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A Bayes factor (BF) is a ratio of two marginal likelihoods: how well the data are predicted by one hypothesis versus a competing one, averaged over each hypothesis’s prior. Unlike a p value, which only ever measures how surprising the data would be if the null hypothesis were true, a Bayes factor can quantify evidence pointing toward either hypothesis — including the null. This guide gives the two evidence scales researchers use to interpret a Bayes factor’s magnitude (Jeffreys, 1961, and Kass & Raftery, 1995), a reproducible worked example showing both directions of evidence, and the reporting conventions that keep a Bayesian analysis reviewable.
What a Bayes Factor Actually Compares
For two competing hypotheses H₁ and H₀ and observed data D, the Bayes factor BF₁₀ is:
BF₁₀ = P(D | H₁) / P(D | H₀)
Each term is a marginal likelihood: the probability of the observed data under that hypothesis, averaged over every parameter value the hypothesis’s prior distribution assigns weight to — not the likelihood at a single best-fit parameter, the way a maximum-likelihood ratio test works. That averaging is why the choice of prior belongs to the model, and why two analysts using different priors for the same hypothesis can get different Bayes factors from identical data — a sensitivity that has no equivalent in a standard p-value calculation, where the null is a single, fully specified point.
A subscript convention matters when reading published Bayes factors: BF₁₀ expresses evidence for H₁ over H₀, while BF₀₁ = 1 / BF₁₀ expresses the same comparison from the opposite direction. A paper reporting “BF₀₁ = 4” is reporting evidence for the null, not a typo — always check which subscript order a source is using before reading the number.
Bayes factors are used the same way in Bayesian adaptive trial designs and in exploratory or confirmatory research generally; see CASRAI’s comparison of frequentist versus Bayesian approaches for how the two inferential frameworks differ more broadly, and the guide to Markov Chain Monte Carlo for how the marginal likelihoods behind more complex Bayes factors are typically estimated when no closed form exists.
The Jeffreys (1961) Scale
Harold Jeffreys proposed the first widely used verbal scale for Bayes factor magnitude in Theory of Probability (1961), built on half-order-of-magnitude bands. The version most commonly reproduced in applied Bayesian statistics texts is:
| BF₁₀ | Evidence for H₁ |
|---|---|
| 1 – 3.16 | Not worth more than a bare mention |
| 3.16 – 10 | Substantial |
| 10 – 31.6 | Strong |
| 31.6 – 100 | Very strong |
| > 100 | Decisive |
The same bands apply symmetrically to evidence for H₀: a BF₁₀ of 0.08 (equivalently BF₀₁ = 12.5) is “strong” evidence for the null on the identical scale, just read from the other direction. Some later authors relabel these five bands (“anecdotal,” “moderate,” “strong,” “very strong,” “extreme” is one common variant, used by default in the JASP software package) without changing the underlying numeric thresholds — check which label set a specific paper or software output is using before comparing across sources.
The Kass and Raftery (1995) Scale
Robert Kass and Adrian Raftery, writing in the Journal of the American Statistical Association (1995), proposed a coarser four-band scale, framed in terms of 2 ln(BF₁₀) — a transformation chosen to put the Bayes factor on the same additive, deviance-like scale as a likelihood-ratio test statistic:
| 2 ln(BF₁₀) | BF₁₀ | Evidence for H₁ |
|---|---|---|
| 0 – 2 | 1 – 2.7 | Not worth more than a bare mention |
| 2 – 6 | 2.7 – 20.1 | Positive |
| 6 – 10 | 20.1 – 148.4 | Strong |
| > 10 | > 148.4 | Very strong |
Kass and Raftery’s scale is coarser than Jeffreys’s (four bands instead of five) and uses different numeric breakpoints, so a Bayes factor of 15 reads as “strong” on Jeffreys’s scale but only “positive” on Kass and Raftery’s — neither scale is more “correct” than the other, but a page or paper should name which one it is using, and so should you when you report a result.
Worked Example: A Coin-Fairness Test
The clearest fully closed-form illustration compares a point null to a diffuse alternative on binomial data. For n flips with k heads:
- H₀ (fair coin): p = 0.5 exactly. P(D | H₀) = C(n,k) × 0.5ⁿ.
- H₁ (unknown fairness): p ∼ Uniform(0, 1), i.e. a flat Beta(1, 1) prior. Integrating the binomial likelihood against that prior gives P(D | H₁) = 1 / (n + 1), a well-known identity that holds for any k — a flat prior spreads its predictive mass evenly across every possible outcome count.
BF₁₀ = P(D | H₁) / P(D | H₀) = [1 / (n + 1)] / [C(n,k) × 0.5ⁿ]. Three cases, computed directly from that formula (script and full output available on request; every figure below was generated by evaluating the closed-form expression, not estimated or assumed):
| Data | BF₁₀ | BF₀₁ | Jeffreys | Kass–Raftery |
|---|---|---|---|---|
| 18 heads / 20 flips | 262.80 | 0.0038 | Decisive (for H₁) | Very strong (for H₁) |
| 15 heads / 20 flips | 3.22 | 0.31 | Substantial (for H₁) | Positive (for H₁) |
| 255 heads / 500 flips | 0.062 | 16.17 | Strong (for H₀) | Positive (for H₀) |
The third row is the point worth sitting with. 255/500 is 51% heads — barely off center, and in the same neighborhood as the 15/20 (75%) result that produced only “substantial” evidence against the null. But with five times the data pinning the estimate tightly to 0.5, the Bayes factor swings to “strong” evidence for the null — the flat H₁ prior keeps spreading its 1/(n+1) predictive mass thinner as n grows, so it increasingly loses to a null that predicts an outcome near 50% precisely. That is a real, quantified statement of support for the fair-coin hypothesis, not merely an absence of evidence against it.
Why a Bayes Factor Can Support the Null and a P-Value Cannot
This is the structural difference that matters most in practice. A p value is computed entirely under the assumption that the null hypothesis is true — it is P(data this extreme or more | H₀), a single conditional probability with no term for any alternative. That construction gives a p value exactly two possible verdicts: the data are surprising enough under H₀ to reject it, or they are not surprising enough, in which case the honest conclusion is “fail to reject H₀,” never “H₀ is supported.” A large p value is consistent with a true null, but it is equally consistent with a true, non-null effect that the study lacked the power to detect — the p value alone cannot distinguish those two explanations, because it never evaluated the alternative at all.
A Bayes factor evaluates both hypotheses as marginal likelihoods and compares them directly, so it produces one number that is intrinsically symmetric: BF₁₀ below 1 is, by construction, evidence for H₀ over H₁, on the identical scale used to express evidence the other direction. That is not a special case or an edge condition — it is what the ratio does whenever the data fit the null’s prediction better than the alternative’s. The 255/500 case above is exactly this: a real, gradable, reportable statement that the data favor the fair-coin hypothesis, expressed on the same Jeffreys/Kass–Raftery bands used for the opposite conclusion. A p-value-only analysis has no comparable move available — it can decline to reject the null, but it cannot report evidence for it.
Reporting a Bayes Factor Correctly
- State the direction explicitly. Report both the value and which hypothesis it favors (“BF₁₀ = 0.06, i.e. BF₀₁ ≈ 16, strong evidence for the null”), not a bare number — a reader who assumes the wrong subscript order will draw the opposite conclusion.
- Name the priors used for both hypotheses, including the alternative’s prior distribution and any scale parameter (a default Cauchy prior with a stated scale, in common software packages, versus a domain-informed prior) — the Bayes factor is a property of the specified models, not of the data alone, and a different reasonable prior can move the number, sometimes across a scale boundary. Reporting a Bayes factor with no stated prior is analogous to reporting a p value with no stated alpha or test.
- A Bayes factor is not a posterior probability. BF₁₀ = 3 does not mean H₁ is three times more likely to be true. It means the data shifted the odds in favor of H₁ by a factor of 3 — the actual posterior odds are BF₁₀ × (prior odds), so the same Bayes factor implies very different final conclusions depending on how plausible each hypothesis was considered before seeing the data.
- Cite the scale you used. “Substantial” on Jeffreys’s scale and “positive” on Kass and Raftery’s cover overlapping but not identical numeric ranges — name the source (Jeffreys, 1961, or Kass & Raftery, 1995) alongside the verbal label.
Common Misinterpretations to Avoid
- Treating “not worth more than a bare mention” as proof of no effect. A Bayes factor near 1 means the data did not discriminate between the two hypotheses well — often a sample-size problem, the Bayesian analogue of an underpowered frequentist test — not evidence that the hypotheses are equally true.
- Reading the verbal labels as fixed, universal thresholds for a publication decision. Jeffreys’s and Kass and Raftery’s bands were proposed as interpretive guides, the same status the 0.05 significance convention has for p values — useful shorthand, not a law of nature; a pre-registered analysis plan should state the threshold in advance rather than picking whichever label sounds best after seeing the result.
- Confusing a Bayes factor with a likelihood ratio at fixed parameter values. A Bayes factor integrates over each hypothesis’s full prior; a likelihood ratio test compares likelihoods at two specific parameter estimates. They coincide only when both hypotheses are themselves simple, fully specified points.
Frequently Asked Questions
What is a good Bayes factor?
There is no single “good” value — interpretation depends on which scale you are using (Jeffreys or Kass–Raftery, see above) and on the research context. A BF₁₀ of 3–10 is commonly described as the threshold where evidence starts being worth reporting on its own, but a pre-registered study should state its intended threshold before collecting data, the same discipline expected of a p-value alpha level.
Can a Bayes factor be negative?
No. A Bayes factor is a ratio of two probabilities, so it is always strictly positive. A value less than 1 means the data favor the denominator hypothesis, not that the ratio has gone negative.
Is a Bayes factor the same as an odds ratio?
No, though they are related. A Bayes factor is the factor by which the data update prior odds into posterior odds (posterior odds = Bayes factor × prior odds). An odds ratio, in epidemiology or logistic regression, compares the odds of an outcome between two groups — a different quantity answering a different question.
How is a Bayes factor different from a p value?
A p value is calculated entirely under the null hypothesis and can only ever lead to rejecting or failing to reject it. A Bayes factor evaluates both the null and an alternative and can express graded evidence toward either one, including the null — see “Why a Bayes Factor Can Support the Null” above.
Does the choice of prior change the Bayes factor?
Yes, for the alternative hypothesis’s prior specifically — this is often called the Bayes factor’s prior sensitivity. A point-null hypothesis has no free prior to vary, but the alternative’s prior (its shape and scale) is a real analytic choice, and reporting a Bayes factor without naming it is incomplete reporting.
