Dynamic causal modeling

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Bayesian model reduction

Bayesian model reduction ^[1][2] is a method for computing the evidence and parameters of Bayesian models which differ in the specification of their priors. A full model is fitted to the available data using standard approaches. Then, hypotheses are tested by defining one or more 'reduced' models with alternative priors. A reduced model generally has more restrictive priors than the full model, which in the limit will switch off certain parameters. The evidence and parameters of the reduced models can then be computed from the evidence and estimated (posterior) parameters of the full model. If the priors and posteriors are normally distributed, then there is an analytic solution which can be computed rapidly. Bayesian model reduction has multiple scientific and engineering applications, including rapidly scoring the evidence for large numbers of models and facilitating the estimation of hierarchical models (Parametric Empirical Bayes).

Theory

Consider some model with parameters ${\displaystyle \theta }$  and a prior probability density on those parameters ${\displaystyle p(\theta )}$ . The posterior belief about ${\displaystyle \theta }$  after seeing the data is given by Bayes rule:

${\displaystyle {\begin{aligned}p(\theta |y)&={\frac {p(y|\theta )p(\theta )}{p(y)}}\\p(y)&=\int p(y|\theta )p(\theta )d\theta \end{aligned}}}$

1

The second line of Equation 1 is the model evidence, which is the probability of observing the data given the model. In practice, the posterior cannot usually be computed analytically due to the integral over parameters. Therefore, the posteriors are estimated using approaches such as MCMC sampling or variational Bayes. A reduced model can then be defined with an alternative set of priors ${\displaystyle {\tilde {p}}(\theta )}$ :

${\displaystyle {\begin{aligned}{\tilde {p}}(\theta |y)&={\frac {p(y|\theta ){\tilde {p}}(\theta )}{{\tilde {p}}(y)}}\\{\tilde {p}}(y)&=\int p(y|\theta ){\tilde {p}}(\theta )d\theta \end{aligned}}}$

2

The objective of Bayesian model reduction is to compute the posterior ${\displaystyle {\tilde {p}}(\theta |y)}$  and evidence ${\displaystyle {\tilde {p}}(y)}$  of the reduced model from the posterior ${\displaystyle p(\theta |y)}$  and evidence ${\displaystyle p(y)}$  of the full model. Combining Equation 1 and Equation 2 and re-arranging, the reduced posterior ${\displaystyle {\tilde {p}}(\theta |y)}$  can be expressed as the product of the full posterior, the ratio of priors and the ratio of evidences:

${\displaystyle {\begin{aligned}{\frac {{\tilde {p}}(\theta |y){\tilde {p}}(y)}{p(\theta |y)p(y)}}&={\frac {p(y|\theta ){\tilde {p}}(\theta )}{p(y|\theta )p(\theta )}}\\\Rightarrow {\tilde {p}}(\theta |y)&=p(\theta |y){\frac {{\tilde {p}}(\theta )}{p(\theta )}}{\frac {p(y)}{{\tilde {p}}(y)}}\end{aligned}}}$

3

The evidence for the reduced model is obtained by integrating over the parameters of each side of the equation:

${\displaystyle \int {\tilde {p}}(\theta |y)d\theta =\int p(\theta |y){\frac {{\tilde {p}}(\theta )}{p(\theta )}}{\frac {p(y)}{{\tilde {p}}(y)}}d\theta =1}$

4

And by re-arrangement:

${\displaystyle {\begin{aligned}1&=\int p(\theta |y){\frac {{\tilde {p}}(\theta )}{p(\theta )}}{\frac {p(y)}{{\tilde {p}}(y)}}d\theta \\&={\frac {p(y)}{{\tilde {p}}(y)}}\int p(\theta |y){\frac {{\tilde {p}}(\theta )}{p(\theta )}}d\theta \\\Rightarrow {\tilde {p}}(y)&=p(y)\int p(\theta |y){\frac {{\tilde {p}}(\theta )}{p(\theta )}}d\theta \end{aligned}}}$

5

Gaussian priors and posteriors

Under Gaussian prior and posterior densities, as are used in the context of variational Bayes, Bayesian model reduction has a simple analytical expression. Given normal densities for the priors and posteriors:

${\displaystyle {\begin{aligned}p(\theta )&=N(\theta ;\mu _{0},\Sigma _{0})\\{\tilde {p}}(\theta )&=N(\theta ;{\tilde {\mu }}_{0},{\tilde {\Sigma }}_{0})\\p(\theta |y)&=N(\theta ;\mu ,\Sigma )\\{\tilde {p}}(\theta |y)&=N(\theta ;{\tilde {\mu }},{\tilde {\Sigma }})\\\end{aligned}}}$

6

Where the tilde symbol (~) indicates quantities relating to the reduced model and subscript zero - such as ${\displaystyle \mu _{0}}$  - indicates parameters of the priors. For convenience we also define precision matrices, which are simply the inverse of each covariance matrix:

${\displaystyle {\begin{aligned}\Pi &=\Sigma ^{-1}\\\Pi _{0}&=\Sigma _{0}^{-1}\\{\tilde {\Pi }}&={\tilde {\Sigma }}^{-1}\\{\tilde {\Pi }}_{0}&={\tilde {\Sigma }}_{0}^{-1}\\\end{aligned}}}$

7

We also assume that the free energy of the full model ${\displaystyle F}$  has been computed, which is a lower bound on the log model evidence: ${\displaystyle F\approx \ln {p(y)}}$ . The reduced model's free energy ${\displaystyle {\tilde {F}}}$  and parameters ${\displaystyle ({\tilde {\mu }},{\tilde {\Sigma }})}$  are then given by the expressions:

${\displaystyle {\begin{aligned}{\tilde {F}}&={\frac {1}{2}}\ln |{\tilde {\Pi }}_{0}\cdot \Pi \cdot {\tilde {\Sigma }}\cdot \Sigma _{0}|\\&-{\frac {1}{2}}(\mu ^{T}\Pi \mu +{\tilde {\mu }}_{0}^{T}{\tilde {\Pi }}_{0}{\tilde {\mu }}_{0}-\mu _{0}^{T}\Pi _{0}\mu _{0}-{\tilde {\mu }}^{T}{\tilde {\Pi }}{\tilde {\mu }})+F\\{\tilde {\mu }}&={\tilde {\Sigma }}(\Pi \mu +{\tilde {\Pi }}_{0}{\tilde {\mu }}_{0}-\Pi _{0}\mu _{0})\\{\tilde {\Sigma }}&=(\Pi +{\tilde {\Pi }}_{0}-\Pi _{0})^{-1}\\\end{aligned}}}$

8

Applications

Neuroimaging

Bayesian model reduction was initially developed for use in neuroimaging analysis ^[1][3], in the context of modelling brain connectivity, as part of the Dynamic causal modelling framework (originally referred to as post-hoc Bayesian model selection). Dynamic causal models (DCMs) are differential equation models of brain dynamics, which predict the neural dynamics and neuroimaging timeseries which would be expected given a particular structure of neural connections. Parameters of these models are estimated using variational Bayes under the Laplace approximation, which means that the priors and posteriors are normally distributed. Bayesian model reduction enables the evidence (free energy) for competing models to be rapidly computed, in order to test hypotheses. These models can be specified manually by the experimenter, or searched automatically to .

This approach was subsequently generalised and applied to other forms of Bayesian models, for example hierarchical models of group effects ^[2].

Neurobiology

Software implementations

Bayesian model reduction is implemented in the Statistical Parametric Mapping toolbox in the Matlab function spm_log_evidence_reduce.m .

References

1. Friston, Karl; Penny, Will (2011-06). "Post hoc Bayesian model selection". NeuroImage. 56 (4): 2089–2099. doi:10.1016/j.neuroimage.2011.03.062. ISSN 1053-8119. PMC 3112494. PMID 21459150. {{cite journal}}: Check date values in: |date= (help)
2. Friston, Karl J.; Litvak, Vladimir; Oswal, Ashwini; Razi, Adeel; Stephan, Klaas E.; van Wijk, Bernadette C.M.; Ziegler, Gabriel; Zeidman, Peter (2016-03). "Bayesian model reduction and empirical Bayes for group (DCM) studies". NeuroImage. 128: 413–431. doi:10.1016/j.neuroimage.2015.11.015. ISSN 1053-8119. PMC 4767224. PMID 26569570. {{cite journal}}: Check date values in: |date= (help)
3. Rosa, M.J.; Friston, K.; Penny, W. (2012-06). "Post-hoc selection of dynamic causal models". Journal of Neuroscience Methods. 208 (1): 66–78. doi:10.1016/j.jneumeth.2012.04.013. ISSN 0165-0270. PMC 3401996. PMID 22561579. {{cite journal}}: Check date values in: |date= (help)