Gold Standard · Judgement specialist guide
Find the nearest question, then follow the evidence
Undirected graphical models are useful when neighbouring variables constrain one another without a natural causal direction. Their clarity depends on a valid factorisation, understood potentials and inference whose convergence and limits are visible.
Some uncertainty is easier to express as mutual compatibility than as a directed chain. Neighbouring pixels tend to agree, adjacent sites in a spatial field interact, and linked labels in a structured prediction problem constrain one another. A Markov random field represents this pattern with an undirected graph.
The graph is only the beginning. Clique potentials need not be probabilities, the partition function matters, zeros complicate factorisation, and loopy message passing may not converge. This guide focuses on model meaning, factor graphs, exact and approximate inference, and sensitivity. It remains separate from the directed diagnostic work of Bayesian networks and from decision or policy models.
Choose a reading route
Read the model
Undirected graphs, Markov properties and factors.ROUTE 2 · CHAPTERS 4–5
Build potentials
Energy, normalisation and factor graphs.ROUTE 3 · CHAPTERS 6–7
Work the field
Infer a fictional noisy grid.ROUTE 4 · CHAPTERS 8–10
Compute carefully
Trees, loops, learning and sensitivity.ROUTE 5 · CHAPTERS 11–12
Transfer and audit
Changed context, limits and reproducibility.
Open the full chapter index · Use the field check · Return to the Judgement root
Chapter contents
Chapters 1–3 · Undirected graph and probability grammar
Chapters 4–6 · Potentials, factors and a worked field
Chapters 7–9 · Inference, learning and robustness
Chapters 10–12 · Transfer, governance and audit
CHAPTER 1 OF 12 · Read the model
1. Choose an undirected model for symmetric dependence
A Markov random field represents a joint distribution using an undirected graph. An absent edge encodes conditional-independence structure under stated conditions, while an edge records direct dependence in the graph rather than a causal arrow. This is useful when no defensible direction exists or when compatibility is naturally mutual.
Write the variables, population and query before choosing the graph. Ask whether direction is essential, whether the system is static or sequential and whether an undirected local-neighbour model fits the mechanism. State explicitly that edges are probabilistic dependencies unless a separate causal argument exists.
A fictional image-cleaning model links adjacent pixel labels because neighbouring regions tend to agree. The edge does not say the left pixel causes the right pixel. It says the joint model rewards compatible local assignments.
Changed-context transfer check
Transfer the distinction to a classroom seating map where neighbouring distraction reports are associated. Explain why the model should not identify one student as the cause.
Limit and misconception
An undirected graph can approximate dependence while missing long-range or directional processes. Symmetry is a modelling choice, not a fact guaranteed by appearance.
Evidence practice
Keep a purpose statement, variable dictionary, query and justification for undirected structure. Label every field as observed, computed, assumed or chosen. Give the record to a second reader and ask them to reproduce the result without the original author explaining what was meant. Differences reveal hidden definitions, missing versions or decision rules that still need to be made explicit.
Deliberately test for this failure: using an undirected edge as a causal claim or choosing the model only because direction is inconvenient. Record the earliest point at which the process can detect it, the correction applied and whether later conclusions change. A method is transferable only when its controls work on an altered case, not merely on the example used to teach it.
CHAPTER 2 OF 12 · Read the model
2. Understand local, pairwise and global Markov properties
Markov properties connect graph separation to conditional independence. Locally, a node is independent of non-neighbours given its neighbours. Pairwise and global forms express related statements, with equivalence requiring conditions such as positivity. The conditioning set is part of the claim.
For each important query, identify separating sets and state the implied independence. Check the distributional conditions under which graph factorisation is being used. Test small counterexamples and zero-probability states rather than assuming every textbook equivalence carries over.
On a chain A—B—C, the graph asserts that A and C are conditionally independent given B under the model. Marginally they can remain associated. If some joint configurations are impossible, extra care is needed before invoking factorisation results.
Changed-context transfer check
Change the chain into a cycle and identify which conditioning sets separate two non-adjacent nodes.
Limit and misconception
Conditional independence is a model assertion that can fail in data or under selection. A missing edge is not proof of no relationship in reality.
Evidence practice
Keep the separation analysis, positivity conditions, tests and observed deviations. Label every field as observed, computed, assumed or chosen. Give the record to a second reader and ask them to reproduce the result without the original author explaining what was meant. Differences reveal hidden definitions, missing versions or decision rules that still need to be made explicit.
Deliberately test for this failure: saying two nodes are independent without naming the conditioning set. Record the earliest point at which the process can detect it, the correction applied and whether later conclusions change. A method is transferable only when its controls work on an altered case, not merely on the example used to teach it.
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CHAPTER 3 OF 12 · Read the model
3. Factorise over cliques without over-reading the graph
For a strictly positive distribution, the Hammersley–Clifford result connects the graph’s Markov properties to factorisation over cliques. The joint distribution is proportional to a product of non-negative potential functions. Potentials encode compatibility, not standalone marginal probabilities.
List maximal cliques or chosen factors and write the unnormalised product. Check factor scopes and double counting. State positivity and support constraints. Separate graph structure from the numerical potential parameterisation so reviewers can challenge each independently.
A four-node grid uses pairwise factors favouring equal neighbouring labels and unary factors reflecting local observations. Multiplying one edge factor by two changes the unnormalised weight, but only the normalised joint distribution has probabilistic meaning.
| Element | Meaning | Common mistake |
|---|---|---|
| Node | Random variable and state space | Vague or changing state definition |
| Edge | Direct undirected dependence in the model | Causal influence |
| Factor | Local compatibility function | Marginal probability |
| Partition function | Global normalising constant | Optional scaling detail |
Changed-context transfer check
Write a factorisation for a triangle and compare maximal-clique and pairwise formulations. Explain when a three-way interaction cannot be represented by pairwise factors alone.
Limit and misconception
The theorem’s conditions matter. Hard zeros and context-specific constraints may require a more careful support treatment.
Evidence practice
Keep the graph, clique list, factor scopes, support assumptions and factorisation. Label every field as observed, computed, assumed or chosen. Give the record to a second reader and ask them to reproduce the result without the original author explaining what was meant. Differences reveal hidden definitions, missing versions or decision rules that still need to be made explicit.
Deliberately test for this failure: treating each potential as a probability or assuming pairwise edges capture every higher-order interaction. Record the earliest point at which the process can detect it, the correction applied and whether later conclusions change. A method is transferable only when its controls work on an altered case, not merely on the example used to teach it.
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CHAPTER 4 OF 12 · Build potentials
4. Use energy and potentials with interpretable scaling
Many Markov random fields are written as a Gibbs distribution in which lower energy means higher unnormalised probability. Unary terms express local evidence and pairwise or higher-order terms express compatibility. Parameter scale matters because competing terms trade off through the total energy.
Define each feature and parameter with units or interpretation. Compute energy for representative configurations and verify that preferred assignments receive lower energy. Check invariances: adding a constant to all energies changes the partition function but not probabilities, while rescaling can sharply change concentration.
A noisy-grid model balances observed labels against neighbour agreement. An excessive smoothness weight erases a real small region; an excessive observation weight preserves isolated noise. The useful range is found through validation and sensitivity, not visual preference alone.
Changed-context transfer check
Transfer the trade-off to a schedule model balancing local preferences and pairwise compatibility. Identify configurations that expose a badly scaled term.
Limit and misconception
Readable feature names do not make parameters causal or ethically neutral. A compatibility term can encode an undesirable assumption.
Evidence practice
Keep feature definitions, parameter provenance, representative energies and scaling tests. Label every field as observed, computed, assumed or chosen. Give the record to a second reader and ask them to reproduce the result without the original author explaining what was meant. Differences reveal hidden definitions, missing versions or decision rules that still need to be made explicit.
Deliberately test for this failure: tuning parameters until one example looks good without checking other structures or groups. Record the earliest point at which the process can detect it, the correction applied and whether later conclusions change. A method is transferable only when its controls work on an altered case, not merely on the example used to teach it.
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A factor graph is a bipartite graph connecting variable nodes to factor nodes. It makes the factorisation explicit and provides a common view for algorithms across undirected and directed models. It should not be confused with the original Markov graph: factor nodes carry functions and reveal higher-order scope.
Create one variable node per modeled variable and one factor node per local function. Connect a factor only to variables in its scope. Check that the product of all factors equals the intended unnormalised global function and that no factor has been omitted or duplicated.
A three-variable clique can be shown as one factor linked to all three variables. Replacing it with three pairwise factors changes the model unless the higher-order function actually decomposes.
Changed-context transfer check
Convert a small Bayesian-network factorisation into a factor graph and explain why the factor graph alone does not preserve causal direction.
Limit and misconception
Factor graphs unify computation, not semantics. The same graph shape can represent probabilities, codes or other functions.
Evidence practice
Keep the variable list, factor list, scope matrix and equality test for the global function. Label every field as observed, computed, assumed or chosen. Give the record to a second reader and ask them to reproduce the result without the original author explaining what was meant. Differences reveal hidden definitions, missing versions or decision rules that still need to be made explicit.
Deliberately test for this failure: drawing factors as unlabeled connectors or assuming a factor graph proves an undirected causal mechanism. Record the earliest point at which the process can detect it, the correction applied and whether later conclusions change. A method is transferable only when its controls work on an altered case, not merely on the example used to teach it.
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CHAPTER 6 OF 12 · Work the field
6. Worked example: infer labels on a fictional noisy grid
A fictional four-by-four sensor grid measures whether each tile is clear or obstructed. Readings are noisy, and neighbouring tiles usually share a state, but sharp boundaries are possible. Unary factors encode sensor reliability; pairwise factors encourage, but do not force, agreement.
The analyst defines states and likelihood-derived unary scores, selects a pairwise compatibility range from held-out grids and enumerates the tiny model for a benchmark. One isolated obstructed reading is smoothed away under the baseline model. Adding corroborating local evidence restores a small obstruction cluster.
The report shows posterior marginals and the maximum-probability configuration separately. It highlights boundary uncertainty rather than colouring every tile as settled. A physical inspection checks the ambiguous region.
| Component | Fictional role | Check |
|---|---|---|
| Unary factor | Sensor support for each tile state | Calibration and missing readings |
| Pairwise factor | Neighbour compatibility | Boundary preservation |
| Marginal | Uncertainty for one tile | Normalisation and benchmark |
| MAP state | Best whole-grid assignment | Do not confuse with certainty |
Changed-context transfer check
Change the case so diagonal neighbours also interact. Identify the added factors, computational cost and risk of oversmoothing.
Limit and misconception
The grid, sensor values and parameters are fictional. Real safety decisions require validated sensing and competent inspection beyond model output.
Evidence practice
Keep the grid data, unary evidence, pairwise settings, exact benchmark, marginals and inspection result. Label every field as observed, computed, assumed or chosen. Give the record to a second reader and ask them to reproduce the result without the original author explaining what was meant. Differences reveal hidden definitions, missing versions or decision rules that still need to be made explicit.
Deliberately test for this failure: publishing only a crisp best configuration while hiding uncertain marginals and parameter sensitivity. Record the earliest point at which the process can detect it, the correction applied and whether later conclusions change. A method is transferable only when its controls work on an altered case, not merely on the example used to teach it.
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CHAPTER 7 OF 12 · Work the field
7. Distinguish marginal inference, MAP inference and the partition function
Marginal inference asks for probabilities of variables or subsets. Maximum a posteriori inference seeks the highest-probability joint assignment. Computing likelihood or comparing models may require the partition function. These tasks can have different algorithms and complexity.
Write the target before running an algorithm. Validate marginals sum to one, compare MAP energy with small enumerations and state whether the partition function is exact, approximated or cancelled in a ratio. Never substitute the easiest computation for the question.
In the grid, the MAP assignment labels a boundary tile clear, while its marginal is 0.53 clear and 0.47 obstructed. Reporting only MAP hides near indifference that matters for inspection.
Changed-context transfer check
Transfer the distinction to a document-labeling problem where several joint assignments have similar probability.
Limit and misconception
A high-probability joint configuration can still have low absolute probability when many alternatives exist.
Evidence practice
Keep the declared target, algorithm, normalisation test and exact small-case comparison. Label every field as observed, computed, assumed or chosen. Give the record to a second reader and ask them to reproduce the result without the original author explaining what was meant. Differences reveal hidden definitions, missing versions or decision rules that still need to be made explicit.
Deliberately test for this failure: calling a MAP label a 100% probability or using unnormalised scores as calibrated marginals. Record the earliest point at which the process can detect it, the correction applied and whether later conclusions change. A method is transferable only when its controls work on an altered case, not merely on the example used to teach it.
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On a tree-structured factor graph, sum-product messages can compute exact marginals by passing local summaries from leaves inward and back outward. Max-product variants address MAP-like tasks. Correctness depends on factor definitions, message scheduling and normalisation.
Choose a root, initialise leaf messages, multiply incoming messages with the local factor and sum out non-target variables. Run inward and outward passes, then form beliefs and normalise. Compare with enumeration on a small tree and test zero messages and extreme scales.
A chain of five binary variables yields the same marginals by enumeration and sum-product, while requiring smaller intermediate objects. Log-domain computation prevents underflow when potentials are tiny.
Changed-context transfer check
Add one edge to create a loop. Explain why the same schedule is no longer guaranteed exact.
Limit and misconception
Tree exactness applies to the specified model. It does not validate graph structure, parameters or calibration.
Evidence practice
Keep messages, schedule, normalisers, log-domain controls and enumeration benchmark. Label every field as observed, computed, assumed or chosen. Give the record to a second reader and ask them to reproduce the result without the original author explaining what was meant. Differences reveal hidden definitions, missing versions or decision rules that still need to be made explicit.
Deliberately test for this failure: reusing tree guarantees after the graph acquires a cycle. Record the earliest point at which the process can detect it, the correction applied and whether later conclusions change. A method is transferable only when its controls work on an altered case, not merely on the example used to teach it.
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CHAPTER 9 OF 12 · Compute carefully
9. Treat loopy belief propagation as an approximation
Applying message passing to graphs with cycles can work well, oscillate or converge to inaccurate beliefs. Convergence is neither guaranteed nor sufficient for correctness. Damping, scheduling and initialisation can change results.
Record update order, damping, tolerance, maximum iterations and initial messages. Run several initialisations, inspect residuals and compare with exact inference on reduced subgraphs or alternative approximations. Report convergence status and between-run variation.
On the fictional grid, synchronous updates oscillate at strong neighbour coupling. Damping produces stable messages, but marginals differ from a small exact benchmark near the boundary. The report widens uncertainty.
Changed-context transfer check
Change the graph to a densely connected clique. Predict why message size or convergence may deteriorate.
Limit and misconception
A stable fixed point does not prove accurate posterior probability. Approximation error is separate from model and data uncertainty.
Evidence practice
Keep algorithm settings, residual history, repeated runs and exact or alternative benchmarks. Label every field as observed, computed, assumed or chosen. Give the record to a second reader and ask them to reproduce the result without the original author explaining what was meant. Differences reveal hidden definitions, missing versions or decision rules that still need to be made explicit.
Deliberately test for this failure: publishing converged-looking beliefs without testing accuracy or dependence on initialisation. Record the earliest point at which the process can detect it, the correction applied and whether later conclusions change. A method is transferable only when its controls work on an altered case, not merely on the example used to teach it.
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CHAPTER 10 OF 12 · Transfer and audit
10. Test sensitivity to graph, potentials and support
Robust judgement requires checking whether the conclusion survives plausible changes in edge structure, parameter scale, observation reliability and allowed configurations. Undirected models can change sharply near strong coupling or when a hard constraint removes support.
Vary influential parameters over evidence-based ranges, remove disputed edges, add plausible higher-order factors and relax hard zeros. Track changes in marginals, MAP configurations, ranking and uncertainty. Identify reversal boundaries and link them to data or review priorities.
The obstruction cluster remains under reasonable sensor calibration but disappears when smoothness rises above a narrow boundary. The team prioritises empirical calibration of neighbour coupling before deployment.
Changed-context transfer check
Transfer the test to a text model whose label changes when one learned edge is removed. Explain what further evidence is needed.
Limit and misconception
Sensitivity cannot rescue a target that is inappropriate or data collected under unfair conditions.
Evidence practice
Keep parameter ranges, structural alternatives, support variants and conclusion changes. Label every field as observed, computed, assumed or chosen. Give the record to a second reader and ask them to reproduce the result without the original author explaining what was meant. Differences reveal hidden definitions, missing versions or decision rules that still need to be made explicit.
Deliberately test for this failure: testing only tiny perturbations chosen to preserve the preferred answer. Record the earliest point at which the process can detect it, the correction applied and whether later conclusions change. A method is transferable only when its controls work on an altered case, not merely on the example used to teach it.
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CHAPTER 11 OF 12 · Transfer and audit
11. Transfer the model without automating sensitive judgement
Undirected models can organise spatial, linguistic and structured data, but transfer requires new variables, features, data and governance. In people-centred settings, neighbour effects can encode social homophily or institutional bias and spread error across connected individuals.
Rebuild the variable and factor dictionary, validate data and subgroup behaviour and review whether relational features are necessary. Use human review for material outcomes and provide correction routes. Avoid models that infer sensitive traits or impose guilt, risk or merit by association.
A school uses a small factor graph to reconcile conflicting timetable constraints, not to infer student character from peers. Staff can inspect each factor and override an infeasible or inequitable result.
Changed-context transfer check
Move the timetable model to another campus with different accessibility needs. Identify factors that must change and governance evidence required.
Limit and misconception
Mathematical symmetry does not create social fairness. Relational data can make harms propagate.
Evidence practice
Keep transfer rationale, data validation, factor review, subgroup tests and human-override record. Label every field as observed, computed, assumed or chosen. Give the record to a second reader and ask them to reproduce the result without the original author explaining what was meant. Differences reveal hidden definitions, missing versions or decision rules that still need to be made explicit.
Deliberately test for this failure: reusing learned neighbour effects across populations or turning association into a high-stakes label. Record the earliest point at which the process can detect it, the correction applied and whether later conclusions change. A method is transferable only when its controls work on an altered case, not merely on the example used to teach it.
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CHAPTER 12 OF 12 · Transfer and audit
12. Audit graph, factors, inference and claims together
A reproducible result requires the graph version, state spaces, factors, parameters, observations, inference code, numerical settings and claim wording. Reviewers must know whether a number is exact, approximate or learned and which uncertainty it excludes.
Version model and code, preserve input snapshots and run benchmark cases after every change. Test factor scopes, normalisation, impossible states, disconnected components and numerical underflow. Require a claim register distinguishing compatibility, association, prediction and causation.
After a software update changes message scheduling, the fictional grid produces slightly different loopy beliefs. Regression tests detect the shift and prevent silent replacement of earlier results.
Changed-context transfer check
Design a handover that lets a second analyst reproduce one marginal and one MAP result six months later.
Limit and misconception
Auditability cannot make an ill-chosen graph or unethical application appropriate. It enables challenge, correction and retirement.
Evidence practice
Keep the model package, benchmark suite, numerical diagnostics, claim register and change log. Label every field as observed, computed, assumed or chosen. Give the record to a second reader and ask them to reproduce the result without the original author explaining what was meant. Differences reveal hidden definitions, missing versions or decision rules that still need to be made explicit.
Deliberately test for this failure: saving only final labels while factor values, settings or graph versions disappear. Record the earliest point at which the process can detect it, the correction applied and whether later conclusions change. A method is transferable only when its controls work on an altered case, not merely on the example used to teach it.
Contents · Previous chapter · Connected articles
Advanced boundary cases
Higher-order factors
Some compatibility applies to groups rather than pairs. Represent the higher-order factor directly or justify a transformation; a pairwise approximation can change the model and inference.
Conditional random fields
Conditional random fields model structured outputs given observed inputs and avoid specifying a distribution over inputs. Their discriminative purpose, feature design and training objective differ from a generative MRF.
Continuous and mixed variables
Gaussian Markov random fields exploit sparse precision structure, while mixed models require suitable factors and normalisation. Preserve units and distributional conditions instead of discretising automatically.
A practical field check
Use this check before trusting a marginal, MAP assignment or learned edge from an undirected graphical model.
- State the query and justify undirected dependence.
- Define variables, states, graph and support.
- Write clique or factor scopes and the unnormalised product.
- Check potentials, energy scaling and partition-function needs.
- Benchmark exact inference on a small case.
- Diagnose loops, convergence and approximation error.
- Test parameter, structure and support sensitivity.
- Archive model, code, settings and bounded claim.
Run the check as a small workbook
Use sheets for variables, graph edges, factors, parameters, observations, inference runs, benchmarks and sensitivity. Keep each model version immutable so a reviewer can reproduce exact and approximate results separately.
| Gate | Pass condition | Stop condition |
|---|---|---|
| Meaning | Edges and factors have explicit semantics | Compatibility presented as causation |
| Probability | Support and normalisation are understood | Unnormalised score called probability |
| Inference | Exactness or approximation is tested | Convergence treated as correctness |
| Transfer | Relational features are justified and governed | Sensitive labels spread by association |
Chapter-by-chapter evidence audit
| Chapter | Evidence to keep | Failure to catch |
|---|---|---|
| 1. Choose an undirected model for symmetric dependence | A purpose statement, variable dictionary, query and justification for undirected structure | Using an undirected edge as a causal claim or choosing the model only because direction is inconvenient |
| 2. Understand local, pairwise and global Markov properties | The separation analysis, positivity conditions, tests and observed deviations | Saying two nodes are independent without naming the conditioning set |
| 3. Factorise over cliques without over-reading the graph | The graph, clique list, factor scopes, support assumptions and factorisation | Treating each potential as a probability or assuming pairwise edges capture every higher-order interaction |
| 4. Use energy and potentials with interpretable scaling | Feature definitions, parameter provenance, representative energies and scaling tests | Tuning parameters until one example looks good without checking other structures or groups |
| 5. Translate the model into a factor graph | The variable list, factor list, scope matrix and equality test for the global function | Drawing factors as unlabeled connectors or assuming a factor graph proves an undirected causal mechanism |
| 6. Worked example: infer labels on a fictional noisy grid | The grid data, unary evidence, pairwise settings, exact benchmark, marginals and inspection result | Publishing only a crisp best configuration while hiding uncertain marginals and parameter sensitivity |
| 7. Distinguish marginal inference, MAP inference and the partition function | The declared target, algorithm, normalisation test and exact small-case comparison | Calling a MAP label a 100% probability or using unnormalised scores as calibrated marginals |
| 8. Use exact message passing on trees | Messages, schedule, normalisers, log-domain controls and enumeration benchmark | Reusing tree guarantees after the graph acquires a cycle |
| 9. Treat loopy belief propagation as an approximation | Algorithm settings, residual history, repeated runs and exact or alternative benchmarks | Publishing converged-looking beliefs without testing accuracy or dependence on initialisation |
| 10. Test sensitivity to graph, potentials and support | Parameter ranges, structural alternatives, support variants and conclusion changes | Testing only tiny perturbations chosen to preserve the preferred answer |
| 11. Transfer the model without automating sensitive judgement | Transfer rationale, data validation, factor review, subgroup tests and human-override record | Reusing learned neighbour effects across populations or turning association into a high-stakes label |
| 12. Audit graph, factors, inference and claims together | The model package, benchmark suite, numerical diagnostics, claim register and change log | Saving only final labels while factor values, settings or graph versions disappear |
Connected articles across the eduKate ecosystem
Start with the governing route
- The Gold Standard of Judgement
- The Gold Standard of Education | Learning That Transfers into Life
- How X Works hub
Continue into verified specialist studies
- Bayesian Networks, Conditional Independence, Diagnostic Inference and Sensitivity — compare directed factorisation and diagnostic updating.
- Influence Diagrams, Information Arcs, Decision Nodes and Backward Induction — add choices and value when the task becomes a decision model.
- Partially Observable Markov Decision Processes, Belief States, Sensing and Policy Value — move to sequential hidden-state decisions and policies.
- Robust Markov Decision Processes, Ambiguity Sets, Policies and Worst-Case Value — study policy robustness rather than static compatibility.
- Value of Information: Is Another Test Worth the Delay? — decide whether another observation is worth collecting.
Questions readers often ask
Does an undirected edge mean mutual causation?
No. It encodes direct dependence in the model, not a causal mechanism.
Are potential functions probabilities?
Not generally. Their product gives an unnormalised score; the partition function normalises the joint distribution.
When is belief propagation exact?
For tree-structured factor graphs under the specified model. Loopy graphs usually require approximation diagnostics.
Does convergence mean loopy beliefs are correct?
No. A fixed point can still differ from exact marginals, so benchmark and sensitivity evidence are needed.
Can a factor graph represent a Bayesian network?
Yes, it can display that factorisation for computation, but it does not preserve causal direction by itself.
Can learned edges be interpreted as causes?
No. They reflect conditional-dependence structure under data and assumptions, not causal proof.
Sources and scope
Wainwright and Jordan: Graphical Models, Exponential Families, and Variational Inference — comprehensive primary technical treatment of Markov random fields, factorisation and approximate inference.
Kschischang, Frey and Loeliger: Factor Graphs and the Sum-Product Algorithm — foundational paper on factor graphs and generic message passing.
IEEE DOI record for Factor Graphs and the Sum-Product Algorithm — stable bibliographic record for the foundational paper.
Ravikumar, Wainwright and Lafferty: High-Dimensional Ising Model Selection — primary research on structure learning and its conditions.
All grids, sensors, schedules and model outputs in the worked examples are fictional. This guide is educational and does not replace qualified statistical review, safety inspection or responsible governance for consequential applications.
“Gold standard” describes the educational benchmark explained in this guide. It is not an official accreditation, legal opinion, policy determination or professional certification.
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