Bayes Factors#

Bayes factors are central quantities in both Bayesian hypothesis testing and Bayesian model comparison. A Bayes factor allows to assess which of two hypotheses or two competing models is in favour of the other.

Bayes Factors in Bayesian Hypothesis Testing#

In the situation of classical hypothesis testing, there is one unknown parameter \(\theta\in\Theta\) lying in a given set \(\Theta\). The set \(\Theta\) can be decomposed as \(\Theta=\Theta_0\cup\Theta_1\) such that \(\Theta_0\cap\Theta_1=\emptyset\). The null hypothesis \(H_0\) states that \(H_0:~\theta\in\Theta_0\) and the alternative hypothesis is given by \(H_1:~\theta\in\Theta_1\). In a Bayesian setting, unknown parameters are assumed to follow a distribution. Therefore, probabilities are assigned to the hypotheses given by the prior likelihoods

\[ \begin{align*} \pi_0=\mathbb{P}(\theta\in\Theta_0),\qquad \pi_1=\mathbb{P}(\theta\in\Theta_1), \end{align*} \]

such that \(\pi_0+\pi_1=1\). Taking into account a data set \(x\), the corresponding posterior likelihoods are defined by

\[ \begin{align*} p_0=\mathbb{P}(\theta\in\Theta_0|x),\qquad p_1=\mathbb{P}(\theta\in\Theta_1|x), \end{align*} \]

where again \(p_0+p_1=1\). The definition of the posterior likelihoods is based on Bayes’ theorem. The Bayes factor \(B\) in favour of \(H_0\) against \(H_1\) is then defined by

(11)#\[ \begin{align} B=\frac{p_0/p_1}{\pi_0/\pi_1}=\frac{p_0\pi_1}{p_1\pi_0}. \end{align} \]

Thus, a Bayes factor represents the ratio of the posterior odds \(p_0/p_1\) and the prior odds \(\pi_0/\pi_1\). It reflects the evidence in the data in favour of \(H_0\) as opposed to \(H_1\). By substituting the above definitions into Equation (11), we obtain by applying Bayes’ theorem

(12)#\[ \begin{align} B=\frac{p_0/p_1}{\pi_0/\pi_1}=\frac{\mathbb{P}(\theta\in\Theta_0|x)/\mathbb{P}(\theta\in\theta_0)}{\mathbb{P}(\theta\in\Theta_1|x)/\mathbb{P}(\theta\in\theta_1)} =\frac{\mathbb{P}(x|\theta\in\Theta_0)}{\mathbb{P}(x|\theta\in\Theta_1)} =\frac{\pi(x|\theta\in\Theta_0)}{\pi(x|\theta\in\Theta_1)}, \end{align} \]

where \(\pi(x|\theta\in\Theta_i)=\mathbb{P}(x|\theta\in\Theta_i)\), for \(i\in\{0,1\}\), denotes the likelihood of alternative \(i\).

The informative value of \(B\) clearly depends on its magnitude. The following table provides established thresholds to interpret a concrete value of \(B\) [Kass and Raftery, 1995]:



Evidence against \(H_1\)



not worth more than a bare mention









very strong

The application of Bayes factors has several advantages compared with the approach of classical hypothesis testing. Instead of considering point estimates, parameter values are assumed to underlie a dristibution and a Bayes factor accounts for the whole distribution of the parameter space. Moreover, a Bayes factor is able to provide evidence in favour of \(H_0\) as well as against \(H_0\). However, by definition, Bayes factors only serve as a quantity for pairwise comparisons of alternatives. Besides, as for Bayesian methods in general, Bayes factors can be sensitive to the choice of prior distributions [Morey et al., 2016].

The concrete form of \(\Theta_0\) and \(\Theta_1\) affects the corresponding Bayes factor and the evaluation of a hypothesis test. A detailed account on this topic can be found in [Lee, 2012].

Bayes Factors for Model Selection#

Bayes factors cannot only be used to assess the plausibility of parameter values for a given model (or, e.g., for a distribution), but also for competing inherently different models [Hug, 2014]. Therefore, the focus is now on selecting one model among competing models \(M_1,\dots,M_m\) parameterized by parameters \(\theta_i\in\Theta_i\), for \(i=1,\dots,m\) that best explains the data. Analogous to Equation (12), the Bayes factor in favour of \(M_k\) over \(M_l\) is defined by

\[ \begin{align*} B_{kl}=\frac{\pi(x|M_k)}{\pi(x|M_l)},\quad k,l\in\{1,\dots,m\}. \end{align*} \]

The quantity \(\pi(x|M_j)\), for \(j\in\{1,\dots,m\}\), is called the marginal likelihood of model \(M_j\) that can be written as

(13)#\[ \begin{align} \pi(x|M_j)=\int_{\Theta_j}\pi(x|\theta_j, M_j)\pi(\theta_j|M_j)\mathrm{d}\theta_j. \end{align} \]

Equation (13) reflects that the marginal likelihood \(\pi(x|M_j)\) defines the likelihood of \(x\) for model \(M_j\) where the impact of the parameter \(\theta_j\) is marginalized (integrated) out. More formally, the marginal likelihood for model \(M_j\) is defined as the integral over the parameter space \(\Theta_j\) of the likelihood of \(M_j\) times the prior distribution of \(M_j\).

In order to determine Bayes factors, the integrals from Equation (13) need to be computed for the models of interest. Since the integral for model \(M_j\) is defined over the whole parameter space \(\Theta_j\), an analytic solution is, in practice, not feasible in many cases. However, various estimation techniques have been developed to address this issue. Different procedures are reviewed in [Friel and Wyse, 2012]. In the last decade, the thermodynamic integration approach has been established as the state-of-the-art method to approximate marginal likelihoods. Its application goes back to [Friel and Pettitt, 2008, Lartillot and Philippe, 2006].

In contrast to this article of a Bayesian model selection method, the article model selection provides an introduction to model selection tools for frequentist statistics.



Nial Friel and Anthony N. Pettitt. Marginal likelihood estimation via power posteriors. Journal of the Royal Statistical Society. Series B (Statistical Methodology), 70:589–607, 2008.


Nial Friel and Jason Wyse. Estimating the evidence – a review. Statistica Neerlandica, 66:288–308, 11 2012.


Sabine C. Hug. From low-dimensional model selection to high-dimensional inference: tailoring Bayesian methods to biological dynamical systems. TU München (PhD Thesis), 2014.


Robert E. Kass and Adrian E. Raftery. Bayes factors. Journal of the American Statistical Association, 90:773–795, 1995.


Nicolas Lartillot and Hervé Philippe. Computing Bayes factors using thermodynamic integration. Systematic Biology, 55:195–207, 4 2006.


Peter M. Lee. Bayesian Statistics: An Introduction. Wiley, 4 edition, 2012.


Richard D. Morey, Jan Willem Romeijn, and Jeffrey N. Rouder. The philosophy of Bayes factors and the quantification of statistical evidence. Journal of Mathematical Psychology, 72:6–18, 2016.


Julian Wäsche


Tamadur Albaraghtheh