Bayesian Networks:

Directed Acyclic Graph (DAG) representation of joint distribution $P(X_1..X_n)$.

Structure: Nodes = random variables, directed edges = direct dependencies (parent → child conditional). Each node has Conditional Probability Table (CPT): $P(X_i|Parents(X_i))$ for discrete, or density for continuous.

Joint factorization: $P(X_1..X_n)=\prod_i P(X_i|Pa(X_i))$ (chain rule with d-separation conditional independencies). Example: $Rain → Sprinkler → Wet Grass$, $Rain → Wet Grass$.

Inference: Compute $P(query|evidence)$ via enumeration, Variable Elimination, Belief Propagation, Junction Tree (exact exponential), or approximate MCMC/Likelihood Weighting. Bayes rule propagates evidence.

Learning:

• Parameter learning: MLE/MAP estimate CPTs from data ($\hat P(x_i|pa)=count(x_i,pa)/count(pa)$).
• Structure learning: Search over DAGs scoring via BIC/MDL, NP-hard.

Advantages: Compact, interpretable causal model, handles missing data, combines prior knowledge, supports both causal and diagnostic reasoning (e.g., $P(Disease|Symptom)$).

Uses: Medical diagnosis (ALARM network), fault detection, speech recognition (HMM is dynamic BN), decision support (Influence Diagrams).

Example: $P(Burglary)=0.001$, $P(Earthquake)=0.002$, $P(Alarm|B,E)$ CPT, $P(JohnCalls|Alarm)$ — can query $P(B|A)$ via Bayes net inference.

Limitations: DAG cannot have cycles, eliciting CPTs expensive, inference NP-hard for dense nets. For undirected cyclic dependencies use Markov Random Fields.