Statistics How To

Bayes Factor: Simple Definition

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A Bayes factor is the ratio of the likelihood of one particular hypothesis to the likelihood of another. It can be interpreted as a measure of the strength of evidence in favor of one theory among two competing theories.

That’s because the Bayes factor gives us a way to evaluate the data in favor of a null hypothesis, and to use external information to do so. It tells us what the weight of the evidence is in favor of a given hypothesis.

Unpacking the Bayes Factor

When we are comparing two hypotheses, H1 (the alternate hypothesis) and H0 (the null hypothesis), the Bayes Factor is often written as B10. It can be defined mathematically as
bayes factor
The Schwarz criterion is one of the easiest ways to calculate rough approximation of the Bayes Factor.

Interpreting Bayes Factors

A Bayes Factor can be any positive number. One of the most common interpretations is this one—first proposed by Harold Jeffereys (1961) and slightly modified by Lee and Wagenmakers in 2013:

If B10 is… then you have…
> 100 Extreme evidence for H1
30 – 100 Very strong evidence for H1
10 – 30 Strong evidence for H1
3 – 10 Moderate evidence for H1
1 – 3 Anecdotal evidence for H1
1 No evidence
1/3 – 1 Anecdotal evidence for H1
1/3 – 1/10 Moderate evidence for H1
1/10 – 1/30 Strong evidence for H1
1/30 – 1/100 Very strong evidence for H1
< 1/100 Extreme evidence for H1

References

Kass & Raftery, Bayes Factors. Journal of the American Statistical Association
Vol. 90, No. 430 (Jun., 1995), pp. 773-795
Retrieved from http://www.jstor.org/stable/2291091 on March 31, 2018

Lavine & Schervish. Bayes Factors: What They Are and What They Are Not. The American Statistician, Vol. 53, No. 2 (May, 1999), pp. 119-122. Retrieved from http://www.jstor.org/stable/2685729 on March 31, 2018.

Beard, Dienes, Muirhead, & West. Using Bayes factors for testing hypotheses about intervention effectiveness in addictions research. Addiction. 2016 Dec; 111(12): 2230–2247. Retrieved from https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5111611/ on March 31, 2018

Morey, Richard. What is a Bayes Factor? Bayes Factor, February 2014. Retrieved from http://bayesfactor.blogspot.com/2014/02/the-bayesfactor-package-this-blog-is.html on March 31, 2018

FelixS, What does a B-factor feel like?. R-Blog, January 29, 2015. Retrieved from https://www.r-bloggers.com/what-does-a-bayes-factor-feel-like/ on March 31, 2018

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