Bayes factor

<h2 id="definition">Definition</h2>
The Bayes factor is the ratio of two marginal likelihoods; that is, the <a href="/facts/Likelihood_function/1L8P4NBX">likelihoods</a> of two statistical models integrated over the <a href="/facts/Prior_probability/JQKAD4o0">prior probabilities</a> of their parameters.<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a>
The <a href="/facts/Posterior_probability/fojsMD0D">posterior probability</a> 
 
 
 
 Pr
 (
 M
 
 |
 
 D
 )
 
 
 {\displaystyle \Pr(M|D)}
 
 of a model M given data D is given by <a href="/facts/Bayes%27_theorem/wQqFB2Dt">Bayes' theorem</a>:

Pr
        (
        M
        
          |
        
        D
        )
        =
        
          
            
              Pr
              (
              D
              
                |
              
              M
              )
              Pr
              (
              M
              )
            
            
              Pr
              (
              D
              )
            
          
        
        .
      
    
    {\displaystyle \Pr(M|D)={\frac {\Pr(D|M)\Pr(M)}{\Pr(D)}}.}

The key data-dependent term 
 
 
 
 Pr
 (
 D
 
 |
 
 M
 )
 
 
 {\displaystyle \Pr(D|M)}
 
 represents the probability that some data are produced under the assumption of the model M; evaluating it correctly is the key to Bayesian model comparison.
Given a <a href="/facts/Model_selection/R11QK2GC">model selection</a> problem in which one wishes to choose between two models on the basis of observed data D, the plausibility of the two different models M1 and M2, parametrised by model parameter vectors 
 
 
 
 
 θ
 
 1
 
 
 
 
 {\displaystyle \theta _{1}}
 
 and 
 
 
 
 
 θ
 
 2
 
 
 
 
 {\displaystyle \theta _{2}}
 
, is assessed by the Bayes factor K given by

K
        =
        
          
            
              Pr
              (
              D
              
                |
              
              
                M
                
                  1
                
              
              )
            
            
              Pr
              (
              D
              
                |
              
              
                M
                
                  2
                
              
              )
            
          
        
        =
        
          
            
              ∫
              Pr
              (
              
                θ
                
                  1
                
              
              
                |
              
              
                M
                
                  1
                
              
              )
              Pr
              (
              D
              
                |
              
              
                θ
                
                  1
                
              
              ,
              
                M
                
                  1
                
              
              )
              
              d
              
                θ
                
                  1
                
              
            
            
              ∫
              Pr
              (
              
                θ
                
                  2
                
              
              
                |
              
              
                M
                
                  2
                
              
              )
              Pr
              (
              D
              
                |
              
              
                θ
                
                  2
                
              
              ,
              
                M
                
                  2
                
              
              )
              
              d
              
                θ
                
                  2
                
              
            
          
        
        =
        
          
            
              
                Pr
                (
                
                  M
                  
                    1
                  
                
                
                  |
                
                D
                )
                Pr
                (
                D
                )
              
              
                Pr
                (
                
                  M
                  
                    1
                  
                
                )
              
            
            
              
                Pr
                (
                
                  M
                  
                    2
                  
                
                
                  |
                
                D
                )
                Pr
                (
                D
                )
              
              
                Pr
                (
                
                  M
                  
                    2
                  
                
                )
              
            
          
        
        =
        
          
            
              Pr
              (
              
                M
                
                  1
                
              
              
                |
              
              D
              )
            
            
              Pr
              (
              
                M
                
                  2
                
              
              
                |
              
              D
              )
            
          
        
        
          
            
              Pr
              (
              
                M
                
                  2
                
              
              )
            
            
              Pr
              (
              
                M
                
                  1
                
              
              )
            
          
        
        .
      
    
    {\displaystyle K={\frac {\Pr(D|M_{1})}{\Pr(D|M_{2})}}={\frac {\int \Pr(\theta _{1}|M_{1})\Pr(D|\theta _{1},M_{1})\,d\theta _{1}}{\int \Pr(\theta _{2}|M_{2})\Pr(D|\theta _{2},M_{2})\,d\theta _{2}}}={\frac {\frac {\Pr(M_{1}|D)\Pr(D)}{\Pr(M_{1})}}{\frac {\Pr(M_{2}|D)\Pr(D)}{\Pr(M_{2})}}}={\frac {\Pr(M_{1}|D)}{\Pr(M_{2}|D)}}{\frac {\Pr(M_{2})}{\Pr(M_{1})}}.}

When the two models have equal prior probability, so that 
 
 
 
 Pr
 (
 
 M
 
 1
 
 
 )
 =
 Pr
 (
 
 M
 
 2
 
 
 )
 
 
 {\displaystyle \Pr(M_{1})=\Pr(M_{2})}
 
, the Bayes factor is equal to the ratio of the posterior probabilities of M1 and M2. If instead of the Bayes factor integral, the likelihood corresponding to the <a href="/facts/Maximum_likelihood/0Yq2dpQD">maximum likelihood estimate</a> of the parameter for each statistical model is used, then the test becomes a classical <a href="/facts/Likelihood-ratio_test/oFqLh3R0">likelihood-ratio test</a>. Unlike a likelihood-ratio test, this Bayesian model comparison does not depend on any single set of parameters, as it integrates over all parameters in each model (with respect to the respective priors). An advantage of the use of Bayes factors is that it automatically, and quite naturally, includes a penalty for including too much model structure.<a class="footnote-ref" id="fnref:12" href="#fn:12">12</a> It thus guards against <a href="/facts/Overfitting/5xnFLcMg">overfitting</a>. For models where an explicit version of the likelihood is not available or too costly to evaluate numerically, <a href="/facts/Approximate_Bayesian_computation/pY4He1HY">approximate Bayesian computation</a> can be used for model selection in a Bayesian framework,<a class="footnote-ref" id="fnref:13" href="#fn:13">13</a>
with the caveat that approximate-Bayesian estimates of Bayes factors are often biased.<a class="footnote-ref" id="fnref:14" href="#fn:14">14</a>
Other approaches are:

<ul><li>to treat model comparison as a <a href="/facts/Decision_theory/TbXcQX0v">decision problem</a>, computing the expected value or cost of each model choice;</li>
<li>to use <a href="/facts/Minimum_message_length/GqS4Tydg">minimum message length</a> (MML).</li>
<li>to use <a href="/facts/Minimum_description_length/em8tXqjf">minimum description length</a> (MDL).</li></ul>
<h2 id="interpretation">Interpretation</h2>
A value of K > 1 means that M1 is more strongly supported by the data under consideration than M2. Note that classical <a href="/facts/Hypothesis_testing/yv9RPF6U">hypothesis testing</a> gives one hypothesis (or model) preferred status (the 'null hypothesis'), and only considers evidence against it. The fact that a Bayes factor can produce evidence for and not just against a null hypothesis is one of the key advantages of this analysis method.<a class="footnote-ref" id="fnref:15" href="#fn:15">15</a> 
<a href="/facts/Harold_Jeffreys/yWPSFpPB">Harold Jeffreys</a> gave a scale (Jeffreys' scale) for interpretation of 
 
 
 
 K
 
 
 {\displaystyle K}
 
:<a class="footnote-ref" id="fnref:16" href="#fn:16">16</a>

<table><tbody><tr><th>K</th><th>dHart</th><th>bits</th><th>Strength of evidence</th></tr><tr><td>< 100</td><td>< 0</td><td>< 0</td><td>Negative (supports M2)</td></tr><tr><td>100 to 101/2</td><td>0 to 5</td><td>0 to 1.6</td><td>Barely worth mentioning</td></tr><tr><td>101/2 to 101</td><td>5 to 10</td><td>1.6 to 3.3</td><td>Substantial</td></tr><tr><td>101 to 103/2</td><td>10 to 15</td><td>3.3 to 5.0</td><td>Strong</td></tr><tr><td>103/2 to 102</td><td>15 to 20</td><td>5.0 to 6.6</td><td>Very strong</td></tr><tr><td>> 102</td><td>> 20</td><td>> 6.6</td><td>Decisive</td></tr></tbody></table>
The second column gives the corresponding weights of evidence in <a href="/facts/Hartley_(unit)/HHnoxYMR">decihartleys</a> (also known as <a href="/facts/Deciban/HHnoxYMR">decibans</a>); <a href="/facts/Bit/S972NSaD">bits</a> are added in the third column for clarity. The table continues in the other direction, so that, for example, 
 
 
 
 K
 ≤
 
 10
 
 −
 2
 
 
 
 
 {\displaystyle K\leq 10^{-2}}
 
 is decisive evidence for 
 
 
 
 
 M
 
 2
 
 
 
 
 {\displaystyle M_{2}}
 
.
An alternative table, widely cited, is provided by Kass and Raftery (1995):<a class="footnote-ref" id="fnref:17" href="#fn:17">17</a>

<table><tbody><tr><th>log10 K</th><th>K</th><th>Strength of evidence</th></tr><tr><td>0 to 1/2</td><td>1 to 3.2</td><td>Not worth more than a bare mention</td></tr><tr><td>1/2 to 1</td><td>3.2 to 10</td><td>Substantial</td></tr><tr><td>1 to 2</td><td>10 to 100</td><td>Strong</td></tr><tr><td>> 2</td><td>> 100</td><td>Decisive</td></tr></tbody></table>
According to <a href="/facts/I._J._Good/nRWnP9KP">I. J. Good</a>, the <a href="/facts/Just-noticeable_difference/r3KeemTF">just-noticeable difference</a> of humans in their everyday life, when it comes to a change <a href="/facts/Bayesian_probability/Cvn7WWfG">degree of belief</a> in a hypothesis, is about a factor of 1.3x, or 1 deciban, or 1/3 of a bit, or from 1:1 to 5:4 in odds ratio.<a class="footnote-ref" id="fnref:18" href="#fn:18">18</a>

<h2 id="example">Example</h2>
Suppose we have a <a href="/facts/Random_variable/TwTBXnLT">random variable</a> that produces either a success or a failure. We want to compare a model M1 where the probability of success is q = 1⁄2, and another model M2 where q is unknown and we take a <a href="/facts/Prior_distribution/JQKAD4o0">prior distribution</a> for q that is <a href="/facts/Uniform_distribution_(continuous)/XbnlVljT">uniform</a> on [0,1]. We take a sample of 200, and find 115 successes and 85 failures. The likelihood can be calculated according to the <a href="/facts/Binomial_distribution/UMoFMjDj">binomial distribution</a>:

(
              
              
                200
                115
              
              
                )
              
            
          
          
            q
            
              115
            
          
          (
          1
          −
          q
          
            )
            
              85
            
          
        
        .
      
    
    {\displaystyle {{200 \choose 115}q^{115}(1-q)^{85}}.}

Thus we have for M1

P
        (
        X
        =
        115
        ∣
        
          M
          
            1
          
        
        )
        =
        
          
            
              (
            
            
              200
              115
            
            
              )
            
          
        
        
          
            (
            
              
                1
                2
              
            
            )
          
          
            200
          
        
        ≈
        0.006
      
    
    {\displaystyle P(X=115\mid M_{1})={200 \choose 115}\left({1 \over 2}\right)^{200}\approx 0.006}

whereas for M2 we have

P
        (
        X
        =
        115
        ∣
        
          M
          
            2
          
        
        )
        =
        
          ∫
          
            0
          
          
            1
          
        
        
          
            
              (
            
            
              200
              115
            
            
              )
            
          
        
        
          q
          
            115
          
        
        (
        1
        −
        q
        
          )
          
            85
          
        
        d
        q
        =
        
          
            1
            201
          
        
        ≈
        0.005
      
    
    {\displaystyle P(X=115\mid M_{2})=\int _{0}^{1}{200 \choose 115}q^{115}(1-q)^{85}dq={1 \over 201}\approx 0.005}

The ratio is then 1.2, which is "barely worth mentioning" even if it points very slightly towards M1.
A <a href="/facts/Frequentist/UcNgebQc">frequentist</a> <a href="/facts/Statistical_hypothesis_testing/yv9RPF6U">hypothesis test</a> of M1 (here considered as a <a href="/facts/Null_hypothesis/C8NwSE9J">null hypothesis</a>) would have produced a very different result. Such a test says that M1 should be rejected at the 5% significance level, since the probability of getting 115 or more successes from a sample of 200 if q = 1⁄2 is 0.02, and as a two-tailed test of getting a figure as extreme as or more extreme than 115 is 0.04. Note that 115 is more than two standard deviations away from 100. Thus, whereas a <a href="/facts/Frequentist/UcNgebQc">frequentist</a> <a href="/facts/Statistical_hypothesis_testing/yv9RPF6U">hypothesis test</a> would yield <a href="/facts/Statistical_significance/PMxjdlAu">significant results</a> at the 5% significance level, the Bayes factor hardly considers this to be an extreme result. Note, however, that a non-uniform prior (for example one that reflects the fact that you expect the number of success and failures to be of the same order of magnitude) could result in a Bayes factor that is more in agreement with the frequentist hypothesis test.
A classical <a href="/facts/Likelihood-ratio_test/oFqLh3R0">likelihood-ratio test</a> would have found the <a href="/facts/Maximum_likelihood/0Yq2dpQD">maximum likelihood</a> estimate for q, namely 
 
 
 
 
 
 
 q
 ^
 
 
 
 =
 
 
 115
 200
 
 
 =
 0.575
 
 
 {\displaystyle {\hat {q}}={\frac {115}{200}}=0.575}
 
, whence

P
          (
          X
          =
          115
          ∣
          
            M
            
              2
            
          
          )
          =
          
            
              
                
                  (
                
                
                  200
                  115
                
                
                  )
                
              
            
            
              
                
                  
                    q
                    ^
                  
                
              
              
                115
              
            
            (
            1
            −
            
              
                
                  q
                  ^
                
              
            
            
              )
              
                85
              
            
          
          ≈
          0.06
        
      
    
    {\displaystyle \textstyle P(X=115\mid M_{2})={{200 \choose 115}{\hat {q}}^{115}(1-{\hat {q}})^{85}}\approx 0.06}

(rather than averaging over all possible q). That gives a likelihood ratio of 0.1 and points towards M2.
M2 is a more complex model than M1 because it has a free parameter which allows it to model the data more closely. The ability of Bayes factors to take this into account is a reason why <a href="/facts/Bayesian_inference/tEpK3zLx">Bayesian inference</a> has been put forward as a theoretical justification for and generalisation of <a href="/facts/Occam%27s_razor/KG1Bx0J9">Occam's razor</a>, reducing <a href="/facts/Type_I_error/aazbF3fq">Type I errors</a>.<a class="footnote-ref" id="fnref:19" href="#fn:19">19</a>
On the other hand, the modern method of <a href="/facts/Relative_likelihood/Ur6INqE2">relative likelihood</a> takes into account the number of free parameters in the models, unlike the classical likelihood ratio. The relative likelihood method could be applied as follows. Model M1 has 0 parameters, and so its <a href="/facts/Akaike_information_criterion/RjKKgcnF">Akaike information criterion</a> (AIC) value is 
 
 
 
 2
 ⋅
 0
 −
 2
 ⋅
 ln
 ⁡
 (
 0.005956
 )
 ≈
 10.2467
 
 
 {\displaystyle 2\cdot 0-2\cdot \ln(0.005956)\approx 10.2467}
 
. Model M2 has 1 parameter, and so its AIC value is 
 
 
 
 2
 ⋅
 1
 −
 2
 ⋅
 ln
 ⁡
 (
 0.056991
 )
 ≈
 7.7297
 
 
 {\displaystyle 2\cdot 1-2\cdot \ln(0.056991)\approx 7.7297}
 
. Hence M1 is about 
 
 
 
 exp
 ⁡
 
 (
 
 
 
 7.7297
 −
 10.2467
 
 2
 
 
 )
 
 ≈
 0.284
 
 
 {\displaystyle \exp \left({\frac {7.7297-10.2467}{2}}\right)\approx 0.284}
 
 times as probable as M2 to minimize the information loss. Thus M2 is slightly preferred, but M1 cannot be excluded.

<h2 id="see-also">See also</h2>
<ul>
<li>Mathematics portal</li></ul>
<ul><li><a href="/facts/Akaike_information_criterion/RjKKgcnF">Akaike information criterion</a></li>
<li><a href="/facts/Approximate_Bayesian_computation/pY4He1HY">Approximate Bayesian computation</a></li>
<li><a href="/facts/Bayesian_information_criterion/NHspnLrM">Bayesian information criterion</a></li>
<li><a href="/facts/Deviance_information_criterion/s2HhaVvc">Deviance information criterion</a></li>
<li><a href="/facts/Lindley%27s_paradox/BM4rbCTr">Lindley's paradox</a></li>
<li><a href="/facts/Minimum_message_length/GqS4Tydg">Minimum message length</a></li>
<li><a href="/facts/Model_selection/R11QK2GC">Model selection</a></li>
<li><a href="/facts/E-values/7oPWyL0T">E-Value</a></li></ul>
Statistical ratios
<ul><li><a href="/facts/Odds_ratio/7WcmubfN">Odds ratio</a></li>
<li><a href="/facts/Relative_risk/IQm0lJtR">Relative risk</a></li></ul>

<h2 id="further-reading">Further reading</h2>
<ul><li>Bernardo, J.; Smith, A. F. M. (1994). Bayesian Theory. John Wiley. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-471-92416-4.</li>
<li>Denison, D. G. T.; Holmes, C. C.; Mallick, B. K.; Smith, A. F. M. (2002). Bayesian Methods for Nonlinear Classification and Regression. John Wiley. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-471-49036-9.</li>
<li>Dienes, Z. (2019). How do I know what my theory predicts? Advances in Methods and Practices in Psychological Science <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1177%2F2515245919876960">10.1177/2515245919876960</a></li>
<li>Duda, Richard O.; Hart, Peter E.; Stork, David G. (2000). "Section 9.6.5". Pattern classification (2nd ed.). Wiley. pp. 487–489. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-471-05669-3.</li>
<li>Gelman, A.; Carlin, J.; Stern, H.; Rubin, D. (1995). Bayesian Data Analysis. London: <a href="/facts/Chapman_%26_Hall/roCcaakR">Chapman & Hall</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-412-03991-5.</li>
<li><a href="/facts/Edwin_Thompson_Jaynes/PU536IkK">Jaynes, E. T.</a> (1994), <a href="http://omega.math.albany.edu:8008/JaynesBook.html">Probability Theory: the logic of science</a>, chapter 24.</li>
<li>Kadane, Joseph B.; Dickey, James M. (1980). "Bayesian Decision Theory and the Simplification of Models". In Kmenta, Jan; Ramsey, James B. (eds.). Evaluation of Econometric Models. New York: Academic Press. pp. 245–268. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-12-416550-8.</li>
<li>Lee, P. M. (2012). Bayesian Statistics: an introduction. Wiley. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9781118332573.</li>
<li>Richard, Mark; Vecer, Jan (2021). <a href="https://doi.org/10.3390%2Frisks9020031">"Efficiency Testing of Prediction Markets: Martingale Approach, Likelihood Ratio and Bayes Factor Analysis"</a>. Risks. 9 (2): 31. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.3390%2Frisks9020031">10.3390/risks9020031</a>. <a href="/facts/Hdl_(identifier)/rdebSxmC">hdl</a>:<a href="https://hdl.handle.net/10419%2F258120">10419/258120</a>.</li>
<li>Winkler, Robert (2003). Introduction to Bayesian Inference and Decision (2nd ed.). Probabilistic. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-9647938-4-9.</li></ul>
<h2 id="external-links">External links</h2>
<ul><li><a href="http://bayesfactorpcl.r-forge.r-project.org/">BayesFactor</a> —an R package for computing Bayes factors in common research designs</li>
<li><a href="http://www.lifesci.sussex.ac.uk/home/Zoltan_Dienes/inference/Bayes.htm">Bayes factor calculator</a> — Online calculator for informed Bayes factors</li>
<li><a href="http://pcl.missouri.edu/bayesfactor">Bayes Factor Calculators</a> <a href="https://web.archive.org/web/20150507195546/http://pcl.missouri.edu/bayesfactor">Archived</a> 2015-05-07 at the <a href="/facts/Wayback_Machine/nmQ3a6JC">Wayback Machine</a> —web-based version of much of the BayesFactor package</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Morey, Richard D.; Romeijn, Jan-Willem; Rouder, Jeffrey N. (2016). "The philosophy of Bayes factors and the quantification of statistical evidence". Journal of Mathematical Psychology. 72: 6–18. doi:10.1016/j.jmp.2015.11.001. <a href="https://doi.org/10.1016%2Fj.jmp.2015.11.001" target="_blank">https://doi.org/10.1016%2Fj.jmp.2015.11.001</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Lesaffre, Emmanuel; Lawson, Andrew B. (2012). "Bayesian hypothesis testing". Bayesian Biostatistics. Somerset: John Wiley & Sons. pp. 72–78. doi:10.1002/9781119942412.ch3. ISBN 978-0-470-01823-1. <a href="978-0-470-01823-1" target="_blank">978-0-470-01823-1</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Ly, Alexander; et al. (2020). "The Bayesian Methodology of Sir Harold Jeffreys as a Practical Alternative to the P Value Hypothesis Test". Computational Brain & Behavior. 3 (2): 153–161. doi:10.1007/s42113-019-00070-x. hdl:2066/226717. <a href="https://doi.org/10.1007%2Fs42113-019-00070-x" target="_blank">https://doi.org/10.1007%2Fs42113-019-00070-x</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Llorente, Fernando; et al. (2023). "Marginal likelihood computation for model selection and hypothesis testing: an extensive review". SIAM Review. to appear: 3–58. arXiv:2005.08334. doi:10.1137/20M1310849. S2CID 210156537. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Congdon, Peter (2014). "Estimating model probabilities or marginal likelihoods in practice". Applied Bayesian Modelling (2nd ed.). Wiley. pp. 38–40. ISBN 978-1-119-95151-3. <a href="978-1-119-95151-3" target="_blank">978-1-119-95151-3</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Chib, Siddhartha (1995). "Marginal Likelihood from the Gibbs Output". Journal of the American Statistical Association. 90 (432): 1313–1321. doi:10.2307/2291521. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Chib, Siddhartha; Jeliazkov, Ivan (2001). "Marginal Likelihood from the Metropolis–Hastings Output". Journal of the American Statistical Association. 96 (453): 270–281. doi:10.1198/016214501750332848. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Koop, Gary (2003). "Model Comparison: The Savage–Dickey Density Ratio". Bayesian Econometrics. Somerset: John Wiley & Sons. pp. 69–71. ISBN 0-470-84567-8. <a href="0-470-84567-8" target="_blank">0-470-84567-8</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Wagenmakers, Eric-Jan; Lodewyckx, Tom; Kuriyal, Himanshu; Grasman, Raoul (2010). "Bayesian hypothesis testing for psychologists: A tutorial on the Savage–Dickey method" (PDF). Cognitive Psychology. 60 (3): 158–189. doi:10.1016/j.cogpsych.2009.12.001. PMID 20064637. S2CID 206867662. <a href="http://www.ejwagenmakers.com/2010/WagenmakersEtAlCogPsy2010.pdf" target="_blank">http://www.ejwagenmakers.com/2010/WagenmakersEtAlCogPsy2010.pdf</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Ibrahim, Joseph G.; Chen, Ming-Hui; Sinha, Debajyoti (2001). "Model Comparison". Bayesian Survival Analysis. Springer Series in Statistics. New York: Springer. pp. 246–254. doi:10.1007/978-1-4757-3447-8_6. ISBN 0-387-95277-2. <a href="0-387-95277-2" target="_blank">0-387-95277-2</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
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<li id="fn:17">Robert E. Kass & Adrian E. Raftery (1995). "Bayes Factors" (PDF). Journal of the American Statistical Association. 90 (430): 791. doi:10.2307/2291091. JSTOR 2291091. <a href="http://www.andrew.cmu.edu/user/kk3n/simplicity/KassRaftery1995.pdf" target="_blank">http://www.andrew.cmu.edu/user/kk3n/simplicity/KassRaftery1995.pdf</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></li>
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<li id="fn:19">Sharpening Ockham's Razor On a Bayesian Strop <a href="http://www.stat.duke.edu/~berger/papers/ockham.html" target="_blank">http://www.stat.duke.edu/~berger/papers/ockham.html</a> <a href="#fnref:19" class="footnote-back-ref">↩</a></li>
</ol>

Bayes factor open-in-new

Bayes factor