Gaussian: It is a bell curve characterized by mean and variance. It's is unimodal and symmetric. The area under the Gaussian adds up to 1.

Variance: measure of uncertainty. Large covariance = more spread = more uncertain.

Bayes Rule

Localization:

Involves "move" (motion) step and "sense" (measurement) step.

Motion (move) : First the robot moves. We use convolution to get the probability that robot moved to the current grid location. We use Bayes Rule (given previous location, find probability of being in this current grid location).

Measurement (sense) : Then robot senses the environment. We use products to get the probability that the sensor measurement is correct. Measurement applies theorem of total probability (sum of:  probability that sensor measurement is correct given it's a hit, prob that sensor is correct given it's a miss).

*Side note: for grid based localization method (histogram method), the memory increases exponentially with number of state variables (x,y,z, theta,row, pitch, yaw, etc)