Bayes Factor Calculator | BF₁₀, Prior/Posterior Odds & Jeffreys Scale
Compute the Bayes factor BF₁₀ from a likelihood ratio, t-statistic, or chi-square statistic. Updates prior odds to posterior odds, interprets evidence strength on the Jeffreys scale (decisive, strong, moderate, anecdotal), and computes posterior probability of H₁.
Jeffreys Scale Reference
| BF₁₀ | Classification | Interpretation |
|---|---|---|
| > 100 | Decisive | Overwhelming evidence |
| 30 – 100 | Very Strong | Very strong evidence |
| 10 – 30 | Strong | Strong evidence |
| 3 – 10 | Moderate | Moderate evidence |
| 1 – 3 | Anecdotal | Weak / anecdotal evidence |
| < 1 | Favors H₀ | Evidence against H₁ |
What Is the Bayes Factor Calculator | BF₁₀, Prior/Posterior Odds & Jeffreys Scale?
The Bayes factor is a Bayesian alternative to the classical p-value for hypothesis testing. While a p-value answers "what is the probability of observing data this extreme if H₀ is true?", the Bayes factor answers "how much more (or less) does the data support H₁ versus H₀?".
- ▸BF₁₀ > 1: data support H₁ over H₀.
- ▸BF₁₀ = 1: data are equally consistent with both hypotheses.
- ▸BF₁₀ < 1: data support H₀ over H₁.
- ▸BF₁₀ = 10: the data are 10× more likely under H₁ than H₀.
- ▸The Jeffreys scale classifies BF₁₀ into verbal categories from "anecdotal" to "decisive".
Unlike p-values, Bayes factors can provide evidence for the null hypothesis (BF₁₀ < 1 means evidence for H₀), quantify how strongly data favor one hypothesis, and can be updated with new data by multiplying sequential Bayes factors.
Formula
The Bayes factor quantifies the relative evidence that data provide for one hypothesis over another. It connects prior beliefs to posterior beliefs via Bayes' theorem.
BF₁₀ = P(data|H₁) / P(data|H₀)
Likelihood ratio: how much more likely the data are under H₁ than H₀.
Posterior Odds = BF₁₀ × Prior Odds
Prior odds × BF = posterior odds. This is Bayes' theorem in odds form.
BF₁₀ = √(n/2) × (1 + t²/n)^(−n/2)
Approximation under unit-information prior. Matches JZS Bayes factor closely.
BF₁₀ ≥ −e · p · ln(p) for p < 1/e
Lower bound on evidence; actual BF with a proper prior is always ≥ this.
How to Use
- 1
Select the input mode tab: Likelihood Ratio, t-Test Result, Chi-Square, or p-Value.
- 2
Enter the prior odds P(H₁)/P(H₀) — default is 1 for equal prior probability of H₀ and H₁.
- 3
In Likelihood Ratio mode enter LR directly. In t-Test mode enter the t-statistic and sample size n. In Chi-Square mode enter χ² and degrees of freedom. In p-Value mode enter the classical p-value.
- 4
Click Compute Bayes Factor to get BF₁₀, BF₀₁, posterior odds, P(H₁|data), and Jeffreys classification.
- 5
Read the Jeffreys classification label and verbal description for an intuitive interpretation of evidence strength.
- 6
Consult the reference table at the bottom to understand where your BF falls on the full Jeffreys scale.
- 1
Select the input mode
Choose how your evidence is expressed: as a likelihood ratio, from a t-test result (t-statistic and sample size), from a chi-square test, or directly from a p-value.
- 2
Set prior odds (optional)
Prior odds = P(H₁)/P(H₀). Default is 1 (equal priors). Enter a value less than 1 if H₁ is a priori unlikely, or greater than 1 if H₁ is expected.
- 3
Enter the test statistic
Depending on mode: enter the likelihood ratio, t-statistic and sample size, chi-square and df, or a p-value.
- 4
Click Compute Bayes Factor
The calculator returns BF₁₀, BF₀₁, posterior odds, P(H₁|data), and the Jeffreys classification.
- 5
Interpret the result
Use the built-in Jeffreys scale table to interpret the strength of evidence. BF₁₀ > 10 is strong evidence, > 100 is decisive.
Example Calculation
Example 1 | From a t-Test
A study finds t = 2.5 with n = 30. Using the Wetzels approximation with equal prior odds:
Example 2 | From a p-Value
A classical test returns p = 0.03. The minimum Bayes factor (Sellke) is:
This famous result shows that p = 0.03 provides at best weak evidence — and may not even favor H₁ with certain priors.
Example 3 | Sequential Updating
Study 1 gives BF₁₀ = 5.0. Study 2 independently gives BF₁₀ = 4.0. Combined evidence:
Understanding Bayes Factor | BF₁₀, Prior/Posterior Odds & Jeffreys Scale
Bayes Factors in Scientific Practice
Bayes factors have become increasingly important in psychology, medicine, and the physical sciences as researchers grapple with replication crises and the limitations of p-value-based inference. Journals in psychology (Psychological Science, PLOS ONE) now commonly request or recommend Bayes factors alongside traditional p-values.
The Jeffreys Scale in Context
- ▸BF₁₀ between 1 and 3: anecdotal — barely worth more than a mention.
- ▸BF₁₀ between 3 and 10: moderate — the conventional threshold for "some evidence" in many fields.
- ▸BF₁₀ between 10 and 30: strong — generally sufficient for a confident conclusion.
- ▸BF₁₀ between 30 and 100: very strong — robust across many prior assumptions.
- ▸BF₁₀ > 100: decisive — overwhelming evidence, comparable to scientific certainty.
When to Use Each Input Mode
- ▸Likelihood Ratio: use when you have a statistical model and can compute P(data|H₁) and P(data|H₀) directly.
- ▸t-Test: use after a standard two-group or one-sample t-test. Requires only the t-statistic and sample size.
- ▸Chi-Square: use for goodness-of-fit and independence tests. The minimum-BF approximation scales with (χ² − df).
- ▸p-Value: use to convert a reported p-value into the minimum possible Bayes factor — useful for interpreting published literature.
Frequently Asked Questions
What is the difference between a p-value and a Bayes factor?
A p-value is the probability of observing data at least as extreme as the observed data, assuming H₀ is true. It does not measure the probability that H₀ is true, nor does it quantify how much data support H₁. A Bayes factor directly answers how much more probable the data are under H₁ versus H₀. Unlike p-values, Bayes factors can provide positive evidence for H₀ (BF < 1), are not affected by stopping rules, and can be multiplied across independent studies for sequential updating.
How do prior odds affect the result?
Prior odds represent your belief about the relative probability of H₁ vs H₀ before seeing the data. With prior odds = 1 (equal priors), the posterior odds equal the Bayes factor. With prior odds = 0.1 (H₁ is ten times less likely a priori), even a BF₁₀ = 10 only brings posterior odds to 1 — still no reason to favor H₁. In exploratory research, conservative priors (< 1) are appropriate; in pre-registered confirmatory studies, equal priors are a common choice.
What does the Wetzels t-test approximation assume?
The Wetzels approximation computes BF₁₀ = √(n/2) × (1 + t²/n)^(−n/2). It assumes a unit-information prior (a normal prior on the effect size calibrated to provide one unit of Fisher information). This is a closed-form approximation to the JZS Bayes factor with a Cauchy prior. It performs well for moderate to large sample sizes. For very small samples (n < 10), a full Bayesian analysis with an explicit prior is more appropriate.
Why does a p-value of 0.05 not correspond to BF₁₀ ≈ 20?
This is one of the most important findings of the Bayesian critique of null hypothesis significance testing. The Sellke et al. minimum BF for p = 0.05 is only about 2.46 — barely "anecdotal" on the Jeffreys scale. A p-value of 0.05 means "data this extreme occur 5% of the time under H₀", which is a much weaker statement than "there is a 95% chance H₁ is true". The disconnect between p-values and Bayes factors is a central argument for Bayesian approaches to inference.
Can Bayes factors be multiplied across independent studies?
Yes — this is one of the key advantages of Bayes factors over p-values. If two independent studies test the same hypothesis and yield BF₁₀ = 5 and BF₁₀ = 4 respectively, the combined Bayes factor is simply 5 × 4 = 20 (strong evidence). This sequential updating property arises naturally from Bayes' theorem: the posterior from Study 1 becomes the prior for Study 2. With p-values, combining results requires meta-analytic methods and is less transparent.
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