Recursive Bayesian Estimation - Model

Model

The true state is assumed to be an unobserved Markov process, and the measurements are the observed states of a Hidden Markov Model (HMM). The following picture presents a Bayesian Network of a 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.

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.

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

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 the sum (integral) of the products 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, over all possible .

The probability distribution of update is proportional to the product of the measurement likelihood and the predicted state.

p(\textbf{x}_k|\textbf{z}_{k}) = \frac{p(\textbf{z}_k|\textbf{x}_k) p(\textbf{x}_k|\textbf{z}_{k-1})}{p(\textbf{z}_k|\textbf{z}_{k-1})} = \alpha\,p(\textbf{z}_k|\textbf{x}_k) p(\textbf{x}_k|\textbf{z}_{k-1})

The denominator

is constant relative to, so we can always substitute it for a coefficient, which can usually be ignored in practice. The numerator can be calculated and then simply normalized, since its integral must be unitary.

Read more about this topic:  Recursive Bayesian Estimation

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