, where Stanley F. Schmidt took interest in this work. This led to its adoption by the Apollo programme and other projects in aerospace and defence. The discrete-time version of the filter derived by Kalman is now known as the **Kalman filter**.

The general solutions are, however, infinite-dimensional and not easily applicable. In practice, numerical approximations are employed. **Particle filters** constitute a particularly important class of such approximations. These methods are sometimes referred to as **sequential Monte Carlo (SMC)**, a term coined by Liu and Chen. The Monte Carlo techniques requisite for particle filtering date back to the work of Hammersley and Morton. **Sequential importance sampling (SIS)** dates back to the work of Mayne and Handschin. The important resampling step was added by Gordon, Salmond, and Smith, based on an idea by Rubin, to obtain the first **sequential importance resampling (SIR)** filter, which, in our experience, remains the most popular particle filtering algorithm used in practice.

**Markov chain Monte Carlo (MCMC)** takes its origin from the work of Nicholas Metropolis, Marshall N. Rosenbluth, Arianna W. Rosenbluth, Edward Teller, and Augusta H. Teller at Los Alamos on simulating a liquid in equilibrium with its gas phase. The discovery came when its authors realized that, instead of simulating the exact dynamics, they could simulate a certain Markov chain with the same equilibrium distribution.

In this course we consider the theory and practice of Kalman and particle filtering and employ **probabilistic programming (PP)** languages, such as the veteran BUGS/WinBUGS/OpenBUGS and the more recent PyStan, PyMC3, and PyMC4 to perform MCMC analysis.

- Teacher: Paul Bilokon