Kalman filter

Before learning Kalman filter, it’s essential to understand Bayes’ theorem, as the Kalman filter is fundamentally based on it.

Table of Contents

Bayes’ Theorem

The Bayes’ Theorem is an approach to statistical inference, where it is used to invert the probability of observations given a model configuration.

Statement of theorem

Bayes’ theorem is stated mathmatically as the following equation:

\[P(A|B) = \frac{P(B|A)P(A)}{P(B)}\]

where $A$ and $B$ are events and $P(B) \neq 0$.

  • $P(A B)$ is a conditional probability; the probability of event $A$ occurring given that $B$ is true.
  • $P(B A)$ is also a conditional probability; the probability of event $B$ occurring given that $A$ is true.

Proof

For events (Discrete)

\[P(A|B) = \frac{P(A \cap B)}{P(B)},\ \text{if} \ P(B) \neq 0\]

where $P(A \cap B)$ is the probability of both $A$ and $B$ being true. Similarly,

\[P(B|A) = \frac{P(A \cap B)}{P(A)},\ \text{if} \ P(A) \neq 0\]
Solving for $P(A \cap B)$ and substituting into the above expression for $P(A B)$

Bayes’ Filter

What is Kalman filter?

The Kalman Filter assumes a linear system with Gaussian noise.

State Transition Model

\[x_k = Fx_{k-1} + w_k\]
  • $x_k$: State at time $k$
  • $F$: State transition matrix

  • $w_k$: Process noise, Gaussian with mean $0$ and convariance $Q(w_k ~ N(0, Q))$, representing model uncertainty.