Bayesian Networks (BN) vs Markov Random Fields (MRF):

Both are Probabilistic Graphical Models representing joint distribution $p(x_1..x_n)$ via graph factorization.

Bayesian Networks — Directed Acyclic Graph (DAG):

• Nodes = random variables, directed edges = direct dependencies (parent → child = causal).
• Factorization: $p(x)=\prod_{i} p(x_i | Pa(x_i))$ (chain rule with conditional independence via d-separation).
• Need Conditional Probability Tables (CPTs) for discrete nodes.
• Represents causal / asymmetric dependencies, natural for generative processes (e.g., disease → symptom).
• Inference: exact (variable elimination, junction tree) NP-hard; approx MCMC.
• Learning: DAG structure learning NP-hard, parameter learning via MLE.

Markov Random Fields — Undirected Graph:

• Nodes linked by undirected edges, factorization over cliques via potentials $\psi_C(x_C)$: $p(x)=\frac1Z \prod_{C} \psi_C(x_C)$ where $Z=\sum_x \prod_C \psi_C$ (partition function, global normalization).
• Represents symmetric / non-causal correlations, spatial interactions (e.g., pixels in image — neighboring pixels similar, Ising model).
• Conditional independence via graph separation.
• No CPTs, more flexible for cyclic dependencies.

Trade-offs:

Aspect	BN	MRF
Edge meaning	Causal, directed	Correlation, undirected
Normalization	Local (each CPT sums to 1)	Global $Z$ — expensive
Cycles	Cannot handle directed cycles	Handles cycles naturally
Interpretability	High (causal)	Lower
Learning complexity	Structure NP-hard but local norm easy	Potential learning hard, $Z$ intractable
Inference	Slightly cheaper if DAG sparse	Often costlier due to $Z$
Use cases	Diagnosis, causal reasoning, time series (DBN)	Image segmentation, stereo, spatial stats

Choosing: Use BN when direction/causality known, need generative sampling, DAG natural. Use MRF when dependencies symmetric, lattice-like, or cyclic correlations (images, social networks). Many distributions can be represented by either with transformations (moralization), but efficiency differs.