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

Example

 

Example priors. In a 'full' model, left, a parameter has a Gaussian prior with mean 0 and standard deviation 0.5. In a 'reduced' model, right, the same parameter has prior mean zero and standard deviation 1/1000. Bayesian model reduction enables the evidence and parameter(s) of the reduced model to be derived from the evidence and parameter(s) of the full model.

Consider a model with a parameter ${\displaystyle \theta }$  and Gaussian prior ${\displaystyle p(\theta )=N(0,0.5^{2})}$ , which is the Normal distribution with mean zero and standard deviation 0.5 (illustrated in the Figure, left). This prior says that without any data, this parameter is expected to have value zero, but we are willing to entertain positive or negative values (with a 99% confidence interval [-1.16 1.16]). This model is fitted to the data to provide an estimate of the parameter ${\displaystyle q(\theta )}$  and the model evidence ${\displaystyle p(y)}$ .

To assess whether the parameter contributed to the model evidence, i.e. whether we have learnt anything about this parameter from the data, an alternative 'reduced' model is specified in which the parameter has a prior with a smaller variance: e.g. ${\displaystyle {\tilde {p}}_{0}=N(0,1000^{2})}$ . This is illustrated in the Figure (right). This prior effectively 'switches off' the parameter, saying that we are almost certain that it has value zero. The parameter ${\displaystyle {\tilde {q}}(\theta )}$  and evidence ${\displaystyle {\tilde {p}}(y)}$  for this reduced model are rapidly computed from the full model using Bayesian model reduction. The hypothesis that the parameter contributed to the model is then tested by computing the Bayes factor, which is the ratio of model evidences:

${\displaystyle BF={\frac {p(y)}{{\tilde {p}}(y)}}}$ The larger this value, the greater the evidence for the model which included the parameter as a free parameter.

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 (where it was originally referred to as post-hoc Bayesian model selection^[1]). Dynamic causal models (DCMs) are differential equation models of brain dynamics ^[4]. The experimenter specifies multiple competing models which differ in their priors - i.e. in the set of parameters which are fixed at their prior expectation of zero. Having fitted a single 'full' model with all parameters of interest informed by the data, Bayesian model reduction enables the evidence and parameters for competing models to be rapidly computed, in order to test hypotheses. These models can be specified manually by the experimenter, or searched over automatically, in order to 'prune' any redundant parameters which do not contribute to the evidence.

Bayesian model reduction was subsequently generalised and applied to other forms of Bayesian models, for example Parametric Empirical Bayes (PEB) models of group effects^[2]. Here, it is used to compute the evidence and parameters for any given level of a hierarchical model under constraints (empirical priors) imposed from the level above.

Neurobiology

Bayesian model reduction has been used to explain functions of the brain. Just as it is used to eliminate redundant parameters from models of experimental data, it has been proposed ^[5] that the brain eliminates redundant parameters of internal models of the world while offline, using a similar process. This may be implemented biologically using synaptic pruning, for example during sleep ^[6].

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)
4. Friston, K.J.; Harrison, L.; Penny, W. (2003-08). "Dynamic causal modelling". NeuroImage. 19 (4): 1273–1302. doi:10.1016/s1053-8119(03)00202-7. ISSN 1053-8119. {{cite journal}}: Check date values in: |date= (help)
5. Friston, Karl J.; Lin, Marco; Frith, Christopher D.; Pezzulo, Giovanni; Hobson, J. Allan; Ondobaka, Sasha (2017-10). "Active Inference, Curiosity and Insight". Neural Computation. 29 (10): 2633–2683. doi:10.1162/neco_a_00999. ISSN 0899-7667. {{cite journal}}: Check date values in: |date= (help)
6. Tononi, Giulio; Cirelli, Chiara (2006-02). "Sleep function and synaptic homeostasis". Sleep Medicine Reviews. 10 (1): 49–62. doi:10.1016/j.smrv.2005.05.002. ISSN 1087-0792. {{cite journal}}: Check date values in: |date= (help)