Notation

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

State Evolution

$$p(x_t | x_{0:t-1} z_{1:t-1}, u_{1:t})$$

State Transition Probability

$$p(x_t | x_{0:t-1} z_{1:t-1}, u_{1:t}) = p ( x_t | x_{t-1}, u_t)$$

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

Measurement Probability

$$p(z_t | x_{0:t}, z_{1:t-1}, u_{1:t}) = p(z_t | x_t)$$

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