---
kind: concept
slug: bayesian-network
title: Bayesian network
summary: >-
  A Bayesian belief network, either as an anomaly model with parameters averaged over their posterior and fitted on similar past cases (one clinical paper, best AUC-ROC 0.81 on 100 cases), or as a dependency graph over a dataset's attributes (one fairness survey of 15 datasets). No code for the first; no variance reported.
tags: [probabilistic-model, anomaly-detection, clinical-decision-support, fairness, dataset-analysis]
papers: ["2110.00530", "2605.04664"]
related: [anomaly-detection, bayesian-neural-network, nearest-neighbor-search, mahalanobis-distance, algorithmic-fairness, dataset-curation]
status: draft
updated_at: 2026-11-01
---

## Overview

A Bayesian belief network (BBN) is a probabilistic model with a structure S and
parameters θ. Two papers on this wiki use it, for different jobs.

In the first job the parameters are random: they carry a prior, and learning
from data updates it. A prediction averages over all parameter settings rather
than using one fitted set [arxiv:2605.04664 p. 2]. Learning is simplified by
two assumptions the paper states, parameter independence and conjugate priors,
so the posterior keeps the prior's form [arxiv:2605.04664 p. 2]. This averaging
is the step the anomaly-test variants below share.

The paper uses the model for a conditional anomaly test. For a case, the target
attribute is one decision A(x) and C(x) holds the other attributes. The statistic
is the predictive probability p(A(x) | C(x), M), and the decision is flagged when
it falls below a fixed threshold [arxiv:2605.04664 p. 2]. The model is fitted on
similar past cases only. Scoring against all cases is biased when the target
sits in a small subpopulation, because the routine decisions of that group look
unusual against the majority [arxiv:2605.04664 p. 2].

In the second job the network is learned over every attribute of a tabular
dataset, and its edges are read as dependencies. No predictive probability is
computed from it [arxiv:2110.00530 §2.2].

Design choices sit on top of these models: which past cases count as similar,
which model to fit, and, for the dependency graph, how to discretise and how
strongly to penalise edges. The anomaly paper crosses three selections (all
cases, Mahalanobis distance, weighted Mahalanobis distance) with two models (a
BBN and Naive Bayes) [arxiv:2605.04664 tab. 2]. A Bayesian neural network is a
different object, with weights as random variables: see
[Bayesian neural network](/concepts/bayesian-neural-network).

## Variants

### BBN on all past cases

The structure is learned greedily with a maximal marginal-likelihood score, on
2187 cases, and then held fixed. Parameters are re-learned for each set of cases
used [arxiv:2605.04664 p. 4]. With all cases, the table reports AUC-ROC 0.81 and
AUC-PR 0.56, the best AUC-ROC of any row [arxiv:2605.04664 tab. 2]. Card:
[Evidence-based anomaly detection in clinical domains](/papers/2605.04664).

### BBN on similar cases

Take the k = 40 past cases nearest the target, under a Mahalanobis or weighted
Mahalanobis distance over C(x), and learn the parameters from them alone
[arxiv:2605.04664 p. 3] [arxiv:2605.04664 p. 4]. Reported AUC-ROC is 0.79 with
Mahalanobis selection and 0.80 with weighted selection; AUC-PR is 0.52 and 0.56
[arxiv:2605.04664 tab. 2]. The paper's explanation for the gap to the all-cases
BBN is model complexity together with small samples: a smaller sample fits the
target better but estimates the parameters less accurately [arxiv:2605.04664
p. 5].

### Naive Bayes

A simpler model with Hospitalization as the class variable
[arxiv:2605.04664 p. 4]. It serves as the
baseline [arxiv:2605.04664 p. 5]. Its reported AUC-ROC and AUC-PR rise with each
selection step: all cases 0.74 and 0.42, Mahalanobis 0.77 and 0.49, weighted
0.80 and 0.60 [arxiv:2605.04664 tab. 2]. Weighted selection gives the highest
AUC-PR in the table.

### Dependency graph over dataset attributes

Learn a directed graph over all attributes of a dataset, with the class
attribute constrained to be a leaf. The graph then shows which features bear on
the class [arxiv:2110.00530 §2.2] [arxiv:2110.00530 eq. 2]. Numerical attributes
are discretised before learning, with bins set per dataset [arxiv:2110.00530
§2.2]. The paper reads a direct or indirect edge from a protected attribute to
the class as a sign that the data may be biased [arxiv:2110.00530 §2.2]. The
authors apply this to each of 15 tabular fairness datasets as part of a
dataset survey; the same card reports only logistic-regression scores, not
network outputs [arxiv:2110.00530 §4.1]. Card:
[A survey on datasets for fairness-aware machine learning](/papers/2110.00530).

## When to use which

| Situation | Use | Why |
|---|---|---|
| Enough past cases to fit on the whole database, and ranking by AUC-ROC matters | BBN on all cases | Highest AUC-ROC in the paper (0.81) [arxiv:2605.04664 tab. 2]; the paper ties the BBN's drop on small neighbourhoods to sample size [arxiv:2605.04664 p. 5] |
| Few similar cases per target (the paper uses k = 40) and AUC-PR matters | Naive Bayes with weighted Mahalanobis selection | AUC-PR 0.60, against 0.52 to 0.56 for BBN on similar cases [arxiv:2605.04664 tab. 2] |
| Choosing between BBN and Naive Bayes on similar cases | Test both, on your data | On weighted selection the two are close on AUC-ROC (0.80 each), and NB leads on AUC-PR [arxiv:2605.04664 tab. 2] |
| Setting an alert threshold or an expected precision | Nothing from this paper | The headline precision uses an enriched test set, and the corrected figure's derivation is not shown [arxiv:2605.04664 p. 5] |
| Checking a tabular dataset for links between a protected attribute and the class, before training | Dependency graph over attributes | The paper uses this kind of screen on 15 datasets [arxiv:2110.00530 §2.2] [arxiv:2110.00530 tab. 1]; read edges as dependencies, not as causal effects |
| Needing a bias verdict or a fairness score from the network | Nothing from this paper | The paper's fairness numbers come from logistic regression, not from the network [arxiv:2110.00530 §4.1] |

The anomaly evidence is one retrospective study: binary attributes, one target
decision, 2287 PORT patients, and an expert panel as gold standard
[arxiv:2605.04664 p. 3]. Transfer to other settings is untested. The results are
one leave-one-out run over 100 cases with 23 positives, with no variance reported
[arxiv:2605.04664 p. 4]. Differences such as 0.56 against 0.52 in AUC-PR should
not be read as significant.

The dependency-graph evidence is one survey. Its objective includes a penalty,
but the paper gives no value for it and names no search algorithm
[arxiv:2110.00530 §2.2]. Learned edges therefore depend on choices the paper
does not fix, so an edge is a prompt for inspection rather than a finding.

## Implementation sketch

Pseudocode for the predictive statistic only, written from the text. The paper
gives no pseudocode.

```pseudocode
# Predictive probability of a decision under a BBN  [arxiv:2605.04664 p. 2]
# Input:  past_cases: rows over (decision, C(x) attributes), e.g. the k
#                     nearest cases, or all cases  [p. 3] [p. 4]
#         structure: fixed BBN structure S, learned beforehand  [p. 4]
#         prior: prior over parameters; family not stated  [p. 2]
# Output: p(A(x) | C(x)), averaged over parameter settings

def predictive_probability(past_cases, target_other, structure, prior):
    # Parameter independence and conjugate priors reduce learning to
    # updating sufficient statistics  [p. 2]
    posterior = update_sufficient_stats(prior, structure, past_cases)
    # Average over θ rather than plugging in one estimate  [p. 2]
    return posterior_predictive(posterior, structure, target_other)
```

Points the paper leaves open, each one an implementer's choice:

- **Prior family and hyperparameters.** Not stated [p. 2]. The predictive
  probability depends on this choice.
- **Attribute weights.** The paper says the weights come from a Wilcoxon
  rank-sum statistic per attribute, but not how that becomes a weight, nor
  whether the statistic used the target case [arxiv:2605.04664 p. 3]
  [arxiv:2605.04664 p. 4]. Compute it on training cases only, or the weights
  leak the outcome.
- **Covariance.** The sample behind Σ is not stated. With 40 neighbours and 18
  binary attributes the estimate may be ill-conditioned, and the paper discusses
  no regularisation [arxiv:2605.04664 p. 3].
- **Threshold p_ε.** Not reported for any setting; the paper sweeps it over
  [0, 1] to draw its curves [arxiv:2605.04664 p. 4].
- **Rejection rule.** The 2-standard-deviation rule for dropping a subpopulation
  is described, but the paper does not say it was used [arxiv:2605.04664 p. 3].
- **Dependency graph: penalty and discretisation.** The penalty value and the
  search algorithm are not stated, and the bins are set per dataset
  [arxiv:2110.00530 §2.2]. Fix and report both before reading the edges.

## Contradictions

No disagreement between the two papers, because they answer different
questions. They do learn structure differently: the anomaly paper uses a greedy
search with a maximal marginal-likelihood score [arxiv:2605.04664 p. 4], while
the survey uses a penalised objective with a leaf constraint on the class and
names no algorithm [arxiv:2110.00530 §2.2]. Neither reports a comparison of
structure learners.

Within the anomaly paper, the ranking depends on the metric: BBN on all cases
has the best AUC-ROC (0.81) and Naive Bayes with weighted selection has the best
AUC-PR (0.60) [arxiv:2605.04664 tab. 2]. The headline precision of 0.5 at 0.53
recall [arxiv:2605.04664 p. 1] and the corrected estimate of 0.32 precision at
0.3 recall on the full PORT data [arxiv:2605.04664 p. 5] are not on the same set.
The authors say precision may be overstated on the enriched set, and do not
reconcile the two figures.

## Papers

- [A survey on datasets for fairness-aware machine learning](/papers/2110.00530):
  learns a Bayesian network over each of 15 tabular datasets, with the class as a
  leaf, to show links between protected attributes and the outcome.
- [Evidence-based anomaly detection in clinical domains](/papers/2605.04664):
  flags a clinical decision as anomalous when a BBN or Naive Bayes model, fitted
  on the most similar past patients, gives it low predictive probability.