Probabilistic Graphical Models and Structured Inference
Structure in uncertainty
Why graphical models matter
Graphical models represent joint distributions using graphs that encode conditional independence. They make complex distributions more interpretable and often more tractable.
Model families
Bayesian networks
- Directed acyclic graphs
- Represent causal or generative structure
Markov random fields
- Undirected graphs
- Represent symmetric interactions
What does a graph encode in a probabilistic graphical model?
Edges capture probabilistic dependencies and independencies among variables.
Correct answer: Conditional dependence structure
What is conditional independence?
This property enables factorization and efficient inference.
Correct answer: Two variables are independent given a third variable or set of variables.
Inference
Exact inference includes variable elimination and junction tree methods. Approximate inference includes belief propagation, Markov chain Monte Carlo, and variational methods when exact computation is intractable.
Causal hint
Graph structure can be used not only for inference but also for causal reasoning, though causality requires assumptions beyond correlation alone.
Which method is an approximate inference technique?
Monte Carlo methods approximate expectations by random sampling.
Correct answer: Monte Carlo sampling
Inference workflow
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1
Specify variables and factorization.
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2
Choose exact or approximate inference based on graph size.
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3
Compute marginals, conditionals, or MAP estimates.
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4
Validate approximations against held-out or synthetic benchmarks.
Why do graphical models help with tractability?
Factorization lowers the complexity of inference relative to naive enumeration.
Correct answer: They exploit conditional independence to factorize joint distributions and reduce computation.