Editing Probabilistic Machine Learning (section)

== <span style="color: #FFFFFF;">Understanding</span> ==
**Why probabilistic ML?** Point predictions discard crucial information. When a medical AI says "positive for cancer" with 51% confidence, that's categorically different from 99% confidence — but a non-probabilistic classifier treats both identically. Probabilistic models express this uncertainty explicitly.

**Sources of uncertainty**: (1) **Aleatoric** (irreducible): inherent randomness in the data-generating process. Even with infinite data, some outcomes are unpredictable — e.g., quantum effects, chaotic systems. (2) **Epistemic** (reducible): uncertainty due to limited knowledge. With more data, the model becomes more certain. Good probabilistic models distinguish these two types.

**Probabilistic graphical models** encode joint distributions over many variables efficiently using conditional independence assumptions. A Bayesian network for medical diagnosis might have nodes for symptoms, diseases, and test results, with edges encoding conditional dependencies. Inference algorithms (variable elimination, belief propagation) compute posterior probabilities of unobserved variables.

**Deep probabilistic models**: VAEs combine deep learning with variational inference. The encoder maps inputs to a distribution over latent codes (not a point); the decoder maps sampled latent codes back to reconstructions. This enables generation (sample from the latent space) and uncertainty quantification. Normalizing flows model complex distributions by composing simple invertible transformations with analytically tractable Jacobians.

**Conformal prediction** provides distribution-free prediction sets with guaranteed coverage: given user-specified error rate α, the prediction set contains the true label with probability ≥ 1-α, regardless of the underlying distribution. This is a practical tool for adding rigorous uncertainty quantification to any classifier.
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