Recursive Bayesian estimation

Recursive Bayesian estimation is a general probabilistic approach for estimating an unknown probability density function recursively over time using incoming measurements and a mathematical process model.

Model

The true state $x$ is assumed to be an unobserved Markov process, and the measurements $z$ are the observed states of a Hidden Markov Model (HMM).

Because of the Markov assumption, the probability of the current true state given the immediately previous one is conditionally independent of the other earlier states.

:$p(\; extbf\{x\}\_k|\; extbf\{x\}\_\{k-1\},\; extbf\{x\}\_\{k-2\},dots,\; extbf\{x\}\_0)\; =\; p(\; extbf\{x\}\_k|\; extbf\{x\}\_\{k-1\}\; )$

Similarly, the measurement at the "k"-th timestep is dependent only upon the current state, so is conditionally independent of all other states given the current state.

:$p(\; extbf\{z\}\_k|\; extbf\{x\}\_k,\; extbf\{x\}\_\{k-1\},dots,\; extbf\{x\}\_\{0\})\; =\; p(\; extbf\{z\}\_k|\; extbf\{x\}\_\{k\}\; )$

Using these assumptions the probability distribution over all states of the HMM can be written simply as:

:$p(\; extbf\{x\}\_0,dots,\; extbf\{x\}\_k,\; extbf\{z\}\_1,dots,\; extbf\{z\}\_k)\; =\; p(\; extbf\{x\}\_0)prod\_\{i=1\}^k\; p(\; extbf\{z\}\_i|\; extbf\{x\}\_i)p(\; extbf\{x\}\_i|\; extbf\{x\}\_\{i-1\})$

However, when using the Kalman filter to estimate the state x, the probability distribution of interest is associated with the current states conditioned on the measurements up to the current timestep. (This is achieved by marginalising out the previous states and dividing by the probability of the measurement set.)

This leads to the "predict" and "update" steps of the Kalman filter written probabilistically. The probability distribution associated with the predicted state is product of the probability distribution associated with the transition from the ("k" - 1)-th timestep to the "k"-th and the probability distribution associated with the previous state, with the true state at ("k" - 1) integrated out.

:$p(\; extbf\{x\}\_k|\; extbf\{Z\}\_\{k-1\})\; =\; int\; p(\; extbf\{x\}\_k\; |\; extbf\{x\}\_\{k-1\})\; p(\; extbf\{x\}\_\{k-1\}\; |\; extbf\{Z\}\_\{k-1\}\; )\; ,\; d\; extbf\{x\}\_\{k-1\}$

The probability distribution of updated is proportional to the product of the measurement likelihood and the predicted state.:$p(\; extbf\{x\}\_k|\; extbf\{Z\}\_\{k\})\; =\; frac\{p(\; extbf\{Z\}\_k|\; extbf\{x\}\_k)\; p(\; extbf\{x\}\_k|\; extbf\{Z\}\_\{k-1\})\}\{p(\; extbf\{Z\}\_k|\; extbf\{Z\}\_\{k-1\})\}\; =\; alpha,p(\; extbf\{Z\}\_k|\; extbf\{x\}\_k)\; p(\; extbf\{x\}\_k|\; extbf\{Z\}\_\{k-1\})$

The denominator:$p(\; extbf\{Z\}\_k|\; extbf\{Z\}\_\{k-1\})\; =\; int\; p(\; extbf\{Z\}\_k|\; extbf\{x\}\_k)\; p(\; extbf\{x\}\_k|\; extbf\{Z\}\_\{k-1\})\; d\; extbf\{x\}\_k$is constant relative to $x$, so we can always substitute it for a coefficient $alpha$, which can usually be ignored in practice. The numerator can be calculated and then simply normalized, since its integral must be unitary.

Applications

* Kalman filter, a recursive Bayesian filter for multivariate normal distributions
* Particle filter, a sequential Monte Carlo (SMC) based technique, which models the PDF using a set of discrete points
* Grid-based estimators, which subdivide the PDF into a discrete grid

equential Bayesian filtering

Sequential Bayesian filtering is the extension of the Bayesian estimation for the case when the observed value changes in time. It is a method to estimate the real value of an observed variable that evolves in time. The method is named filtering when we estimate the current value given past observations, smoothing when estimating past value given present and past measures, and prediction when estimating a probable future value.

The notion of Sequential Bayesian filtering is extensively used in control and robotics.

External links

* [http://citeseer.ist.psu.edu/504843.html A Tutorial on Particle Filters for On-line Non-linear/Non-Gaussian Bayesian Tracking] , IEEE Transactions on Signal Processing (2001)
* [http://julien.diard.free.fr/articles/CIRAS03.pdf A survey of probabilistic models, using the Bayesian Programming methodology as a unifying framework]