Bayes Factor: Guide to Bayesian Hypothesis Testing & Interpretation

What Is a Bayes Factor?

The Bayes Factor sits at the heart of Bayesian hypothesis testing. To understand it, start with a basic question any researcher faces after collecting data: given what I observed, which of my two competing explanations does the data favor, and by how much? A p-value answers a different question — it tells you how unlikely your data would be if the null hypothesis were true, but it says nothing about the alternative. The Bayes Factor compares both hypotheses head-to-head against the same data.

Formally, the marginal likelihood P(D | H) is the probability of the observed data integrated across all possible values of the model's parameters, weighted by the prior distribution on those parameters. This integration is what separates the Bayes Factor from a simple likelihood ratio, which evaluates models only at their best-fit parameter values.

Harold Jeffreys developed the framework in the 1930s and 1940s, publishing it fully in his 1961 book Theory of Probability. His goal was a method that could confirm a null effect, not merely fail to reject it — something classical significance testing cannot do. Researchers in psychology, medicine, and data science have returned to Bayes Factors with growing frequency because they address real inferential needs that frequentist tools leave unmet. The broader context for this sits in the Bayes theorem and conditional probability topics on Statistics Fundamentals.

The Bayes Factor Formula

The formula derives directly from Bayes' theorem applied to competing models. Start with the relationship between prior odds, posterior odds, and the Bayes Factor:

The marginal likelihood for each hypothesis is computed by integrating the likelihood function over the prior distribution of the model's parameters:

Understanding Marginal Likelihoods

The marginal likelihood is often called the "model evidence" because it measures how well a hypothesis predicts the observed data overall — not just at the best-fit parameter value. A model that is flexible enough to fit many possible datasets will not be rewarded as much as a model that makes precise predictions that happen to match what you observed.

This is why the Bayes Factor naturally penalizes overly complex models. If H₁ specifies that a wide range of effect sizes are plausible, its marginal likelihood is spread thin across many possible predictions. A more focused prior — one that predicts more precisely what you found — yields higher marginal likelihood. This automatic complexity penalty is a feature, not a bug: it implements Occam's Razor mathematically.

The Savage-Dickey Density Ratio

For nested models — where H₀ is a special case of H₁ with a parameter fixed to a specific value, such as zero — the Bayes Factor has an elegant simplification known as the Savage-Dickey density ratio:

This ratio compares the prior density and posterior density at the null value θ₀. When the data shift probability mass away from θ₀, the posterior density at θ₀ becomes smaller than the prior density. Consequently, BF₁₀ exceeds 1, indicating evidence in favor of H₁. In contrast, when the data concentrate probability mass around θ₀, the posterior density at θ₀ becomes larger than the prior density. Consequently, BF₁₀ is less than 1, indicating evidence in favor of H₀. This computational approach is known as the Savage–Dickey density ratio and is used in the default Bayesian t-tests implemented in JASP.

BF10 vs BF01: Which Direction to Report

BF₁₀ and BF₀₁ measure the same evidence; they just flip the direction of the comparison. BF₁₀ quantifies how much more strongly the data support H₁ over H₀. BF₀₁ quantifies how much more strongly the data support H₀ over H₁. They are exact mathematical reciprocals:

Convention in most research fields is to report BF₁₀ when results favor the alternative and BF₀₁ when results favor the null, always choosing the direction greater than 1 for clarity. Some journals and the JASP statistical software report both. When reading research papers, check which subscript the authors used before interpreting the magnitude.

The Log Bayes Factor Scale

Because Bayes Factors can range from near zero to very large numbers, researchers working with extreme evidence often use the natural logarithm, written log(BF). On the log scale, values above zero favor H₁, values below zero favor H₀, and the magnitude indicates strength symmetrically in both directions. A log(BF₁₀) of 2.3 corresponds to BF₁₀ = e^2.3 ≈ 10 — strong evidence for the alternative. Log Bayes Factors also add naturally when combining independent pieces of evidence, making them useful in meta-analytic settings.

The Jeffreys Scale: Interpreting Bayes Factor Strength

Harold Jeffreys proposed a graduated classification of evidence strength in 1961. The scale has been refined and relabeled by various researchers since then, but the numerical thresholds remain standard across most research fields. The table below combines the original Jeffreys thresholds with the labels used by JASP and the University of Amsterdam's Bayesian statistics group.

How to Calculate a Bayes Factor: 6 Steps

Bayes Factor Worked Examples

The two examples below follow the 6-step process. Calculations use the standard Cauchy prior with scale r = 0.707, matching the default in JASP and R's BayesFactor package. Both examples show the full reasoning chain from raw data to an interpretable conclusion.

Example 1 — Bayesian Independent-Samples T-Test (Psychology)

Example 2 — Bayesian A/B Test Confirming a Null Effect (Digital Marketing)

Bayes Factor vs P-Value: A Direct Comparison

The p-value and the Bayes Factor answer fundamentally different questions. A p-value answers: "If H₀ is true, how likely is it that I would see data at least as extreme as mine?" A Bayes Factor answers: "Given the data I actually observed, how much more evidence do I have for H₁ than for H₀?" These are not the same thing, and conflating them causes real errors in scientific reasoning.

One practical consequence is that, with a very large sample, even a trivially small effect can yield p < 0.05. For example, a drug that lowers systolic blood pressure by only 0.3 mmHg may be declared statistically significant if the sample size is sufficiently large. A Bayes factor can behave differently: if the observed effect is much smaller than the effect sizes predicted under H₁, the evidence may favor H₀, yielding BF₁₀ < 1. In this way, Bayesian inference evaluates not only whether an effect differs from zero but also whether the observed magnitude is consistent with the predictions of the competing hypotheses.

Computing Bayes Factors in R and Python

R: The BayesFactor Package

The BayesFactor package by Morey and Rouder is the standard tool for Bayesian t-tests, ANOVA, and regression in R. Install it once with install.packages("BayesFactor").

library(BayesFactor)

# Example data: control and intervention group scores
control <- c(48, 52, 54, 50, 55, 49, 53, 51)
intervention <- c(57, 61, 59, 63, 58, 60, 62, 64)

# Run Bayesian independent-samples t-test
# rscale = 0.707 is the default "medium" Cauchy prior
bf_result <- ttestBF(x = control, y = intervention, rscale = 0.707)

# Print the result
print(bf_result)
# Output: BF10 value + posterior probability of H1

# Extract the numeric BF10 value
bf_value <- extractBF(bf_result)$bf
cat("BF10 =", bf_value, "\nBF01 =", 1/bf_value, "\n")

# One-sample test: is the mean different from 0?
bf_one_sample <- ttestBF(x = control, mu = 50, rscale = 0.707)
print(bf_one_sample)

# Bayesian Pearson correlation test
x <- rnorm(50, mean = 5, sd = 2)
y <- x * 0.6 + rnorm(50, mean = 0, sd = 1.5)

bf_corr <- correlationBF(y = y, x = x)
print(bf_corr)
# Interprets evidence for non-zero correlation vs rho = 0

Python: Pingouin Library

The pingouin library provides Bayesian t-tests directly. Install with pip install pingouin.

import pingouin as pg
import numpy as np

# Generate example data
np.random.seed(42)
control = np.random.normal(52, 8, 40)
intervention = np.random.normal(58, 8, 42)

# Bayesian independent-samples t-test
# Returns BF10 directly in the output table
result = pg.ttest(intervention, control, correction=True)
print(result[["BF10", "dof", "p-val", "cohen-d"]])

# One-sample Bayesian t-test
result_one = pg.ttest(control, 50)
print(result_one[["BF10", "T", "p-val"]])

# Convert BF10 to BF01
bf10 = result["BF10"].values[0]
bf01 = 1 / bf10
print(f"BF10 = {bf10:.2f}, BF01 = {bf01:.4f}")

Bayes Factor Calculator

This calculator implements the marginal likelihood ratio for a one-sample or two-sample scenario using a Cauchy prior on the standardized effect size. Enter your t-statistic, sample sizes, and choose the prior scale to compute BF₁₀ directly. For a full Bayesian t-test with raw data, use the R code above.

Real-World Applications of the Bayes Factor

How to Report a Bayes Factor in Research

Academic reporting of Bayes Factors requires four elements: the specific statistic, its numerical value, the prior specification, and the qualitative interpretation. The examples below follow APA 7th edition style as adapted for Bayesian reporting per Keysers, Gazzola, and Wagenmakers (2020).

Bayes Factor Cheat Sheet

Frequently Asked Questions

Related Statistical Concepts

The Bayes Factor connects to a broader set of topics in Bayesian and frequentist statistics. The links below cover the foundational concepts that underpin Bayes Factor analysis, all from Statistics Fundamentals: