Probability
This page is a reference manual. The formal language of world models is probability: belief states, observation noise, stochastic dynamics, and uncertainty quantification all rest on the minimal set below.
Minimal knowledge setâ
- Conditional distributions and Bayes: â the entire content of the observation update.
- Gaussian distributions and covariance: the Kalman filter, the stochastic state of RSSM, and 3D Gaussian Splatting are all Gaussian; be able to read the geometric meaning of mean/covariance.
- Markov property: â the mathematical statement of "the state is a sufficient statistic for the future."
- Latent variables and marginalization: â VAEs, belief states, and particle filters all share this structure.
- Expectations and Monte Carlo estimation: CEM/MPPI planners and gradient estimation in policy gradients both approximate expectations by sampling.
When to consultâ
| Main-track module | Probability used |
|---|---|
| Track A 02 Observation, State and POMDP | Bayes, belief states, marginalization |
| Track A 05/06 Latent Dynamics / RSSM | Latent-variable models, KL divergence, ELBO |
| Track A 09 Planning | Sampling-based estimation, expected return |
| Track A 12 / Track B 06 | Uncertainty (aleatoric vs epistemic), calibration |
Best external resourcesâ
- Probabilistic Machine Learning: Basics (Kevin Murphy, probml.github.io/pml-book), Chapters 2â4: free PDF, ML perspective.
- Seeing Theory (seeing-theory.brown.edu): interactive visualizations â the fastest way to rebuild intuition.
- CIS6280 L02 History, Foundations, Probabilistic Formulation (PDF): a demonstration of probabilistic formulation in the world-model context.
Self-checkâ
You are ready when you can derive a conditional distribution from a joint, explain the orientation and axes of a covariance-matrix ellipse, and articulate the relationship between the Markov property and "compressing history."
Nextâ
Back to the main tracks: Track A Module 02 or Track B Module 06