Bayesian Logic

Origin. Multiple traditions: Bayesian networks (Pearl, 1988), Bayesian logic programs (Kersting, 2000), BLOG (Milch, 2005). Combines first-order logic with Bayesian probability. Uncertain relational domains. Foundation for probabilistic programming and statistical relational learning.

Models. Probability over relational structures. First-order logic: relations, quantifiers, certain truth. Bayesian: probability distributions, conditioning. Combined: distributions over possible worlds with relational structure. Query: P(φ | evidence).

Formalism.

Bayesian networks: DAG with nodes = random variables. Edges = direct dependencies. P(X₁,...,Xₙ) = ∏ᵢ P(Xᵢ | Parents(Xᵢ))

Bayesian Logic Programs (BLPs): Combine logic programs with Bayesian networks. Clauses with conditional probabilities. Ground to Bayesian network for inference.

BLOG (Bayesian Logic): First-order probabilistic language. Unknown number of objects (open-universe). type Person; origin(Person) ~ Poisson(10); random Boolean Friends(Person, Person) ~ Bernoulli(0.1);

Probabilistic relational models: Classes, attributes, relations with CPDs. Aggregation: how to combine evidence from multiple related objects.

Inference:

Exact: variable elimination, junction trees
Approximate: MCMC, importance sampling, variational
Lifted: exploit symmetry, avoid grounding

Learning: Parameter learning: MLE, EM Structure learning: search over models

Symbols.

Symbol	Unicode	Name	Meaning
P(·\|·)	—	Conditional	Conditional probability
~	U+223C	Distributed	Drawn from
∏	U+220F	Product	Chain rule
⊥	U+22A5	Independent	Statistical independence
Parents	—	Parents	Graph parents
CPD	—	CPD	Conditional distribution

Metatheory. Inference is #P-hard in general. Lifted inference: polynomial for certain structures. Exchangeability: De Finetti's theorem. Consistency: as data grows, inference converges. Open vs closed world affects semantics.

Applies to. Knowledge graphs. Natural language understanding. Biological networks. Social network analysis. Robotics (SLAM). Citation analysis. Medical diagnosis.

Limitations. Scalability to large domains. Inference approximation quality. Structure learning difficulty. Mixing logic and probability subtle. Tool ecosystem fragmented. Specification requires expertise.