A Markov random field has a clique of three variables {A, B, C} with a single joint potential ψ(A,B,C). How does this appear in a factor graph?
AAs a triangle of three variable nodes with edges between them — the same structure as the MRF clique
BAs one factor node connected by edges to three variable nodes (A, B, C)
CAs three separate factor nodes, one per variable, each connected to the others
DAs a single variable node labeled ABC representing the joint state
In a factor graph, every factor gets its own explicit square node. A single potential ψ(A,B,C) becomes one factor node connected to all three variable nodes — a star pattern, not a triangle. This explicitness is the key advantage: an MRF clique might contain one factor or many factors, and the graph topology alone doesn't tell you which. Factor graphs resolve this by making each factor a visible node.
Question 2 Multiple Choice
On a tree-structured factor graph, what does the sum-product algorithm guarantee?
AApproximate marginals that converge after enough iterations
BExact joint distribution over all variables in the graph
CExact marginal distributions for all variables, computed in a single forward-backward message-passing pass
DThe most probable assignment of all variables via dynamic programming
On trees (no cycles), the sum-product algorithm computes exact marginals in a single pass: messages flow from leaves inward, then back outward, and each variable's marginal is the product of all incoming messages. There is no approximation and no iteration needed. The max-product algorithm does the analogous computation for the most probable configuration. Loops are the source of the exactness problem — on loopy graphs, belief propagation is approximate.
Question 3 True / False
Loopy belief propagation on a factor graph with cycles usually fails to converge and can seldom produce useful results.
TTrue
FFalse
Answer: False
False. While loopy belief propagation is not guaranteed to converge or give exact marginals, in practice it often converges and produces excellent approximate results. It is the backbone of modern error-correcting codes (LDPC, turbo codes) and computer vision algorithms. The lack of a convergence guarantee is a theoretical limitation, not a practical one in many important applications.
Question 4 True / False
A factor graph can represent any distribution expressible as either a Bayesian network or a Markov random field.
TTrue
FFalse
Answer: True
True. Factor graphs are a universal representation for graphical models. Any Bayesian network (directed, with CPTs) and any Markov random field (undirected, with potential functions) can be converted to a factor graph by creating one factor node for each conditional probability table or potential function. This universality is what makes factor graphs the preferred representation for unifying inference algorithms across model types.
Question 5 Short Answer
What ambiguity in Markov random field representations do factor graphs resolve, and how do they resolve it?
Think about your answer, then reveal below.
Model answer: In a Markov random field, a clique in the undirected graph could correspond to a single factor over all variables in the clique, or to a product of several smaller factors — the graph topology alone cannot distinguish these cases. Factor graphs resolve this by giving each factor its own explicit node in a bipartite graph. If P(a,b,c) = f₁(a,b) × f₂(b,c), the factor graph has two factor nodes (f₁, f₂), not one — making the factorization completely unambiguous.
This disambiguation matters for inference efficiency. If a clique factor actually decomposes into smaller subfactors, the message-passing algorithm can exploit that decomposition to reduce computation. Treating it as a single large factor when smaller factors exist wastes computation. Factor graphs make the true factorization explicit, so inference algorithms can always operate at the finest granularity available.