Core building blocks · the mechanism everything reuses

Probability for Robotics: a robot that knows what it doesn't know.

A robot never knows its exact state: sensors are noisy, motion slips, the world is hidden. So it carries a belief, a probability distribution over where it might be, and updates that belief with Bayes' rule every time it moves and senses. That single idea, the recursive Bayes filter, is the engine under the Kalman filter, the EKF/UKF, particle filters, HMMs, and SLAM. This module builds it from scratch and climbs to the research frontier: the free-energy principle, the Bayesian brain, energy-based models, and learned differentiable filters, with real math, step-through proofs, visualizations, worked numbers, and code.

∑ real math 🔓 step-through proofs 🔬 visualizations 💻 implementations 📚 cited

Open the concept deck

How to read this

Foundations → filters → SLAM → the research frontier.

Guided lesson: one idea per step, plain words first, the picture beside it.
The math: each equation with a plain-English reading and a symbol legend.
Proof, step by step: unfold each derivation one move at a time.
Visualization & a worked numerical example: the algorithm, run with real numbers.
Implementation: compact, readable code, plus exercises and the citation.

The deck

Pick a concept.

Ordered as a curriculum, so start at the top if probability is new. Foundations → the filter zoo → advanced estimation & SLAM → research.

✦ Beautiful math

Two beliefs fuse by adding their certainty

\[ \frac{1}{\sigma^2} \;=\; \frac{1}{\sigma_1^2} + \frac{1}{\sigma_2^2} \]

Flip every variance into its reciprocal (its precision, or how loudly it claims to know) and merging two guesses becomes pure addition. The fused estimate is always sharper than either parent, and the louder voice automatically pulls harder. This one line is the seed that grows into the Kalman update and the information filter; the whole module is really this trick wearing different clothes.

✦ Beautiful math

The Kalman gain is just a trust ratio

\[ K \;=\; \bar{\Sigma}\,H^{\!\top}\!\left(H\bar{\Sigma}H^{\!\top} + R\right)^{-1} \]

A robot weighs two stories: what it predicted, and what it just measured. The gain K is the dial that asks one question, who is less uncertain right now, and blends accordingly. Trust a crisp sensor (small R) and K swings toward the measurement; trust a confident prediction and it swings back. No tuning, no magic: the optimal compromise falls straight out of the variances.

✦ Beautiful math

One quantity is filtering, the brain, and the ELBO

\[ F \;=\; D_{\mathrm{KL}}\!\big(q \,\|\, p(x\mid o)\big) - \ln p(o) \;=\; -\,\mathrm{ELBO} \]

Free energy is the gap between the belief you hold, q, and the truth you cannot see, plus your surprise at the data. Shrink it and three things happen at once: your filter becomes Bayes-optimal, a brain minimizes prediction error, and a deep model maximizes its ELBO. The same scalar quietly unifies robotics, neuroscience, and machine learning.

Try it · the conceptual heart

The Kalman update, by hand.

Every filter in this room (Kalman, EKF, UKF, the information filter, even predictive coding) reduces to one move: fuse two Gaussians by their precision. Drag the sliders and watch the prediction and the measurement combine into a sharper posterior.

1-D Kalman update: drag to fuse a prediction with a measurement

The robot predicts its position is 20 with variance σ̄² = 4 (blue). A sensor reads z with measurement noise R_z (clay). The fused posterior (teal) is the precision-weighted blend, always sharper than either input.

Measurement noise R_z2.0

Sensor reading z23.0

References

The canon.

The standard text is Thrun, Burgard & Fox, Probabilistic Robotics (MIT Press, 2005). Each concept lists its primary citation; read the sources.

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Where this gets used.

Reinforcement Learning & Control World Models Vision-Language-Action Deep Learning All missions