1.15 Joint, marginal and conditional distributions
You can reason about several random variables at once.
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Real problems involve several random variables at once, and joint, marginal and conditional distributions are how they stay straight. Conditional independence is the load-bearing assumption behind Naive Bayes and graphical models, and the multivariate normal is the workhorse case. It sits late in the probability run because it composes everything earlier. The classic confusion is conflating independence with conditional independence — two variables can be dependent yet independent given a third, and vice versa.
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Joint densities and marginalization
The joint distribution describes several quantities together, and summing or integrating one away recovers the distribution of the rest. Marginalisation is the operation used most in probabilistic modelling.
Conditional distributions and the chain rule
Conditioning on one variable and factoring a joint distribution into a product of conditionals. This factorisation is what makes large probabilistic models expressible at all.
Independence vs. conditional independence
Two variables can be dependent overall and independent once a third is known, or the reverse. This distinction is the foundation of graphical models and of thinking clearly about confounding.
Multivariate normal and its covariance structure
The multivariate normal is the workhorse of continuous multivariate modelling, and its covariance matrix encodes every dependency in it. Its geometry is worth picturing rather than memorising.
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