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b/dev/_sources/cite.md.txt index 8911db777..89977bd8f 100644 --- a/dev/_sources/cite.md.txt +++ b/dev/_sources/cite.md.txt @@ -2,10 +2,43 @@ If you are using the *pyhgf* package for your research, we ask you to cite the following paper in the final publication: -* Mathys, C. (2011). A Bayesian foundation for individual learning under uncertainty. Frontiers in Human Neuroscience, 5. https://doi.org/10.3389/fnhum.2011.00039 -* Mathys, C. D., Lomakina, E. I., Daunizeau, J., Iglesias, S., Brodersen, K. H., Friston, K. J., & Stephan, K. E. (2014). Uncertainty in perception and the hierarchical Gaussian filter. Frontiers in Human Neuroscience, 8. https://doi.org/10.3389/fnhum.2014.00825 +> Legrand, N., Weber, L., Waade, P. T., Daugaard, A. H. M., Khodadadi, M., Mikuš, N., & Mathys, C. (2024). pyhgf: A neural network library for predictive coding (Version 1). arXiv. https://doi.org/10.48550/ARXIV.2410.09206 -*In BibTeX format:* +```text +@misc{https://doi.org/10.48550/arxiv.2410.09206, + doi = {10.48550/ARXIV.2410.09206}, + url = {https://arxiv.org/abs/2410.09206}, + author = {Legrand, Nicolas and Weber, Lilian and Waade, Peter Thestrup and Daugaard, Anna Hedvig Møller and Khodadadi, Mojtaba and Mikuš, Nace and Mathys, Chris}, + keywords = {Neural and Evolutionary Computing (cs.NE), Artificial Intelligence (cs.AI), Machine Learning (cs.LG), Neurons and Cognition (q-bio.NC), FOS: Computer and information sciences, FOS: Computer and information sciences, FOS: Biological sciences, FOS: Biological sciences}, + title = {pyhgf: A neural network library for predictive coding}, + publisher = {arXiv}, + year = {2024}, + copyright = {Creative Commons Attribution 4.0 International} +} +``` + +If your application is using the generalised Hierarchical Gaussian Filer, we also ask you to cite the following publication: + +> Weber, L. A., Waade, P. T., Legrand, N., Møller, A. H., Stephan, K. E., & Mathys, C. (2023). The generalized Hierarchical Gaussian Filter (Version 2). arXiv. https://doi.org/10.48550/ARXIV.2305.10937 + +```text +@misc{https://doi.org/10.48550/arxiv.2305.10937, + doi = {10.48550/ARXIV.2305.10937}, + url = {https://arxiv.org/abs/2305.10937}, + author = {Weber, Lilian Aline and Waade, Peter Thestrup and Legrand, Nicolas and Møller, Anna Hedvig and Stephan, Klaas Enno and Mathys, Christoph}, + keywords = {Neural and Evolutionary Computing (cs.NE), Neurons and Cognition (q-bio.NC), FOS: Computer and information sciences, FOS: Computer and information sciences, FOS: Biological sciences, FOS: Biological sciences}, + title = {The generalized Hierarchical Gaussian Filter}, + publisher = {arXiv}, + year = {2023}, + copyright = {Creative Commons Attribution 4.0 International} +} +``` + +If you want to refere to the fundational description of the Hierarchical Gaussian Filter, or other important mathematical derivations, please refer to the following publications: + +> Mathys, C. (2011). A Bayesian foundation for individual learning under uncertainty. Frontiers in Human Neuroscience, 5. https://doi.org/10.3389/fnhum.2011.00039 + +> Mathys, C. D., Lomakina, E. I., Daunizeau, J., Iglesias, S., Brodersen, K. H., Friston, K. J., & Stephan, K. E. (2014). Uncertainty in perception and the hierarchical Gaussian filter. Frontiers in Human Neuroscience, 8. https://doi.org/10.3389/fnhum.2014.00825 ```text @article{2011:mathys, diff --git a/dev/_sources/index.md.txt b/dev/_sources/index.md.txt index 4372a8e35..6d8b78ef7 100644 --- a/dev/_sources/index.md.txt +++ b/dev/_sources/index.md.txt @@ -5,7 +5,7 @@ hgf -PyHGF is a Python library for creating and manipulating dynamic probabilistic networks for predictive coding. These networks approximate Bayesian inference by optimizing beliefs through the diffusion of predictions and precision-weighted prediction errors. The network structure remains flexible during message-passing steps, allowing for dynamic adjustments. They can be used as a biologically plausible cognitive model in computational neuroscience or as a generalization of Bayesian filtering for designing efficient, modular decision-making agents. The default implementation supports the generalized Hierarchical Gaussian Filters (gHGF, Weber et al., 2024), but the framework is designed to be adaptable to other algorithms. Built on top of JAX, the core functions are differentiable and JIT-compiled where applicable. The library is optimized for modularity and ease of use, allowing seamless integration with other libraries in the ecosystem for Bayesian inference and optimization. Additionally, a binding with an implementation in Rust is under active development, which will further enhance flexibility during inference. +PyHGF is a Python library for creating and manipulating dynamic probabilistic networks for predictive coding. These networks approximate Bayesian inference by optimizing beliefs through the diffusion of predictions and precision-weighted prediction errors. The network structure remains flexible during message-passing steps, allowing for dynamic adjustments. They can be used as a biologically plausible cognitive model in computational neuroscience or as a generalization of Bayesian filtering for designing efficient, modular decision-making agents. The default implementation supports the generalized Hierarchical Gaussian Filters (gHGF, Weber et al., 2024), but the framework is designed to be adaptable to other algorithms. Built on top of JAX, the core functions are differentiable and JIT-compiled where applicable. The library is optimized for modularity and ease of use, allowing seamless integration with other libraries in the ecosystem for Bayesian inference and optimization. Additionally, a binding with an implementation in Rust is under active development, which will further enhance flexibility during inference. You can find the method paper describing the toolbox [here](https://arxiv.org/abs/2410.09206) and the method paper describing the gHGF, which is the main framework currently supported by the toolbox [here](https://arxiv.org/abs/2305.10937). * 📖 [API Documentation](https://ilabcode.github.io/pyhgf/api.html) * ✏️ [Tutorials and examples](https://ilabcode.github.io/pyhgf/learn.html) @@ -49,7 +49,7 @@ Generalized Hierarchical Gaussian Filters (gHGF) are specific instances of dynam You can find a deeper introduction on how does the gHGF works under the following link: -* 🎓 [Introduction to the Hierarchical Gaussian Filter](https://ilabcode.github.io/pyhgf/notebooks/0.2-Theory.html#theory) +* 🎓 [Introduction to the Hierarchical Gaussian Filter](https://ilabcode.github.io/pyhgf/notebooks/0.1-Theory.html#theory) ### Model fitting @@ -100,10 +100,12 @@ This implementation of the Hierarchical Gaussian Filter was inspired by the orig ## References -1. Mathys, C. (2011). A Bayesian foundation for individual learning under uncertainty. In Frontiers in Human Neuroscience (Vol. 5). Frontiers Media SA. https://doi.org/10.3389/fnhum.2011.00039 -2. Mathys, C. D., Lomakina, E. I., Daunizeau, J., Iglesias, S., Brodersen, K. H., Friston, K. J., & Stephan, K. E. (2014). Uncertainty in perception and the hierarchical Gaussian filter. Frontiers in Human Neuroscience, 8. https://doi.org/10.3389/fnhum.2014.00825 -3. Weber, L. A., Waade, P. T., Legrand, N., Møller, A. H., Stephan, K. E., & Mathys, C. (2023). The generalized Hierarchical Gaussian Filter (Version 1). arXiv. https://doi.org/10.48550/ARXIV.2305.10937 -4. Frässle, S., Aponte, E. A., Bollmann, S., Brodersen, K. H., Do, C. T., Harrison, O. K., Harrison, S. J., Heinzle, J., Iglesias, S., Kasper, L., Lomakina, E. I., Mathys, C., Müller-Schrader, M., Pereira, I., Petzschner, F. H., Raman, S., Schöbi, D., Toussaint, B., Weber, L. A., … Stephan, K. E. (2021). TAPAS: An Open-Source Software Package for Translational Neuromodeling and Computational Psychiatry. In Frontiers in Psychiatry (Vol. 12). Frontiers Media SA. https://doi.org/10.3389/fpsyt.2021.680811 +1. Legrand, N., Weber, L., Waade, P. T., Daugaard, A. H. M., Khodadadi, M., Mikuš, N., & Mathys, C. (2024). pyhgf: A neural network library for predictive coding (Version 1). arXiv. https://doi.org/10.48550/ARXIV.2410.09206 +2. Mathys, C. (2011). A Bayesian foundation for individual learning under uncertainty. In Frontiers in Human Neuroscience (Vol. 5). Frontiers Media SA. https://doi.org/10.3389/fnhum.2011.00039 +3. Mathys, C. D., Lomakina, E. I., Daunizeau, J., Iglesias, S., Brodersen, K. H., Friston, K. J., & Stephan, K. E. (2014). Uncertainty in perception and the hierarchical Gaussian filter. Frontiers in Human Neuroscience, 8. https://doi.org/10.3389/fnhum.2014.00825 +4. Weber, L. A., Waade, P. T., Legrand, N., Møller, A. H., Stephan, K. E., & Mathys, C. (2023). The generalized Hierarchical Gaussian Filter (Version 2). arXiv. https://doi.org/10.48550/ARXIV.2305.10937 +5. Frässle, S., Aponte, E. A., Bollmann, S., Brodersen, K. H., Do, C. T., Harrison, O. K., Harrison, S. J., Heinzle, J., Iglesias, S., Kasper, L., Lomakina, E. I., Mathys, C., Müller-Schrader, M., Pereira, I., Petzschner, F. H., Raman, S., Schöbi, D., Toussaint, B., Weber, L. A., … Stephan, K. E. (2021). TAPAS: An Open-Source Software Package for Translational Neuromodeling and Computational Psychiatry. In Frontiers in Psychiatry (Vol. 12). Frontiers Media SA. https://doi.org/10.3389/fpsyt.2021.680811 +6. Iglesias, S., Kasper, L., Harrison, S. J., Manka, R., Mathys, C., & Stephan, K. E. (2021). Cholinergic and dopaminergic effects on prediction error and uncertainty responses during sensory associative learning. In NeuroImage (Vol. 226, p. 117590). Elsevier BV. https://doi.org/10.1016/j.neuroimage.2020.117590 ```{toctree} --- diff --git a/dev/cite.html b/dev/cite.html index fdebcae05..02fd9869d 100644 --- a/dev/cite.html +++ b/dev/cite.html @@ -449,11 +449,44 @@

How to cite?#

If you are using the pyhgf package for your research, we ask you to cite the following paper in the final publication:

-
    -

  • Mathys, C. (2011). A Bayesian foundation for individual learning under uncertainty. Frontiers in Human Neuroscience, 5. https://doi.org/10.3389/fnhum.2011.00039

    • -

  • Mathys, C. D., Lomakina, E. I., Daunizeau, J., Iglesias, S., Brodersen, K. H., Friston, K. J., & Stephan, K. E. (2014). Uncertainty in perception and the hierarchical Gaussian filter. Frontiers in Human Neuroscience, 8. https://doi.org/10.3389/fnhum.2014.00825

    • -
-

In BibTeX format:

+
+

Legrand, N., Weber, L., Waade, P. T., Daugaard, A. H. M., Khodadadi, M., Mikuš, N., & Mathys, C. (2024). pyhgf: A neural network library for predictive coding (Version 1). arXiv. https://doi.org/10.48550/ARXIV.2410.09206

+
+
@misc{https://doi.org/10.48550/arxiv.2410.09206,
+  doi = {10.48550/ARXIV.2410.09206},
+  url = {https://arxiv.org/abs/2410.09206},
+  author = {Legrand,  Nicolas and Weber,  Lilian and Waade,  Peter Thestrup and Daugaard,  Anna Hedvig Møller and Khodadadi,  Mojtaba and Mikuš,  Nace and Mathys,  Chris},
+  keywords = {Neural and Evolutionary Computing (cs.NE),  Artificial Intelligence (cs.AI),  Machine Learning (cs.LG),  Neurons and Cognition (q-bio.NC),  FOS: Computer and information sciences,  FOS: Computer and information sciences,  FOS: Biological sciences,  FOS: Biological sciences},
+  title = {pyhgf: A neural network library for predictive coding},
+  publisher = {arXiv},
+  year = {2024},
+  copyright = {Creative Commons Attribution 4.0 International}
+}
+
+
+

If your application is using the generalised Hierarchical Gaussian Filer, we also ask you to cite the following publication:

+
+

Weber, L. A., Waade, P. T., Legrand, N., Møller, A. H., Stephan, K. E., & Mathys, C. (2023). The generalized Hierarchical Gaussian Filter (Version 2). arXiv. https://doi.org/10.48550/ARXIV.2305.10937

+
+
@misc{https://doi.org/10.48550/arxiv.2305.10937,
+  doi = {10.48550/ARXIV.2305.10937},
+  url = {https://arxiv.org/abs/2305.10937},
+  author = {Weber,  Lilian Aline and Waade,  Peter Thestrup and Legrand,  Nicolas and Møller,  Anna Hedvig and Stephan,  Klaas Enno and Mathys,  Christoph},
+  keywords = {Neural and Evolutionary Computing (cs.NE),  Neurons and Cognition (q-bio.NC),  FOS: Computer and information sciences,  FOS: Computer and information sciences,  FOS: Biological sciences,  FOS: Biological sciences},
+  title = {The generalized Hierarchical Gaussian Filter},
+  publisher = {arXiv},
+  year = {2023},
+  copyright = {Creative Commons Attribution 4.0 International}
+}
+
+
+

If you want to refere to the fundational description of the Hierarchical Gaussian Filter, or other important mathematical derivations, please refer to the following publications:

+
+

Mathys, C. (2011). A Bayesian foundation for individual learning under uncertainty. Frontiers in Human Neuroscience, 5. https://doi.org/10.3389/fnhum.2011.00039

+
+
+

Mathys, C. D., Lomakina, E. I., Daunizeau, J., Iglesias, S., Brodersen, K. H., Friston, K. J., & Stephan, K. E. (2014). Uncertainty in perception and the hierarchical Gaussian filter. Frontiers in Human Neuroscience, 8. https://doi.org/10.3389/fnhum.2014.00825

+
@article{2011:mathys,
 abstract = {Computational learning models are critical for understanding mechanisms of adaptive behavior. However, the two major current frameworks, reinforcement learning (RL) and Bayesian learning, both have certain limitations. For example, many Bayesian models are agnostic of inter-individual variability and involve complicated integrals, making online learning difficult. Here, we introduce a generic hierarchical Bayesian framework for individual learning under multiple forms of uncertainty (e.g., environmental volatility and perceptual uncertainty). The model assumes Gaussian random walks of states at all but the first level, with the step size determined by the next highest level. The coupling between levels is controlled by parameters that shape the influence of uncertainty on learning in a subject-specific fashion. Using variational Bayes under a mean-field approximation and a novel approximation to the posterior energy function, we derive trial-by-trial update equations which (i) are analytical and extremely efficient, enabling real-time learning, (ii) have a natural interpretation in terms of RL, and (iii) contain parameters representing processes which play a key role in current theories of learning, e.g., precision-weighting of prediction error. These parameters allow for the expression of individual differences in learning and may relate to specific neuromodulatory mechanisms in the brain. Our model is very general: it can deal with both discrete and continuous states and equally accounts for deterministic and probabilistic relations between environmental events and perceptual states (i.e., situations with and without perceptual uncertainty). These properties are illustrated by simulations and analyses of empirical time series. Overall, our framework provides a novel foundation for understanding normal and pathological learning that contextualizes RL within a generic Bayesian scheme and thus connects it to principles of optimality from probability theory.},
 author = {Mathys, Christoph D.},
diff --git a/dev/index.html b/dev/index.html
index 6b7c80966..c5cf88e9d 100644
--- a/dev/index.html
+++ b/dev/index.html
@@ -427,7 +427,7 @@
 

 
PyHGF: A Neural Network Library for Predictive Coding#

 hgf
-
PyHGF is a Python library for creating and manipulating dynamic probabilistic networks for predictive coding. These networks approximate Bayesian inference by optimizing beliefs through the diffusion of predictions and precision-weighted prediction errors. The network structure remains flexible during message-passing steps, allowing for dynamic adjustments. They can be used as a biologically plausible cognitive model in computational neuroscience or as a generalization of Bayesian filtering for designing efficient, modular decision-making agents. The default implementation supports the generalized Hierarchical Gaussian Filters (gHGF, Weber et al., 2024), but the framework is designed to be adaptable to other algorithms. Built on top of JAX, the core functions are differentiable and JIT-compiled where applicable. The library is optimized for modularity and ease of use, allowing seamless integration with other libraries in the ecosystem for Bayesian inference and optimization. Additionally, a binding with an implementation in Rust is under active development, which will further enhance flexibility during inference.

+
PyHGF is a Python library for creating and manipulating dynamic probabilistic networks for predictive coding. These networks approximate Bayesian inference by optimizing beliefs through the diffusion of predictions and precision-weighted prediction errors. The network structure remains flexible during message-passing steps, allowing for dynamic adjustments. They can be used as a biologically plausible cognitive model in computational neuroscience or as a generalization of Bayesian filtering for designing efficient, modular decision-making agents. The default implementation supports the generalized Hierarchical Gaussian Filters (gHGF, Weber et al., 2024), but the framework is designed to be adaptable to other algorithms. Built on top of JAX, the core functions are differentiable and JIT-compiled where applicable. The library is optimized for modularity and ease of use, allowing seamless integration with other libraries in the ecosystem for Bayesian inference and optimization. Additionally, a binding with an implementation in Rust is under active development, which will further enhance flexibility during inference. You can find the method paper describing the toolbox here and the method paper describing the gHGF, which is the main framework currently supported by the toolbox here.

 

 

@@ -514,10 +514,12 @@ 
Acknowledgments
 
References#

 
    
+
  1. Legrand, N., Weber, L., Waade, P. T., Daugaard, A. H. M., Khodadadi, M., Mikuš, N., & Mathys, C. (2024). pyhgf: A neural network library for predictive coding (Version 1). arXiv. https://doi.org/10.48550/ARXIV.2410.09206
    2. 
 
  2. Mathys, C. (2011). A Bayesian foundation for individual learning under uncertainty. In Frontiers in Human Neuroscience (Vol. 5). Frontiers Media SA. https://doi.org/10.3389/fnhum.2011.00039
    3. 
 
  3. Mathys, C. D., Lomakina, E. I., Daunizeau, J., Iglesias, S., Brodersen, K. H., Friston, K. J., & Stephan, K. E. (2014). Uncertainty in perception and the hierarchical Gaussian filter. Frontiers in Human Neuroscience, 8. https://doi.org/10.3389/fnhum.2014.00825
    4. 
-
  4. Weber, L. A., Waade, P. T., Legrand, N., Møller, A. H., Stephan, K. E., & Mathys, C. (2023). The generalized Hierarchical Gaussian Filter (Version 1). arXiv. https://doi.org/10.48550/ARXIV.2305.10937
    5. 
+
  5. Weber, L. A., Waade, P. T., Legrand, N., Møller, A. H., Stephan, K. E., & Mathys, C. (2023). The generalized Hierarchical Gaussian Filter (Version 2). arXiv. https://doi.org/10.48550/ARXIV.2305.10937
    6. 
 
  6. Frässle, S., Aponte, E. A., Bollmann, S., Brodersen, K. H., Do, C. T., Harrison, O. K., Harrison, S. J., Heinzle, J., Iglesias, S., Kasper, L., Lomakina, E. I., Mathys, C., Müller-Schrader, M., Pereira, I., Petzschner, F. H., Raman, S., Schöbi, D., Toussaint, B., Weber, L. A., … Stephan, K. E. (2021). TAPAS: An Open-Source Software Package for Translational Neuromodeling and Computational Psychiatry. In Frontiers in Psychiatry (Vol. 12). Frontiers Media SA. https://doi.org/10.3389/fpsyt.2021.680811
    7. 
+
  7. Iglesias, S., Kasper, L., Harrison, S. J., Manka, R., Mathys, C., & Stephan, K. E. (2021). Cholinergic and dopaminergic effects on prediction error and uncertainty responses during sensory associative learning. In NeuroImage (Vol. 226, p. 117590). Elsevier BV. https://doi.org/10.1016/j.neuroimage.2020.117590
    8. 
 

 

 

diff --git a/dev/notebooks/0.1-Theory.html b/dev/notebooks/0.1-Theory.html
index aa1fc21ad..60baa907f 100644
--- a/dev/notebooks/0.1-Theory.html
+++ b/dev/notebooks/0.1-Theory.html
@@ -921,21 +921,21 @@ 
System configuration
-
Last updated: Tue Oct 15 2024
+
Last updated: Sun Oct 20 2024
 
 Python implementation: CPython
 Python version       : 3.12.7
 IPython version      : 8.28.0
 
-pyhgf : 0.0.0.post1.dev0+7bf4ebd
+pyhgf : 0.0.0.post1.dev0+5332083
 jax   : 0.4.31
 jaxlib: 0.4.31
 
-matplotlib: 3.9.2
 sys       : 3.12.7 (main, Oct  1 2024, 15:17:55) [GCC 11.4.0]
+matplotlib: 3.9.2
 seaborn   : 0.13.2
-IPython   : 8.28.0
 numpy     : 1.26.0
+IPython   : 8.28.0
 
 Watermark: 2.5.0
 

diff --git a/dev/notebooks/0.2-Creating_networks.html b/dev/notebooks/0.2-Creating_networks.html
index 064edf5d6..5b60cbe06 100644
--- a/dev/notebooks/0.2-Creating_networks.html
+++ b/dev/notebooks/0.2-Creating_networks.html
@@ -778,7 +778,7 @@ 
Continuous value coupling
-../_images/6db13be5fd35a236666123ad7ad5029e06a298bc37412103aa94e1a00998eb34.png
+../_images/5925149e4af8226fd70aec5be4017a0706347b4e1b0f8258d976c52474f7f3bb.png
@@ -1125,21 +1125,21 @@

System configuration

-
Last updated: Tue Oct 15 2024
+
Last updated: Sun Oct 20 2024
 
 Python implementation: CPython
 Python version       : 3.12.7
 IPython version      : 8.28.0
 
-pyhgf : 0.0.0.post1.dev0+7bf4ebd
+pyhgf : 0.0.0.post1.dev0+5332083
 jax   : 0.4.31
 jaxlib: 0.4.31
 
-pyhgf     : 0.0.0.post1.dev0+7bf4ebd
 IPython   : 8.28.0
+pyhgf     : 0.0.0.post1.dev0+5332083
+matplotlib: 3.9.2
 seaborn   : 0.13.2
 numpy     : 1.26.0
-matplotlib: 3.9.2
 sys       : 3.12.7 (main, Oct  1 2024, 15:17:55) [GCC 11.4.0]
 
 Watermark: 2.5.0
diff --git a/dev/notebooks/0.3-Generalised_filtering.html b/dev/notebooks/0.3-Generalised_filtering.html
index 303d49ca5..4389fa062 100644
--- a/dev/notebooks/0.3-Generalised_filtering.html
+++ b/dev/notebooks/0.3-Generalised_filtering.html
@@ -990,23 +990,23 @@ 
System configuration
 

 

-
Last updated: Tue Oct 15 2024
+
Last updated: Sun Oct 20 2024
 
 Python implementation: CPython
 Python version       : 3.12.7
 IPython version      : 8.28.0
 
-pyhgf : 0.0.0.post1.dev0+7bf4ebd
+pyhgf : 0.0.0.post1.dev0+5332083
 jax   : 0.4.31
 jaxlib: 0.4.31
 
-seaborn   : 0.13.2
-IPython   : 8.28.0
 numpy     : 1.26.0
-pyhgf     : 0.0.0.post1.dev0+7bf4ebd
+IPython   : 8.28.0
 jax       : 0.4.31
-matplotlib: 3.9.2
 sys       : 3.12.7 (main, Oct  1 2024, 15:17:55) [GCC 11.4.0]
+matplotlib: 3.9.2
+seaborn   : 0.13.2
+pyhgf     : 0.0.0.post1.dev0+5332083
 
 Watermark: 2.5.0
 

diff --git a/dev/notebooks/1.1-Binary_HGF.html b/dev/notebooks/1.1-Binary_HGF.html
index 2ca85ca3d..ee588edd3 100644
--- a/dev/notebooks/1.1-Binary_HGF.html
+++ b/dev/notebooks/1.1-Binary_HGF.html
@@ -52,7 +52,7 @@
     
     
     
-    
+    
     
     
     
@@ -829,8 +829,8 @@ 
Sampling
 
NUTS: [tonic_volatility_2]
 

 

-

-

+

+

 
Sampling 2 chains for 1_000 tune and 1_000 draw iterations (2_000 + 2_000 draws total) took 5 seconds.
 

 

@@ -846,7 +846,7 @@ 
Sampling
 

 

 

-../_images/a16f2a0e803af1687d9302e019a91fd3ef85cda1d405663f49b57c6e441736d0.png
+../_images/98e8df62f5c6626197fa1eadacb20ea921409801b2f8c0ced3aca1d0606b9dad.png
 

 

 

@@ -883,7 +883,7 @@ 
Using the learned parameters
-../_images/999a4d53dc2b173e3709e3182e41fc89cee354bf195ebe403a332cdb04deeabe.png
+../_images/5870548da7580e87dc96976cef7e986f89a58846775237d170818f378e026c25.png
 
 
 

@@ -893,7 +893,7 @@ 
Using the learned parameters
-
Array(202.52966, dtype=float32)
+
Array(202.53049, dtype=float32)
 

 

 

@@ -945,7 +945,7 @@ 
Visualizing the model
-../_images/2ec19856911096fbe9c3234ef83ab3600af54719c813af4a3f819412cb3c0342.svg
+../_images/e23b939a310486df3afa5ada7aac1a37266da7c16855b27f94e2ecf820ea7441.svg
 

 

 
@@ -971,8 +971,8 @@ 
Sampling#
 
NUTS: [tonic_volatility_2, tonic_volatility_3]
 

 

-

-

+

+

 
Sampling 2 chains for 1_000 tune and 1_000 draw iterations (2_000 + 2_000 draws total) took 8 seconds.
 

 

@@ -990,7 +990,7 @@ 
Sampling#
 

 
 

-../_images/7448dbd5b01f734252295940ca521bb79a7e9ce7d7c02c23790922cf8dee12ad.png
+../_images/8099d16c3bd69dac48d6783186b28bf51cfef46af4cd393a0bd717606ce1fba3.png
 

 
 
@@ -1028,7 +1028,7 @@ 
Using the learned parameters
-../_images/0dc8b87353233283107f25343d965d4499efe5da957a578eb62db1f545510ca5.png
+../_images/094f6f56f04bccb6978a4c6cf2ec5dcd4ef6c2bfe284697bedea2bf9ac881c29.png
 
 
 

@@ -1038,7 +1038,7 @@ 
Using the learned parameters
-
Array(203.00299, dtype=float32)
+
Array(203.01106, dtype=float32)
 

 

 

@@ -1056,23 +1056,23 @@ 
System configuration
 

 

-
Last updated: Tue Oct 15 2024
+
Last updated: Sun Oct 20 2024
 
 Python implementation: CPython
 Python version       : 3.12.7
 IPython version      : 8.28.0
 
-pyhgf : 0.0.0.post1.dev0+7bf4ebd
+pyhgf : 0.0.0.post1.dev0+5332083
 jax   : 0.4.31
 jaxlib: 0.4.31
 
-jax       : 0.4.31
+matplotlib: 3.9.2
 IPython   : 8.28.0
+pyhgf     : 0.0.0.post1.dev0+5332083
 sys       : 3.12.7 (main, Oct  1 2024, 15:17:55) [GCC 11.4.0]
-matplotlib: 3.9.2
-pyhgf     : 0.0.0.post1.dev0+7bf4ebd
-pymc      : 5.17.0
+jax       : 0.4.31
 arviz     : 0.20.0
+pymc      : 5.17.0
 
 Watermark: 2.5.0
 

diff --git a/dev/notebooks/1.2-Categorical_HGF.html b/dev/notebooks/1.2-Categorical_HGF.html
index 677c30c36..4ebc6fe00 100644
--- a/dev/notebooks/1.2-Categorical_HGF.html
+++ b/dev/notebooks/1.2-Categorical_HGF.html
@@ -587,7 +587,7 @@ 
Simulating a dataset
 

 

-../_images/ccabe62c371c8a61f698b189a7d306ae989b26daefcf2efd7b822a60eedc82e1.png
+../_images/8b9a0887ecd366513b02fe235b9bad8dbe01366a68f2120300c7ff0b08d56f01.png
 

 

 

@@ -701,7 +701,7 @@ 
Fitting the model forwards
-../_images/13a7d0df5117064a404a2f22cd754aea275a0131f6a9215e253cd7cdf0f40c52.png
+../_images/c356c335832d4a4938f04a7fc61c1b84bdb88239b5f356c2492d457da8dee69c.png
 

 

 
@@ -849,24 +849,24 @@ 
System configuration
 

 

-
Last updated: Tue Oct 15 2024
+
Last updated: Sun Oct 20 2024
 
 Python implementation: CPython
 Python version       : 3.12.7
 IPython version      : 8.28.0
 
-pyhgf : 0.0.0.post1.dev0+7bf4ebd
+pyhgf : 0.0.0.post1.dev0+5332083
 jax   : 0.4.31
 jaxlib: 0.4.31
 
-pyhgf     : 0.0.0.post1.dev0+7bf4ebd
-numpy     : 1.26.0
-pytensor  : 2.25.5
 IPython   : 8.28.0
-seaborn   : 0.13.2
+sys       : 3.12.7 (main, Oct  1 2024, 15:17:55) [GCC 11.4.0]
 jax       : 0.4.31
+pyhgf     : 0.0.0.post1.dev0+5332083
+pytensor  : 2.25.5
+seaborn   : 0.13.2
 matplotlib: 3.9.2
-sys       : 3.12.7 (main, Oct  1 2024, 15:17:55) [GCC 11.4.0]
+numpy     : 1.26.0
 
 Watermark: 2.5.0
 

diff --git a/dev/notebooks/1.3-Continuous_HGF.html b/dev/notebooks/1.3-Continuous_HGF.html
index 1eafaef94..2913c37f9 100644
--- a/dev/notebooks/1.3-Continuous_HGF.html
+++ b/dev/notebooks/1.3-Continuous_HGF.html
@@ -52,7 +52,7 @@
     
     
     
-    
+    
     
     
     
@@ -854,7 +854,7 @@ 
Visualizing the model
 

 

-../_images/276f245ebd9fd4bcae5e926f51a69ac0c4a8b7994776641b7ca80207aba0e073.svg
+../_images/0747ca4d2db1d73075db0036fa2d7adb40e86a06d3c587c73103c1d2a5698bb6.svg
 

 

 
@@ -880,9 +880,9 @@ 
Sampling
 
NUTS: [tonic_volatility_1]
 

 

-

-

-
Sampling 2 chains for 1_000 tune and 1_000 draw iterations (2_000 + 2_000 draws total) took 7 seconds.
+

+

+
Sampling 2 chains for 1_000 tune and 1_000 draw iterations (2_000 + 2_000 draws total) took 8 seconds.
 

 

 
We recommend running at least 4 chains for robust computation of convergence diagnostics
@@ -898,11 +898,11 @@ 
Sampling
 

 

 

-
/tmp/ipykernel_3231/2516081684.py:2: UserWarning: The figure layout has changed to tight
+
/tmp/ipykernel_3199/2516081684.py:2: UserWarning: The figure layout has changed to tight
   plt.tight_layout()
 

 

-../_images/f01e0b345faa7d40cf93694ad661cb50d11e69236c738a27444fde267493537c.png
+../_images/c035746bbc7aef9b1dca655ff4f1d108058bc532b0510447e7ff9f43631b2ecb.png
 

 

 
@@ -937,7 +937,7 @@ 
Using the learned parameters
-../_images/5b4a39bc05248c24783aa1a3b0e2e1aaa3489663bbd310b271920d48f264a721.png
+../_images/4ab50732bcd05d082d0f892f89a63f5c6ae73cfdee140fe0cbb3e2c05c383d79.png
 

 

 

@@ -947,7 +947,7 @@ 
Using the learned parameters
-
Array(-1106.1428, dtype=float32)
+
Array(-1106.0854, dtype=float32)
 

 

 

@@ -997,7 +997,7 @@ 
Visualizing the model
-../_images/276f245ebd9fd4bcae5e926f51a69ac0c4a8b7994776641b7ca80207aba0e073.svg
+../_images/0747ca4d2db1d73075db0036fa2d7adb40e86a06d3c587c73103c1d2a5698bb6.svg
 

 

 
@@ -1023,11 +1023,14 @@ 
Sampling#
 
NUTS: [tonic_volatility_1]
 

 

-

-

+

+

 
Sampling 2 chains for 1_000 tune and 1_000 draw iterations (2_000 + 2_000 draws total) took 9 seconds.
 

 

+
There were 2 divergences after tuning. Increase `target_accept` or reparameterize.
+

+

 
We recommend running at least 4 chains for robust computation of convergence diagnostics
 

 

@@ -1040,7 +1043,7 @@ 
Sampling#
 

 

 

-../_images/500ef73e4ec08b5c2df34c9a8ae09ec09a75357f36cbc3542f842e229f3d5ca1.png
+../_images/56f1af6cdd1d35f9ce4bf5f3dc1eb12c54cc009a01e4e017dbc7fe5595483c51.png
 

 

 
@@ -1074,7 +1077,7 @@ 
Using the learned parameters
-../_images/0fd03b0a8d1b44a6793a813994bb9e24c446e29152fc568e9b8f863cb92c3b30.png
+../_images/30be98f8c9532ee6f9b6c3320fb762b3ed55bcee208da11ea485bf7b5b85ebbd.png
 

 

 

@@ -1084,7 +1087,7 @@ 
Using the learned parameters
-
Array(-1118.0038, dtype=float32)
+
Array(-1118.0156, dtype=float32)
 

 

 

@@ -1102,22 +1105,22 @@ 
System configuration
 

 

-
Last updated: Tue Oct 15 2024
+
Last updated: Sun Oct 20 2024
 
 Python implementation: CPython
 Python version       : 3.12.7
 IPython version      : 8.28.0
 
-pyhgf : 0.0.0.post1.dev0+7bf4ebd
+pyhgf : 0.0.0.post1.dev0+5332083
 jax   : 0.4.31
 jaxlib: 0.4.31
 
-sys       : 3.12.7 (main, Oct  1 2024, 15:17:55) [GCC 11.4.0]
-matplotlib: 3.9.2
+IPython   : 8.28.0
+pyhgf     : 0.0.0.post1.dev0+5332083
 arviz     : 0.20.0
+matplotlib: 3.9.2
+sys       : 3.12.7 (main, Oct  1 2024, 15:17:55) [GCC 11.4.0]
 pymc      : 5.17.0
-pyhgf     : 0.0.0.post1.dev0+7bf4ebd
-IPython   : 8.28.0
 jax       : 0.4.31
 
 Watermark: 2.5.0
diff --git a/dev/notebooks/2-Using_custom_response_functions.html b/dev/notebooks/2-Using_custom_response_functions.html
index 12f269a87..bf0f8f0a3 100644
--- a/dev/notebooks/2-Using_custom_response_functions.html
+++ b/dev/notebooks/2-Using_custom_response_functions.html
@@ -52,7 +52,7 @@
     
     
     
-    
+    
     
     
     
@@ -890,8 +890,8 @@ 
Recovering HGF parameters from the observed behaviors
NUTS: [tonic_volatility_2]
 

 

-

-

+

+

 
Sampling 2 chains for 1_000 tune and 1_000 draw iterations (2_000 + 2_000 draws total) took 5 seconds.
 

 

@@ -940,19 +940,19 @@ 
Recovering HGF parameters from the observed behaviors
     
       tonic_volatility_2
-      -3.95
-      0.494
-      -4.862
-      -3.07
-      0.018
-      0.013
-      797.0
-      1302.0
+      -3.948
+      0.505
+      -4.905
+      -3.049
+      0.016
+      0.011
+      1003.0
+      954.0
       1.0
     
   
 
-
../_images/4291acfb28941c400868b537e1194b2b7dffd8cdcb65f3839e462e19feffd434.png
+
../_images/6ee48f0814a36c6d2a092e7f3d7190e52df355f5b763e277222f198582801bd9.png
 

 

 
The results above indicate that given the responses provided by the participant, the most likely values for the parameter \(\omega_2\) are between -4.9 and -3.1, with a mean at -3.9 (you can find slightly different values if you sample different actions from the decisions function). We can consider this as an excellent estimate given the sparsity of the data, and the complexity of the model.

@@ -980,24 +980,24 @@ 
System configuration
 
 

-
Last updated: Tue Oct 15 2024
+
Last updated: Sun Oct 20 2024
 
 Python implementation: CPython
 Python version       : 3.12.7
 IPython version      : 8.28.0
 
-pyhgf : 0.0.0.post1.dev0+7bf4ebd
+pyhgf : 0.0.0.post1.dev0+5332083
 jax   : 0.4.31
 jaxlib: 0.4.31
 
-IPython   : 8.28.0
-pyhgf     : 0.0.0.post1.dev0+7bf4ebd
+pyhgf     : 0.0.0.post1.dev0+5332083
 jax       : 0.4.31
-numpy     : 1.26.0
 pymc      : 5.17.0
+IPython   : 8.28.0
+arviz     : 0.20.0
+numpy     : 1.26.0
 sys       : 3.12.7 (main, Oct  1 2024, 15:17:55) [GCC 11.4.0]
 matplotlib: 3.9.2
-arviz     : 0.20.0
 
 Watermark: 2.5.0
 

diff --git a/dev/notebooks/3-Multilevel_HGF.html b/dev/notebooks/3-Multilevel_HGF.html
index 548bf5410..aba6ccad5 100644
--- a/dev/notebooks/3-Multilevel_HGF.html
+++ b/dev/notebooks/3-Multilevel_HGF.html
@@ -52,7 +52,7 @@
     
     
     
-    
+    
     
     
     
@@ -806,7 +806,7 @@ 
Plot the computational graph
-../_images/50995f53cefc3c3203ab5161a4f9dd7d4207dff120246367c1eae6dc86853532.svg
+../_images/afd058794b250ecaff14b62d2d33f561b49859d0078579d33aadea7a6f8bff67.svg
 

 

 
@@ -832,12 +832,12 @@ 
Sampling
 
NUTS: [mu_volatility, sigma_volatility, volatility, mu_temperature, sigma_temperature, inverse_temperature]
 

 

-

-

+

+

 
Sampling 2 chains for 1_000 tune and 1_000 draw iterations (2_000 + 2_000 draws total) took 46 seconds.
 

 

-
There were 1 divergences after tuning. Increase `target_accept` or reparameterize.
+
There were 2 divergences after tuning. Increase `target_accept` or reparameterize.
 

 

 
We recommend running at least 4 chains for robust computation of convergence diagnostics
@@ -869,7 +869,7 @@ 
Visualization of the posterior distributions
-../_images/16c6ada155f288747dee4bff6b2f604c656055b9d1bcb0b91b45bb3f16c69305.png
+../_images/ba4ce698e4bee20f9cd17a2747f7c5b03142dd97a82b583cd4961c4197aca871.png
 

 

 
The reference values on both posterior distributions indicate the mean of the distribution used for simulation.

@@ -896,8 +896,8 @@ 
Model comparison
Computed from 2000 posterior samples and 3200 observations log-likelihood matrix.
 
          Estimate       SE
-elpd_loo -1684.50    25.67
-p_loo       18.29        -
+elpd_loo -1684.31    25.63
+p_loo       18.01        -
 
 There has been a warning during the calculation. Please check the results.
 ------
@@ -923,25 +923,25 @@ 
System configuration
 

 

-
Last updated: Tue Oct 15 2024
+
Last updated: Sun Oct 20 2024
 
 Python implementation: CPython
 Python version       : 3.12.7
 IPython version      : 8.28.0
 
-pyhgf : 0.0.0.post1.dev0+7bf4ebd
+pyhgf : 0.0.0.post1.dev0+5332083
 jax   : 0.4.31
 jaxlib: 0.4.31
 
-pytensor  : 2.25.5
-arviz     : 0.20.0
-matplotlib: 3.9.2
 IPython   : 8.28.0
+pytensor  : 2.25.5
 sys       : 3.12.7 (main, Oct  1 2024, 15:17:55) [GCC 11.4.0]
-seaborn   : 0.13.2
 numpy     : 1.26.0
 pymc      : 5.17.0
-pyhgf     : 0.0.0.post1.dev0+7bf4ebd
+arviz     : 0.20.0
+seaborn   : 0.13.2
+matplotlib: 3.9.2
+pyhgf     : 0.0.0.post1.dev0+5332083
 
 Watermark: 2.5.0
 

diff --git a/dev/notebooks/4-Parameter_recovery.html b/dev/notebooks/4-Parameter_recovery.html
index 17e7523b1..63bb96fb1 100644
--- a/dev/notebooks/4-Parameter_recovery.html
+++ b/dev/notebooks/4-Parameter_recovery.html
@@ -50,7 +50,7 @@
     
     
     
-    
+    
     
     
     
@@ -676,9 +676,9 @@ 
Inference from the simulated behaviours
NUTS: [censored_volatility, inverse_temperature]
 

 

-

-

-
Sampling 2 chains for 1_000 tune and 1_000 draw iterations (2_000 + 2_000 draws total) took 74 seconds.
+

+

+
Sampling 2 chains for 1_000 tune and 1_000 draw iterations (2_000 + 2_000 draws total) took 58 seconds.
 

 

 
There were 1997 divergences after tuning. Increase `target_accept` or reparameterize.
@@ -751,7 +751,7 @@ 
Visualizing parameters recovery
-../_images/f833a2bd70b73c6b7e43c91941b53493e3082768f1a7edb10e51a769deb3798b.png
+../_images/d8696f77ad78ab88a09d5a8e2003e056348250ad4d8fa2437cd42a5e9f090db8.png
 

 

 
@@ -766,25 +766,25 @@ 
System configuration
 

 

-
Last updated: Tue Oct 15 2024
+
Last updated: Sun Oct 20 2024
 
 Python implementation: CPython
 Python version       : 3.12.7
 IPython version      : 8.28.0
 
-pyhgf : 0.0.0.post1.dev0+7bf4ebd
+pyhgf : 0.0.0.post1.dev0+5332083
 jax   : 0.4.31
 jaxlib: 0.4.31
 
-pymc      : 5.17.0
 arviz     : 0.20.0
+pyhgf     : 0.0.0.post1.dev0+5332083
 jax       : 0.4.31
-numpy     : 1.26.0
+pymc      : 5.17.0
 IPython   : 8.28.0
-matplotlib: 3.9.2
-pyhgf     : 0.0.0.post1.dev0+7bf4ebd
-seaborn   : 0.13.2
+numpy     : 1.26.0
 sys       : 3.12.7 (main, Oct  1 2024, 15:17:55) [GCC 11.4.0]
+seaborn   : 0.13.2
+matplotlib: 3.9.2
 
 Watermark: 2.5.0
 

diff --git a/dev/notebooks/5-Non_linear_value_coupling.html b/dev/notebooks/5-Non_linear_value_coupling.html
index e861ef15b..720fc71fb 100644
--- a/dev/notebooks/5-Non_linear_value_coupling.html
+++ b/dev/notebooks/5-Non_linear_value_coupling.html
@@ -978,22 +978,22 @@ 
System configuration
 

 

-
Last updated: Tue Oct 15 2024
+
Last updated: Sun Oct 20 2024
 
 Python implementation: CPython
 Python version       : 3.12.7
 IPython version      : 8.28.0
 
-pyhgf : 0.0.0.post1.dev0+7bf4ebd
+pyhgf : 0.0.0.post1.dev0+5332083
 jax   : 0.4.31
 jaxlib: 0.4.31
 
-pyhgf     : 0.0.0.post1.dev0+7bf4ebd
+jax       : 0.4.31
 numpy     : 1.26.0
-seaborn   : 0.13.2
 matplotlib: 3.9.2
-jax       : 0.4.31
+pyhgf     : 0.0.0.post1.dev0+5332083
 IPython   : 8.28.0
+seaborn   : 0.13.2
 sys       : 3.12.7 (main, Oct  1 2024, 15:17:55) [GCC 11.4.0]
 
 Watermark: 2.5.0
diff --git a/dev/notebooks/Example_1_Heart_rate_variability.html b/dev/notebooks/Example_1_Heart_rate_variability.html
index 3778a57fe..a319585fc 100644
--- a/dev/notebooks/Example_1_Heart_rate_variability.html
+++ b/dev/notebooks/Example_1_Heart_rate_variability.html
@@ -50,7 +50,7 @@
     
     
     
-    
+    
     
     
     
@@ -574,16 +574,16 @@ 
Loading and preprocessing physiological recording
Downloading ECG channel:   0%|          | 0/2 [00:00<?, ?it/s]
 

 

-
Downloading ECG channel:  50%|█████     | 1/2 [00:00<00:00,  1.49it/s]
+
Downloading ECG channel:  50%|█████     | 1/2 [00:00<00:00,  1.10it/s]
 

 

-
Downloading Respiration channel:  50%|█████     | 1/2 [00:00<00:00,  1.49it/s]
+
Downloading Respiration channel:  50%|█████     | 1/2 [00:00<00:00,  1.10it/s]
 

 

-
Downloading Respiration channel: 100%|██████████| 2/2 [00:01<00:00,  1.40it/s]
+
Downloading Respiration channel: 100%|██████████| 2/2 [00:01<00:00,  1.00s/it]
 

 

-
Downloading Respiration channel: 100%|██████████| 2/2 [00:01<00:00,  1.41it/s]
+
Downloading Respiration channel: 100%|██████████| 2/2 [00:01<00:00,  1.01it/s]
 

 

 

@@ -657,7 +657,7 @@ 
Model#<
 

 

 

-../_images/8fc9c210ab62f5dd9a3be3ce533d3c8c7ee05351b66a0f94da4f8a26be10ae39.svg
+../_images/8f02834ccc45a1fbb4bd4aabfac2d75a89a86b3ef90438ded772ac7bf43de688.svg
 

 

 

@@ -680,9 +680,9 @@ 
Model#<
 
NUTS: [tonic_volatility_2]
 

 

-

-

-
Sampling 2 chains for 1_000 tune and 1_000 draw iterations (2_000 + 2_000 draws total) took 10 seconds.
+

+

+
Sampling 2 chains for 1_000 tune and 1_000 draw iterations (2_000 + 2_000 draws total) took 11 seconds.
 

 

 
We recommend running at least 4 chains for robust computation of convergence diagnostics
@@ -697,7 +697,7 @@ 
Model#<
 

 

 

-../_images/79f571b81671e4d1a5a330dea7a300a737c0184361362bece44293e16c758c6c.png
+../_images/b08e712ef31245f2159c140642c43c12b2835a62eace4eb23381383e1f5c2e59.png
 

 

 

@@ -729,7 +729,7 @@ 
Model#<
 

 

 

-../_images/8da2eae63f79146baec13996b7f83f3f12f7c09283b70bd98903dd078f7ac15d.png
+../_images/70c1ec0f520a7179c16b0742faad8dfa6c01828ad307eb237c7db994189d4043.png
 

 

 
@@ -744,24 +744,24 @@ 
System configuration
 

 

-
Last updated: Tue Oct 15 2024
+
Last updated: Sun Oct 20 2024
 
 Python implementation: CPython
 Python version       : 3.12.7
 IPython version      : 8.28.0
 
-pyhgf : 0.0.0.post1.dev0+7bf4ebd
+pyhgf : 0.0.0.post1.dev0+5332083
 jax   : 0.4.31
 jaxlib: 0.4.31
 
+numpy     : 1.26.0
 IPython   : 8.28.0
-sys       : 3.12.7 (main, Oct  1 2024, 15:17:55) [GCC 11.4.0]
-systole   : 0.3.0
-arviz     : 0.20.0
+pyhgf     : 0.0.0.post1.dev0+5332083
 pymc      : 5.17.0
 matplotlib: 3.9.2
-pyhgf     : 0.0.0.post1.dev0+7bf4ebd
-numpy     : 1.26.0
+sys       : 3.12.7 (main, Oct  1 2024, 15:17:55) [GCC 11.4.0]
+arviz     : 0.20.0
+systole   : 0.3.0
 
 Watermark: 2.5.0
 

diff --git a/dev/notebooks/Example_2_Input_node_volatility_coupling.html b/dev/notebooks/Example_2_Input_node_volatility_coupling.html
index 4ce092dda..3b9113930 100644
--- a/dev/notebooks/Example_2_Input_node_volatility_coupling.html
+++ b/dev/notebooks/Example_2_Input_node_volatility_coupling.html
@@ -694,21 +694,21 @@ 
System configuration
 

 

-
Last updated: Tue Oct 15 2024
+
Last updated: Sun Oct 20 2024
 
 Python implementation: CPython
 Python version       : 3.12.7
 IPython version      : 8.28.0
 
-pyhgf : 0.0.0.post1.dev0+7bf4ebd
+pyhgf : 0.0.0.post1.dev0+5332083
 jax   : 0.4.31
 jaxlib: 0.4.31
 
-seaborn   : 0.13.2
+pyhgf     : 0.0.0.post1.dev0+5332083
 sys       : 3.12.7 (main, Oct  1 2024, 15:17:55) [GCC 11.4.0]
-matplotlib: 3.9.2
-pyhgf     : 0.0.0.post1.dev0+7bf4ebd
 numpy     : 1.26.0
+matplotlib: 3.9.2
+seaborn   : 0.13.2
 IPython   : 8.28.0
 
 Watermark: 2.5.0
diff --git a/dev/notebooks/Example_3_Multi_armed_bandit.html b/dev/notebooks/Example_3_Multi_armed_bandit.html
index 688718e82..1ab83d054 100644
--- a/dev/notebooks/Example_3_Multi_armed_bandit.html
+++ b/dev/notebooks/Example_3_Multi_armed_bandit.html
@@ -52,7 +52,7 @@
     
     
     
-    
+    
     
     
     
@@ -1085,8 +1085,8 @@ 
Bayesian inference
NUTS: [omega]
 

 

-

-

+

+

 
Sampling 2 chains for 1_000 tune and 1_000 draw iterations (2_000 + 2_000 draws total) took 7 seconds.
 

 

@@ -1108,7 +1108,7 @@ 
Bayesian inference
-../_images/0c276c5cfb50b309f07a39e05a3ba3cd13c8b5b60aafcdd907fe5fcd49f5f34c.png
+../_images/6c1a22605a4997a45cfb410047699b80ee28a2f56f4c2d59e9f7a5753a844883.png
 

 

 
@@ -1123,26 +1123,26 @@ 
System configuration
 

 

-
Last updated: Tue Oct 15 2024
+
Last updated: Sun Oct 20 2024
 
 Python implementation: CPython
 Python version       : 3.12.7
 IPython version      : 8.28.0
 
-pyhgf : 0.0.0.post1.dev0+7bf4ebd
+pyhgf : 0.0.0.post1.dev0+5332083
 jax   : 0.4.31
 jaxlib: 0.4.31
 
-pytensor  : 2.25.5
-jax       : 0.4.31
-sys       : 3.12.7 (main, Oct  1 2024, 15:17:55) [GCC 11.4.0]
-IPython   : 8.28.0
 arviz     : 0.20.0
-pyhgf     : 0.0.0.post1.dev0+7bf4ebd
-numpy     : 1.26.0
-pymc      : 5.17.0
 matplotlib: 3.9.2
+pyhgf     : 0.0.0.post1.dev0+5332083
+pymc      : 5.17.0
+jax       : 0.4.31
+pytensor  : 2.25.5
+IPython   : 8.28.0
 seaborn   : 0.13.2
+sys       : 3.12.7 (main, Oct  1 2024, 15:17:55) [GCC 11.4.0]
+numpy     : 1.26.0
 
 Watermark: 2.5.0
 

diff --git a/dev/notebooks/Exercise_1_Introduction_to_the_generalised_hierarchical_gaussian_filter.html b/dev/notebooks/Exercise_1_Introduction_to_the_generalised_hierarchical_gaussian_filter.html
index 2c3538a09..8ef997da6 100644
--- a/dev/notebooks/Exercise_1_Introduction_to_the_generalised_hierarchical_gaussian_filter.html
+++ b/dev/notebooks/Exercise_1_Introduction_to_the_generalised_hierarchical_gaussian_filter.html
@@ -1034,23 +1034,23 @@ 
System configuration
 

 

-
Last updated: Tue Oct 15 2024
+
Last updated: Sun Oct 20 2024
 
 Python implementation: CPython
 Python version       : 3.12.7
 IPython version      : 8.28.0
 
-pyhgf : 0.0.0.post1.dev0+7bf4ebd
+pyhgf : 0.0.0.post1.dev0+5332083
 jax   : 0.4.31
 jaxlib: 0.4.31
 
+matplotlib: 3.9.2
+pyhgf     : 0.0.0.post1.dev0+5332083
 pandas    : 2.2.3
-pyhgf     : 0.0.0.post1.dev0+7bf4ebd
 IPython   : 8.28.0
 seaborn   : 0.13.2
-matplotlib: 3.9.2
-numpy     : 1.26.0
 sys       : 3.12.7 (main, Oct  1 2024, 15:17:55) [GCC 11.4.0]
+numpy     : 1.26.0
 
 Watermark: 2.5.0
 

diff --git a/dev/notebooks/Exercise_2_Bayesian_reinforcement_learning.html b/dev/notebooks/Exercise_2_Bayesian_reinforcement_learning.html
index 8b4dd6b49..a179500fb 100644
--- a/dev/notebooks/Exercise_2_Bayesian_reinforcement_learning.html
+++ b/dev/notebooks/Exercise_2_Bayesian_reinforcement_learning.html
@@ -52,7 +52,7 @@
     
     
     
-    
+    
     
     
     
@@ -726,8 +726,8 @@ 
Parameters optimization
NUTS: [tonic_volatility_2]
 

 

-

-

+

+

 
Sampling 2 chains for 1_000 tune and 1_000 draw iterations (2_000 + 2_000 draws total) took 5 seconds.
 

 

@@ -743,7 +743,7 @@ 
Parameters optimization
 

 

-../_images/36adc3655fa3b1edac899d675d346f794decde0bd2883b967aa5549b6d0c7be0.png
+../_images/3cd13b6a780f84d5824d600f479102e9b62d6c7c5191e89e109d922f954ab434.png
 

 

 

@@ -785,15 +785,15 @@ 
Parameters optimization
     
       tonic_volatility_2
-      -2.558
-      0.39
-      -3.329
-      -1.89
-      0.018
-      0.013
-      488.0
-      404.0
-      1.01
+      -2.583
+      0.402
+      -3.38
+      -1.879
+      0.02
+      0.015
+      377.0
+      452.0
+      1.0
     
   
 
@@ -836,8 +836,8 @@ 
Biased random
NUTS: [bias]
 

 

-

-

+

+

 
Sampling 2 chains for 1_000 tune and 1_000 draw iterations (2_000 + 2_000 draws total) took 2 seconds.
 

 

@@ -853,7 +853,7 @@ 
Biased random
-../_images/bcdae9e6cbcc389e9c8c6218baffd9083d48746ab8b151db2823d7fe6dc7b996.png
+../_images/c9bc147d13c0708e4e9c5a643a942f49a94eddb588e285e5d81da81413067502.png
 

 

 
Assess model fitting, here using leave-one-out cross-validation from the Arviz library.

@@ -925,12 +925,12 @@ 
Rescorla-Wagner{"version_major": 2, "version_minor": 0, "model_id": "ac11ef03023549e391a79062a69e3214"}
/opt/hostedtoolcache/Python/3.12.7/x64/lib/python3.12/multiprocessing/popen_fork.py:66: RuntimeWarning: os.fork() was called. os.fork() is incompatible with multithreaded code, and JAX is multithreaded, so this will likely lead to a deadlock.
+
/opt/hostedtoolcache/Python/3.12.7/x64/lib/python3.12/multiprocessing/popen_fork.py:66: RuntimeWarning: os.fork() was called. os.fork() is incompatible with multithreaded code, and JAX is multithreaded, so this will likely lead to a deadlock.
   self.pid = os.fork()
 

 

 

-
Sampling 2 chains for 1_000 tune and 1_000 draw iterations (2_000 + 2_000 draws total) took 22 seconds.
+
Sampling 2 chains for 1_000 tune and 1_000 draw iterations (2_000 + 2_000 draws total) took 23 seconds.
 

 

 
We recommend running at least 4 chains for robust computation of convergence diagnostics
@@ -945,7 +945,7 @@ 
Rescorla-Wagner
-../_images/f5950061c3cf90e405e403be0dace229921e9a8df24f271be700f0c3fe0d2130.png
+../_images/d17082169a18874a8b4b778dbaf1366dc44976ae773c8d4e8d97d95660018aaa.png
 

 

 
@@ -1036,14 +1036,11 @@ 
Two-level HGF
NUTS: [tonic_volatility_2]
 

 

-

-

+

+

 
Sampling 2 chains for 1_000 tune and 1_000 draw iterations (2_000 + 2_000 draws total) took 5 seconds.
 

 

-
There were 1 divergences after tuning. Increase `target_accept` or reparameterize.
-

-

 
We recommend running at least 4 chains for robust computation of convergence diagnostics
 

 

@@ -1056,7 +1053,7 @@ 
Two-level HGF
-../_images/514cf2b6e324d10af6bbcddc12a6da57216c859078391051ba199b1f0e56b878.png
+../_images/f5c0a7b58c1a364d2906d74c1c317a8dcae6a146a9cfbd3dc3ac08cb1f05b470.png
 

 

 
We have saved the pointwise log probabilities as a variable, here we simply move this variable to the log_likelihoo field of the idata object, so Arviz knows that this can be used later for model comparison.

@@ -1150,12 +1147,12 @@ 
Three-level HGF
NUTS: [tonic_volatility_2]
 

 

-

-

+

+

 
Sampling 2 chains for 1_000 tune and 1_000 draw iterations (2_000 + 2_000 draws total) took 7 seconds.
 

 

-
There were 71 divergences after tuning. Increase `target_accept` or reparameterize.
+
There were 66 divergences after tuning. Increase `target_accept` or reparameterize.
 

 

 
We recommend running at least 4 chains for robust computation of convergence diagnostics
@@ -1170,7 +1167,7 @@ 
Three-level HGF
-../_images/06cd5951c2551d0555a324efb3885633bc610d2e08e6e1d08e3af0d939fa3ce5.png
+../_images/1b5e44337acad6ef76a6d338b3ff2d4ca7ccb2900ec66f94fe260c37dfb9e34b.png
 

 

 
Move pointwise estimate into the likelihood field.

@@ -1243,11 +1240,11 @@ 
Model comparison
-../_images/a7ff42ea62b5cfd75e7e16dfeaf0164812fa63a820ea31d9e721139fd19406ad.png
+../_images/56340da6a6724fa689816d4bb9f702c278ce28f80f94776a2b8f799b1da81ab5.png
 

 

 
Looking at the final result, we can see that the three-level HGF had the best predictive performance on the participant decision, suggesting that higher-level uncertainty is important here to understand the agent’s behaviour.

@@ -1416,7 +1413,7 @@ 
Beliefs trajectories
 
 

-../_images/ea6b742a70636bb323ad1002378d2c4957c72c53daea8171096b9a0c377ec5e0.png
+../_images/7d1d653f89c31dcbff553f9e4e7fa21764fcdcec90588cd21967bb3c91758740.png
 

 

 
The resulting samples show belief trajectories for 10 samples for each model (we are not depicting the biased random here for clarity). The trajectories are highly similar, but we can see that the two and three-level HGF are slightly adjusting their learning rates in a way that was more consistent with the observed behaviours.

@@ -1479,25 +1476,25 @@ 
System configuration
 

 

-
Last updated: Tue Oct 15 2024
+
Last updated: Sun Oct 20 2024
 
 Python implementation: CPython
 Python version       : 3.12.7
 IPython version      : 8.28.0
 
-pyhgf : 0.0.0.post1.dev0+7bf4ebd
+pyhgf : 0.0.0.post1.dev0+5332083
 jax   : 0.4.31
 jaxlib: 0.4.31
 
-matplotlib: 3.9.2
-numpy     : 1.26.0
-pytensor  : 2.25.5
-pymc      : 5.17.0
-IPython   : 8.28.0
 sys       : 3.12.7 (main, Oct  1 2024, 15:17:55) [GCC 11.4.0]
-pyhgf     : 0.0.0.post1.dev0+7bf4ebd
-arviz     : 0.20.0
+pymc      : 5.17.0
 seaborn   : 0.13.2
+pyhgf     : 0.0.0.post1.dev0+5332083
+numpy     : 1.26.0
+arviz     : 0.20.0
+IPython   : 8.28.0
+pytensor  : 2.25.5
+matplotlib: 3.9.2
 
 Watermark: 2.5.0
 

diff --git a/dev/searchindex.js b/dev/searchindex.js
index 2e1cf4139..416688b59 100644
--- a/dev/searchindex.js
+++ b/dev/searchindex.js
@@ -1 +1 @@
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"Inference using MCMC sampling": [[63, "inference-using-mcmc-sampling"]], "Installation": [[57, "installation"]], "Introduction to the Generalised Hierarchical Gaussian Filter": [[59, null]], "Kown mean, unknown precision": [[70, "kown-mean-unknown-precision"]], "Learn": [[58, null]], "Learning parameters with MCMC sampling": [[62, "learning-parameters-with-mcmc-sampling"], [64, "learning-parameters-with-mcmc-sampling"]], "Loading and preprocessing physiological recording": [[69, "loading-and-preprocessing-physiological-recording"]], "Math": [[0, "math"]], "Model": [[0, "model"], [69, "model"]], "Model comparison": [[66, "model-comparison"], [73, "model-comparison"]], "Model fitting": [[57, "model-fitting"]], "Model inversion: the generalized Hierarchical Gaussian Filter": [[72, "model-inversion-the-generalized-hierarchical-gaussian-filter"]], "Modifying the attributes": [[60, "modifying-the-attributes"]], "Modifying the edges": [[60, "modifying-the-edges"]], "Multivariate coupling": 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