Ban and Deciban: Log-Odds Information Units
Cross-reference this discussion with the Hartley unit explainer, the SNR guide, and planning tools such as the Bayes' theorem calculator to manage evidence accumulation across analytics and engineering projects.
Introduction
The ban (from "binary-decimal") and its subunit, the deciban, quantify information as logarithms of odds ratios using base 10. A shift of one ban corresponds to a tenfold change in odds, while one deciban equals one tenth of a ban and represents a modest 100.1 ≈ 1.2589 odds adjustment. These units emerged in early twentieth-century statistical communications to express Bayesian evidence on additive scales, facilitating mental arithmetic in wartime code-breaking and modern decision analysis.
Understanding bans bridges information theory, statistics, and signal processing. Because logarithmic measures convert multiplicative likelihood ratios into additive scores, they streamline sequential analysis, hypothesis testing, and model comparison. This article defines the units formally, reviews their history, and shows how to deploy them in contemporary analytics workflows.
Definition and Mathematical Relationships
Given prior odds Oprior and posterior odds Oposterior, the evidence E measured in bans is E = log10(Oposterior) − log10(Oprior). When working with likelihood ratios Λ = P(data | H1)/P(data | H0), the evidence reduces to E = log10(Λ), mirroring the additive property of decibels in acoustics. Multiplying Λ by 10 increases the evidence by one ban; multiplying by 1.2589 increases it by one deciban.
Conversions to other information units
Because logarithms differ only by constant factors, 1 ban equals log2(10) ≈ 3.3219 bits and loge(10) ≈ 2.3026 nats. These relationships allow analysts to translate evidence between ban-based reports and binary information quantities used in coding theory. Tools like the logarithm base converter automate these transformations while preserving precision.
Additivity and sequential testing
In sequential probability ratio tests (SPRTs), each observation contributes a log-likelihood ratio. Summing decibans simplifies the stopping rule: decision thresholds become fixed additive targets rather than products of likelihoods. This mirrors how signal-to-noise ratios express power gains on logarithmic scales for intuitive engineering control.
Historical Development
Alan Turing and I.J. Good popularised the deciban during World War II while working at Bletchley Park. Their statistical bombe analyses required rapid mental arithmetic to accumulate evidence for or against cipher keys; representing Bayes factors in decibans enabled analysts to add small numbers instead of multiplying probabilities. Turing's declassified reports formalised the terminology, linking code-breaking practice with Bayesian decision theory decades before it was widely adopted in statistics.
Post-war, Good advocated for log-odds metrics in scientific inference, publishing extensively on weight of evidence measured in decibans. The concept influenced the development of logistic regression, which models log-odds linearly, and inspired communication between statisticians and intelligence analysts. Contemporary Bayesian textbooks revisit decibans to illustrate the continuity between historical applications and modern analytics pipelines.
Practical Measurement and Reporting
To compute evidence in bans, start with prior probabilities P(H1) and P(H0), convert to odds, multiply by the likelihood ratio derived from data, and then take base-10 logarithms. Software implementations often operate in natural logs for numerical stability; dividing by ln 10 converts the result to bans. Bayesian A/B testing platforms present posterior odds, credible intervals, and cumulative decibans to inform rollout decisions.
Regulatory submissions in fields such as clinical diagnostics may document evidential strength in decibans to contextualise probability updates for auditors familiar with log-odds interpretation. When communicating with stakeholders who prefer percentage probabilities, combine deciban summaries with visual aids generated by the Bayes' theorem tool to bridge intuition and formal metrics. Maintaining SI-compliant notation—log10 for bans, log10(Λ) for decibans—avoids ambiguity.
Applications and Importance
Data scientists use decibans to set adaptive experiment thresholds: for example, stopping an online test once 10 decibans (odds ≈ 12.6:1) favour a treatment. Intelligence analysts combine evidence from heterogeneous sensors by summing decibans, mirroring wartime precedents. Financial risk teams evaluate fraud alerts using log-odds scoring, translating machine-learning outputs into deciban-based policies that align with regulatory expectations for explainability.
In medicine, Bayesian diagnostic frameworks quantify how lab results update disease odds; deciban thresholds communicate when post-test probabilities justify intervention. Machine-learning practitioners integrate log-odds (logits) into calibration workflows, reinforcing the conceptual link between bans and logistic models. Because the units align with additive decision costs, they support cross-disciplinary collaboration between statisticians, engineers, and policymakers.
Future Directions
As probabilistic programming and automated inference mature, deciban dashboards will provide human-interpretable summaries of complex Bayesian networks. Research into explainable AI leverages ban-based reporting to show cumulative evidence supporting classifications or anomaly detections. Cybersecurity teams increasingly require auditable metrics; ban-based scoring can tie machine decisions to historically grounded evidential scales.
Reviving awareness of bans and decibans ensures that modern analytics retains ties to foundational measurement concepts. By mastering conversions, historical context, and practical reporting, practitioners can integrate log-odds units into contemporary dashboards, experiment planning, and regulatory communication.