Bayesian Model Selection and Averaging Nonlinear Models Bayes - PowerPoint PPT Presentation

Bayesian Model Selection and Averaging Will Penny Model Comparison Model Evidence Complexity Bayesian Model Selection and Averaging Nonlinear Models Bayes factors Example Families FFX Model Will Penny Inference RFX Model Inference Example PXPs SPM short course for M/EEG, Model Averaging London 2015 RFX Parameter Inference FFX Parameter Inference References

Bayesian Model Ten Simple Rules Selection and Averaging Will Penny Model Comparison Model Evidence Complexity Nonlinear Models Bayes factors Example Families FFX Model Inference RFX Model Inference Example PXPs Model Averaging RFX Parameter Inference FFX Parameter Inference References Stephan et al. Neuroimage, 2010

Bayesian Model Model Structure Selection and Averaging Will Penny Model Comparison Model Evidence Complexity Nonlinear Models Bayes factors Example Families FFX Model Inference RFX Model Inference Example PXPs Model Averaging RFX Parameter Inference FFX Parameter Inference References

Bayesian Model Model Evidence Selection and Averaging Will Penny Model Comparison The model evidence is given by integrating out the Model Evidence Complexity dependence on model parameters Nonlinear Models Bayes factors Example � Families p ( y | m ) = p ( y , θ | m ) d θ FFX Model Inference � = p ( y | θ, m ) p ( θ | m ) d θ RFX Model Inference Example PXPs Because we have marginalised over θ the evidence is Model Averaging also known as the marginal likelihood. RFX Parameter Inference For linear Gaussian models there is an analytic FFX Parameter Inference expression for the model evidence. References

Bayesian Model Linear Models Selection and Averaging For Linear Models Will Penny y = Xw + e Model Comparison where X is a design matrix and w are now regression Model Evidence coefficients. For prior mean µ w , prior covariance C w , Complexity Nonlinear Models observation noise covariance C y the posterior distribution Bayes factors Example is given by Families FFX Model S − 1 X T C − 1 y X + C − 1 = Inference w w � � RFX Model X T C − 1 y y + C − 1 m w = S w w µ w Inference Example PXPs Model Averaging RFX Parameter Inference FFX Parameter Inference References

Bayesian Model Model Evidence Selection and Averaging Will Penny The log model evidence comprises sum squared Model Comparison precision weighted prediction errors and Occam factors Model Evidence Complexity Nonlinear Models − 1 y e y − 1 2 log | C y | − N y Bayes factors 2 e T y C − 1 log p ( y | m ) = 2 log 2 π Example Families 1 w e w − 1 2 log | C w | FFX Model 2 e T w C − 1 − Inference | S w | RFX Model Inference where prediction errors are the difference between what Example PXPs is expected and what is observed Model Averaging RFX Parameter e y = y − Xm w Inference FFX Parameter = m w − µ w e w Inference References Bishop, Pattern Recognition and Machine Learning, 2006

Bayesian Model Accuracy and Complexity Selection and Averaging Will Penny The log evidence for model m can be split into an accuracy and a complexity term Model Comparison Model Evidence Complexity log p ( y | m ) = Accuracy ( m ) − Complexity ( m ) Nonlinear Models Bayes factors Example where Families FFX Model Accuracy ( m ) = − 1 y e y − 1 2 log | C y | − N y Inference 2 e T y C − 1 2 log 2 π RFX Model Inference Example and PXPs Model Averaging 1 w e w + 1 2 log | C w | RFX Parameter 2 e T w C − 1 Complexity ( m ) = Inference | S w | FFX Parameter ≈ KL ( prior || posterior ) Inference References The Kullback-Leibler divergence measures the distance between probability distributions.

Bayesian Model Small KL Selection and Averaging Will Penny Model Comparison Model Evidence Complexity Nonlinear Models Bayes factors Example Families FFX Model Inference RFX Model Inference Example PXPs Model Averaging RFX Parameter Inference FFX Parameter Inference References

Bayesian Model Medium KL Selection and Averaging Will Penny Model Comparison Model Evidence Complexity Nonlinear Models Bayes factors Example Families FFX Model Inference RFX Model Inference Example PXPs Model Averaging RFX Parameter Inference FFX Parameter Inference References

Bayesian Model Big KL Selection and Averaging Will Penny Model Comparison Model Evidence Complexity Nonlinear Models Bayes factors Example Families FFX Model Inference RFX Model Inference Example PXPs Model Averaging RFX Parameter Inference FFX Parameter Inference References

Bayesian Model Nonlinear Models Selection and Averaging For nonlinear models, we replace the true posterior with Will Penny the approximate posterior ( m w , S w ), and the previous expression becomes an approximation to the log model Model Comparison Model Evidence evidence called the (negative) Free Energy Complexity Nonlinear Models Bayes factors − 1 y e y − 1 2 log | C y | − N y Example 2 e T y C − 1 F = 2 log 2 π Families FFX Model 1 w e w − 1 2 log | C w | Inference 2 e T w C − 1 − | S w | RFX Model Inference Example where PXPs Model Averaging RFX Parameter e y = y − g ( m w ) Inference e w = m w − µ w FFX Parameter Inference References and g ( m w ) is the DCM prediction. This is used to approximate the model evidence for DCMs. W Penny, Neuroimage, 2011 .

Bayesian Model Bayes rule for models Selection and Averaging A prior distribution over model space p ( m ) (or ‘hypothesis Will Penny space’) can be updated to a posterior distribution after observing data y . Model Comparison Model Evidence Complexity Nonlinear Models Bayes factors Example Families FFX Model Inference RFX Model Inference Example PXPs This is implemented using Bayes rule Model Averaging p ( m | y ) = p ( y | m ) p ( m ) RFX Parameter Inference p ( y ) FFX Parameter Inference where p ( y | m ) is referred to as the evidence for model m and References the denominator is given by � p ( y | m ′ ) p ( m ′ ) p ( y ) = m ′

Bayesian Model Bayes Factors Selection and Averaging Will Penny Model Comparison The Bayes factor for model j versus i is the ratio of model Model Evidence Complexity evidences Nonlinear Models B ji = p ( y | m = j ) Bayes factors Example p ( y | m = i ) Families We have FFX Model Inference B ij = 1 RFX Model B ji Inference Example PXPs Hence Model Averaging RFX Parameter = log p ( y | m = j ) − log p ( y | m = i ) logB ji Inference = F j − F i FFX Parameter Inference References

Bayesian Model Posterior Model Probability Selection and Averaging Will Penny Given equal priors, p ( m = i ) = p ( m = j ) the posterior Model Comparison model probability is Model Evidence Complexity Nonlinear Models p ( y | m = i ) Bayes factors p ( m = i | y ) = Example p ( y | m = i ) + p ( y | m = j ) Families 1 FFX Model = Inference 1 + p ( y | m = j ) p ( y | m = i ) RFX Model Inference 1 Example = PXPs 1 + B ji Model Averaging 1 RFX Parameter = Inference 1 + exp ( log B ji ) FFX Parameter 1 Inference = References 1 + exp ( − log B ij )

Bayesian Model Posterior Model Probability Selection and Averaging Will Penny Model Comparison Model Evidence Complexity Nonlinear Models Hence Bayes factors Example p ( m = i | y ) = σ ( log B ij ) Families FFX Model where is the Bayes factor for model i versus model j and Inference RFX Model 1 Inference σ ( x ) = Example 1 + exp ( − x ) PXPs Model Averaging is the sigmoid function. RFX Parameter Inference FFX Parameter Inference References

Bayesian Model Bayes factors Selection and Averaging The posterior model probability is a sigmoidal function of Will Penny the log Bayes factor Model Comparison Model Evidence Complexity p ( m = i | y ) = σ ( log B ij ) Nonlinear Models Bayes factors Example Families FFX Model Inference RFX Model Inference Example PXPs Model Averaging RFX Parameter Inference FFX Parameter Inference References

Bayesian Model Bayes factors Selection and Averaging The posterior model probability is a sigmoidal function of Will Penny the log Bayes factor Model Comparison Model Evidence p ( m = i | y ) = σ ( log B ij ) Complexity Nonlinear Models Bayes factors Example Families FFX Model Inference RFX Model Inference Example PXPs Model Averaging RFX Parameter Inference FFX Parameter Inference References Kass and Raftery, JASA, 1995 .

Bayesian Model Odds Ratios Selection and Averaging If we don’t have uniform priors one can work with odds Will Penny ratios. Model Comparison Model Evidence The prior and posterior odds ratios are defined as Complexity Nonlinear Models Bayes factors Example p ( m = i ) π 0 = Families ij p ( m = j ) FFX Model Inference p ( m = i | y ) π ij = RFX Model p ( m = j | y ) Inference Example PXPs resepectively, and are related by the Bayes Factor Model Averaging RFX Parameter π ij = B ij × π 0 Inference ij FFX Parameter Inference eg. priors odds of 2 and Bayes factor of 10 leads References posterior odds of 20. An odds ratio of 20 is 20-1 ON in bookmakers parlance.

Bayesian Model Example Selection and Averaging Will Penny Modelling auditory responses with DCM for ERP Model Comparison Model Evidence Complexity Nonlinear Models Bayes factors Example Families FFX Model Inference RFX Model Inference Example PXPs Model Averaging RFX Parameter Inference FFX Parameter Inference References Garrido et al, PNAS, 2007