Odds are an alternative way of expressing probability. Instead of stating that the probability of rain is 0.75, one can say the odds of rain are 3 to 1 — three chances of rain for every one chance of no rain. Mathematically, the odds of an event H are O(H) = P(H) / P(¬H) = P(H) / (1 − P(H)). While probabilities range from 0 to 1, odds range from 0 to infinity, with odds of 1 corresponding to a 50-50 probability.
The odds form of Bayes' theorem expresses belief updating as a simple multiplication: posterior odds equal the likelihood ratio times the prior odds. This formulation is not just algebraically equivalent to the standard form — it is often more intuitive, more computationally convenient, and more naturally suited to sequential evidence accumulation. It is the form that I. J. Good, Alan Turing, and Jack Good used in their cryptanalytic work at Bletchley Park during World War II.
Expanded P(H | E) / P(¬H | E) = [P(E | H) / P(E | ¬H)] × [P(H) / P(¬H)]
Where O(H) → Prior odds
O(H | E) → Posterior odds
LR → Likelihood ratio (also called Bayes factor for simple hypotheses)
The Likelihood Ratio
The likelihood ratio LR = P(E | H) / P(E | ¬H) is the engine of Bayesian updating in the odds framework. It measures the diagnostic strength of the evidence — how much more (or less) probable the evidence is under the hypothesis than under its negation.
LR = 1: Evidence is uninformative (equally probable either way)
LR < 1: Evidence supports ¬H (data less probable under H)
LR = 10: Data are 10× more probable under H — strong evidence for H
LR = 100: Data are 100× more probable under H — very strong evidence
A likelihood ratio of exactly 1 means the evidence does not discriminate between the hypothesis and its alternative — the posterior odds equal the prior odds, and beliefs are unchanged. This is the hallmark of irrelevant evidence. A likelihood ratio of 10 means the evidence is 10 times more probable if H is true, multiplying the prior odds by 10. The beauty of the odds form is that each piece of evidence contributes a multiplicative factor, and these factors compose naturally.
Sequential Evidence Accumulation
When multiple independent pieces of evidence E₁, E₂, …, Eₙ are observed sequentially, the odds form chains naturally:
Log-Odds (Additive) Form log O(H | E₁, …, Eₙ) = Σᵢ log LRᵢ + log O(H)
Weight of Evidence W(E) = log₁₀ LR (measured in "bans")
W(E) = 10 × log₁₀ LR (measured in "decibans")
Taking logarithms converts the multiplicative chain to an additive one. Each piece of evidence contributes a weight of evidence — a term coined by Charles Sanders Peirce and formalized by I. J. Good. The total weight is the sum of individual weights, making it easy to track how evidence accumulates over time.
During World War II, Alan Turing and I. J. Good used the odds form of Bayes' theorem to crack the German Enigma cipher. Each piece of evidence — a known plaintext-ciphertext pair, a probable word, a repeated key pattern — contributed a weight measured in "bans" (named after Banbury, where the special scoring sheets were printed) or "decibans" (one-tenth of a ban, analogous to decibels). Turing's Banburismus procedure was one of the first systematic applications of sequential Bayesian reasoning. The log-odds framework allowed cryptanalysts to accumulate evidence incrementally, deciding when enough weight had been gathered to justify a conclusion.
Odds Ratios in Epidemiology
In epidemiology and medical statistics, the term "odds ratio" has a specific technical meaning that is related to but distinct from the Bayesian odds form. The epidemiological odds ratio compares the odds of exposure among cases to the odds of exposure among controls:
In a 2×2 Table Case Control
Exposed: a b
Unexposed: c d
OR = (a × d) / (b × c)
This epidemiological odds ratio is estimable from case-control studies (where the base rate of disease is not directly observable) and approximates the relative risk when the disease is rare. While its connection to Bayesian updating is indirect, the mathematical structure of odds — ratios of probabilities — is the same. In a Bayesian reanalysis of a case-control study, the odds ratio enters the likelihood function and is updated by the prior to produce a posterior distribution over the odds ratio itself.
Advantages of the Odds Form
Intuitive updating. The odds form makes the mechanism of Bayesian updating transparent: evidence multiplies the odds. Strong evidence (large LR) dramatically shifts beliefs; weak evidence (LR near 1) barely moves them. The metaphor of "weighing evidence" is literal in the log-odds framework.
No normalizing constant. The odds form avoids the denominator P(E) entirely. This makes it particularly convenient for binary hypotheses, where the normalizing constant would require computing P(E | H)P(H) + P(E | ¬H)P(¬H).
Composability. Independent pieces of evidence contribute multiplicatively in odds (additively in log-odds), allowing evidence from different sources to be combined naturally without recomputing from scratch.
Symmetry. Evidence that supports H (LR > 1) and evidence that supports ¬H (LR < 1) are treated symmetrically. The log-likelihood ratio is positive for evidence favoring H and negative for evidence favoring ¬H, with magnitude reflecting strength.
Odds, Logistic Regression, and Log-Odds
The log-odds (logit) function is the link function in logistic regression — one of the most widely used statistical models. The logistic model specifies that the log-odds of the outcome are a linear function of the predictors: log O(Y = 1 | X) = β₀ + β₁X₁ + … + βₚXₚ. Each coefficient βⱼ represents the change in log-odds per unit increase in Xⱼ, and exp(βⱼ) is an odds ratio. This is a direct application of the additive structure of log-odds.
From a Bayesian perspective, logistic regression is naturally fitted by placing priors on the regression coefficients and computing the posterior. Bayesian logistic regression avoids the separation problems that plague maximum likelihood fitting when the data are sparse, and it provides full posterior distributions on odds ratios rather than just point estimates and confidence intervals.
"The likelihood ratio is the unique measure of evidential strength. It tells you how much the evidence should change your mind — no more, no less." — I. J. Good, Probability and the Weighing of Evidence (1950)
From Odds Back to Probabilities
Converting between odds and probabilities is straightforward: P(H) = O(H) / (1 + O(H)). If the posterior odds are 3 to 1, the posterior probability is 3/4 = 0.75. If the posterior odds are 19 to 1, the probability is 19/20 = 0.95. The odds form and the probability form of Bayes' theorem carry identical information — the choice between them is a matter of convenience and clarity.
Example: Sports Betting — Will the Underdog Win?
A sportsbook sets the odds for an upcoming boxing match. The champion has won 8 of her last 10 fights against similar opponents. Before the fight, a pundit estimates the challenger's chance of winning at 25%. Then news breaks that the champion has a hand injury.
Working in Odds Form
Prior odds = P(Win) / P(Lose) = 0.25 / 0.75 = 1 : 3 (or 1/3)
Now the pundit considers: how much more likely is the injury news if the challenger is going to win vs. if the champion will still win despite it?
P(Injury reported | Champion wins) = 0.20 (champion might power through)
Likelihood ratio = 0.60 / 0.20 = 3
= 3 × (1/3) = 1 : 1 (even odds)
Posterior probability = 1 / (1 + 1) = 50%
The injury report tripled the challenger's odds, moving her from a 25% underdog to a coin flip. The odds form makes this update intuitive: the prior odds set the starting point, and the likelihood ratio acts as a simple multiplier.
The odds form of Bayes' theorem is the natural language of gambling and forensic science precisely because the likelihood ratio isolates the evidential strength of new information from the base rate. A bookmaker can say "this news triples the odds" without specifying what the starting odds were. Each bettor can apply that multiplier to their own prior. The odds form turns Bayesian updating into straightforward multiplication — which is why Turing used it to crack Enigma, one deciban at a time.