Bayes' Rule and Belief Updates

A robot rarely knows anything for sure. Bayes' rule tells it how to update a belief when a new sensor reading arrives: weigh what it believed before by how well each possibility explains the reading.

Starting belief the door is open Sensor says open, when it is open Sensor says open, when it is closed The door is really open

The rule

$$P(\text{open} \mid z) = \frac{P(z \mid \text{open})\,P(\text{open})}{P(z \mid \text{open})\,P(\text{open}) + P(z \mid \text{closed})\,P(\text{closed})}$$

The top is the area of the highlighted open part of the square. The bottom is the total highlighted area. The new belief is the share of highlighted area that belongs to "open".

What to notice

One reading from a poor sensor moves the belief only a little. Several readings that agree move it a lot, even if each one is unreliable. That is how robots build confidence from noisy sensors over time. If the sensor's false alarm rate equals its hit rate, readings carry no information and the belief never moves.

This classic door example comes from Thrun, Burgard, and Fox's Probabilistic Robotics. The same rule, applied to every possible robot position at once, gives the Bayes filter, which is the basis of the Kalman filter and the particle filter.