Probability & Statistics

Uncertainty, Bayes, Gaussians — the basis of estimation.

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Why it matters in robotics

Nearly every robotics estimation system — localization, SLAM, sensor fusion, tracking — reasons about noisy, uncertain data using probability. A robot combines what it expected to happen with what a sensor just measured to form an updated belief about the world, and the shape of that uncertainty (its mean and how it spreads and correlates across variables) has to stay easy to compute with as the robot moves and takes new readings, over and over, in real time. This probabilistic foundation is what makes reliable state estimation possible.

What to study

✓Bayes' rule: priors, likelihoods, posteriors, and the normalizing evidence term
✓Gaussian distributions: mean/covariance, marginalization, conditioning, and why products of Gaussians stay Gaussian
✓Random variables, expectation, variance, and conditional/joint distributions
✓The Bayes filter: recursive predict (motion) and update (measurement) state estimation

Where to study

Cheatsheet

Symbols used below

P(A)

probability of event

A

P(A \mid B)

probability of

A

given

B

\mu, \Sigma

mean vector, covariance matrix

x, y

jointly Gaussian random vectors

\Sigma_{xy}

cross-covariance block between

x

and

y

Bayes' rule

$P(x \mid z) = \dfrac{P(z \mid x)\, P(x)}{P(z)}$

posterior $\propto$ likelihood $\times$ prior. $P(z)$ just normalizes so the posterior integrates to 1 — it doesn't depend on $x$ .

Gaussians stay Gaussian

The product of two Gaussian densities is (up to a constant) another Gaussian — this is why a Gaussian prior times a Gaussian likelihood gives a clean, closed-form Gaussian posterior.

Marginals and conditionals of a joint Gaussian are also Gaussian — nothing exotic ever appears downstream.

Gaussian conditioning

For $\begin{bmatrix}x\\y\end{bmatrix} \sim \mathcal{N}\!\left(\begin{bmatrix}\mu_x\\\mu_y\end{bmatrix}, \begin{bmatrix}\Sigma_{xx} & \Sigma_{xy}\\ \Sigma_{yx} & \Sigma_{yy}\end{bmatrix}\right)$ , observing $y$ gives

$\mu_{x\mid y} = \mu_x + \Sigma_{xy}\Sigma_{yy}^{-1}(y - \mu_y)$

$\Sigma_{x\mid y} = \Sigma_{xx} - \Sigma_{xy}\Sigma_{yy}^{-1}\Sigma_{yx}$ — uncertainty can only shrink, never grow.

The Bayes filter cycle

Predict: push the belief through the motion model — uncertainty grows (convolve with process noise).

Update: multiply by the measurement likelihood and renormalize — uncertainty shrinks. Every recursive estimator (Kalman filter, particle filter, histogram filter) is this same two-step cycle.

Uncorrelated ≠ independent

Zero covariance only rules out linear relationship. Two variables can have $\mathrm{Cov}(X,Y) = 0$ while one is an exact nonlinear function of the other.

Independence is the stronger condition: $P(x,y) = P(x)P(y)$ for all values, not just zero linear correlation.

A 2-D Gaussian's covariance sets its ellipse: off-diagonal correlation tilts it; ρ=0 is axis-aligned; |ρ|→1 collapses it toward a line.

	Frequentist	Bayesian
Treats parameters as	Fixed, unknown constants	Random variables with a distribution
Uses prior knowledge?	No — only the observed data	Yes — explicit prior, updated by data
Output	Point estimate + confidence interval	Full posterior distribution
Robotics fit	Simple calibration, hypothesis tests	Recursive estimation (Bayes/Kalman/particle filters)

Every Bayes filter alternates a predict step (convolve the belief with the motion model) and an update step (multiply by the measurement likelihood, then normalize).

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