state and belief
Every robot has two very different versions of "where it is and what's going on." The first is the state: the actual, real-world truth — the robot really is at this spot, facing this way, moving at this speed. The state exists whether or not the robot knows it. The catch is that the robot can never see the state directly; it only gets noisy, indirect hints through its sensors. So a robot can no more read its own true state than you can know the exact temperature of a room just by how warm your skin feels.
What the robot actually carries around instead is a belief: its best current guess about the state, together with how sure it is. The crucial move is that a belief is not a single number but a whole spread of possibilities with weights — "I'm probably right here, fairly likely a little to the left, and very unlikely way over there." In the language of probability this spread is a distribution. A confident robot has a narrow, peaked belief (almost all the weight piled on one spot); a confused robot has a wide, flat belief smeared across many places. Honest uncertainty is built right in.
Keeping these two apart is the foundation of the whole subject. The true state is the thing we wish we knew; the belief is the thing we can compute and keep updating as new sensor readings and motion commands arrive. Estimation is simply the craft of nudging the belief to hug the true state as closely as the noisy evidence allows — never reaching it perfectly, but getting usefully, measurably close.
A delivery robot's true state is "3.20 m down the hallway." Its belief, though, is a little bell-shaped hump centered at 3.05 m and about 30 cm wide — its honest summary of "probably around three meters, give or take." The peak is the best single guess; the width is how unsure it is.
The true state is one exact value; the belief is a weighted spread of guesses around it.
A subtle but vital point: the belief lives inside the robot's computer, while the state lives out in the world. When people loosely say a robot "knows where it is," they really mean its belief is currently narrow and well-centered on the true state.