Bayesian BM25

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The reference implementation of the Bayesian BM25, From Bayesian Inference to Neural Computation, and Vector Scores as Likelihood Ratios papers, by the original author. Converts raw BM25 retrieval scores into calibrated relevance probabilities using Bayesian inference.

Overview

Standard BM25 produces unbounded scores that lack consistent meaning across queries, making threshold-based filtering and multi-signal fusion unreliable. Bayesian BM25 addresses this by applying a sigmoid likelihood model with a composite prior (term frequency + document length normalization) and computing Bayesian posteriors that output well-calibrated probabilities in [0, 1]. A corpus-level base rate prior further improves calibration by 68–77% without requiring relevance labels.

Key capabilities:

• Score-to-probability transform — convert raw BM25 scores into calibrated relevance probabilities via sigmoid likelihood + composite prior + Bayesian posterior
• Base rate calibration — corpus-level base rate prior estimated from score distribution (95th percentile, mixture model, or elbow detection) decomposes the posterior into three additive log-odds terms, reducing expected calibration error by 68–77% without relevance labels
• Parameter learning — batch gradient descent or online SGD with EMA-smoothed gradients and Polyak averaging, with three training modes: balanced (C1), prior-aware (C2), and prior-free (C3)
• Probabilistic fusion — combine multiple probability signals using AND, OR, NOT, and log-odds conjunction with multiplicative confidence scaling, optional per-signal reliability weights (Log-OP), and sparse signal gating (ReLU/Swish/GELU/Softplus activations from Jeong, 2026b, Theorems 6.5.3/6.7.4/6.8.1/Remark 6.5.4) with generalized beta control (Theorem 6.7.6)
• Learnable fusion weights — LearnableLogOddsWeights learns per-signal reliability from labeled data via a Hebbian gradient that is backprop-free, starting from Naive Bayes uniform initialization (Remark 5.3.2); supports optional base_rate additive bias in log-odds space
• Attention-based fusion — AttentionLogOddsWeights learns query-dependent signal weights via attention mechanism (Jeong, 2026b, Section 8), with exact attention pruning via compute_upper_bounds() and prune() (Theorem 8.7.1); supports optional base_rate
• Multi-head attention — MultiHeadAttentionLogOddsWeights creates multiple independent attention heads with different initializations and averages their log-odds for more robust fusion (Remark 8.6, Corollary 8.7.2)
• Neural score calibration — PlattCalibrator (sigmoid) and IsotonicCalibrator (PAVA) convert raw neural model scores into calibrated probabilities for Bayesian fusion (Section 12.2 #5)
• External prior features — prior_fn callable on BayesianProbabilityTransform allows custom document priors to replace the composite prior, enabling features like recency or popularity weighting (Section 12.2 #6)
• Temporal adaptation — TemporalBayesianTransform uses exponential decay to weight recent observations more heavily in fit(), tracking concept drift in non-stationary relevance patterns (Section 12.2 #3)
• Hybrid search — cosine_to_probability() converts vector similarity scores to probabilities for fusion with BM25 signals via weighted log-odds conjunction
• Vector score calibration — VectorProbabilityTransform converts vector distances into calibrated probabilities via likelihood ratio framework: P(R|d) = sigmoid(log(f_R(d) / f_G(d)) + logit(P_base)), with KDE/GMM density estimation, gap detection, and auto-routing (Jeong, 2026c, Theorem 3.1.1); calibrate_with_sample() decouples density estimation from evaluation points for index-aware calibration where ANN neighborhoods inform the density model
• Index-aware density priors — ivf_density_prior() and knn_density_prior() provide density-based prior weights from IVF cell populations and k-NN distances for informing the vector calibration (Jeong, 2026c, Strategy 4.6.2)
• WAND pruning — wand_upper_bound() computes safe Bayesian probability upper bounds for document pruning in top-k retrieval; BlockMaxIndex provides tighter block-level bounds for BMW-style pruning (Section 6.2, Corollary 7.4.2)
• Calibration metrics — expected_calibration_error(), brier_score(), reliability_diagram(), and calibration_report() for evaluating probability quality, with CalibrationReport bundling all metrics into a single diagnostic
• Fusion debugger — FusionDebugger records every intermediate value through the full pipeline (likelihood, prior, posterior, fusion) for transparent inspection, document comparison, and crossover detection; supports hierarchical fusion tracing with AND/OR/NOT composition and gating trace fields
• Multi-field search — MultiFieldScorer maintains separate BM25 indexes per field and fuses field-level probabilities via log-odds conjunction with configurable per-field weights
• Search integration — drop-in scorer wrapping bm25s that returns probabilities instead of raw scores

Adoption

• Apache Lucene — merged as BayesianScoreQuery and LogOddsFusionQuery for probabilistic hybrid search
• MTEB — included as a baseline retrieval model (bb25) for the Massive Text Embedding Benchmark
• txtai — used for BM25 score normalization in hybrid search (normalize="bayesian-bm25")
• Vespa.ai — adopted as an official sample application
• UQA — used as the scoring operator for probabilistic text retrieval and multi-signal fusion in the unified query algebra

Installation

pip install bayesian-bm25

To use the integrated search scorer (requires bm25s):

pip install bayesian-bm25[scorer]

Quick Start

Converting BM25 Scores to Probabilities

import numpy as np
from bayesian_bm25 import BayesianProbabilityTransform

transform = BayesianProbabilityTransform(alpha=1.5, beta=1.0, base_rate=0.01)

scores = np.array([0.5, 1.0, 1.5, 2.0, 3.0])
tfs = np.array([1, 2, 3, 5, 8])
doc_len_ratios = np.array([0.3, 0.5, 0.8, 1.0, 1.5])

probabilities = transform.score_to_probability(scores, tfs, doc_len_ratios)

End-to-End Search with Probabilities

from bayesian_bm25 import BayesianBM25Scorer

corpus_tokens = [
    ["python", "machine", "learning"],
    ["deep", "learning", "neural", "networks"],
    ["data", "visualization", "tools"],
]

scorer = BayesianBM25Scorer(k1=1.2, b=0.75, method="lucene", base_rate="auto")
scorer.index(corpus_tokens, show_progress=False)

doc_ids, probabilities = scorer.retrieve([["machine", "learning"]], k=3)

Multi-Field Search

from bayesian_bm25 import MultiFieldScorer

documents = [
    {"title": ["bayesian", "bm25"], "body": ["probabilistic", "framework", "search"]},
    {"title": ["neural", "networks"], "body": ["deep", "learning", "models"]},
    {"title": ["information", "retrieval"], "body": ["search", "ranking", "relevance"]},
]

scorer = MultiFieldScorer(
    fields=["title", "body"],
    field_weights={"title": 0.4, "body": 0.6},
    k1=1.2, b=0.75, method="lucene",
)
scorer.index(documents, show_progress=False)
doc_ids, probabilities = scorer.retrieve(["bayesian", "search"], k=3)

Combining Multiple Signals

import numpy as np
from bayesian_bm25 import log_odds_conjunction, prob_and, prob_not, prob_or

signals = np.array([0.85, 0.70, 0.60])

prob_and(signals)                # 0.357 (shrinkage problem)
log_odds_conjunction(signals)    # 0.773 (agreement-aware)

# Exclusion query: "python AND NOT java"
p_python, p_java = 0.90, 0.75
prob_and(np.array([p_python, prob_not(p_java)]))  # 0.225

Hybrid Text + Vector Search

import numpy as np
from bayesian_bm25 import cosine_to_probability, log_odds_conjunction

# BM25 probabilities (from Bayesian BM25)
bm25_probs = np.array([0.85, 0.60, 0.40])

# Vector search cosine similarities -> probabilities
cosine_scores = np.array([0.92, 0.35, 0.70])
vector_probs = cosine_to_probability(cosine_scores)  # [0.96, 0.675, 0.85]

# Fuse with reliability weights (BM25 weight=0.6, vector weight=0.4)
stacked = np.stack([bm25_probs, vector_probs], axis=-1)
fused = log_odds_conjunction(stacked, weights=np.array([0.6, 0.4]))

# Fuse with weights and confidence scaling (alpha + weights compose)
fused = log_odds_conjunction(stacked, alpha=0.5, weights=np.array([0.6, 0.4]))

# Gated fusion: ReLU/Swish/GELU/Softplus activation in logit space (Jeong, 2026b, Theorems 6.5-6.8)
fused_relu = log_odds_conjunction(stacked, gating="relu")         # MAP estimation
fused_swish = log_odds_conjunction(stacked, gating="swish")       # Bayes estimation
fused_gelu = log_odds_conjunction(stacked, gating="gelu")         # Gaussian noise model
fused_softplus = log_odds_conjunction(stacked, gating="softplus") # evidence-preserving

# Generalized beta controls gate sharpness (Theorem 6.7.6)
# beta -> 0: x/2 (maximum ignorance), beta=1: standard form, beta -> inf: ReLU
fused_soft = log_odds_conjunction(stacked, gating="swish", gating_beta=0.5)

# Softplus for small datasets: preserves all evidence (Remark 6.5.4)
# softplus(x) > x for all finite x, so consider lower alpha to compensate
fused_sp = log_odds_conjunction(stacked, gating="softplus", gating_beta=2.0)

Vector Score Calibration

import numpy as np
from bayesian_bm25 import VectorProbabilityTransform

# Estimate background distribution from corpus distances
corpus_distances = np.random.normal(0.8, 0.15, size=10000)
vpt = VectorProbabilityTransform.fit_background(corpus_distances, base_rate=0.01)

# Calibrate query-document distances via likelihood ratio
query_distances = np.array([0.3, 0.5, 0.7, 0.9, 1.1])
probabilities = vpt.calibrate(query_distances)

# With BM25 probability weights for informed density estimation
bm25_probs = np.array([0.85, 0.60, 0.40, 0.20, 0.10])
probabilities = vpt.calibrate(query_distances, weights=bm25_probs)

# Explicit KDE or GMM estimation
probabilities_kde = vpt.calibrate(query_distances, method="kde")
probabilities_gmm = vpt.calibrate(query_distances, method="gmm")

# Index-aware density priors for IVF / HNSW indexes
from bayesian_bm25 import ivf_density_prior, knn_density_prior

cell_prior = ivf_density_prior(cell_population=150, avg_population=100)
knn_prior = knn_density_prior(kth_distance=0.5, global_median_kth=0.8)

# Use density prior to inform calibration
probabilities = vpt.calibrate(query_distances, density_prior=np.full(5, cell_prior))

# Index-aware calibration: density estimated from local ANN sample,
# probabilities produced for a separate evaluation set
sample_distances = np.array([0.10, 0.15, 0.20, 0.50, 0.75, 0.80, 0.85])
eval_distances = np.array([0.12, 0.30, 0.70])
probabilities = vpt.calibrate_with_sample(
    eval_distances, sample_distances, weights=bm25_probs[:3],
)

Learning Fusion Weights from Data

import numpy as np
from bayesian_bm25 import LearnableLogOddsWeights

# 3 retrieval signals: BM25, vector search, metadata match
learner = LearnableLogOddsWeights(n_signals=3, alpha=0.0)
# Initial weights are uniform: [0.333, 0.333, 0.333]

# Batch fit from labeled data (probs: m x 3, labels: m)
learner.fit(training_probs, training_labels, learning_rate=0.1)
# Learned weights reflect signal reliability: [0.70, 0.19, 0.11]

# Online refinement from streaming feedback
for probs, label in feedback_stream:
    learner.update(probs, label, learning_rate=0.05, momentum=0.9)

# Inference with Polyak-averaged weights for stability
fused = learner(test_probs, use_averaged=True)

Attention-Based Fusion

import numpy as np
from bayesian_bm25 import AttentionLogOddsWeights

# 2 retrieval signals, 3 query features, per-signal logit normalization
attn = AttentionLogOddsWeights(
    n_signals=2, n_query_features=3, alpha=0.5, normalize=True,
)

# Train on labeled data with query features
# training_probs: (m, 2), training_labels: (m,), query_features: (m, 3)
attn.fit(training_probs, training_labels, query_features,
         learning_rate=0.01, max_iterations=500)

# Query-dependent fusion: weights adapt per query
fused = attn(test_probs, test_features, use_averaged=True)

Multi-Head Attention Fusion

import numpy as np
from bayesian_bm25 import MultiHeadAttentionLogOddsWeights

# 4 heads, 2 signals, 3 query features
mh = MultiHeadAttentionLogOddsWeights(
    n_heads=4, n_signals=2, n_query_features=3, alpha=0.5,
)

# Train all heads on the same data (different init -> different solutions)
mh.fit(training_probs, training_labels, query_features,
       learning_rate=0.01, max_iterations=500)

# Inference: heads produce fused log-odds independently, then average + sigmoid
fused = mh(test_probs, test_features, use_averaged=True)

# Attention pruning: safely eliminate candidates below a threshold
surviving_idx, fused_probs = mh.prune(
    candidate_probs, query_features, threshold=0.5,
    upper_bound_probs=candidate_upper_bounds,
)

Neural Score Calibration

from bayesian_bm25.calibration import PlattCalibrator, IsotonicCalibrator
from bayesian_bm25 import log_odds_conjunction

# Platt scaling: P = sigmoid(a * score + b)
platt = PlattCalibrator()
platt.fit(neural_scores, labels, learning_rate=0.01, max_iterations=1000)
calibrated = platt.calibrate(new_scores)  # output in (0, 1)

# Isotonic regression: non-parametric monotone mapping via PAVA
iso = IsotonicCalibrator()
iso.fit(neural_scores, labels)
calibrated = iso.calibrate(new_scores)

# Combine calibrated neural scores with BM25 probabilities
stacked = np.stack([bm25_probs, calibrated], axis=-1)
fused = log_odds_conjunction(stacked)

Temporal Adaptation

from bayesian_bm25.probability import TemporalBayesianTransform

# Short half-life: adapt quickly to changing relevance patterns
transform = TemporalBayesianTransform(
    alpha=1.0, beta=0.0, decay_half_life=100.0,
)

# Batch fit with timestamps: recent data gets more weight
transform.fit(scores, labels, timestamps=timestamps)

# Online update: timestamp auto-increments, Polyak decay reduces over time
for score, label in feedback_stream:
    transform.update(score, label)

WAND Pruning with Bayesian Upper Bounds

from bayesian_bm25 import BayesianProbabilityTransform

transform = BayesianProbabilityTransform(alpha=1.5, beta=2.0, base_rate=0.01)

# Standard BM25 upper bound per query term
bm25_upper_bound = 5.0

# Bayesian upper bound for safe pruning — any document's actual
# probability is guaranteed to be at most this value
bayesian_bound = transform.wand_upper_bound(bm25_upper_bound)

Debugging the Fusion Pipeline

from bayesian_bm25 import BayesianProbabilityTransform
from bayesian_bm25.debug import FusionDebugger

transform = BayesianProbabilityTransform(alpha=0.45, beta=6.10, base_rate=0.02)
debugger = FusionDebugger(transform)

# Trace a single document through the full pipeline
trace = debugger.trace_document(
    bm25_score=8.42, tf=5, doc_len_ratio=0.60,
    cosine_score=0.74, doc_id="doc-42",
)
print(debugger.format_trace(trace))

# Compare two documents to see which signal drove the rank difference
trace_a = debugger.trace_document(bm25_score=8.42, tf=5, doc_len_ratio=0.60, cosine_score=0.74)
trace_b = debugger.trace_document(bm25_score=5.10, tf=2, doc_len_ratio=1.20, cosine_score=0.88)
comparison = debugger.compare(trace_a, trace_b)
print(debugger.format_comparison(comparison))

# Hierarchical fusion: AND(OR(title, body), vector, NOT(spam))
step1 = debugger.trace_fusion([0.85, 0.70], names=["title", "body"], method="prob_or")
step2 = debugger.trace_not(0.90, name="spam")
step3 = debugger.trace_fusion(
    [step1.fused_probability, 0.80, step2.complement],
    names=["OR(title,body)", "vector", "NOT(spam)"],
    method="prob_and",
)

Evaluating Calibration Quality

import numpy as np
from bayesian_bm25 import (
    expected_calibration_error, brier_score, reliability_diagram, calibration_report,
)

probabilities = np.array([0.9, 0.8, 0.3, 0.1, 0.7, 0.2])
labels = np.array([1.0, 1.0, 0.0, 0.0, 1.0, 0.0])

ece = expected_calibration_error(probabilities, labels)   # lower is better
bs = brier_score(probabilities, labels)                   # lower is better
bins = reliability_diagram(probabilities, labels, n_bins=5)  # (avg_pred, avg_actual, count)

# One-call diagnostic report
report = calibration_report(probabilities, labels)
print(report.summary())   # formatted text with ECE, Brier, and reliability table

Online Learning from User Feedback

from bayesian_bm25 import BayesianProbabilityTransform

transform = BayesianProbabilityTransform(alpha=1.0, beta=0.0)

# Batch warmup on historical data
transform.fit(historical_scores, historical_labels)

# Online refinement from live feedback
for score, label in feedback_stream:
    transform.update(score, label, learning_rate=0.01, momentum=0.95)

# Use Polyak-averaged parameters for stable inference
alpha = transform.averaged_alpha
beta = transform.averaged_beta

Training Modes

from bayesian_bm25 import BayesianProbabilityTransform

transform = BayesianProbabilityTransform(alpha=1.0, beta=0.0)

# C1 (balanced, default): train on sigmoid likelihood
transform.fit(scores, labels, mode="balanced")

# C2 (prior-aware): train on full Bayesian posterior
transform.fit(scores, labels, mode="prior_aware", tfs=tfs, doc_len_ratios=ratios)

# C3 (prior-free): train on likelihood, inference uses prior=0.5
transform.fit(scores, labels, mode="prior_free")

Benchmarks

BEIR Hybrid Search

Evaluated on 5 BEIR datasets using the retrieve-then-evaluate protocol (top-1000 per signal, union candidates, pytrec_eval). Dense encoder: all-MiniLM-L6-v2. BM25: k1=1.2, b=0.75, Lucene variant with Snowball English stemmer.

NDCG@10

Method	ArguAna	FiQA	NFCorpus	SciDocs	SciFact	Average
BM25	36.13	25.31	31.82	15.63	68.02	35.38
Dense	36.98	36.87	31.59	21.64	64.51	38.32
Convex	40.01	37.10	35.60	19.67	73.37	41.15
RRF	39.61	36.85	34.43	20.11	71.43	40.49
Bayesian-OR	0.06	25.52	33.46	15.89	66.95	28.38
Bayesian-LogOdds	0.92	32.59	35.31	18.44	72.00	31.85
Bayesian-LogOdds-Local	39.79	37.19	34.10	19.51	73.80	40.88
Bayesian-LogOdds-BR	2.55	32.82	30.99	18.49	71.80	31.33
Bayesian-Balanced	37.27	40.58	35.73	21.42	72.47	41.50
Bayesian-Balanced-Mix	37.29	40.66	35.70	21.53	72.33	41.50
Bayesian-Balanced-Elbow	37.29	40.56	35.76	21.42	72.46	41.50
Bayesian-Gated-ReLU	35.16	27.54	32.45	17.08	69.01	36.25
Bayesian-Gated-Swish	36.20	27.39	28.66	16.82	68.61	35.54
Bayesian-Gated-GELU	36.07	27.62	30.85	17.08	69.40	36.20
Bayesian-Gated-Swish-B2	35.53	27.65	31.34	17.14	69.38	36.21
Bayesian-Gated-Softplus	34.11	27.05	34.63	16.67	69.08	36.31
Bayesian-Attention	37.05	38.86	34.37	21.05	70.51	40.37
Bayesian-Attn-Norm	37.22	40.53	35.42	21.91	73.24	41.67
Bayesian-Attn-Norm-CV	37.22	40.51	35.37	21.97	72.57	41.53
Bayesian-MultiHead	37.04	39.28	34.31	21.18	70.48	40.46
Bayesian-MultiHead-Norm	37.13	39.05	35.70	21.78	70.59	40.85
Bayesian-MultiField	7.41	--	31.16	15.68	60.06	28.58*
Bayesian-MultiField-Bal	38.40	--	34.49	20.93	66.83	40.16*
Bayesian-Vector-Balanced	27.39	33.67	29.50	18.51	66.06	35.03
Bayesian-Vector-Softplus	22.47	34.43	32.15	18.94	68.56	35.31
Bayesian-Vector-Attn	37.66	39.81	34.82	21.94	71.34	41.11

MAP@10

Method	ArguAna	FiQA	NFCorpus	SciDocs	SciFact	Average
BM25	23.84	19.10	11.76	9.15	63.38	25.45
Dense	24.46	29.14	11.05	12.94	59.59	27.44
Convex	26.76	29.21	13.46	11.79	69.12	30.07
RRF	26.30	28.85	12.84	11.98	66.58	29.31
Bayesian-OR	0.03	19.09	12.41	9.19	61.70	20.49
Bayesian-LogOdds	0.48	25.15	13.40	10.93	67.17	23.43
Bayesian-LogOdds-Local	26.60	29.32	12.31	11.70	69.29	29.84
Bayesian-LogOdds-BR	1.62	25.42	11.50	10.97	67.16	23.33
Bayesian-Balanced	24.61	32.73	13.80	12.85	68.03	30.40
Bayesian-Balanced-Mix	24.62	32.77	13.79	12.93	67.84	30.39
Bayesian-Balanced-Elbow	24.62	32.72	13.80	12.85	68.02	30.40
Bayesian-Gated-ReLU	22.95	21.00	11.67	10.02	64.10	25.95
Bayesian-Gated-Swish	23.86	20.88	10.23	9.85	63.80	25.73
Bayesian-Gated-GELU	23.77	21.04	10.88	10.03	64.60	26.06
Bayesian-Gated-Swish-B2	23.40	21.07	11.10	10.06	64.58	26.04
Bayesian-Gated-Softplus	22.38	20.54	13.04	9.79	64.11	25.97
Bayesian-Attention	24.49	30.96	12.68	12.60	65.92	29.33
Bayesian-Attn-Norm	24.57	32.62	13.40	13.22	68.91	30.54
Bayesian-Attn-Norm-CV	24.58	32.58	13.39	13.24	68.05	30.37
Bayesian-MultiHead	24.48	31.34	12.66	12.70	65.89	29.41
Bayesian-MultiHead-Norm	24.53	31.18	13.79	13.08	66.12	29.74
Bayesian-MultiField	4.76	--	11.45	9.04	55.34	20.15*
Bayesian-MultiField-Bal	25.45	--	13.04	12.57	63.21	28.57*
Bayesian-Vector-Balanced	19.79	26.13	11.13	10.76	60.84	25.73
Bayesian-Vector-Softplus	15.92	27.21	11.35	11.32	63.87	25.93
Bayesian-Vector-Attn	25.32	31.62	13.09	13.18	66.56	29.96

Recall@10

Method	ArguAna	FiQA	NFCorpus	SciDocs	SciFact	Average
BM25	75.04	31.98	14.46	16.34	80.78	43.72
Dense	76.53	44.13	15.50	23.09	78.33	47.52
Convex	81.65	45.04	17.06	20.62	84.89	49.85
RRF	81.65	45.03	16.87	21.15	84.76	49.89
Bayesian-OR	0.14	32.71	15.98	16.76	81.37	29.39
Bayesian-LogOdds	2.42	40.56	17.24	19.40	84.96	32.92
Bayesian-LogOdds-Local	81.37	45.22	16.29	20.42	86.22	49.90
Bayesian-LogOdds-BR	5.69	40.67	15.01	19.32	84.29	33.00
Bayesian-Balanced	77.31	47.61	17.23	22.61	84.83	49.92
Bayesian-Balanced-Mix	77.38	47.61	17.26	22.73	84.83	49.96
Bayesian-Balanced-Elbow	77.38	47.56	17.24	22.63	84.83	49.93
Bayesian-Gated-ReLU	74.04	34.39	16.03	17.79	82.58	44.97
Bayesian-Gated-Swish	75.39	34.21	13.88	17.43	81.91	44.56
Bayesian-Gated-GELU	75.11	34.62	14.85	17.75	82.64	44.99
Bayesian-Gated-Swish-B2	74.04	34.65	15.09	17.83	82.64	44.85
Bayesian-Gated-Softplus	71.48	33.99	17.24	17.22	82.87	44.56
Bayesian-Attention	76.74	46.60	17.09	22.23	83.04	49.14
Bayesian-Attn-Norm	77.24	47.43	17.05	23.24	84.69	49.93
Bayesian-Attn-Norm-CV	77.24	47.50	17.04	23.39	84.71	49.98
Bayesian-MultiHead	76.74	47.04	17.08	22.37	83.04	49.25
Bayesian-MultiHead-Norm	76.96	46.45	17.26	23.20	83.00	49.37
Bayesian-MultiField	16.43	--	14.64	16.68	72.87	30.16*
Bayesian-MultiField-Bal	79.30	--	16.84	22.03	76.63	48.70*
Bayesian-Vector-Balanced	51.21	41.71	13.56	20.16	81.07	41.54
Bayesian-Vector-Softplus	42.75	40.79	15.44	19.45	81.56	39.99
Bayesian-Vector-Attn	76.81	47.98	16.90	23.24	84.36	49.86

*Bayesian-MultiField/Bayesian-MultiField-Bal average over 4 datasets (FiQA corpus lacks title field).

All methods above are zero-shot (no relevance labels required). With --tune, additional supervised methods are evaluated:

Method	ArguAna	FiQA	NFCorpus	SciDocs	SciFact	NDCG@10 Avg
Bayesian-Balanced-Tuned	37.29	40.49	35.65	22.03	72.70	41.63
Bayesian-Hybrid-AND-Tuned	37.13	28.37	34.44	16.82	69.34	37.22
Bayesian-Tuned	0.79	24.76	32.11	15.68	67.67	28.20

Delta vs BM25 (NDCG@10)

Method	Type	Delta
Bayesian-Attn-Norm	zero-shot	+6.28
Bayesian-Attn-Norm-CV	zero-shot	+6.14
Bayesian-Balanced-Mix	zero-shot	+6.12
Bayesian-Balanced-Elbow	zero-shot	+6.12
Bayesian-Balanced	zero-shot	+6.11
Convex	zero-shot	+5.76
Bayesian-Vector-Attn	zero-shot	+5.73
Bayesian-LogOdds-Local	zero-shot	+5.50
Bayesian-MultiHead-Norm	zero-shot	+5.47
RRF	zero-shot	+5.11
Bayesian-MultiHead	zero-shot	+5.08
Bayesian-Attention	zero-shot	+4.99
Dense	zero-shot	+2.94
Bayesian-MultiField-Bal	zero-shot	+2.26*
Bayesian-Gated-Softplus	zero-shot	+0.93
Bayesian-Gated-ReLU	zero-shot	+0.86
Bayesian-Gated-Swish-B2	zero-shot	+0.82
Bayesian-Gated-GELU	zero-shot	+0.82
Bayesian-Gated-Swish	zero-shot	+0.16

*Bayesian-MultiField-Bal delta computed over 4 datasets (FiQA corpus lacks title field).

Vector Calibration Experiments (Jeong, 2026c)

The following experiments evaluate the vector calibration framework from Jeong, 2026c ("Vector Scores as Likelihood Ratios"). All methods use VectorProbabilityTransform with additive log-odds fusion (Theorem 7.1.1).

Calibration baselines (Section 8.2) — monotone transforms that preserve ranking but differ in calibration quality (ECE / Brier / LogLoss, lower is better):

Method	ArguAna ECE	FiQA ECE	NFCorpus ECE	SciDocs ECE	SciFact ECE
Dense-Kappa (global sigmoid)	0.009	0.021	0.231	0.210	0.032
Dense-Arctan	0.186	0.237	0.463	0.132	0.232
Dense-Platt (supervised)	0.065	0.075	0.097	0.165	0.074

Conditional independence penalty (Section 8.4, Stage 6) — compares structurally independent signals (IVF density prior / gap detection) vs cross-modal BM25 weights that violate the CI assumption (Assumption 4.2.1):

Method	ArguAna	FiQA	NFCorpus	SciDocs	SciFact	Average
VPT-DensityPrior (CI-compliant)	1.66	17.76	25.90	12.75	42.12	20.04
VPT-BM25Weights (CI-violating)	0.02	24.38	35.61	13.53	59.95	26.70

The BM25-weighted estimator outperforms the density-prior-only estimator on 4 of 5 datasets despite violating conditional independence — the information gain from cross-modal lexical signal dominates the bias from dependence. On ArguAna, the relationship reverses: VPT-DensityPrior (1.66) outperforms VPT-BM25Weights (0.02) because counter-argument retrieval makes BM25 an adversarial signal — the calibration framework correctly propagates the misleading weights into degraded probabilities. This bidirectional result validates that the likelihood ratio calibration faithfully reflects input signal quality: it amplifies informative weights and exposes harmful ones.

Bandwidth ablation (Section 8.4, Stage 7) — Silverman bandwidth scaling factor c in {0.2, 0.5, 1.0, 2.0} (Remark 4.4.2):

Method	ArguAna	FiQA	NFCorpus	SciDocs	SciFact	Average
VPT-BW-0.2	0.02	28.01	35.32	16.54	65.92	29.16
VPT-BW-0.5	0.02	27.90	35.16	16.49	65.91	29.10
VPT-BW-1.0	0.02	27.95	35.03	16.42	65.33	28.95
VPT-BW-2.0	0.02	27.27	35.36	15.80	63.33	28.36

All bandwidth variants produce 0.02 NDCG@10 on ArguAna, confirming that the KDE estimation with BM25 importance weights inherits the adversarial signal quality observed in Stage 6. On the remaining 4 datasets, narrower bandwidths (c=0.2) slightly improve ranking quality, consistent with the concentration of $f_R$ in high-dimensional spaces (Theorem 3.4.1).

Method descriptions:

Method	Description
BM25	Sparse retrieval via bm25s (Lucene variant)
Dense	Cosine similarity via sentence-transformers
Convex	w * dense_norm + (1-w) * bm25_norm, w=0.5
RRF	Reciprocal Rank Fusion, sum(1/(k + rank)), k=60
Bayesian-OR	Bayesian BM25 probs + cosine probs via prob_or
Bayesian-LogOdds	Bayesian BM25 probs to logit, dense calibrated via logit = alpha * (sim - median), combined
Bayesian-LogOdds-Local	Both raw BM25 and dense calibrated symmetrically via logit = alpha * (score - median), combined
Bayesian-LogOdds-BR	Bayesian-LogOdds with base rate prior
Bayesian-Balanced	balanced_log_odds_fusion: Bayesian BM25 probs and dense sims to logit space, min-max normalize each, combine with equal weights
Bayesian-Balanced-Mix	Bayesian-Balanced with mixture-model base rate estimation
Bayesian-Balanced-Elbow	Bayesian-Balanced with elbow-detection base rate estimation
Bayesian-Gated-ReLU	log_odds_conjunction with ReLU gating in logit space (Jeong, 2026b, Theorem 6.5.3)
Bayesian-Gated-Swish	log_odds_conjunction with Swish gating in logit space (Jeong, 2026b, Theorem 6.7.4)
Bayesian-Gated-GELU	log_odds_conjunction with GELU gating (Jeong, 2026b, Theorem 6.8.1): logit * sigmoid(1.702 * logit)
Bayesian-Gated-Swish-B2	Generalized swish with gating_beta=2.0 (Jeong, 2026b, Theorem 6.7.6)
Bayesian-Gated-Softplus	log_odds_conjunction with softplus gating (Remark 6.5.4): log(1 + exp(logit)), evidence-preserving smooth ReLU
Bayesian-Attention	Query-dependent signal weighting via AttentionLogOddsWeights (Jeong, 2026b, Section 8)
Bayesian-Attn-Norm	Attention with per-signal logit normalization (normalize=True) and 7 features (sparse + dense + cross-signal)
Bayesian-Attn-Norm-CV	Bayesian-Attn-Norm with 5-fold cross-validation (train/test split per query)
Bayesian-MultiHead	4-head MultiHeadAttentionLogOddsWeights, averages log-odds across heads (Remark 8.6)
Bayesian-MultiHead-Norm	Multi-head + logit normalization + 7 features (Corollary 8.7.2)
Bayesian-MultiField	MultiFieldScorer (title + body) with log_odds_conjunction, sparse-only
Bayesian-MultiField-Bal	MultiField probs + dense via balanced_log_odds_fusion
Bayesian-Vector-Balanced	VectorProbabilityTransform-calibrated dense probabilities + BM25 via balanced_log_odds_fusion (Jeong, 2026c, Theorem 3.1.1)
Bayesian-Vector-Softplus	VPT-calibrated dense + BM25 via softplus-gated log_odds_conjunction
Bayesian-Vector-Attn	VPT-calibrated dense + attention with logit normalization + 7 features
Dense-Kappa	Global sigmoid calibration: P = sigmoid(kappa * (beta - d)) with corpus-level parameters (Jeong, 2026c, Section 8.4 Stage 1)
Dense-Arctan	Arctangent normalization: p = (2/pi) * arctan(alpha * s) (Jeong, 2026c, Section 8.2)
Dense-Platt	Supervised Platt scaling: P = sigmoid(a * s + b) with labeled data (Jeong, 2026c, Section 8.2)
VPT-DensityPrior	VPT with gap detection / density prior only (CI-compliant, Jeong, 2026c, Stage 6)
VPT-BM25Weights	VPT with BM25 cross-modal importance weights only (CI-violating, Jeong, 2026c, Stage 6)
VPT-BW-{c}	VPT with bandwidth factor c applied to Silverman bandwidth (Jeong, 2026c, Stage 7)
Bayesian-Balanced-Tuned	Bayesian-Balanced + supervised BayesianProbabilityTransform.fit() + grid search over base_rate and fusion_weight
Bayesian-Hybrid-AND-Tuned	log_odds_conjunction of Bayesian BM25 and dense probs with tuned alpha
Bayesian-Tuned	Sparse-only Bayesian BM25 with tuned alpha, beta, and base_rate (no dense signal)

Why include underperforming methods? The tables above deliberately include methods that underperform BM25. Each failure mode is informative:

• Bayesian-OR (NDCG@10 avg 28.38) — Probabilistic OR assumes signal independence and catastrophically fails on ArguAna (0.06%). This demonstrates why the log-odds conjunction framework (Jeong, 2026b, Section 4) is needed: naive probability combination without logit-space calibration collapses when signal distributions differ.
• Bayesian-Gated-* — Sparse gating (Jeong, 2026b, Theorems 6.5-6.8) is too aggressive for the BEIR hybrid fusion task. These gates are designed for high-dimensional signal spaces where most inputs are noise — in a two-signal (sparse + dense) setting, there is no noise to suppress.
• Bayesian-MultiField (28.58 over 4 datasets) — Sparse-only multi-field search loses to concatenated BM25 because field separation fragments term statistics. However, Bayesian-MultiField-Bal (40.16) recovers most of the gap by fusing with dense embeddings.

Reproduce:

# Zero-shot (35 methods including Jeong, 2026c experiments)
python benchmarks/hybrid_beir.py -d <beir-data-dir>

# With tuning (auto-estimation + supervised learning + grid search)
python benchmarks/hybrid_beir.py -d <beir-data-dir> --tune

# Download BEIR datasets automatically
python benchmarks/hybrid_beir.py -d <beir-data-dir> --download

Requires pip install bayesian-bm25[scorer] sentence-transformers pytrec-eval-0.5 PyStemmer.

Sparse Retrieval

Evaluated on BEIR datasets (NFCorpus, SciFact) with k1=1.2, b=0.75, Lucene BM25. Queries are split 50/50 for training and evaluation. "Batch fit" uses gradient descent on training labels; all other Bayesian methods are unsupervised.

Ranking Quality

Base rate prior is a monotonic transform — it does not change document ordering.

Method	NFCorpus NDCG@10	NFCorpus MAP	SciFact NDCG@10	SciFact MAP
Raw BM25	0.5023	0.4395	0.5900	0.5426
Bayesian (auto)	0.5050	0.4403	0.5791	0.5283
Bayesian (auto) + base rate	0.5050	0.4403	0.5791	0.5283
Bayesian (batch fit)	0.5041	0.4400	0.5826	0.5305
Bayesian (batch fit) + base rate	0.5041	0.4400	0.5826	0.5305
Platt scaling	0.0229	0.0165	0.0000	0.0000
Min-max normalization	0.5023	0.4395	0.5900	0.5426
Batch fit (prior-aware, C2)	0.5066	0.4424	0.5776	0.5236
Batch fit (prior-free, C3)	0.5023	0.4395	0.5880	0.5389

Probability Calibration

Expected Calibration Error (ECE) and Brier score. Lower is better.

Method	NFCorpus ECE	NFCorpus Brier	SciFact ECE	SciFact Brier
Bayesian (no base rate)	0.6519	0.4667	0.7989	0.6635
Bayesian (base_rate=auto)	0.1461 (-77.6%)	0.0619	0.2577 (-67.7%)	0.1308
Bayesian (base_rate=0.001)	0.0081 (-98.8%)	0.0114	0.0354 (-95.6%)	0.0157
Batch fit (no base rate)	0.0093 (-98.6%)	0.0114	0.0103 (-98.7%)	0.0051
Batch fit + base_rate=auto	0.0085 (-98.7%)	0.0096	0.0021 (-99.7%)	0.0013
Platt scaling	0.0186 (-97.1%)	0.0101	0.0188 (-97.7%)	0.0007
Min-max normalization	0.0189 (-97.1%)	0.0105	0.0156 (-98.0%)	0.0009
Batch fit (prior-aware, C2)	0.0892 (-86.3%)	0.0439	0.1427 (-82.1%)	0.0802
Batch fit (prior-free, C3)	0.0029 (-99.6%)	0.0099	0.0058 (-99.3%)	0.0030

Threshold Transfer

F1 scores using the best threshold found on training queries, applied to evaluation queries. Smaller gap indicates better generalization.

Method	NFCorpus Train F1	NFCorpus Test F1	SciFact Train F1	SciFact Test F1
Bayesian (no base rate)	0.1607	0.1511	0.3374	0.2800
Batch fit (no base rate)	0.1577	0.1405	0.2358	0.2294
Batch fit + base_rate=auto	0.1559	0.1403	0.3316	0.3341
Platt scaling	0.0219	0.0193	0.0005	0.0005
Min-max normalization	0.1796	0.1751	0.3526	0.3486
Batch fit (prior-aware, C2)	0.1657	0.1539	0.3370	0.3275
Batch fit (prior-free, C3)	0.1808	0.1758	0.2836	0.2852

Reproduce with python benchmarks/base_rate.py (requires pip install bayesian-bm25[bench]). The base rate benchmark also includes Platt scaling, min-max normalization, and prior-aware/prior-free training mode comparisons.

Additional benchmarks (no external datasets required):

• python benchmarks/learnable_weights.py — learnable weight recovery, fusion quality, online convergence, and timing
• python benchmarks/weighted_fusion.py — weighted vs uniform log-odds fusion across noise scenarios
• python benchmarks/wand_upper_bound.py — WAND upper bound tightness and skip rate analysis
• python benchmarks/gating_functions.py — gating comparison (none/relu/swish/gelu/softplus), beta sensitivity, timing overhead
• python benchmarks/bmw_upper_bound.py — BMW block-max vs global WAND tightness, pruning rate, block size sensitivity
• python benchmarks/neural_calibration.py — Platt vs isotonic calibration accuracy, hybrid fusion quality, timing
• python benchmarks/multi_head_attention.py — multi-head vs single-head quality, pruning safety/efficiency, head diversity

Citation

If you use this work, please cite the following papers:

@preprint{Jeong2026BayesianBM25,
  author    = {Jeong, Jaepil},
  title     = {Bayesian {BM25}: {A} Probabilistic Framework for Hybrid Text
               and Vector Search},
  year      = {2026},
  publisher = {Zenodo},
  doi       = {10.5281/zenodo.18414940},
  url       = {https://doi.org/10.5281/zenodo.18414940}
}

@preprint{Jeong2026BayesianNeural,
  author    = {Jeong, Jaepil},
  title     = {From {Bayesian} Inference to Neural Computation: The Analytical
               Emergence of Neural Network Structure from Probabilistic
               Relevance Estimation},
  year      = {2026},
  publisher = {Zenodo},
  doi       = {10.5281/zenodo.18512411},
  url       = {https://doi.org/10.5281/zenodo.18512411}
}

@preprint{Jeong2026VectorLikelihoodRatios,
  author    = {Jeong, Jaepil},
  title     = {Vector Scores as Likelihood Ratios: {Index-Derived} {Bayesian}
               Calibration for Hybrid Search},
  year      = {2026},
  publisher = {Zenodo},
  doi       = {10.5281/zenodo.19181568},
  url       = {https://doi.org/10.5281/zenodo.19181568}
}

License

This project is licensed under the Apache License 2.0.

Copyright (c) 2023-2026 Cognica, Inc.