Sensory uncertainty in gaussian jumping estimation

[HuSuyi]

Dear Prof. Dr. Mathys

Currently, I am working on my Ph.D. thesis with hierarchical Gaussian filter, which fascinates me totally. I have a question about the update of the sensory uncertainty in the file "tapas_hgf_jget.m":

piuhat(k) = 1/exp(kau *mua(k-1,1) +omu);

Since in this case the precision of the input u is updated after each time step depending on the difference between mux and input u, I would like to ask why piuhat is not also updated with piuhat at the previous time step. I would expect something like:

piuhat(k) = 1 / (1 / piuhat(k-1) + exp(kau *mua(k-1,1) +omu)) );

I would be very grateful if you could help me. Thank you very much for your time.

Best Regards Suyi Hu

[chmathys]

Dear Suyi,

The quick answer is that these are the update equations that you get when you invert the generative model according to the method described in the original HGF paper (Mathys et al., 2011).

The deeper reason for the nature of this update is that u does not depend on its own previous value in the generative model (in the model graph, it’s a diamond, not a hexagon). That’s why the previous value of the precision of the prediction (piuhat) is not relevant for its current value.

Best wishes, Chris

On 2 September 2019 at 10:12:31 am, suyihu (notifications@github.com) wrote:

Dear Prof. Dr. Mathys

Currently, I am working on my Ph.D. thesis with hierarchical Gaussian filter, which fascinates me totally. I have a question about the update of the sensory uncertainty in the file "tapas_hgf_jget.m":

piuhat(k) = 1/exp(kau *mua(k-1,1) +omu);

Since in this case the precision of the input u is updated after each time step depending on the difference between mux and input u, I would like to ask why piuhat is not also updated with piuhat at the previous time step. I would expect something like:

piuhat(k) = 1 / (1 / piuhat(k-1) + exp(kau *mua(k-1,1) +omu)) );

I would be very grateful if you could help me. Thank you very much for your time.

Best Regards Suyi Hu

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[HuSuyi]

Dear Prof. Dr. Mathys

Thank you very much for your quick response. Your explanation makes a lot of sense. I have another question about the inversion of the generative model in this case: Does that mean for p(u(k)| alpha(k), ka, om) I don't have to calculate:

p(u(k) | alpha(k), ka, om) = p(u(k)|u(k-1), alpha(k), ka, om) * q(u(k-1)) ;

p(u(k) | alpha(k), ka, om) = N(u(k); u(k-1), exp(ka alpha(k) + om)) N(u(k-1); u(k-1), sigma_u (k-1)) ;

Since u(k) is not dependent on u(k-1), p(u(k) | alpha(k), ka, om) should be:

p(u(k) | alpha(k), ka, om) = N(u(k); u(k), exp(ka * alpha(k) + om))

Please correct me if I'm wrong.

Thank you very much for your help.

Best Regards

Suyi

[chmathys]

Dear Suyi,

The update equations as we derived them are in the code (in the file tapas_hgf_jget.m). There should be no need to change them.

Best wishes, Chris

On 2 September 2019 at 11:55:03 am, suyihu (notifications@github.com) wrote:

Dear Prof. Dr. Mathys

Thank you very much for your quick response. Your explanation makes a lot of sense. I have another question about the inversion of the generative model in this case: Does that mean for p(u(k)| alpha(k), ka, om) I don't have to calculate:

p(u(k) | alpha(k), ka, om) = p(u(k)|u(k-1), alpha(k), ka, om) * q(u(k-1)) ;

p(u(k) | alpha(k), ka, om) = N(u(k); u(k-1), exp(ka alpha(k) + om)) N(u(k-1); u(k-1), sigma_u (k-1)) ;

Since u(k) is not dependent on u(k-1), p(u(k) | alpha(k), ka, om) should be:

p(u(k) | alpha(k), ka, om) = N(u(k); u(k), exp(ka * alpha(k) + om))

Please correct me if I'm wrong.

Thank you very much for your help.

Best Regards

Suyi

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[HuSuyi]

Dear Prof. Dr. Mathys

Thank you very much for your help.

Best Regards

Suyi