530 heads in 1000 tosses. The exact binomial p-value is 0.06202 — the shape of result usually written up as “approaching significance” — and the Bayes factor is 0.2394, which means the data is 4.18 times MORE likely under the null than under the alternative. The p-value and the evidence point in opposite directions, and this is not an edge case: it is what a p near 0.05 looks like at n = 1000.
530 of 1000 = 0.53000 against a null of 0.5000
BF₁₀ = 0.23942 — moderate evidence for the null
The data is 4.177 times more likely under the null. The exact binomial p-value on the same data is 0.06202. Those point in opposite directions, which is what a p-value near 0.05 looks like at this sample size. Starting from even odds, the posterior probability that the alternative is true is 19.32%.
BF₁₀
0.23942
moderate, favours the null
BF₀₁
4.17685
evidence for the null
p-value
0.06202
exact binomial, two sided
P(H₁) at even odds
19.32%
from a 50/50 starting point
The p-value is 0.06202 and the Bayes factor says the data is 4.18 times more likely under the null. They are not contradicting each other: a p-value measures how far the data sits from the null, and a Bayes factor asks whether that distance is better explained by a specific alternative. At a large n, a small departure from the null is far from it in standard errors and still unremarkable under an alternative that allows any rate at all.
Jeffreys’ bands
Bayes factor strength-of-evidence bands
Bayes factor
Evidence
This data
1 to 3
anecdotal
3 to 10
moderate
← for the null
10 to 30
strong
30 to 100
very strong
above 100
extreme
The bands run in both directions: a factor of 5 is moderate evidence for the alternative and a factor of 1/5 is moderate evidence for the null. That symmetry is the thing a p-value does not have.
How the ratio is built
The two marginal likelihoods the Bayes factor compares
Hypothesis
What it says
Log marginal
H₀
the rate is exactly 0.5000
-691.7176
H₁
the rate is anything, weighted by Beta(1.00, 1.00)
-694.5767
Ratio
log BF₁₀
-1.4296
The alternative pays a price for being vague: spreading belief over every possible rate means very little of it sits near the observed one. A tighter alternative would score better against the same data, which is why the prior is an input here and not a hidden constant.
Closed form, no integration p-value beside it Can support the null Depends on the alternative
What this tool shows
530 successes in 1000 gives p = 0.06202 — “approaching significance” — and a Bayes factor of 0.2394, meaning the data is 4.18 times MORE likely under the null. The p-value points one way and the evidence points the other, and this is not an edge case: it is what a p near 0.05 looks like at n = 1000. Both numbers are computed here from the same data.
Bayes factors for a proportion in closed form — a ratio of two marginal likelihoods, no integration
The exact binomial p-value on the same data, for the comparison that is the point
Jeffreys’ evidence bands running in both directions, which a p-value cannot
The two marginal likelihoods printed, so the ratio can be seen rather than trusted
The alternative’s prior as a visible input, because the answer depends on it
A preset where the evidence favours the null and a p-value can only say “not significant”
Closed form p-value beside it Can support the null Both marginals shown
A Bayes factor compares two specific hypotheses. Change the alternative and it changes.
Updated 13 September 2026 · Works in any browser, no installation
A Bayes factor is how much more likely the data was under one hypothesis than under another. A factor of 10 means the data is ten times more likely under the alternative; a factor of 0.1 means ten times more likely under the null. That symmetry is the property a p-value lacks — a p-value can only ever measure distance from the null, so it cannot distinguish “no evidence of a difference” from “evidence of no difference”.
At a glance
Formula shown
BF₁₀ = P(data | H₁) / P(data | H₀). With a point null at p₀ the denominator is p₀^x(1−p₀)^(n−x), and with a Beta(α, β) alternative the numerator is B(α+x, β+n−x)/B(α, β) — both closed forms, so the ratio needs no integration. Starting from even prior odds, the posterior probability of H₁ is BF₁₀/(1 + BF₁₀). Jeffreys’ bands read the same in both directions: 3 is moderate evidence for the alternative and 1/3 is moderate evidence for the null.
Scenario support
Reporting a null result as evidence rather than as an absence of evidence, deciding whether a borderline p-value means anything, pre-registered analyses where the alternative is specified in advance, replication studies, and any setting where “we found nothing” needs to be distinguished from “we learned nothing”.
Educational estimate
Planning support from the values you enter — not professional advice.
p = 0.062, and the evidence favours the null
The first preset is not a trick. It is what a p-value just outside 0.05 routinely means at a large sample size.
530 of 1000 against a null of 0.5. The exact binomial p-value is 0.06202 — the shape of result usually written up as approaching significance.
The Bayes factor is 0.2394. The data is 4.18 times more likely under the null than under an alternative that allows any rate at all.
They are not contradicting each other. A p-value measures how far the data sits from the null in standard errors; at n = 1000, 3% off 0.5 is a long way in those units. A Bayes factor asks whether a vague alternative explains that distance better, and mostly it does not.
This is the Lindley paradox, and it sharpens with n. The same 53% rate at n = 10,000 gives a Bayes factor of 830,313 — extreme evidence for a difference. The frameworks agree on clear results and part company exactly where decisions are hard.
Evidence FOR the null is a claim p-values cannot make
The fourth preset is the case that motivates Bayes factors in practice more than any other.
55 of 100 gives p = 0.36820. Not significant. A frequentist test stops there, and the honest report is “we failed to reject”.
The Bayes factor is 0.2043 — 4.89 times more likely under the null. That is moderate evidence that there is no difference, not merely an absence of evidence that there is one.
The distinction matters most when it is expensive. A safety study, a replication, a feature that did not work — all need to say something stronger than “we did not find anything”.
A p-value is structurally incapable of it. It is computed under the null and can only measure distance from it, so a large p-value means either the null is true or the study was too small, and it cannot tell you which.
The answer depends on the alternative
The most common objection to Bayes factors is that they are sensitive to the prior on the alternative. The objection is correct and it is also the point.
The null is a single value; the alternative is a distribution over values. The Bayes factor compares the data against both, so how the alternative spreads its belief matters.
A vague alternative pays for its vagueness. Beta(1, 1) spreads belief over every rate from 0 to 1, so very little of it sits near the observed value, and the marginal likelihood is small.
A tighter alternative centred near the observed rate would score far better on exactly the same data — which is why the prior is an input on this page rather than a hidden default.
The discipline is to specify it before seeing the data. A prior chosen afterwards to produce a satisfying Bayes factor is the same error as a p-hacked p-value, in a different notation.
The bands are a convention, and they are symmetric
Jeffreys’ labels — anecdotal, moderate, strong, very strong, extreme — are a reading aid rather than a decision rule.
They have no theoretical basis, exactly as 0.05 has none. Jeffreys proposed them as a rough guide and said so.
What they do have is symmetry. A factor of 5 is moderate evidence for the alternative and a factor of 1/5 is moderate evidence for the null, read off the same table.
A factor between 1/3 and 3 is the useful category p-values lack: the data is uninformative. The second preset sits there at 0.9130, from a study of 100 that looked borderline.
Report the number, not the band. “BF₁₀ = 4.2” is a quantity a reader can combine with their own prior odds; “moderate evidence” is a word.
A Bayes factor is not a probability
The most common misreading is to treat the factor as the probability that the alternative is true. It is the step before that.
Posterior odds = Bayes factor × prior odds. The factor is what the data contributed, and it needs prior odds to become a probability.
Starting from even odds, a factor of 4.2 gives a posterior probability of 81%. Starting from 1-in-20 prior odds, the same factor gives 17%.
Which is a feature, not a gap. The data speaks for itself and the reader supplies their own starting point — two people with different priors can agree completely about what the evidence was worth.
The posterior probability on this page assumes even odds and says so. It is a convenience, not a claim about how likely the hypothesis was before anyone looked.
What a Bayes factor does not solve
Bayes factors fix a specific problem with p-values and inherit several others unchanged.
They do not measure effect size. Extreme evidence for a difference says nothing about whether the difference matters — the third preset has a factor of 830,313 for a gap of three percentage points.
They do not fix a bad study. Selection bias, measurement error and confounding move a Bayes factor exactly as they move a p-value.
They are not immune to multiplicity. Computing twenty Bayes factors and reporting the largest is the same error as reporting the smallest of twenty p-values.
And a point null is often a straw man. Almost no rate is exactly 0.5. Testing against a region of practical equivalence is usually the better question, and a credible interval answers it directly.
Reporting a Bayes factor
Four items, and the second is the one that makes the number reproducible.
Give the direction explicitly. BF₁₀ and BF₀₁ are reciprocals and both get written as “BF”; a factor of 4 means opposite things depending on which.
State the alternative. The prior on H₁ is part of the calculation, and without it the number cannot be recomputed or compared.
Say the prior was set in advance. A prior chosen after seeing the data is p-hacking with different notation.
And give the raw counts. They let a reader compute the factor under their own alternative, which is the main thing a Bayes factor is for.
Method. Both marginal likelihoods are closed forms, so the ratio needs no integration: under a point null the data probability is p₀^x(1−p₀)^(n−x), and under a Beta(α, β) alternative it is B(α+x, β+n−x) divided by B(α, β). The suite checks that closed form against numerical integration of the same integral at three configurations, and asserts the exact value at 5 of 10 under a flat prior, where the marginal is 1/((n+1)·C(n,x)) — the binomial coefficient absent because it cancels in the ratio. The p-value printed beside it is the exact two-sided binomial, summed over every outcome no more likely than the one observed, rather than a normal approximation — which matters because the whole point of the page is a comparison, and comparing against an approximation would blur it. The suite also asserts that the posterior probability follows from the factor exactly, that the factor is always positive and finite, and that evidence against the null never weakens as the data moves away from it. That engine is verified on every change against 106 assertions. The count and the per-case breakdown are published on the formula verification page.
Related calculators
Where this goes next:
Credible IntervalBayesian credible intervals for a proportion with both the highest-density and equal-tailed versions and the prior's weight reported.
p-valueA p-value from a t or z statistic, one- or two-tailed — with a panel that holds an effect fixed and grows the sample, so you can watch significance appear from nothing but n.
Bayesian A/B TestP(B beats A) from a closed form rather than sampling, with expected loss as the stopping rule and a margin of practical interest.
Exact Binomial TestSums every outcome rather than approximating: 60 of 100 against p = 0.5 gives an exact p of 0.056888 where the uncorrected normal approximation gives 0.045500.
Posterior ProbabilitySequential Bayesian updating with the posterior after every batch, the prior's weight reported, and grouping-independence demonstrated.
Bayes' TheoremPosterior probability from a prior, sensitivity and specificity — plus the true and false positives per 100,000, because a 99% accurate test for a 1-in-10,000 condition is right 0.98% of the time and the percentage alone does not make that believable.
An educational tool. A Bayes factor compares two specific hypotheses, and changing the prior on the alternative changes the answer — a vague alternative is penalised for its vagueness. It also measures evidence rather than effect size: extreme evidence for a difference says nothing about whether that difference matters, and a point null of exactly 0.5 is rarely a hypothesis anyone believes.