Editing Bayesian Machine Learning (section)

== <span style="color: #FFFFFF;">Understanding</span> ==
The Bayesian framework offers a coherent solution to a fundamental problem: how should we update our beliefs given evidence? The answer is Bayes' theorem: start with a prior, multiply by the likelihood of the observed data, and normalize to get the posterior.

**The frequentist vs. Bayesian divide**: Frequentist ML (standard deep learning) treats model parameters as fixed unknowns estimated from data. Bayesian ML treats parameters as random variables with distributions — capturing our uncertainty about the true values. A single point estimate (e.g., maximum likelihood) discards this uncertainty information.

**Why uncertainty matters**: A model that outputs "99% confidence: benign tumor" should actually be correct 99% of the time, not just have the highest logit. In medical AI, an overconfident prediction is dangerous. Bayesian methods provide principled calibration.

**Gaussian processes**: A GP is a distribution over functions. Given a kernel (covariance function) and training data, GPs provide exact posterior distributions over function values, including uncertainty bounds. GPs are the backbone of Bayesian optimization: fit a GP to evaluated function values, use the posterior to identify where to evaluate next (acquisition function).

**The computational challenge**: Computing exact posteriors requires integrating over all parameters, which is intractable for neural networks. Solutions: MCMC (exact but slow), variational inference (approximate but fast), Laplace approximation (quadratic approximation around MAP estimate), Monte Carlo Dropout (practical approximation).
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