world state at time \(t\)
measurement data at time \(t\) (e.g. camera images)
control data (change of state in the environment) at time \(t\)

State Evolution

\begin{equation} p(x_t | x_{0:t-1} z_{1:t-1}, u_{1:t}) \end{equation}

State Transition Probability

\begin{equation} p(x_t | x_{0:t-1} z_{1:t-1}, u_{1:t}) = p ( x_t | x_{t-1}, u_t) \end{equation}

The world state at the previous time-step is a sufficient summary of all that happened in previous time-steps.

Measurement Probability

\begin{equation} p(z_t | x_{0:t}, z_{1:t-1}, u_{1:t}) = p(z_t | x_t) \end{equation}

The measurement at time-step \(t\) is often just a noisy projection of the world state at time-step \(t\).