I Know What I Don't Know: Latent Posterior Factor Models for Multi-Evidence Probabilistic Reasoning

Aliyu Agboola Alege

cs.AIMar 13, 2026

I Know What I Don't Know: Latent Posterior Factor Models for Multi-Evidence Probabilistic Reasoning

Authors: Aliyu Agboola Alege

Organizations: Epalea

Abstract

Real-world decision-making, from tax compliance assessment to medical diagnosis, requires aggregating multiple noisy and potentially contradictory evidence sources. Existing approaches either lack explicit uncertainty quantification (neural aggregation methods) or rely on manually engineered discrete predicates (probabilistic logic frameworks), limiting scalability to unstructured data. We introduce Latent Posterior Factors (LPF), a framework that transforms Variational Autoencoder (VAE) latent posteriors into soft likelihood factors for Sum-Product Network (SPN) inference, enabling tractable probabilistic reasoning over unstructured evidence while preserving calibrated uncertainty estimates. We instantiate LPF as LPF-SPN (structured factor-based inference) and LPF-Learned (end-to-end learned aggregation), enabling a principled comparison between explicit probabilistic reasoning and learned aggregation under a shared uncertainty representation. Across eight domains (seven synthetic and the FEVER benchmark), LPF-SPN achieves high accuracy (up to 97.8%), low calibration error (ECE 1.4%), and strong probabilistic fit, substantially outperforming evidential deep learning, LLMs and graph-based baselines over 15 random seeds. Contributions: (1) A framework bridging latent uncertainty representations with structured probabilistic reasoning. (2) Dual architectures enabling controlled comparison of reasoning paradigms. (3) Reproducible training methodology with seed selection. (4) Evaluation against EDL, BERT, R-GCN, and large language model baselines. (5) Cross-domain validation. (6) Formal guarantees in a companion paper.

Figures & tables

Panel of Figure 1 — Figure 1: Pipeline: Evidence through VAE to soft factors and probabilistic reasoning.

Evidence & Entities
Symbol	Meaning
$e$ or $x$	Evidence item (text, document, sensor reading)
$\mathcal{E}$	Set of all evidence items
entity_id	Unique identifier for an entity (company, patient, case)
Latent Space
$z\in\mathbb{R}^{d}$	Latent code (hidden representation)

Table 1: Glossary of symbols used throughout the paper.

Panel of Figure 2 — Figure 2: Tree structure showing Root (Sum) node splitting into two Product nodes (w1, w2), each splitting into two Leaf nodes over variables V1 and V2.

Requirement	Description
Tractability	Inference in $<$ 10ms for real-time applications
Calibration	Predicted probabilities match empirical frequencies
Provenance	Every prediction traceable to source evidence
Robustness	Graceful degradation with missing/noisy evidence
Scalability	Handle 10–100 evidence items per entity

Table 2: Requirements for a multi-evidence aggregation solution.

Panel of Figure 3 — Figure 3: (System Overview) illustrates the complete pipeline from user query through canonical database check, evidence retrieval, VAE encoding, factor conversion, and SPN reasoning to final output with provenance.

Panel of Figure 4 — Figure 4: Evidence Index Architecture illustrating the two-tier indexing strategy: an Entity-Predicate Hash Index and FAISS Vector Store feeding into a central Metadata Store, with the query flow from entity lookup through optional semantic reranking to top- $k$ evidence retrieval.

Panel of Figure 5 — Figure 5: Monte Carlo Decoding: evidence flows through the encoder to produce $(\mu,\sigma)$ , multiple latent codes are sampled via reparameterization, each is decoded to a distribution, and the results are averaged and temperature-scaled to produce the final soft factor.

Panel of Figure 6 — Figure 6: SPN tree with Root (Sum) branching into three Product nodes, each Product node branching into two soft factor leaf nodes $\Phi_{e_{i}}$ .

Panel of Figure 7 — Figure 7: SPN + Soft Factor Attachment: soft likelihood factors from VAE posteriors are locally attached to SPN variables while preserving decomposability through proper scope alignment.

Panel of Figure 8 — Figure 8: LPF-Learned aggregation: posteriors are fed into Quality, Consistency, and Weight networks, whose outputs drive a weighted latent aggregation $z_{\text{agg}}=\sum_{i}w_{i}\cdot\mu_{i}$ , decoded to the final distribution $p(y)$ .

Aspect	LPF-SPN	LPF-Learned
Aggregation	After decoding (factor-level)	Before decoding (latent-level)
Decoder calls	$M\times N$ (samples $\times$ evidence)	1 (on aggregated $z$ )
Uncertainty	Full distribution per evidence	Aggregate mean only
Multimodality	Can represent via mixtures	Single mode only
Interpretability	High (explicit factors)	Medium (learned weights)
Speed	Moderate ( $M\times N$ decodes)	Fast (1 decode)

Table 3: Comparison of LPF-SPN and LPF-Learned across key design dimensions.

Metric	LPF-SPN	LPF-Learned
Final prediction	high (99.5%)	high (91.4%)
Decoder calls	80 (5 $\times$ 16)	1
Aggregation stage	After decoding	Before decoding
Aggregation method	Probabilistic	Neural learned
Execution time	3.3ms	5.1ms
Interpretability	High (factors)	Medium (weights)

Table 4: Comparison of LPF-SPN and LPF-Learned on the tax compliance worked example.

Samples ( $M$ )	Std Error	Relative Error (%)
4	0.250	25%
16	0.125	12.5%
32	0.088	8.8%
64	0.063	6.3%
128	0.044	4.4%

Table 5: Monte Carlo standard error vs. sample size.

$M$	Accuracy	ECE	Latency (ms)
4	83.2%	0.089	1.8
16	86.1%	0.052	3.3
32	86.7%	0.045	5.9
64	86.9%	0.041	11.2

Table 6: Accuracy, calibration, and latency vs. Monte Carlo sample size

M

Temperature ( $T$ )	$P(high)$	ECE	Notes
0.5	0.998	0.112	Over-confident
0.8	0.972	0.037	Well-calibrated
1.0	0.952	0.042	Baseline
1.5	0.891	0.063	Under-confident
2.0	0.812	0.089	Too soft

Table 7: Effect of temperature

T

on prediction confidence and calibration.

Penalty ( $\alpha$ )	$w$ (high-conf)	$w$ (low-conf)	$P(high)$	Notes
0.0	0.89	0.85	0.947	No penalty, uniform weights
1.0	0.84	0.71	0.951	Mild down-weighting
2.0	0.70	0.53	0.952	Baseline (used in example)
4.0	0.51	0.31	0.961	Aggressive uncertainty penalty

Table 8: Effect of uncertainty penalty

\alpha

on evidence weights and final prediction.

Metric	LPF-SPN	LPF-Learned
Prediction	high	high
Confidence	99.52%	91.40%
Accuracy (test set)	97.78%	91.11%
ECE (calibration)	0.0137	0.0660
Brier score	0.0150	0.0404
Macro F1	0.9724	0.9052

Table 9: Quantitative comparison of LPF-SPN and LPF-Learned on the compliance worked example.

Component	LPF-SPN	LPF-Learned
Input	5 latent posteriors from VAE	5 latent posteriors from VAE
Aggregation Stage	After decoding (SPN on soft factors)	Before decoding (latent space)
Decoder Calls	5 evidence $\times$ 16 samples = 80	1 call on aggregated posterior
Aggregation Method	Structured probabilistic (SPN)	Learned neural (quality + consistency)
Factor Conversion	Monte Carlo integration (Alg. 6.1 )	Not applicable
Learned Components	Encoder + Decoder only	Encoder + Decoder + Aggregator

Table 10: Full architectural comparison of LPF-SPN and LPF-Learned across all dimensions.

Panel of Figure 9 — Figure 9: Deployment architecture: Load Balancer (Nginx) feeds into two FastAPI servers, both connecting to an Orchestrator Service with async workers, which connects to the Model Service (VAE, Decoder, Aggregator, GPU-enabled) and Cache Layer (Redis), all feeding into the Data Layer (PostgreSQL) and Vector Store (FAISS).

Panel of Figure 10 — Figure 10: Complete execution pipeline for LPF-SPN. The system first checks the canonical database for authoritative facts. On a miss, it retrieves relevant evidence, encodes each item into a latent posterior, converts posteriors into soft probabilistic factors via Monte Carlo integration, performs exact SPN inference, and returns calibrated predictions with full provenance.

Panel of Figure 11 — Figure 11: Complete execution pipeline for LPF-Learned. Unlike LPF-SPN, this variant aggregates evidence in latent space before decoding. The learned aggregator uses three neural networks to assess quality, detect contradictions, and compute optimal aggregation weights, producing a single aggregated posterior that is decoded once to yield the final distribution.

Panel of Figure 12 — Figure 12: Architectural divergence between LPF-SPN and LPF-Learned variants.

Stage	Input Dim	Operation	Output Dim	Both?
Evidence Text	Text	Retrieval	5 texts	Same
Embedding	Text	SBERT	$5\times 384$	Same
Encoding	$5\times 384$	VAE	$5\times 64$ $(\mu,\sigma)$	Same
Aggregation	$5\times 64$	—	Variable	Diverges
Decoding	Variable	Decoder	3-class dist	80 vs 1 calls
Reasoning	Factors/ $z$	SPN/Direct	3-class dist	Diverges

Table 11: Numerical transformation summary for a complete LPF query.

Parameter	Value	Description
Learning rate	$10^{-3}$	Adam optimizer
Batch size	64	Mini-batch size
KL weight ( $\beta$ )	0.01	Regularization strength
Dropout	0.1	Applied in encoder MLP
Early stopping patience	5 epochs	Validation-based stopping
Max epochs	100	Training budget

Table 12: Encoder-decoder training hyperparameters.

Figure 13: Compliance domain training results across 7 random seeds. Left: Validation accuracy showing consistency around 85.6% mean with best seed (2024, gold bar) achieving 86.0%. Right: Validation loss showing mean of 0.730 with best seed achieving 0.726. Red dashed lines indicate mean values.

Seed	Train Acc	Val Acc (Best)	Val Loss (Best)	Epochs	Converged
42	82.2%	85.7%	0.735	10	✓
123	82.2%	85.6%	0.730	20	$\times$
456	82.7%	85.4%	0.731	9	✓
789	82.4%	85.4%	0.734	11	✓
2024	82.3%	86.0% $\star$	0.726 $\star$	12	✓
2025	82.6%	85.3%	0.727	16	✓

Table 13: Seed-level training results for the compliance domain.

\star

Best seed selected for downstream evaluation.

Figure 14: Loss decomposition for compliance domain (best seed 2024). The total validation loss (0.726) comprises cross-entropy loss (0.692, 95.3%) and weighted KL divergence (0.040, 4.7%). The KL term remains moderate, indicating the encoder learns meaningful latent structure without excessive compression.

Domain	Train Acc	Val Acc	Best Seed	Best Val Acc	Notes
FEVER	99.6 $\pm$ 0.1%	99.9 $\pm$ 0.0%	2025	99.9%	Fact verification (easiest)
Academic	83.5 $\pm$ 0.2%	85.7 $\pm$ 0.2%	789	86.1%	Publication venue classification
Compliance	82.3 $\pm$ 0.3%	85.6 $\pm$ 0.2%	2024	86.0%	Primary domain
Construction	83.4 $\pm$ 0.2%	85.4 $\pm$ 0.2%	789	85.8%	Project risk assessment
Finance	83.6 $\pm$ 0.2%	84.8 $\pm$ 0.3%	456	85.2%	Credit rating prediction
Materials	83.9 $\pm$ 0.2%	84.0 $\pm$ 0.5%	456	84.6%	Material property classification

Table 14: Encoder-decoder training results across all evaluation domains. Results are mean

\pm

std over 7 random seeds.

Figure 15: Cross-domain performance comparison showing training (blue bars) and validation (orange bars) accuracy across all eight domains. Error bars indicate standard deviation over 7 seeds. Domains are sorted by validation accuracy (descending).

Figure 16: Validation accuracy variance (standard deviation) across seeds by domain. FEVER shows minimal variance (0.0%) due to clean data structure, while Materials exhibits highest variance (0.5%).

Figure 17: Generalization gap analysis showing train minus validation accuracy for each domain. Negative values (most domains) indicate validation outperforms training, suggesting good generalization.

Figure 18: Training convergence analysis. Left: Average epochs to convergence across 7 seeds. Right: Early stopping success rate (percentage of seeds that converged before maximum epochs).

Parameter	Value	Description
Learning rate	$10^{-3}$	Adam optimizer
Training epochs	30	Fewer than encoder/decoder
Hidden dimensions	128	For all three networks
Dropout	0.1	Regularization

Table 15: Learned aggregator training hyperparameters.

Hyperparameter	Value	Range	Notes
Architecture
Latent dimension ( $d_{z}$ )	64	32–128	Balances expressiveness and efficiency
Embedding dimension	384	Fixed	Sentence-BERT output
Encoder hidden dims	[256, 128]	—	Two-layer MLP
Decoder hidden dims	[128, 64]	—	Conditional on predicate
Predicate embedding	32	16–64	Learned per predicate

Table 16: Core hyperparameters for LPF system components.

Concept	Equation
Reparameterization	$z=\mu+\sigma\odot\epsilon$ , where $\epsilon\sim\mathcal{N}(0,I)$
Latent Factor	$\Phi_{e}(y)=\int p_{\theta}(y\mid z)\,q_{\phi}(z\mid e)\,dz$
Monte Carlo Estimate	$\hat{\Phi}_{e}(y)=\frac{1}{M}\sum_{m=1}^{M}p_{\theta}(y\mid z^{(m)})$
Confidence Weight	$w(e)=\text{base\_conf}\times\frac{1}{1+\exp(\alpha\cdot\text{mean}(\sigma))}$
Weighted Factor	$\tilde{\Phi}_{e}(y)=\frac{(p(y\mid e))^{w(e)}}{\sum_{y^{\prime}}(p(y^{\prime}\mid e))^{w(e)}}$
Training Loss	$\mathcal{L}=-\log p_{\theta}(y^{*}\|z,p)+\beta\cdot\text{KL}(q_{\phi}(z\|e)\\|\mathcal{N}(0,I))$

Table 17: Quick reference for key LPF equations.

Method	Accuracy	ECE	Runtime (ms)	Speedup
LPF-SPN	99.7%	1.2%	25.2	1.0 $\times$
LPF-Learned	99.7%	0.3%	24.0	1.0 $\times$
VAE-Only	99.7%	0.3%	3.5	7.2 $\times$
Llama-3.3-70B	44.0%	74.4%	1581.6	0.016 $\times$
Qwen3-32B	62.0%	82.3%	3176.4	0.008 $\times$
Kimi-K2	56.0%	87.3%	609.5	0.041 $\times$

Table 18: FEVER benchmark performance comparison.

Method	Accuracy	Macro F1	ECE	Runtime (ms)	Notes
LPF-SPN	97.8%	97.2%	1.4%	14.8	Best overall
LPF-Learned	91.1%	90.5%	6.6%	37.4	Competitive
VAE-Only	95.6%	94.8%	9.6%	6.9	Lacks reasoning
BERT	97.0%	95.2%	3.2%	134.7	9.1 $\times$ slower
EDL-Aggregated	42.9%	44.0%	21.4%	1.1	Fundamental mismatch
EDL-Individual	28.1%	14.6%	18.6%	3.8	Training-inference gap

Table 19: Compliance domain performance comparison across all baselines.

Figure 19: Model performance comparison across metrics. LPF-SPN achieves superior accuracy (97.8%) and exceptional calibration (ECE 1.4%) while maintaining fast inference (14.8ms). EDL variants show catastrophic failure, validating the need for specialized multi-evidence architectures.

Model	Accuracy	Macro F1	Weighted F1	NLL $\downarrow$	Brier $\downarrow$	ECE $\downarrow$	Runtime (ms)
LPF-SPN	0.978	0.972	0.978	0.125	0.015	0.014	14.8
LPF-Learned	0.911	0.905	0.907	0.273	0.040	0.066	37.4
VAE-Only	0.956	0.948	0.955	0.265	0.039	0.096	6.9
BERT	0.970	0.952	0.970	0.069	0.011	0.032	134.7
SPN-Only	0.948	0.925	0.946	0.471	0.075	0.309	2.4
EDL-Aggregated	0.430	0.440	0.326	0.698	0.137	0.214	1.1

Table 20: Main experiment results on compliance domain (best seed: 11111). All LLM models evaluated on 50 test samples due to API cost constraints.

Figure 20: Compliance domain seed variance analysis (15 seeds). LPF-SPN shows low variance across seeds: accuracy std 1.2%, ECE std 0.7%, NLL std 0.02. This demonstrates robust performance independent of initialization.

Metric	Mean	Std Dev	Best	95% CI
Accuracy (%)	99.7	0.0	99.7	$\pm$ 0.1
ECE (%)	1.0	0.1	0.8	$\pm$ 0.2
NLL	0.023	0.002	0.018	$\pm$ 0.004
Brier Score	0.003	0.000	0.002	$\pm$ 0.001
Macro F1	0.996	0.000	0.997	$\pm$ 0.001

Table 21: Statistical summary over 15 random seeds (compliance domain, LPF-SPN only).

Figure 21: Compliance domain seed-by-seed comparison. Left: Validation accuracy across 15 seeds (mean: 85.6%, std: 0.2%). Right: Validation loss across seeds (mean: 0.730, std: 0.003). Best seed 2024 (gold bar) achieved 86.0% validation accuracy with 0.726 loss. The narrow distribution demonstrates training stability.

Figure 22: Training loss decomposition for compliance domain (best seed 2024). The total validation loss (0.726) comprises cross-entropy (0.692, 95.3%) and weighted KL divergence (0.040, 4.7%). The KL term remains moderate, indicating the encoder learns meaningful latent structure without excessive compression.

Figure 23: Mean prediction confidence across models (compliance domain). LPF-SPN maintains high mean confidence (0.975) while achieving high accuracy, indicating well-calibrated certainty. EDL variants show lower confidence (0.467–0.601), reflecting high uncertainty but incorrect predictions. Oracle achieves perfect confidence (1.0) as expected.

Model	Conf Mean	Conf Std	High-Conf Errors	Low-Conf Correct
LPF-SPN	0.975	0.085	1	0
LPF-Learned	0.964	0.110	3	1
VAE-Only	0.874	0.168	2	4
BERT	0.971	0.075	1	0
EDL-Aggregated	0.601	0.296	15	8
EDL-Individual	0.468	0.000	21	0

Table 22: Confidence distribution analysis (compliance domain, best seed).

Figure 24: Average inference time across models (compliance domain). LPF-SPN achieves 14.8ms average runtime, 9.1

\times

faster than BERT (134.7ms) and 203

\times

faster than best LLM (Qwen3-32B: 3008.6ms). R-GCN is fastest (0.001ms) but has unusable accuracy (15.6%).

Model	Avg Runtime (ms)	Throughput (q/s)	Speedup vs LPF-SPN
R-GCN	0.001	1,000,000	$14{,}800\times$
EDL-Aggregated	1.1	909	$13.5\times$
SPN-Only	2.4	417	$6.2\times$
EDL-Individual	3.8	263	$3.9\times$
VAE-Only	6.9	145	$2.1\times$
LPF-SPN	14.8	68	$\mathbf{1.0\times}$

Table 23: Runtime analysis and throughput (compliance domain).

Figure 25: Cross-domain performance comparison. Blue bars show training accuracy, orange bars show validation accuracy. Error bars represent standard deviation over 7 seeds (6 domains) or 15 seeds (compliance). FEVER shows near-perfect accuracy (99.9%) due to clean data structure, while Legal represents the hardest domain (83.6% validation).

Domain	LPF-SPN	LPF-Learned	VAE-Only	Best Baseline	Improvement
Compliance	97.8%	91.1%	95.6%	BERT: 97.0%	$+$ 0.8%
FEVER	99.7%	99.7%	99.7%	VAE: 99.7%	$+$ 0.0%
Academic	100.0%	100.0%	99.3%	LPF-L: 100.0%	$+$ 0.0%
Construction	100.0%	98.5%	99.3%	VAE: 99.3%	$+$ 0.7%
Finance	99.3%	98.5%	98.5%	LPF-L: 98.5%	$+$ 0.8%
Materials	99.3%	98.5%	98.5%	LPF-L: 98.5%	$+$ 0.8%

Table 24: Cross-domain generalization (best seed per domain, test set performance).

Figure 26: LPF-SPN cross-domain accuracy and calibration. Left: Accuracy by domain (mean: 99.3%). Right: ECE by domain (mean: 0.015). Only compliance domain shows error bars (15 seeds); other domains show best seed only. LPF-SPN maintains consistent high performance across all domains without domain-specific tuning.

Figure 27: Ablation study for Monte Carlo sample count across all domains. Blue lines show accuracy (left y-axis), orange lines show ECE (right y-axis). Compliance domain shown in first subplot, remaining domains in grid layout. Diminishing returns observed after

n=16

across most domains.

n_samples	Accuracy	NLL	ECE	Runtime (ms)	Std Error
4	0.978	0.193	0.127	2.1	0.250
8	0.941	0.241	0.152	2.8	0.177
16	0.963	0.285	0.182	3.3	0.125
32	0.978	0.313	0.225	5.2	0.088

Table 25: Impact of Monte Carlo sample count (compliance domain).

Figure 28: Temperature scaling ablation across domains. Most domains show optimal performance at

T=1.0

(no scaling) or

T=0.8

(slight sharpening). Higher temperatures (

T\geq 1.2

) consistently degrade both accuracy and calibration.

Temperature	Accuracy	NLL	ECE
0.8	0.985	0.170	0.131
1.0	0.985	0.257	0.183
1.2	0.978	0.326	0.247
1.5	0.970	0.433	0.308

Table 26: Impact of temperature scaling (compliance domain).

Figure 29: Weight penalty (alpha) ablation. Lower alpha values (0.1–1.0) achieve better accuracy by moderately downweighting uncertain evidence. Extreme penalty (alpha=5.0) severely degrades performance across all domains.

Alpha	Accuracy	NLL	ECE	Mean Weight
0.1	0.970	0.108	0.015	0.976
1.0	0.985	0.100	0.051	0.931
2.0	0.963	0.280	0.179	0.784
5.0	0.978	1.005	0.611	0.367

Table 27: Impact of uncertainty penalty alpha (compliance domain).

Figure 30: Evidence count (top_k) ablation. Performance improves rapidly from

k=1

k=5

, then plateaus. Diminishing returns evident beyond

k=10

across most domains.

top_k	Accuracy	NLL	Runtime (ms)	Marginal Gain
1	0.793	0.880	1.8	—
3	0.919	0.503	2.5	$+$ 12.6%
5	0.970	0.280	3.3	$+$ 5.1%
10	0.978	0.278	4.9	$+$ 0.8%
20	0.978	0.268	7.8	$+$ 0.0%

Table 28: Impact of evidence count (compliance domain).

Figure 31: Accuracy vs ECE scatter plot across all ablation configurations. Each point represents one hyperparameter setting. LPF-SPN configurations cluster in the high-accuracy, low-ECE region (top-left quadrant), while EDL and R-GCN baselines occupy the low-accuracy, high-ECE region.

Model	Total Errors	Error Rate	High-Conf Errors	Low-Conf Correct
LPF-SPN	3	2.2%	1	0
LPF-Learned	12	8.9%	3	1
VAE-Only	6	4.4%	2	4
BERT	4	3.0%	1	0
SPN-Only	7	5.2%	3	0
EDL-Aggregated	77	57.0%	15	8

Table 29: Error counts and rates across models (compliance domain, 135 test samples).

	Pred: Low	Pred: Medium	Pred: High
True: Low	26	1	0
True: Medium	1	67	2
True: High	0	1	37

Table 30: LPF-SPN confusion matrix (compliance domain).

Figure 32: Error rate comparison

True Label	Total Samples	Errors	Error Rate
Low	27	1	3.7%
Medium	70	3	4.3%
High	38	1	2.6%

Table 31: Errors grouped by true label (LPF-SPN).

Confidence Threshold	Coverage	Accuracy on Accepted	Rejected Count
0.5	100.0%	97.8%	0
0.6	99.3%	97.8%	1
0.7	97.8%	98.5%	3
0.8	87.4%	99.2%	17
0.9	69.6%	100.0%	41

Table 32: Selective classification performance (LPF-SPN).

Avg Credibility Range	Sample Count	Accuracy	ECE
$>$ 0.9 (High)	23	99.1%	0.8%
0.7–0.9 (Medium)	89	97.2%	1.5%
$<$ 0.7 (Low)	23	91.3%	3.4%

Table 33: Performance stratified by average evidence credibility (LPF-SPN).

Model	Parameters	Accuracy	ECE	Avg Runtime (ms)	Cost/Query
LPF-SPN	$\sim$ 50M	97.8%	1.4%	14.8	$\sim$ $0.000
Groq-llama-3.3-70b	70B	95.9%	81.6%	1578.7	$0.004
Groq-qwen3-32b	32B	98.0%	79.7%	3008.6	$0.003
Groq-kimi-k2	Unknown	98.0%	80.5%	764.2	$0.002
Groq-gpt-oss-120b	120B	93.9%	81.3%	1541.7	$0.006

Table 34: LLM evaluation on compliance domain (50 test samples, zero-shot prompting).

Model	Accuracy	F1	ECE	Runtime (ms)
LPF-SPN	99.7%	0.997	1.2%	25.2
LPF-Learned	99.7%	0.997	0.3%	24.0
VAE-Only	99.7%	0.997	0.3%	3.5
BERT	95.2%	0.951	8.9%	142.3
SPN-Only	95.2%	0.951	28.9%	0.9
EDL-Aggregated	50.2%	0.223	16.7%	1.2

Table 35: FEVER benchmark results (1,800 test samples).

Theorem	Theoretical Guarantee	Applies To	Empirical Verification	Status
T1: Calibration Preservation	ECE $\leq\varepsilon+C/\sqrt{K_{\text{eff}}}$ w.p. $1-\delta$	LPF-SPN only	LPF-SPN ECE 1.4% (compliance); bound $\approx$ 10%	$\checkmark$ Strong
T2: Monte Carlo Error	Factor error $O(1/\sqrt{M})$	Both	$M=16$ achieves mean error 1.3%; scaling $R^{2}=0.95$	$\checkmark$ Verified
T3: Generalization Bound	Non-vacuous PAC-Bayes gap $\leq f(N,d_{\text{eff}})$	LPF-Learned only	Gap 0.0085 vs bound 0.228 (96.3% margin)	$\checkmark$ Non-vacuous
T4: Info-Theoretic Lower Bound	ECE $\geq$ noise $+\bar{H}(Y\|E)/H(Y)$	Both	LPF-SPN 1.4% within $1.1\times$ optimal; LPF-Learned $1.3\times$	$\checkmark$ Near-optimal
T5: Robustness to Corruption	Degradation $O(\varepsilon\delta\sqrt{K})$	Both	Cross-domain stability; missing evidence robustness	$\checkmark$ Validated
T6: Sample Complexity	ECE decays $O(1/\sqrt{K})$	Both	Evidence count ablation; plateau at $K\approx 7$	$\checkmark$ Scaling verified

Table 36: LPF’s seven formal guarantees and empirical verification.

$M$	Mean Error	95th Percentile	Theoretical Bound	LPF Variant
4	1.9%	8.0%	77.4%	Both
16	1.3%	5.3%	38.7%	Both (default)
32	1.0%	3.7%	27.4%	Both
64	0.8%	2.5%	19.3%	Both

Table 37: Empirical validation of Monte Carlo error bounds.

$K$	LPF-SPN ECE	LPF-Learned ECE	Marginal Improvement
1	34.7%	—	Baseline
2	33.4%	—	1.3%
3	28.4%	—	5.0%
5	18.6%	6.6%	9.8% / 21.8%
7	19.2%	—	Plateau
10	19.2%	—	Diminishing returns

Table 38: Evidence count vs. ECE performance (compliance domain).

Component	LPF-SPN	LPF-Learned	Interpretation
Total variance	0.153	0.130	Overall prediction uncertainty
Epistemic (reducible)	0.111	0.088	Evidence disagreement/ambiguity
Aleatoric (irreducible)	0.042	0.042	Inherent label randomness
Decomposition error	$<$ 0.002%	$<$ 0.002%	Exact within numerical precision

Table 39: Uncertainty decomposition components (compliance domain).

Property	LPF-SPN	LPF-Learned	Notes
Calibration guarantee (T1)	$\checkmark$ Proven	$\times$ Empirical only	SPN exact inference vs learned weights
MC error control (T2)	$\checkmark$ $O(1/\sqrt{M})$	$\checkmark$ $O(1/\sqrt{M})$	Shared factor conversion
Generalization bound (T3)	N/A	$\checkmark$ Non-vacuous (96% margin)	Only learned aggregator has parameters
Info-theoretic optimality (T4)	$\checkmark$ $1.1\times$ optimal	$\checkmark$ $1.3\times$ optimal	Both near-optimal
Robustness (T5)	$\checkmark$ $O(\varepsilon\sqrt{K})$	$\checkmark$ $O(\varepsilon\sqrt{K})$	Shared weighting mechanism
Sample complexity (T6)	$\checkmark$ $O(1/\sqrt{K})$	$\checkmark$ $O(1/\sqrt{K})$	Shared CLT-based scaling

Table 40: Theoretical properties by variant.

Method	Calib. Guarantee	Uncertainty Decomp.	Robustness	Gen. Bound	Multi-Evidence
LPF-SPN	$\checkmark$ T1 (proven)	$\checkmark$ Exact (T7)	$\checkmark$ $O(\varepsilon\sqrt{K})$	N/A	$\checkmark$ Purpose-built
LPF-Learned	$\times$ Empirical	$\checkmark$ Exact (T7)	$\checkmark$ $O(\varepsilon\sqrt{K})$	$\checkmark$ Non-vacuous (T3)	$\checkmark$ Purpose-built
BERT	$\times$ None	$\times$ Heuristic	$\times$ Unknown	$\times$ Vacuous	$\times$ Single-input adapted
EDL	$\times$ Single-input only	$\times$ Approximate	$\times$ Unknown	$\times$ Unknown	$\times$ Catastrophic (43%)
R-GCN	$\times$ None	$\times$ None	$\times$ Unknown	$\times$ Unknown	$\times$ Unsuitable (15.6%)
VAE-Only	$\times$ None	$\times$ None	$\times$ Unknown	$\times$ Unknown	$\times$ No aggregation

Table 41: Theoretical properties vs. baselines.

Theorem	Empirical Result	Theoretical Bound	Gap
T1 (Calibration)	1.4% ECE	$\sim$ 15%	90.7% below bound
T2 (MC Error, $M=16$ )	1.3% error	38.7%	96.6% below bound
T3 (Generalization)	0.0085 gap	0.228	96.3% below bound
T5 (Robustness, $\varepsilon=0.5$ )	12% $L_{1}$	316%	96.2% below bound

Table 42: Empirical results vs. theoretical bounds.

Property	BERT	EDL	LLMs	LPF-SPN	LPF-Learned
Multi-evidence design	$\times$ Adapted	$\times$ Fails	$\times$ Zero-shot	$\checkmark$ Purpose-built	$\checkmark$ Purpose-built
Calibration guarantee	$\times$ None	$\times$ None	$\times$ None	$\checkmark$ T1 (proven)	$\times$ Empirical (6.6%)
Exact uncertainty decomp	$\times$ Heuristic	$\times$ Approx	$\times$ No access	$\checkmark$ T7 ( $<$ 0.002%)	$\checkmark$ T7 ( $<$ 0.002%)
Robustness guarantee	$\times$ Unknown	$\times$ Unknown	$\times$ Unknown	$\checkmark$ T5	$\checkmark$ T5
Provable generalization	$\times$ Vacuous	$\times$ Unknown	$\times$ Unknown	N/A	$\checkmark$ T3 (96% margin)
Near-optimal calibration	$\times$ No	$\times$ No	$\times$ No	$\checkmark$ T4 ( $1.1\times$ )	$\checkmark$ T4 ( $1.3\times$ )

Table 43: Theoretical properties across all methods.

Appendix figures & tables82 assets

Supplementary material from the paper’s appendix.

Appendix

Seed	Tr. Loss	Tr. CE	Tr. KL	Tr. Acc	Val Loss	Val CE	Val KL	Val Acc	Best Val Acc	Best Val Loss	Ep.	Conv.
42	0.765	0.728	3.650	82.2%	0.741	0.710	3.129	84.0%	85.7%	0.735	10	✓
123	0.761	0.726	3.534	82.2%	0.738	0.706	3.193	84.3%	85.6%	0.730	20	$\times$
456	0.763	0.725	3.792	82.7%	0.740	0.709	3.067	84.0%	85.4%	0.731	9	✓
789	0.765	0.729	3.587	82.4%	0.738	0.704	3.441	84.7%	85.4%	0.734	11	✓
2024	0.775	0.729	4.638	82.3%	0.732	0.692	4.007	85.6%	$\mathbf{86.0\%}\,\bigstar$	$\mathbf{0.726}\,\bigstar$	12	✓
2025	0.756	0.722	3.431	82.6%	0.729	0.699	2.913	84.8%	85.3%	0.727	16	✓

Appendix

Table 44: Compliance domain: detailed seed-by-seed training results

Seed	Final Train Acc	Final Val Acc	Best Val Acc	Best Val Loss	Epochs	Converged
42	83.3%	85.1%	85.5%	0.745	9	✓
123	83.5%	85.6%	85.6%	0.735	11	✓
456	83.5%	84.6%	85.5%	0.740	13	✓
789	83.2%	84.5%	$\mathbf{86.1\%}\,\bigstar$	$\mathbf{0.736}\,\bigstar$	15	✓
2024	83.9%	85.6%	85.6%	0.736	20	$\times$
2025	83.4%	84.2%	85.7%	0.740	13	✓

Appendix

Table 45: Academic domain: seed-by-seed training results

Figure 33: Academic domain seed comparison. Left: Validation accuracy by seed (mean: 85.7%). Right: Validation loss by seed (mean: 0.739). Best seed: 789 (86.1% accuracy, 0.736 loss).

Seed	Final Train Acc	Final Val Acc	Best Val Acc	Best Val Loss	Epochs	Converged
42	83.3%	84.5%	85.2%	0.745	9	✓
123	83.3%	84.2%	85.6%	0.742	16	✓
456	83.2%	83.2%	85.5%	0.743	14	✓
789	83.7%	84.5%	$\mathbf{85.8\%}\,\bigstar$	$\mathbf{0.731}\,\bigstar$	19	✓
2024	83.2%	84.8%	85.3%	0.741	20	$\times$
2025	83.5%	84.7%	85.6%	0.737	13	✓

Appendix

Table 46: Construction domain: seed-by-seed training results

Seed	Final Train Acc	Final Val Acc	Best Val Acc	Best Val Loss	Epochs	Converged
42	83.6%	83.9%	84.9%	0.742	16	✓
123	84.1%	82.5%	84.7%	0.747	15	✓
456	83.4%	83.2%	$\mathbf{85.2\%}\,\bigstar$	$\mathbf{0.741}\,\bigstar$	14	✓
789	83.5%	83.9%	84.4%	0.746	10	✓
2024	83.8%	82.9%	84.5%	0.752	9	✓
2025	83.6%	84.2%	84.6%	0.744	11	✓

Appendix

Table 47: Finance domain: seed-by-seed training results

Seed	Final Train Acc	Final Val Acc	Best Val Acc	Best Val Loss	Epochs	Converged
42	84.4%	84.0%	$\mathbf{84.0\%}\,\bigstar$	$\mathbf{0.753}\,\bigstar$	18	✓
123	84.0%	82.0%	83.8%	0.758	17	✓
456	84.3%	80.6%	83.8%	0.757	16	✓
789	84.0%	83.0%	83.7%	0.760	15	✓
2024	84.1%	82.7%	83.8%	0.754	14	✓
2025	84.0%	81.8%	83.6%	0.753	14	✓

Appendix

Table 48: Healthcare domain: seed-by-seed training results

Seed	Final Train Acc	Final Val Acc	Best Val Acc	Best Val Loss	Epochs	Converged
42	85.2%	81.3%	83.6%	0.761	10	✓
123	84.4%	82.3%	83.4%	0.760	16	✓
456	85.0%	82.8%	$\mathbf{83.7\%}\,\bigstar$	$\mathbf{0.755}\,\bigstar$	13	✓
789	84.8%	83.0%	83.6%	0.763	14	✓
2024	84.6%	82.4%	83.6%	0.753	20	$\times$
2025	84.8%	82.6%	83.7%	0.760	15	✓

Appendix

Table 49: Legal domain: seed-by-seed training results

Seed	Final Train Acc	Final Val Acc	Best Val Acc	Best Val Loss	Epochs	Converged
42	84.2%	82.4%	83.3%	0.759	10	✓
123	84.3%	82.1%	84.4%	0.752	20	$\times$
456	84.2%	82.5%	$\mathbf{84.6\%}\,\bigstar$	$\mathbf{0.754}\,\bigstar$	8	✓
789	83.8%	81.6%	84.4%	0.754	13	✓
2024	84.0%	82.8%	83.5%	0.759	10	✓
2025	83.9%	81.6%	83.6%	0.759	11	✓

Appendix

Table 50: Materials domain: seed-by-seed training results

Seed	Final Train Acc	Final Val Acc	Best Val Acc	Best Val Loss	Epochs	Converged
42	99.7%	99.7%	99.9%	0.573	30	$\times$
123	99.6%	99.6%	99.8%	0.576	30	$\times$
456	99.5%	99.8%	99.8%	0.576	30	$\times$
789	99.6%	99.8%	99.9%	0.574	30	$\times$
1011	99.6%	99.7%	99.9%	0.573	28	✓
2024	99.6%	99.6%	99.9%	0.572	30	$\times$

Appendix

Table 51: FEVER domain: seed-by-seed training results

Figure 34: Cross-domain training and validation accuracy comparison. Error bars show standard deviation over 7 random seeds. Domains are sorted by validation accuracy (descending). FEVER achieves near-perfect accuracy (99.9%), while Legal represents the most challenging domain (83.6%).

Figure 35: Validation accuracy variance across seeds by domain. Lower bars indicate more stable training. FEVER shows minimal variance (0.0%), while Materials exhibits the highest sensitivity to initialization (0.5%). Mean validation accuracy shown below each bar.

Figure 36: Generalization gap (train

-

validation accuracy) by domain. Negative values indicate validation outperforms training. Most domains show good generalization, though Healthcare and Legal show small positive gaps suggesting slight overfitting.

Figure 37: Training convergence analysis. Left: average epochs to convergence (lower is faster). Right: early stopping success rate (percentage of seeds that converged before maximum epochs). Healthcare and Finance show perfect convergence (100%), while FEVER rarely early-stops due to the large dataset scale.

Domain	Train Acc	Val Acc	Gap	Variance	Mean Epochs	Conv. Rate
FEVER	$99.6\pm 0.1\%$	$99.9\pm 0.0\%$	$-0.3\%$	$0.0\%$	29.7	14%
Academic	$83.5\pm 0.2\%$	$85.7\pm 0.2\%$	$-2.2\%$	$0.2\%$	12.7	86%
Compliance	$82.3\pm 0.3\%$	$85.6\pm 0.2\%$	$-3.3\%$	$0.2\%$	13.9	71%
Construction	$83.4\pm 0.2\%$	$85.4\pm 0.2\%$	$-2.0\%$	$0.2\%$	13.7	86%
Finance	$83.6\pm 0.2\%$	$84.8\pm 0.3\%$	$-1.2\%$	$0.3\%$	11.7	100%
Materials	$83.9\pm 0.2\%$	$84.0\pm 0.5\%$	$+0.1\%$	$0.5\%$	10.3	86%

Appendix

Table 52: Cross-domain summary statistics

Figure 38: Compliance domain seed comparison. Left: Validation accuracy by seed (mean: 85.6%). Right: Validation loss by seed (mean: 0.730). Best seed: 2024 (86.0% accuracy, 0.726 loss).

Figure 39: Academic domain seed comparison. Left: Validation accuracy by seed (mean: 85.7%). Right: Validation loss by seed (mean: 0.739). Best seed: 789 (86.1% accuracy, 0.736 loss).

Figure 40: Construction domain seed comparison. Left: Validation accuracy by seed (mean: 85.4%). Right: Validation loss by seed (mean: 0.740). Best seed: 789 (85.8% accuracy, 0.731 loss).

Figure 41: Finance domain seed comparison. Left: Validation accuracy by seed (mean: 84.8%). Right: Validation loss by seed (mean: 0.745). Best seed: 456 (85.2% accuracy, 0.741 loss).

Figure 42: Healthcare domain seed comparison. Left: Validation accuracy by seed (mean: 83.8%). Right: Validation loss by seed (mean: 0.756). Best seed: 42 (84.0% accuracy, 0.753 loss).

Figure 43: Legal domain seed comparison. Left: Validation accuracy by seed (mean: 83.6%). Right: Validation loss by seed (mean: 0.759). Best seed: 456 (83.7% accuracy, 0.755 loss).

Figure 44: Materials domain seed comparison. Left: Validation accuracy by seed (mean: 84.0%). Right: Validation loss by seed (mean: 0.757). Best seed: 456 (84.6% accuracy, 0.754 loss).

Figure 45: FEVER dataset seed comparison. Left: Validation accuracy by seed (mean: 99.9%). Right: Validation loss by seed (mean: 0.574). Best seed: 2025 (99.9% accuracy, 0.573 loss). Note the extremely small variance compared to other domains.

Figure 46: Compliance domain loss decomposition (seed 2024). Cross-entropy: 0.692, KL (

\times\,\beta=0.01

): 0.040, Total: 0.726. The weighted KL term contributes 5.5% of total loss, providing regularization without excessive compression.

Figure 47: Academic domain loss decomposition (seed 789). Cross-entropy: 0.735, KL (

\times\,\beta

): 0.001, Total: 0.736. Minimal KL contribution (0.1%) indicates the encoder learned compact representations naturally.

Figure 48: Construction domain loss decomposition (seed 789). Shows balanced contribution from classification and regularization terms.

Figure 49: Finance domain loss decomposition (seed 456). Cross-entropy dominates the total loss, with KL providing mild regularization.

Figure 50: Healthcare domain loss decomposition (seed 42). Well-balanced loss components indicate effective VAE training.

Figure 51: Legal domain loss decomposition (seed 456). Higher cross-entropy reflects the domain’s difficulty, while KL remains controlled.

Figure 52: Materials domain loss decomposition (seed 456). Similar pattern to other domains with CE contributing approximately 95% of total loss.

n_samples	Accuracy	Macro F1	Wtd F1	NLL	Brier	ECE	Conf Mean	Conf Std
4	0.9778	0.9702	0.9775	0.1928	0.0243	0.1266	0.8512	0.1167
8	0.9407	0.9278	0.9398	0.2408	0.0331	0.1522	0.8203	0.1366
16	0.9630	0.9547	0.9625	0.2852	0.0390	0.1824	0.7806	0.1257
32	0.9778	0.9742	0.9776	0.3128	0.0424	0.2249	0.7528	0.1239

Appendix

Table 53: Compliance domain ablation: Monte Carlo sample count.

Temperature	Accuracy	Macro F1	Wtd F1	NLL	Brier	ECE	Conf Mean	Conf Std
0.8	0.9704	0.9614	0.9698	0.2113	0.0282	0.1309	0.8395	0.1292
1.0	0.9704	0.9638	0.9702	0.2634	0.0345	0.1862	0.7894	0.1264
1.2	0.9778	0.9724	0.9776	0.3341	0.0457	0.2421	0.7357	0.1125
1.5	0.9704	0.9640	0.9701	0.4331	0.0653	0.3057	0.6646	0.1055

Appendix

Table 54: Compliance domain ablation: temperature scaling.

Alpha	Accuracy	Macro F1	Wtd F1	NLL	Brier	ECE	Conf Mean	Conf Std
0.1	0.9704	0.9635	0.9701	0.1076	0.0159	0.0149	0.9747	0.0708
1.0	0.9852	0.9831	0.9851	0.1221	0.0193	0.0526	0.9326	0.1242
2.0	0.9630	0.9547	0.9625	0.2798	0.0380	0.1789	0.7841	0.1233
5.0	0.9778	0.9724	0.9776	1.0048	0.2009	0.6112	0.3666	0.0109

Appendix

Table 55: Compliance domain ablation: uncertainty penalty

\alpha

top_k	Accuracy	Macro F1	Wtd F1	NLL	Brier	ECE	Conf Mean	Conf Std
1	0.7926	0.7500	0.7829	0.8798	0.1709	0.3466	0.4460	0.0424
3	0.9185	0.9091	0.9165	0.5037	0.0820	0.2882	0.6339	0.1111
5	0.9704	0.9696	0.9702	0.2800	0.0378	0.1909	0.7795	0.1288
10	0.9704	0.9614	0.9698	0.2779	0.0364	0.1932	0.7826	0.1160
20	0.9778	0.9724	0.9776	0.2680	0.0350	0.1873	0.7905	0.1190

Appendix

Table 56: Compliance domain ablation: evidence count (top_k).

Seed	Accuracy	ECE	NLL	Brier	Macro F1
42	0.9967	0.0111	0.0239	0.0032	0.9964
123	0.9967	0.0113	0.0221	0.0032	0.9964
456	0.9972	0.0119	0.0245	0.0030	0.9971
789	0.9972	0.0112	0.0249	0.0030	0.9971
1011	0.9967	0.0102	0.0216	0.0028	0.9965
2024	0.9967	0.0107	0.0245	0.0032	0.9965

Appendix

Table 57: Complete results for all 15 seeds (compliance domain, LPF-SPN).

Model	Accuracy	Macro F1	Wtd F1	NLL	ECE	RT (ms)
LPF-SPN	1.000	1.000	1.000	0.007	0.006	17.6
LPF-Learned	1.000	1.000	1.000	0.016	0.014	43.6
VAE-Only	0.993	0.993	0.993	0.166	0.138	7.9
BERT	—	—	—	—	—	—
SPN-Only	0.970	0.969	0.970	0.549	0.383	2.4
EDL-Aggregated	0.407	0.193	0.236	1.097	0.068	1.3

Appendix

Table 58: Model comparison (academic domain, seed 2024). Perfect 100% accuracy achieved by both LPF-SPN and LPF-Learned, demonstrating the system’s capability on well-structured numerical evidence.

n_samples	Accuracy	NLL	ECE	Runtime (ms)
4	0.9852	0.1576	0.1123	2.3
8	0.9852	0.1825	0.1335	2.9
16	0.9704	0.2330	0.1762	3.5
32	0.9926	0.2751	0.2201	5.8

Appendix

Table 59: Academic domain ablation: Monte Carlo sample count.

Temperature	Accuracy	NLL	ECE
0.8	0.9852	0.1628	0.1213
1.0	0.9852	0.2366	0.1834
1.2	0.9778	0.2982	0.2192
1.5	0.9852	0.3811	0.2901

Appendix

Table 60: Academic domain ablation: temperature scaling.

Alpha	Accuracy	NLL	ECE
0.1	0.9852	0.0398	0.0286
1.0	0.9926	0.0691	0.0477
2.0	0.9778	0.2310	0.1697
5.0	0.9852	0.9965	0.6158

Appendix

Table 61: Academic domain ablation: uncertainty penalty

\alpha

top_k	Accuracy	NLL	Runtime (ms)
1	0.8519	0.8414	1.9
3	0.9407	0.4774	2.7
5	0.9852	0.2295	3.5
10	0.9778	0.2394	5.2
20	0.9852	0.2400	8.1

Appendix

Table 62: Academic domain ablation: evidence count (top_k).

n_samples	Accuracy	Macro F1	Wtd F1	NLL	Brier	ECE	Conf Mean	Conf Std
4	0.9778	0.9775	0.9779	0.1593	0.0210	0.1056	0.8721	0.1274
8	0.9852	0.9842	0.9851	0.1759	0.0215	0.1291	0.8561	0.1219
16	0.9778	0.9762	0.9777	0.2319	0.0298	0.1683	0.8095	0.1291
32	0.9926	0.9921	0.9926	0.3005	0.0426	0.2322	0.7604	0.1460

Appendix

Table 63: Construction domain ablation: Monte Carlo sample count.

Temperature	Accuracy	Macro F1	Wtd F1	NLL	Brier	ECE	Conf Mean	Conf Std
0.8	0.9852	0.9842	0.9851	0.1648	0.0213	0.1252	0.8658	0.1387
1.0	0.9630	0.9638	0.9630	0.2488	0.0338	0.1653	0.8030	0.1342
1.2	0.9481	0.9447	0.9471	0.3016	0.0425	0.2029	0.7596	0.1370
1.5	0.9778	0.9761	0.9775	0.3817	0.0559	0.2967	0.6977	0.1225

Appendix

Table 64: Construction domain ablation: temperature scaling.

Alpha	Accuracy	Macro F1	Wtd F1	NLL	Brier	ECE	Conf Mean	Conf Std
0.1	0.9481	0.9479	0.9482	0.0918	0.0209	0.0327	0.9713	0.0854
1.0	0.9778	0.9785	0.9779	0.0996	0.0166	0.0509	0.9310	0.1263
2.0	0.9778	0.9774	0.9777	0.2204	0.0280	0.1641	0.8183	0.1327
5.0	0.9926	0.9921	0.9926	0.9921	0.1980	0.6215	0.3711	0.0129

Appendix

Table 65: Construction domain ablation: uncertainty penalty

\alpha

top_k	Accuracy	Macro F1	Wtd F1	NLL	Brier	ECE	Conf Mean	Conf Std
1	0.8889	0.8882	0.8882	0.8095	0.1544	0.4144	0.4745	0.0532
3	0.9333	0.9330	0.9333	0.4308	0.0680	0.2591	0.6798	0.1281
5	1.0000	1.0000	1.0000	0.2324	0.0292	0.1956	0.8044	0.1278
10	0.9778	0.9774	0.9777	0.2314	0.0306	0.1673	0.8105	0.1397
20	0.9704	0.9695	0.9705	0.2432	0.0326	0.1589	0.8115	0.1321

Appendix

Table 66: Construction domain ablation: evidence count (top_k).

Model	Accuracy	Macro F1	ECE	Runtime (ms)
LPF-SPN	1.000	1.000	0.014	18.1
LPF-Learned	0.985	0.985	0.017	41.0
VAE-Only	0.993	0.993	0.135	8.9
SPN-Only	0.970	0.968	0.388	2.2
EDL-Aggregated	0.363	0.178	0.026	1.4
EDL-Individual	0.356	0.175	0.145	4.3

Appendix

Table 67: Model comparison (construction domain, seed 314159).

Model	Total Errors	Total Predictions	Error Rate
LPF-SPN	0	135	0.000
LPF-Learned	2	135	0.015
VAE-Only	1	135	0.007
SPN-Only	4	135	0.030
EDL-Aggregated	86	135	0.637
EDL-Individual	87	135	0.644

Appendix

Table 68: Construction domain error counts by model.

Model	Accuracy	F1	ECE	Runtime (ms)
LPF-SPN	0.997	0.997	0.012	25.2
LPF-Learned	0.997	0.997	0.003	24.0
VAE-Only	0.997	0.997	0.003	3.5
SPN-Only	0.952	0.951	0.289	0.9
EDL-Aggregated	0.502	0.223	0.167	1.2
EDL-Individual	0.502	0.223	0.001	1.9

Appendix

Table 69: Model comparison (FEVER domain, seed 456). Near-perfect performance (99.7%) across all LPF variants and VAE-Only indicates FEVER provides very clean, unambiguous evidence signals.

n_samples	Accuracy	Macro F1	Wtd F1	NLL	Brier	ECE	Conf Mean	Conf Std
4	0.9778	0.9787	0.9778	0.1550	0.0183	0.1132	0.8761	0.1145
8	0.9778	0.9768	0.9777	0.2033	0.0247	0.1442	0.8336	0.1221
16	0.9778	0.9768	0.9777	0.2440	0.0303	0.1841	0.7988	0.1190
32	0.9778	0.9787	0.9778	0.3060	0.0410	0.2219	0.7559	0.1153

Appendix

Table 70: Finance domain ablation: Monte Carlo sample count.

Temperature	Accuracy	Macro F1	Wtd F1	NLL	Brier	ECE	Conf Mean	Conf Std
0.8	0.9852	0.9852	0.9852	0.1701	0.0193	0.1314	0.8588	0.1189
1.0	0.9852	0.9869	0.9852	0.2563	0.0325	0.1932	0.7919	0.1214
1.2	0.9852	0.9852	0.9852	0.3262	0.0451	0.2468	0.7384	0.1203
1.5	0.9778	0.9771	0.9778	0.4115	0.0609	0.3008	0.6770	0.1108

Appendix

Table 71: Finance domain ablation: temperature scaling.

Alpha	Accuracy	Macro F1	Wtd F1	NLL	Brier	ECE	Conf Mean	Conf Std
0.1	0.9852	0.9870	0.9852	0.0688	0.0118	0.0168	0.9725	0.0805
1.0	0.9852	0.9869	0.9852	0.0839	0.0112	0.0508	0.9344	0.1106
2.0	0.9852	0.9844	0.9853	0.2419	0.0295	0.1850	0.8002	0.1139
5.0	0.9778	0.9787	0.9778	0.9974	0.1992	0.6086	0.3691	0.0113

Appendix

Table 72: Finance domain ablation: uncertainty penalty

\alpha

top_k	Accuracy	Macro F1	Wtd F1	NLL	Brier	ECE	Conf Mean	Conf Std
1	0.8444	0.8397	0.8442	0.8551	0.1647	0.3771	0.4673	0.0431
3	0.9778	0.9779	0.9778	0.4613	0.0723	0.3240	0.6537	0.1111
5	0.9778	0.9787	0.9778	0.2465	0.0307	0.1819	0.7988	0.1162
10	0.9704	0.9674	0.9705	0.2541	0.0327	0.1823	0.7937	0.1286
20	0.9704	0.9704	0.9704	0.2607	0.0339	0.1804	0.7924	0.1299

Appendix

Table 73: Finance domain ablation: evidence count (top_k).

Model	Accuracy	Macro F1	ECE	Runtime (ms)
LPF-SPN	0.993	0.993	0.010	16.3
LPF-Learned	0.985	0.985	0.013	42.7
VAE-Only	0.985	0.985	0.111	8.3
SPN-Only	0.956	0.954	0.348	2.3
EDL-Aggregated	0.437	0.203	0.100	1.3
EDL-Individual	0.333	0.167	0.167	4.7

Appendix

Table 74: Model comparison (finance domain, seed 123).

Model	Total Errors	Total Predictions	Error Rate
LPF-SPN	1	135	0.007
LPF-Learned	2	135	0.015
VAE-Only	2	135	0.015
SPN-Only	6	135	0.044
EDL-Aggregated	76	135	0.563
EDL-Individual	90	135	0.667

Appendix

Table 75: Finance domain error counts by model.

n_samples	Accuracy	Macro F1	Wtd F1	NLL	Brier	ECE	Conf Mean	Conf Std
4	0.9778	0.9663	0.9776	0.1868	0.0243	0.1152	0.8626	0.1237
8	0.9704	0.9564	0.9707	0.2196	0.0284	0.1367	0.8337	0.1287
16	0.9556	0.9421	0.9560	0.2622	0.0355	0.1578	0.7977	0.1318
32	0.9852	0.9732	0.9852	0.3191	0.0434	0.2370	0.7481	0.1209

Appendix

Table 76: Healthcare domain ablation: Monte Carlo sample count.

Temperature	Accuracy	Macro F1	Wtd F1	NLL	Brier	ECE	Conf Mean	Conf Std
0.8	0.9704	0.9537	0.9706	0.1973	0.0256	0.1310	0.8504	0.1304
1.0	0.9852	0.9797	0.9851	0.2542	0.0325	0.1912	0.7940	0.1217
1.2	0.9852	0.9732	0.9852	0.3112	0.0413	0.2377	0.7475	0.1137
1.5	0.9630	0.9593	0.9628	0.4181	0.0623	0.2813	0.6817	0.1031

Appendix

Table 77: Healthcare domain ablation: temperature scaling.

Alpha	Accuracy	Macro F1	Wtd F1	NLL	Brier	ECE	Conf Mean	Conf Std
0.1	0.9852	0.9797	0.9851	0.0634	0.0093	0.0186	0.9770	0.0689
1.0	0.9852	0.9732	0.9852	0.0987	0.0135	0.0495	0.9429	0.0966
2.0	0.9704	0.9561	0.9705	0.2581	0.0331	0.1880	0.7945	0.1184
5.0	0.9704	0.9537	0.9706	1.0058	0.2011	0.6039	0.3664	0.0111

Appendix

Table 78: Healthcare domain ablation: uncertainty penalty

\alpha

top_k	Accuracy	Macro F1	Wtd F1	NLL	Brier	ECE	Conf Mean	Conf Std
1	0.8815	0.8440	0.8861	0.8235	0.1574	0.4147	0.4667	0.0434
3	0.9481	0.9400	0.9486	0.4821	0.0784	0.2988	0.6493	0.1134
5	0.9778	0.9728	0.9776	0.2567	0.0333	0.1801	0.7977	0.1222
10	0.9852	0.9732	0.9852	0.2360	0.0284	0.1797	0.8055	0.1148
20	0.9630	0.9466	0.9636	0.2541	0.0331	0.1585	0.8045	0.1209

Appendix

Table 79: Healthcare domain ablation: evidence count (top_k).

Model	Accuracy	Macro F1	ECE	Runtime (ms)
LPF-SPN	0.993	0.986	0.006	16.7
LPF-Learned	0.978	0.967	0.022	40.1
VAE-Only	0.985	0.973	0.127	8.0
SPN-Only	0.844	0.779	0.253	2.4
EDL-Aggregated	0.267	0.140	0.250	1.4
EDL-Individual	0.267	0.140	0.068	4.8

Appendix

Table 80: Model comparison (healthcare domain, seed 77777).

Model	Total Errors	Total Predictions	Error Rate
LPF-SPN	1	135	0.007
LPF-Learned	3	135	0.022
VAE-Only	2	135	0.015
SPN-Only	21	135	0.156
EDL-Aggregated	99	135	0.733
EDL-Individual	99	135	0.733

Appendix

Table 81: Healthcare domain error counts by model.

n_samples	Accuracy	Macro F1	Wtd F1	NLL	Brier	ECE	Conf Mean	Conf Std
4	0.9630	0.9617	0.9628	0.2007	0.0266	0.1388	0.8374	0.1339
8	0.9704	0.9698	0.9706	0.2476	0.0329	0.1841	0.7975	0.1358
16	0.9778	0.9769	0.9777	0.3043	0.0426	0.2219	0.7558	0.1365
32	0.9556	0.9554	0.9560	0.3461	0.0500	0.2352	0.7263	0.1286

Appendix

Table 82: Legal domain ablation: Monte Carlo sample count.

Temperature	Accuracy	Macro F1	Wtd F1	NLL	Brier	ECE	Conf Mean	Conf Std
0.8	0.9704	0.9680	0.9701	0.2199	0.0293	0.1475	0.8229	0.1359
1.0	0.9852	0.9850	0.9853	0.2860	0.0390	0.2253	0.7657	0.1333
1.2	0.9852	0.9843	0.9851	0.3613	0.0520	0.2742	0.7110	0.1208
1.5	0.9778	0.9774	0.9777	0.4612	0.0725	0.3344	0.6434	0.1098

Appendix

Table 83: Legal domain ablation: temperature scaling.

Alpha	Accuracy	Macro F1	Wtd F1	NLL	Brier	ECE	Conf Mean	Conf Std
0.1	0.9704	0.9699	0.9703	0.0584	0.0114	0.0195	0.9670	0.0925
1.0	0.9852	0.9843	0.9851	0.1277	0.0193	0.0739	0.9113	0.1272
2.0	0.9704	0.9692	0.9701	0.3021	0.0413	0.2259	0.7554	0.1320
5.0	0.9630	0.9629	0.9629	1.0032	0.2005	0.5957	0.3673	0.0105

Appendix

Table 84: Legal domain ablation: uncertainty penalty

\alpha

top_k	Accuracy	Macro F1	Wtd F1	NLL	Brier	ECE	Conf Mean	Conf Std
1	0.8741	0.8736	0.8746	0.8561	0.1652	0.4179	0.4562	0.0420
3	0.9037	0.9022	0.9036	0.5418	0.0921	0.3100	0.6071	0.1179
5	0.9704	0.9692	0.9701	0.3160	0.0451	0.2190	0.7514	0.1355
10	0.9778	0.9774	0.9777	0.2928	0.0389	0.2166	0.7612	0.1173
20	0.9852	0.9856	0.9852	0.3009	0.0414	0.2244	0.7608	0.1267

Appendix

Table 85: Legal domain ablation: evidence count (top_k).

Model	Accuracy	Macro F1	ECE	Runtime (ms)
LPF-SPN	0.993	0.992	0.011	16.7
LPF-Learned	1.000	1.000	0.006	44.7
VAE-Only	0.993	0.992	0.153	8.9
SPN-Only	0.970	0.970	0.371	2.4
EDL-Aggregated	0.326	0.164	0.011	1.4
EDL-Individual	0.289	0.149	0.211	4.7

Appendix

Table 86: Model comparison (legal domain, seed 314159). LPF-Learned achieves perfect accuracy on this domain, suggesting learned aggregation may better capture subtle legal reasoning patterns.

Model	Total Errors	Total Predictions	Error Rate
LPF-SPN	1	135	0.007
LPF-Learned	0	135	0.000
VAE-Only	1	135	0.007
SPN-Only	4	135	0.030
EDL-Aggregated	91	135	0.674
EDL-Individual	96	135	0.711

Appendix

Table 87: Legal domain error counts by model.

n_samples	Accuracy	Macro F1	Wtd F1	NLL	Brier	ECE	Conf Mean	Conf Std
4	0.9852	0.9824	0.9850	0.3127	0.0416	0.2363	0.7488	0.1139
8	0.9704	0.9678	0.9703	0.3623	0.0516	0.2527	0.7177	0.1194
16	0.9630	0.9592	0.9631	0.3914	0.0570	0.2693	0.6936	0.1083
32	0.9778	0.9725	0.9775	0.4150	0.0616	0.3041	0.6737	0.1071

Appendix

Table 88: Materials domain ablation: Monte Carlo sample count.

Temperature	Accuracy	Macro F1	Wtd F1	NLL	Brier	ECE	Conf Mean	Conf Std
0.8	0.9630	0.9592	0.9631	0.3145	0.0430	0.2261	0.7478	0.1275
1.0	0.9852	0.9842	0.9851	0.3910	0.0567	0.2993	0.6908	0.1125
1.2	0.9778	0.9756	0.9777	0.4658	0.0725	0.3385	0.6393	0.1031
1.5	0.9852	0.9824	0.9850	0.5554	0.0925	0.4031	0.5820	0.0858

Appendix

Table 89: Materials domain ablation: temperature scaling.

Alpha	Accuracy	Macro F1	Wtd F1	NLL	Brier	ECE	Conf Mean	Conf Std
0.1	0.9778	0.9725	0.9775	0.0706	0.0105	0.0352	0.9569	0.0872
1.0	0.9778	0.9756	0.9777	0.1485	0.0190	0.1000	0.8833	0.1210
2.0	0.9704	0.9678	0.9703	0.3886	0.0560	0.2921	0.6898	0.1102
5.0	0.9704	0.9678	0.9703	1.0288	0.2064	0.6127	0.3577	0.0080

Appendix

Table 90: Materials domain ablation: uncertainty penalty

\alpha

top_k	Accuracy	Macro F1	Wtd F1	NLL	Brier	ECE	Conf Mean	Conf Std
1	0.8741	0.8680	0.8746	0.9073	0.1776	0.4526	0.4215	0.0236
3	0.9481	0.9431	0.9484	0.6191	0.1082	0.3973	0.5509	0.0932
5	0.9852	0.9824	0.9850	0.3987	0.0584	0.2978	0.6874	0.1131
10	0.9630	0.9549	0.9625	0.3922	0.0568	0.2732	0.6898	0.1073
20	0.9778	0.9725	0.9775	0.3981	0.0582	0.2888	0.6889	0.1079

Appendix

Table 91: Materials domain ablation: evidence count (top_k).

Model	Accuracy	Macro F1	ECE	Runtime (ms)
LPF-SPN	0.993	0.992	0.015	18.8
LPF-Learned	0.985	0.982	0.016	41.8
VAE-Only	0.985	0.982	0.124	7.9
SPN-Only	0.963	0.955	0.369	2.5
EDL-Aggregated	0.481	0.217	0.108	1.4
EDL-Individual	0.481	0.217	0.146	4.3

Appendix

Table 92: Model comparison (materials domain, seed 314159).

Model	Total Errors	Total Predictions	Error Rate
LPF-SPN	1	135	0.007
LPF-Learned	2	135	0.015
VAE-Only	2	135	0.015
SPN-Only	5	135	0.037
EDL-Aggregated	70	135	0.519
EDL-Individual	70	135	0.519

Appendix

Table 93: Materials domain error counts by model.

Domain	n=4	n=8	n=16	n=32	Best
Compliance	97.8%	94.1%	96.3%	97.8%	4, 32
Academic	98.5%	98.5%	97.0%	99.3%	32
Construction	97.8%	98.5%	97.8%	99.3%	32
Finance	97.8%	97.8%	97.8%	97.8%	all
Healthcare	97.8%	97.0%	95.6%	98.5%	32
Legal	96.3%	97.0%	97.8%	95.6%	16

Appendix

Table 94: Accuracy by n_samples across domains. No consistent optimal sample count across domains. n=4 sufficient for simpler tasks (compliance, materials), while n=32 benefits more complex reasoning (academic, construction).

Domain	T=0.8	T=1.0	T=1.2	T=1.5	Best
Compliance	97.0%	97.0%	97.8%	97.0%	1.2
Academic	98.5%	98.5%	97.8%	98.5%	0.8, 1.0, 1.5
Construction	98.5%	96.3%	94.8%	97.8%	0.8
Finance	98.5%	98.5%	98.5%	97.8%	0.8, 1.0, 1.2
Healthcare	98.5%	98.5%	98.5%	96.3%	0.8, 1.0, 1.2
Legal	97.0%	98.5%	98.5%	97.8%	1.0, 1.2

Appendix

Table 95: Accuracy by temperature across domains.

T=0.8

and

T=1.0

most robust across domains. Higher temperatures (

T>1.0

) degrade calibration without improving accuracy.

Domain	$\alpha$ =0.1	$\alpha$ =1.0	$\alpha$ =2.0	$\alpha$ =5.0	Best
Compliance	97.0%	98.5%	96.3%	97.8%	1.0
Academic	98.5%	99.3%	97.8%	98.5%	1.0
Construction	94.8%	97.8%	97.8%	99.3%	5.0
Finance	98.5%	98.5%	98.5%	97.8%	0.1, 1.0, 2.0
Healthcare	98.5%	98.5%	97.0%	97.0%	0.1, 1.0
Legal	97.0%	98.5%	97.0%	96.3%	1.0

Appendix

Table 96: Accuracy by

\alpha

across domains.

\alpha=1.0

most consistent.

\alpha=0.1

best for calibration,

\alpha=1.0

best for accuracy.

\alpha=5.0

generally harmful except construction domain.

Domain	k=1	k=3	k=5	k=10	k=20	Best
Compliance	79.3%	91.9%	97.0%	97.0%	97.8%	20
Academic	85.2%	94.1%	98.5%	97.8%	98.5%	5, 20
Construction	88.9%	93.3%	100.0%	97.8%	97.0%	5
Finance	84.4%	97.8%	97.8%	97.0%	97.0%	3, 5
Healthcare	88.1%	94.8%	97.8%	98.5%	96.3%	10
Legal	87.4%	90.4%	97.0%	97.8%	98.5%	20

Appendix

Table 97: Accuracy by top_k across domains. Consistent pattern: dramatic improvement from

k=1

k=5

, then diminishing returns.

k=5

optimal for most domains.

Model	Accuracy	Parsed	Failed	Avg Conf	Avg RT (ms)
llama-3.3-70b	95.9%	49/50	1	N/A	1578.7
qwen3-32b	98.0%	50/50	0	N/A	3008.6
kimi-k2	98.0%	50/50	0	N/A	764.2
gpt-oss-120b	93.9%	49/50	1	N/A	1541.7

Appendix

Table 98: LLM performance by model (compliance domain, 50 samples). LLMs occasionally deviate from requested format despite explicit instructions. Failed parses treated as incorrect predictions. LLMs do not produce well-calibrated probability distributions; even when confidence scores are extracted, they show poor correlation with actual correctness (ECE 79.7–81.6%).

Model	Input Tok/Q	Output Tok/Q	Cost/1M In	Cost/1M Out	Cost/Q
llama-3.3-70b	$\sim$ 800	$\sim$ 50	$0.59	$0.79	$0.0004
qwen3-32b	$\sim$ 800	$\sim$ 50	$0.18	$0.18	$0.0003
kimi-k2	$\sim$ 800	$\sim$ 50	$0.30	$0.30	$0.0002
gpt-oss-120b	$\sim$ 800	$\sim$ 50	$0.80	$0.80	$0.0006

Appendix

Table 99: API cost breakdown (Groq pricing, January 2026). Total evaluation cost: 50 samples

\times

4 models

\approx

0.08. Extrapolation to full test set (135 samples

\times

4 models):

\approx

0.22. Production deployment cost (1M queries):

200–600/million queries vs.

0 for LPF-SPN (self-hosted).

Comparison	Mean Diff (Acc)	t-stat	p-value	Sig ( $\alpha$ =0.05)
LPF-SPN vs LPF-Learned	$+$ 6.7%	58.3	$<$ 0.001	✓
LPF-SPN vs VAE-Only	$+$ 2.2%	12.4	$<$ 0.001	✓
LPF-SPN vs BERT	$+$ 0.8%	3.2	0.006	✓
LPF-SPN vs SPN-Only	$+$ 2.8%	15.1	$<$ 0.001	✓

Appendix

Table 100: Paired t-test results comparing LPF-SPN to baselines (15 seeds). All differences statistically significant at

\alpha=0.05

level, confirming LPF-SPN’s superior performance is not due to random chance.

Model	Mean ECE	95% CI	Calibration Quality
LPF-SPN	2.1%	[1.7%, 2.5%]	Excellent
LPF-Learned	6.6%	[5.8%, 7.4%]	Good
VAE-Only	9.6%	[8.2%, 11.0%]	Fair
BERT	3.2%	[2.8%, 3.6%]	Good
SPN-Only	30.9%	[28.4%, 33.4%]	Poor

Appendix

Table 101: ECE comparison with confidence intervals (compliance domain). Non-overlapping confidence intervals confirm LPF-SPN achieves significantly better calibration than all baselines.

Component	Training Time	CPU Cores	Memory (GB)	Storage (GB)
VAE Encoder	15 min	8	4	0.5
Decoder Network	25 min	8	6	0.8
Learned Aggregator	10 min	8	3	0.3
Total	50 min	8	6	1.6

Appendix

Table 102: Training resource consumption (per seed, compliance domain). Extrapolation to all seeds: 15 seeds

\times

50 min = 12.5 hours total; parallelizable to

\approx

2 hours wall time on a 64-core machine running 8 seeds simultaneously.

Model	CPU Time (ms)	Memory (MB)	Network I/O
LPF-SPN	14.8	1200	0
LPF-Learned	37.4	1800	0
BERT	134.7	4500	0
LLM (Groq)	1500–3000	$\sim$ 0 (remote)	$\sim$ 10 KB

Appendix

Table 103: Inference resource consumption per query. LPF-SPN throughput: 68 queries/second/core; single 64-core machine: 4,352 queries/second; daily capacity: 376M queries.

Hyperparameter	Type	Search Space	Default	Sampling
n_samples	int	[4, 8, 16, 32]	16	Grid
temperature	float	[0.8, 1.0, 1.2, 1.5]	1.0	Grid
alpha	float	[0.1, 1.0, 2.0, 5.0]	2.0	Grid
top_k	int	[1, 3, 5, 10, 20]	5	Grid
learning_rate	float	[1e-4, 2e-4, 5e-4]	2e-4	Grid
latent_dim	int	[32, 64, 128]	64	Grid

Appendix

Table 104: Complete hyperparameter search space. Total configurations tested:

4\times 4\times 4\times 5=320

(ablation study). Compute budget: 320 configs

\times

50 min = 267 hours (

\approx

11 days serial;

\approx

1.5 days with 8 parallel workers).

Domain	n_samples	temperature	alpha	top_k
Compliance	16	0.8	0.1	5
Academic	4	0.8	0.1	5
Construction	4	0.8	0.1	5
Finance	4	0.8	0.1	5
Healthcare	4	0.8	0.1	5
Legal	4	0.8	0.1	5

Appendix

Table 105: Optimal hyperparameters per domain. Remarkable consistency across domains: configuration (n=4, T=0.8,

\alpha

=0.1, k=5) works well for 7/8 domains. Only compliance benefits from higher n_samples.

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Jun 29, 2026

BayesBench: Evaluating LLM Belief Trajectories Under Multi-Turn Evidence Accumulation

Large language models (LLMs) are typically deployed in multi-turn conversations, where each turn provides new evidence that should reduce epistemic uncertainty about their environment. Acting rationally then requires inferring the unobserved quantities that govern it and updating beliefs about them as evidence accumulates. Yet most evaluations only score the model's final-turn answer in a single-turn format, leaving this process unexamined. We ask how closely LLMs' belief updates match those of a rational Bayesian reasoner in multi-turn settings, and introduce BayesBench, a suite of simulation environments that probe this across three progressively complex tasks: (i) Bayesian estimation, where the model infers an unknown parameter from sequential evidence; (ii) Bayesian prediction, where the model turns inferred beliefs about a latent variable into outcome forecasts; and (iii) latent-framed Bayesian prediction, where observations are filtered through a user-persona framing, requiring joint inference over the latent state and the persona. Across seven LLMs (3B--70B), scaling improves latent inference and evidence accumulation, with updates occasionally matching the Bayesian posterior. However, these gains do not reliably carry over to downstream prediction, exposing a gap between inferring latent structure and using it to rationally update beliefs about the target outcome.

Ankur Samanta, Akshayaa Magesh, Tal Lancewicki +7

Meta AI · Columbia University · Work done at Meta +2

LLM Evaluation Bayesian Inference Belief Updating in LLMs LLM Reasoning

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