Posterior Regimes and Latent Deception: Variational Bayesian Inference in Hidden Markov Models for Sequential Fraud Detection in Financial Transactions

Joseph Uririoghene Obukofe; Anthony O'Hare; Chioma Sandra Dike

cs.LGSep 27, 2026

Posterior Regimes and Latent Deception: Variational Bayesian Inference in Hidden Markov Models for Sequential Fraud Detection in Financial Transactions

Authors: Joseph Uririoghene Obukofe, Anthony O'Hare, Chioma Sandra Dike

Organizations: Computing Science and Mathematics, Faculty of Natural Sciences, University of Stirling, Stirling FK9 4LA, Scotland, United Kingdom · University of Nottingham, Nottingham, NG7 2RD, United Kingdom

Abstract

We present a three-tier progression of Hidden Markov Models: maximum-likelihood (Baum-Welch), variational Bayesian (VBEM), and a neural variational extension (Neural VBEM), that model each customer's transaction history as a trajectory through a small number of latent behavioural regimes, one of which is empirically identified as fraud-associated. The Neural VBEM HMM replaces the fixed Gaussian-multinomial emission family with a learned encoder, compressing a 741-dimensional transaction representation into a 64-dimensional latent space in which the VBEM HMM's posterior operates; a UMAP projection of this space reveals that the discovered regimes are not discrete clusters but ordered segments of a single continuous behavioural manifold, with confirmed fraud concentrated at its extreme. We show that the model's natural output, that is, the posterior probability of regime membership, is routinely mistaken for a fraud probability, and quantify the resulting miscalibration (the regime-membership interpretation error, MRIE); a corrected posterior-predictive score, closes most of this gap. We further distinguish batch (smoothed) inference, which uses look-ahead unavailable at deployment time, from filtered (forward-only) inference, and report both. On IEEE-CIS transaction data, the neural tier achieves a 14.4 $\times$ fraud enrichment in its identified regime; while its AUPRC trails a discriminative XGBoost baseline, we show this gap is structural and not incidental, and argue the model is best positioned as a calibrated triage and interpretability layer rather than a drop-in ranking replacement.

Figures & tables

Model	$K^{*}$	Occupancy	Fraud rate	Enrichment
Baum-Welch HMM	8	1.77%	19.4%	$5.48\times$
VBEM HMM	7	3.78%	25.3%	$7.16\times$
Neural VBEM HMM ( $d_{z}=64$ )	9	3.8%	50.9%	$14.43\times$

Table 1: Selected model order and fraud-state diagnostics for each HMM tier under batch posteriors for engineered features.

Model	Score	AUPRC	KS	ECE
Baum-Welch HMM	raw / batch	0.0775	0.2667	0.0414
	corrected / batch	0.0853	0.3883	$\approx 0$
	corrected / filtered	0.0853	0.3883	$\approx 0$
VBEM HMM	raw / batch	0.0667	0.2151	0.0685
	corrected / batch	0.0806	0.2841	$\approx 0$
	corrected / filtered	0.0806	0.2841	$\approx 0$

Table 2: HMM tier predictive performance under three scoring conditions. Raw/batch is

\gamma_{t}(k^{*})

, smoothed posteriors; corrected/batch and corrected/filtered use

\mathcal{F}_{\mathrm{corrected}}=\sum_{k}\gamma_{t}(k)\eta_{k}

( Section 3.6 ) under smoothed and forward-only posteriors respectively.

Model	AUPRC	KS	ECE
Isolation Forest	0.098	0.365	0.458
Neural VBEM HMM	0.199	0.471	0.049
XGBoost	0.514	0.563	0.019

Table 3: Predictive performance against non-sequential baselines on held-out evaluation set. Neural VBEM figure is the raw/batch score,

\gamma_{t}(k^{*})

; see note below.

Figure 1 : UMAP projection of the Neural VBEM HMM ( $d_{z}=64$ , $K^{*}=9$ ) latent embeddings. Panel (i): each transaction coloured by its hard state assignment, fraud-associated state $k^{*}=4$ highlighted in red. Panel (ii): the same projection coloured by ground-truth fraud label. Confirmed fraud concentrates at the same terminal tip, density diminishing continuously toward the legitimate-dominated body.

Appendix figures & tables18 assets

Supplementary material from the paper’s appendix.

Appendix

Symbol	Form	Description
$\mathcal{D}$	Set	Full dataset of $N$ labelled transactions.
$N$	Scalar integer	Total number of transactions.
$U$	Scalar integer	Total number of unique customers.
$u$	Index	Customer index, $u=1,\ldots,U$ .
$n$	Index	Global transaction index, $n=1,\ldots,N$ .
$t$	Index	Chronological index within customer $u$ ’s sequence, $t=1,\ldots,T_{u}$ .

Appendix

Table 4 : Dataset and Sequence Notation

Symbol	Form	Description
$K$ , $K^{*}$	Scalar integers	Candidate and selected model order.
$k,i,j$	State indices	General, from-, and to-state indices over $1,\ldots,K$ .
$q_{u,t}$ , $\mathbf{q}_{u}$	Latent, Latent path	Hidden state at $t$ ; full hidden state sequence.
$k^{*}$	Scalar integer	Index of the fraud-associated state, identified post-training as $\arg\max_{k}\eta_{k}$ .
$\bm{\theta}$	Parameter set	All HMM parameters: $\bm{\pi},A,\bm{\mu}_{k},\bm{\Sigma}_{k},\bm{\pi}_{k}^{(j)}$ .
$\bm{\pi}$ , $\pi_{k}$	Vector, scalar	Initial state distribution; probability of starting in state $k$ .

Appendix

Table 5 : Hidden Markov Model Notation

Symbol	Form	Description
$\alpha_{t}(k)$ , $\bm{\alpha}$	Scalar, matrix	Forward variable; full forward matrix for a sequence.
$\beta_{t}(k)$ , $\bm{\beta}$	Scalar, matrix	Backward variable; full backward matrix for a sequence.
$\gamma_{t}(k)$ , $\bm{\gamma}$	Scalar, matrix	Batch (smoothed) state-occupancy posterior; full matrix for a sequence.
$\gamma^{\mathrm{filt}}_{t}(k)$	Scalar	Filtered (forward-only) state-occupancy posterior: conditions on $\mathbf{x}_{u,1:t}$ only, no backward pass.
$\xi_{t}(i,j)$ , $\bm{\xi}$	Scalar, tensor	Transition posterior; full tensor for a sequence.
$\odot$	Operator	Element-wise multiplication.

Appendix

Table 6 : Forward-Backward Variables

Symbol	Form	Description
$p(\mathbf{X}_{u}\mid\theta)$ , $p(\mathbf{X}_{u},\mathbf{q}_{u}\mid\theta)$	Scalars	Marginal and joint sequence probabilities.
$p(\bm{\theta})$ , $p(\bm{\theta}\mid\mathbf{X})$ , $p(\mathbf{X})$	Distributions, scalar	Prior, true posterior, and model evidence.
$q(\cdot)$ , $q_{i}(\theta_{i})$	Distributions	Variational posterior; its mean-field factors.
$D_{\mathrm{KL}}(p\\|q)$	Functional	Kullback-Leibler divergence.
$\mathbb{E}_{q}[\cdot]$	Operator	Expectation under $q$ .
$\mathcal{L}(q)$	Scalar	Evidence Lower Bound.

Appendix

Table 7 : Variational Bayesian Inference Notation

Symbol	Form	Description
$f_{\theta}$	Function	Neural encoder, $f_{\theta}:\mathbb{R}^{F}\to\mathbb{R}^{d_{z}}$ .
$\mathbf{z}_{u,t}$ , $d_{z}$ , $d_{h}$	Vector, integers	Latent embedding; latent and hidden dimensions ( $d_{z}\in\{32,64,128\}$ , $d_{h}=512$ ).
$d_{e_{j}}$ , $E_{j}$	Integer, matrix	Embedding dimension and table for categorical feature $j$ .
$\hat{\mathbf{x}}$	Vector	Normalised continuous features fed to the encoder MLP (LayerNorm output).
$\mathrm{LN}(\cdot)$ , $\mathrm{GELU}(\cdot)$	Operator, activation	Layer Normalisation; Gaussian Error Linear Unit.
$\mathbf{h}^{\mathrm{cont}}$ , $\mathbf{h}^{\mathrm{cat}}$ , $\mathbf{h}^{\mathrm{fused}}$	Vectors	Continuous-pathway, categorical-pathway, and fused hidden representations.

Appendix

Table 8 : Neural VBEM Notation

Symbol	Form	Description
$\mathbf{m}_{0}^{\mathrm{p}},\mathbf{S}_{0}^{\mathrm{p}}$	Vector, matrix	Population-level NIW prior (moment-matched).
$\hat{\mathbf{m}}_{k}^{(u)},\hat{\mathbf{S}}_{k}^{(u)}$	Vector, matrix	Customer- $u$ converged posterior statistics feeding the population prior.
$\mathcal{F}(\mathbf{x}_{u^{\prime},t}\mid\mathbf{X}_{u^{\prime},1:t})$	Scalar	Cold-start fraud score for new customer $u^{\prime}$ .

Appendix

Table 9 : Hierarchical Prior Extension

Symbol	Form	Description
$\mathcal{K}$ , $\mathcal{K}^{*}$	Sets	Candidate model orders $\{2,\ldots,10\}$ ; the eligible subset.
$N^{(K)}_{k}$ , $\mathrm{o}^{(K)}_{k}$	Scalars	Effective count and fractional occupancy of state $k$ under order $K$ .
$F^{(K)}_{k}$ , $\eta^{(K)}_{k}$	Scalars	Fraud-weighted effective count; empirical fraud rate of state $k$ under order $K$ .
$k^{*(K)}$ , $\mathrm{e}^{(K)}_{k}$	Integer, scalar	Fraud-associated state under order $K$ ; its enrichment ratio.
$\bar{\eta}$ , $\delta$	Scalars	Population fraud rate ( $\approx 3.5\%$ ); parsimony tolerance ( $=0.5$ ).
$\mathrm{f}$	Notation	Shorthand for “fraud”.

Appendix

Table 10 : State Diagnostics and Model Selection

Symbol	Form	Description
$\mathbf{c}_{u,t}$	Vector, $\Delta^{K-1}$	Transaction consequence: the real-time belief state, updated one transaction at a time from $\mathbf{c}_{u,t-1}$ ; coincides with $\gamma^{\mathrm{filt}}_{t}(\cdot)$ .
$\mathcal{F}_{\mathrm{batch}}(\mathbf{x}_{u,t})$	Scalar	Batch fraud score, $\gamma_{t}(k^{*})$ .
$\mathcal{F}_{\mathrm{real}}(\mathbf{x}_{u,t})$	Scalar	Real-time fraud score, $\mathbf{c}_{u,t}(k^{*})$ .
$\mathcal{F}_{\mathrm{corrected}}(\mathbf{x}_{u,t})$	Scalar	Posterior-predictive corrected fraud score, $\sum_{k}\gamma_{t}(k)\eta_{k}$ .
$\mathrm{MRIE}$	Scalar	Mean Regime Interpretation Error, $\mathrm{o}_{k^{}}(1-\eta_{k^{}})$ .

Appendix

Table 11 : Inference and Fraud Scoring

K	Occup. (p)	Occup. (e)	F. Rate (p)	F. Rate (e)	Enrich. (p)	Enrich. (e)
2	$0.242$	$0.242$	$0.082$	$0.083$	$2.337$	$2.343$
3	$0.219$	$0.109$	$0.090$	$0.137$	$2.536$	$3.877$
4	$0.073$	$0.103$	$0.162$	$0.134$	$4.596$	$3.790$
5	$0.041$	$0.057$	$0.163$	$0.195$	$4.618$	$5.513$
6	$0.045$	$0.044$	$0.216$	$0.224$	$6.125$	$6.358$
7	$0.040$	$0.030$	$0.219$	$0.211$	$6.194$	$5.973$

Appendix

Table 12: Baum-Welch HMM: fraud-state occupancy, fraud rate, and enrichment under batch (smoothed) posteriors,

K\in\{2,\ldots,10\}

, processed (p) and engineered (e) features.

K	Occup. (p)	Occup. (e)	F. Rate (p)	F. Rate (e)	Enrich. (p)	Enrich. (e)
2	$0.242$	$0.242$	$0.082$	$0.083$	$2.337$	$2.343$
3	$0.219$	$0.109$	$0.090$	$0.137$	$2.536$	$3.877$
4	$0.073$	$0.103$	$0.162$	$0.134$	$4.596$	$3.790$
5	$0.041$	$0.057$	$0.163$	$0.195$	$4.618$	$5.513$
6	$0.045$	$0.044$	$0.216$	$0.224$	$6.125$	$6.358$
7	$0.040$	$0.030$	$0.219$	$0.211$	$6.194$	$5.974$

Appendix

Table 13: Baum-Welch HMM: fraud-state occupancy, fraud rate, and enrichment under filtered (forward-only) posteriors,

K\in\{2,\ldots,10\}

, processed (p) and engineered (e) features.

K	Occup. (p)	Occup. (e)	F. Rate (p)	F. Rate (e)	Enrich. (p)	Enrich. (e)
2	$0.242$	$0.242$	$0.082$	$0.083$	$2.337$	$2.344$
3	$0.085$	$0.110$	$0.135$	$0.136$	$3.824$	$3.858$
4	$0.071$	$0.104$	$0.161$	$0.134$	$4.549$	$3.784$
5	$0.050$	$0.052$	$0.134$	$0.213$	$3.809$	$6.043$
6	$0.026$	$0.046$	$0.152$	$0.220$	$4.311$	$6.242$
7	$0.056$	$0.038$	$0.186$	$0.253$	$5.261$	$7.158$

Appendix

Table 14: VBEM HMM: fraud-state occupancy, fraud rate, and enrichment under batch (smoothed) posteriors,

K\in\{2,\ldots,10\}

, processed (p) and engineered (e) features.

K	Occup. (p)	Occup. (e)	F. Rate (p)	F. Rate (e)	Enrich. (p)	Enrich. (e)
2	$0.242$	$0.242$	$0.082$	$0.083$	$2.337$	$2.344$
3	$0.085$	$0.110$	$0.135$	$0.136$	$3.824$	$3.858$
4	$0.071$	$0.104$	$0.161$	$0.134$	$4.549$	$3.784$
5	$0.050$	$0.052$	$0.134$	$0.213$	$3.809$	$6.043$
6	$0.026$	$0.046$	$0.152$	$0.220$	$4.311$	$6.242$
7	$0.056$	$0.038$	$0.186$	$0.253$	$5.261$	$7.159$

Appendix

Table 15: VBEM HMM: fraud-state occupancy, fraud rate, and enrichment under filtered (forward-only) posteriors,

K\in\{2,\ldots,10\}

, processed (p) and engineered (e) features.

K	Occup. (p)	Occup. (e)	F. Rate (p)	F. Rate (e)	Enrich. (p)	Enrich. (e)
2	$0.705$	$0.441$	$0.049$	$0.055$	$1.386$	$1.560$
3	$0.229$	$0.145$	$0.136$	$0.198$	$3.860$	$5.617$
4	$0.179$	$0.128$	$0.165$	$0.204$	$4.675$	$5.778$
5	$0.130$	$0.119$	$0.215$	$0.187$	$6.079$	$5.300$
6	$0.059$	$0.122$	$0.340$	$0.206$	$9.644$	$5.843$
7	$0.053$	$0.075$	$0.363$	$0.253$	$10.292$	$7.157$

Appendix

Table 16: Neural VBEM HMM (

d_{z}=32

): fraud-state occupancy, fraud rate, and enrichment under batch posteriors,

K\in\{2,\ldots,10\}

, processed (p) and engineered (e) features.

K	Occup. (p)	Occup. (e)	F. Rate (p)	F. Rate (e)	Enrich. (p)	Enrich. (e)
2	$0.391$	$0.633$	$0.044$	$0.054$	$1.240$	$1.530$
3	$0.384$	$0.106$	$0.074$	$0.243$	$2.100$	$6.891$
4	$0.120$	$0.083$	$0.215$	$0.307$	$6.083$	$8.686$
5	$0.116$	$0.085$	$0.226$	$0.285$	$6.411$	$8.074$
6	$0.065$	$0.074$	$0.279$	$0.314$	$7.913$	$8.887$
7	$0.063$	$0.074$	$0.275$	$0.239$	$7.794$	$6.776$

Appendix

Table 17: Neural VBEM HMM (

d_{z}=64

): fraud-state occupancy, fraud rate, and enrichment under batch posteriors,

K\in\{2,\ldots,10\}

, processed (p) and engineered (e) features.

K	Occup. (p)	Occup. (e)	F. Rate (p)	F. Rate (e)	Enrich. (p)	Enrich. (e)
2	$0.360$	$0.448$	$0.053$	$0.059$	$1.493$	$1.677$
3	$0.254$	$0.181$	$0.051$	$0.089$	$1.442$	$2.521$
4	$0.111$	$0.113$	$0.237$	$0.244$	$6.700$	$6.903$
5	$0.071$	$0.125$	$0.264$	$0.208$	$7.489$	$5.888$
6	$0.066$	$0.082$	$0.247$	$0.313$	$6.986$	$8.864$
7	$0.072$	$0.088$	$0.206$	$0.274$	$5.847$	$7.774$

Appendix

Table 18: Neural VBEM HMM (

d_{z}=128

): fraud-state occupancy, fraud rate, and enrichment under batch posteriors,

K\in\{2,\ldots,10\}

, processed (p) and engineered (e) features.

Model	$K$	Occupancy	Fraud rate	Enrichment
Baum-Welch HMM	8	$4.3\%$	$22.3\%$	$6.31\times$
VBEM HMM	-	-	-	-
Neural VBEM ( $d_{z}=32$ )	8	$4.4\%$	$42.5\%$	$12.03\times$
Neural VBEM ( $d_{z}=64$ )	10	$3.4\%$	$48.1\%$	$13.61\times$
Neural VBEM ( $d_{z}=128$ )	10	$4.6\%$	$30.3\%$	$8.59\times$

Appendix

Table 19: Fraud-state diagnostics under processed features, by tier, at the model order given (

K

is the eligible/selected order where one exists).

Model	$K^{*}$	Occupancy	Fraud rate	Enrichment
Baum-Welch HMM	8	$1.8\%$	$19.4\%$	$5.48\times$
VBEM HMM	7	$3.8\%$	$25.3\%$	$7.16\times$
Neural VBEM ( $d_{z}=32$ )	8	$3.7\%$	$46.1\%$	$13.05\times$
Neural VBEM ( $d_{z}=64$ )	9	$3.8\%$	$50.9\%$	$\mathbf{14.43\times}$
Neural VBEM ( $d_{z}=128$ )	9	$3.8\%$	$55.1\%$	$15.59\times$

Appendix

Table 20: Fraud-state diagnostics under engineered features, by tier, at each tier’s selected

K^{*}

( Table 1 ).

Encoder architecture
Component	Value
Hidden dimension $d_{h}$	512 (fixed)
Latent dimension $d_{z}$	$\{32,64,128\}$ (swept)
Categorical embedding dim. $d_{e_{j}}$	$\min(50,\max(2,\lfloor(C_{j}+1)/2\rfloor))$
Continuous pathway	LayerNorm $\to$ 2-layer MLP, GELU
Dropout	0.1

Appendix

Table 21: Encoder architecture and training hyperparameters, Neural VBEM HMM.

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