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+mu_volatility->volatility + + + + + +sigma_volatility + +sigma_volatility +~ +HalfNormal + + + +sigma_volatility->volatility + + + + + +sigma_temperature + +sigma_temperature +~ +HalfNormal + + + +inverse_temperature + +inverse_temperature +~ +LogNormal + + + +sigma_temperature->inverse_temperature + + + + + +mu_temperature + +mu_temperature +~ +Normal + + + +mu_temperature->inverse_temperature + + + + + +pointwise_loglikelihood + +pointwise_loglikelihood +~ +Deterministic + + + +volatility->pointwise_loglikelihood + + + + + +log_likelihood + +log_likelihood +~ +CustomDist_log_likelihood + + + +volatility->log_likelihood + + + + + +inverse_temperature->pointwise_loglikelihood + + + + + +inverse_temperature->log_likelihood + + + + + \ No newline at end of file diff --git a/dev/_images/acbf6fbb2ca082f097ee859d4d468b99bced871ccca24ef9f1f162c2d526acf5.png b/dev/_images/acbf6fbb2ca082f097ee859d4d468b99bced871ccca24ef9f1f162c2d526acf5.png new file mode 100644 index 000000000..7c431c9a2 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-inverse_temperature->log_likelihood - - - - - -pointwise_loglikelihood - -pointwise_loglikelihood -~ -Deterministic - - - -inverse_temperature->pointwise_loglikelihood - - - - - -volatility->log_likelihood - - - - - -volatility->pointwise_loglikelihood - - - - - \ No newline at end of file diff --git a/dev/_images/d795c16f780b3629d7892222d25649d814af2b4887a94bfe8a42519157ee1c95.png b/dev/_images/d795c16f780b3629d7892222d25649d814af2b4887a94bfe8a42519157ee1c95.png new file mode 100644 index 000000000..b300c2a97 Binary files /dev/null and b/dev/_images/d795c16f780b3629d7892222d25649d814af2b4887a94bfe8a42519157ee1c95.png differ diff --git a/dev/_images/dfd5447e2f78b946df72f1f7169ec6d98555857bcc7e2db1bfdc805be5fcb4fc.png b/dev/_images/dfd5447e2f78b946df72f1f7169ec6d98555857bcc7e2db1bfdc805be5fcb4fc.png deleted file mode 100644 index 32342eebd..000000000 Binary files a/dev/_images/dfd5447e2f78b946df72f1f7169ec6d98555857bcc7e2db1bfdc805be5fcb4fc.png and /dev/null differ diff --git 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+Normal + + + hgf_loglike hgf_loglike ~ Potential - - -tonic_volatility_2 - -tonic_volatility_2 -~ -Uniform - - + -tonic_volatility_2->hgf_loglike +tonic_volatility_3->hgf_loglike - - -tonic_volatility_3 + + +tonic_volatility_2 -tonic_volatility_3 +tonic_volatility_2 ~ -Normal +Uniform - + -tonic_volatility_3->hgf_loglike +tonic_volatility_2->hgf_loglike diff --git a/dev/_images/e471bdfe6e97fd35d7668e61f2864bc512c2551617b3837477c2925b4d3a7abc.png b/dev/_images/e471bdfe6e97fd35d7668e61f2864bc512c2551617b3837477c2925b4d3a7abc.png deleted file mode 100644 index 3b461d5b5..000000000 Binary files a/dev/_images/e471bdfe6e97fd35d7668e61f2864bc512c2551617b3837477c2925b4d3a7abc.png and /dev/null differ diff --git a/dev/_images/e4a0ac1e1d1a1c73a4ea74ca9c2684ff9ab30c93926efc4d3621dd368b9cab39.png b/dev/_images/e4a0ac1e1d1a1c73a4ea74ca9c2684ff9ab30c93926efc4d3621dd368b9cab39.png deleted file mode 100644 index 675948660..000000000 Binary files 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df4b352ef..000000000 --- a/dev/_images/logo.svg +++ /dev/null @@ -1,373 +0,0 @@ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - image/svg+xml - - - - - - - - - - - - - - F - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - diff --git a/dev/_images/trajectories.png b/dev/_images/trajectories.png index 7df45a2ff..516bdb54d 100644 Binary files a/dev/_images/trajectories.png and b/dev/_images/trajectories.png differ diff --git a/dev/_sources/index.md.txt b/dev/_sources/index.md.txt index d892c4500..ec85757d1 100644 --- a/dev/_sources/index.md.txt +++ b/dev/_sources/index.md.txt @@ -1,10 +1,10 @@ -![png](./images/logo.svg) - [![pre-commit](https://img.shields.io/badge/pre--commit-enabled-brightgreen?logo=pre-commit&logoColor=white)](https://github.com/pre-commit/pre-commit) [![license](https://img.shields.io/badge/License-GPL%20v3-blue.svg)](https://github.com/ilabcode/pyhgf/blob/master/LICENSE) [![codecov](https://codecov.io/gh/ilabcode/pyhgf/branch/master/graph/badge.svg)](https://codecov.io/gh/ilabcode/pyhgf) [![black](https://img.shields.io/badge/code%20style-black-000000.svg)](https://github.com/psf/black) [![mypy](http://www.mypy-lang.org/static/mypy_badge.svg)](http://mypy-lang.org/) [![Imports: isort](https://img.shields.io/badge/%20imports-isort-%231674b1?style=flat&labelColor=ef8336)](https://pycqa.github.io/isort/) [![pip](https://badge.fury.io/py/pyhgf.svg)](https://badge.fury.io/py/pyhgf) # PyHGF: A Neural Network Library for Predictive Coding -PyHGF is a Python library to create and manipulate dynamic hierarchical probabilistic networks for predictive coding. The networks can approximate Bayesian inference and optimize beliefs through the diffusion of predictions and precision-weighted prediction errors, and their structure is flexible during both observation and inference. These systems can serve as biologically plausible cognitive models for computational psychiatry and reinforcement learning or as a generalisation of Bayesian filtering to arbitrarily sized dynamic graphical structures for signal processing or decision-making agents. The default implementation supports the generalisation and nodalisation of the Hierarchical Gaussian Filters for predictive coding (gHGF, Weber et al., 2024), but the framework is flexible enough to support any possible algorithm. The library is written on top of [JAX](https://jax.readthedocs.io/en/latest/jax.html), the core functions are derivable and JIT-able whenever feasible and it is possible to sample free parameters from a network under given observations. It is conceived to facilitate manipulation and modularity, so the user can focus on modeling while interfacing smoothly with other libraries in the ecosystem for Bayesian inference or optimization. A binding with an implementation in [Rust](https://www.rust-lang.org/) - that will provide full flexibility on structures during inference - is also under active development. +hgf + +PyHGF is a Python library to create and manipulate dynamic probabilistic networks for predictive coding. The networks can approximate Bayesian inference and optimize beliefs through the diffusion of predictions and precision-weighted prediction errors, and their structure is flexible during both observation and inference. These systems can serve as biologically plausible cognitive models for computational psychiatry and reinforcement learning or as a generalisation of Bayesian filtering to arbitrarily sized dynamic graphical structures for signal processing or decision-making agents. The default implementation supports the generalisation and nodalisation of the Hierarchical Gaussian Filters for predictive coding (gHGF, Weber et al., 2024), but the framework is flexible enough to support any possible algorithm. The library is written on top of [JAX](https://jax.readthedocs.io/en/latest/jax.html), the core functions are derivable and JIT-able whenever feasible and it is possible to sample free parameters from a network under given observations. It is conceived to facilitate manipulation and modularity, so the user can focus on modeling while interfacing smoothly with other libraries in the ecosystem for Bayesian inference or optimization. A binding with an implementation in [Rust](https://www.rust-lang.org/) - that will provide full flexibility on structures during inference - is also under active development. * 📖 [API Documentation](https://ilabcode.github.io/pyhgf/api.html) * ✏️ [Tutorials and examples](https://ilabcode.github.io/pyhgf/learn.html) @@ -27,7 +27,7 @@ pip install “git+https://github.com/ilabcode/pyhgf.git” ### How does it work? -A dynamic hierarchical probabilistic network can be defined as a tuple containing the following variables: +Dynamic networks can be defined as a tuple containing the following variables: * The attributes (dictionary) that store each node's states and parameters (e.g. value, precision, learning rates, volatility coupling, ...). * The edges (tuple) that lists, for each node, the indexes of the parents and children. @@ -44,9 +44,9 @@ You can find a deeper introduction to how to create and manipulate networks unde ### The Generalized Hierarchical Gaussian Filter -Generalized Hierarchical Gaussian Filters (gHGF) are specific instances of dynamic probabilistic networks where each node encodes a Gaussian distribution that inherits its value (mean) and volatility (variance) from its parent. The presentation of a new observation at the lowest level of the hierarchy (i.e., the input node) triggers a recursive update of the nodes' belief (i.e., posterior distribution) through top-down predictions and bottom-up precision-weighted prediction errors. The resulting probabilistic network operates as a Bayesian filter, and a decision function can parametrize actions/decisions given the current beliefs. By comparing those behaviours with actual outcomes, a surprise function can be optimized over a set of free parameters. The Hierarchical Gaussian Filter for binary and continuous inputs was first described in Mathys et al. (2011, 2014), and later implemented in the Matlab HGF Toolbox (part of [TAPAS](https://translationalneuromodeling.github.io/tapas) (Frässle et al. 2021). +Generalized Hierarchical Gaussian Filters (gHGF) are specific instances of dynamic networks where node encodes a Gaussian distribution that can inherit its value (mean) and volatility (variance) from other nodes. The presentation of a new observation at the lowest level of the hierarchy (i.e., the input node) triggers a recursive update of the nodes' belief (i.e., posterior distribution) through top-down predictions and bottom-up precision-weighted prediction errors. The resulting probabilistic network operates as a Bayesian filter, and a response function can parametrize actions/decisions given the current beliefs. By comparing those behaviours with actual outcomes, a surprise function can be optimized over a set of free parameters. The Hierarchical Gaussian Filter for binary and continuous inputs was first described in Mathys et al. (2011, 2014), and later implemented in the Matlab HGF Toolbox (part of [TAPAS](https://translationalneuromodeling.github.io/tapas) (Frässle et al. 2021). -You can find a deeper introduction on how does the HGF works under the following link: +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) @@ -64,9 +64,8 @@ u, y = load_data("binary") # Create a two-level binary HGF from scratch hgf = ( Network() - .add_nodes(kind="binary-input") - .add_nodes(kind="binary-state", value_children=0) - .add_nodes(kind="continuous-state", value_children=1) + .add_nodes(kind="binary-state") + .add_nodes(kind="continuous-state", value_children=0) ) # add new observations @@ -96,7 +95,7 @@ print(f"Sum of surprises = {surprise.sum()}") ## Acknowledgments -This implementation of the Hierarchical Gaussian Filter was inspired by the original [Matlab HGF Toolbox](https://translationalneuromodeling.github.io/tapas). A Julia implementation with similar aims is also available [here](https://github.com/ilabcode/HGF.jl). +This implementation of the Hierarchical Gaussian Filter was inspired by the original [Matlab HGF Toolbox](https://translationalneuromodeling.github.io/tapas). A Julia implementation is also available [here](https://github.com/ilabcode/HGF.jl). ## References diff --git a/dev/index.html b/dev/index.html index e812fdf15..9bd5459b5 100644 --- a/dev/index.html +++ b/dev/index.html @@ -423,11 +423,11 @@
-

png

-

pre-commit license codecov black mypy Imports: isort pip

+

pre-commit license codecov black mypy Imports: isort pip

PyHGF: A Neural Network Library for Predictive Coding#

-

PyHGF is a Python library to create and manipulate dynamic hierarchical probabilistic networks for predictive coding. The networks can approximate Bayesian inference and optimize beliefs through the diffusion of predictions and precision-weighted prediction errors, and their structure is flexible during both observation and inference. These systems can serve as biologically plausible cognitive models for computational psychiatry and reinforcement learning or as a generalisation of Bayesian filtering to arbitrarily sized dynamic graphical structures for signal processing or decision-making agents. The default implementation supports the generalisation and nodalisation of the Hierarchical Gaussian Filters for predictive coding (gHGF, Weber et al., 2024), but the framework is flexible enough to support any possible algorithm. The library is written on top of JAX, the core functions are derivable and JIT-able whenever feasible and it is possible to sample free parameters from a network under given observations. It is conceived to facilitate manipulation and modularity, so the user can focus on modeling while interfacing smoothly with other libraries in the ecosystem for Bayesian inference or optimization. A binding with an implementation in Rust - that will provide full flexibility on structures during inference - is also under active development.

+hgf +

PyHGF is a Python library to create and manipulate dynamic probabilistic networks for predictive coding. The networks can approximate Bayesian inference and optimize beliefs through the diffusion of predictions and precision-weighted prediction errors, and their structure is flexible during both observation and inference. These systems can serve as biologically plausible cognitive models for computational psychiatry and reinforcement learning or as a generalisation of Bayesian filtering to arbitrarily sized dynamic graphical structures for signal processing or decision-making agents. The default implementation supports the generalisation and nodalisation of the Hierarchical Gaussian Filters for predictive coding (gHGF, Weber et al., 2024), but the framework is flexible enough to support any possible algorithm. The library is written on top of JAX, the core functions are derivable and JIT-able whenever feasible and it is possible to sample free parameters from a network under given observations. It is conceived to facilitate manipulation and modularity, so the user can focus on modeling while interfacing smoothly with other libraries in the ecosystem for Bayesian inference or optimization. A binding with an implementation in Rust - that will provide full flexibility on structures during inference - is also under active development.

  • 📖 API Documentation

  • ✏️ Tutorials and examples

    • @@ -447,7 +447,7 @@

    Installation

    How does it work?#

    -

    A dynamic hierarchical probabilistic network can be defined as a tuple containing the following variables:

    +

    Dynamic networks can be defined as a tuple containing the following variables:

    • The attributes (dictionary) that store each node’s states and parameters (e.g. value, precision, learning rates, volatility coupling, …).

    • The edges (tuple) that lists, for each node, the indexes of the parents and children.

      • @@ -462,8 +462,8 @@

      How does it work?

      The Generalized Hierarchical Gaussian Filter#

      -

      Generalized Hierarchical Gaussian Filters (gHGF) are specific instances of dynamic probabilistic networks where each node encodes a Gaussian distribution that inherits its value (mean) and volatility (variance) from its parent. The presentation of a new observation at the lowest level of the hierarchy (i.e., the input node) triggers a recursive update of the nodes’ belief (i.e., posterior distribution) through top-down predictions and bottom-up precision-weighted prediction errors. The resulting probabilistic network operates as a Bayesian filter, and a decision function can parametrize actions/decisions given the current beliefs. By comparing those behaviours with actual outcomes, a surprise function can be optimized over a set of free parameters. The Hierarchical Gaussian Filter for binary and continuous inputs was first described in Mathys et al. (2011, 2014), and later implemented in the Matlab HGF Toolbox (part of TAPAS (Frässle et al. 2021).

      -

      You can find a deeper introduction on how does the HGF works under the following link:

      +

      Generalized Hierarchical Gaussian Filters (gHGF) are specific instances of dynamic networks where node encodes a Gaussian distribution that can inherit its value (mean) and volatility (variance) from other nodes. The presentation of a new observation at the lowest level of the hierarchy (i.e., the input node) triggers a recursive update of the nodes’ belief (i.e., posterior distribution) through top-down predictions and bottom-up precision-weighted prediction errors. The resulting probabilistic network operates as a Bayesian filter, and a response function can parametrize actions/decisions given the current beliefs. By comparing those behaviours with actual outcomes, a surprise function can be optimized over a set of free parameters. The Hierarchical Gaussian Filter for binary and continuous inputs was first described in Mathys et al. (2011, 2014), and later implemented in the Matlab HGF Toolbox (part of TAPAS (Frässle et al. 2021).

      +

      You can find a deeper introduction on how does the gHGF works under the following link:

      @@ -480,9 +480,8 @@

      Model fitting# Create a two-level binary HGF from scratch hgf = ( Network() - .add_nodes(kind="binary-input") - .add_nodes(kind="binary-state", value_children=0) - .add_nodes(kind="continuous-state", value_children=1) + .add_nodes(kind="binary-state") + .add_nodes(kind="continuous-state", value_children=0) ) # add new observations @@ -510,7 +509,7 @@

      Model fitting

      Acknowledgments#

      -

      This implementation of the Hierarchical Gaussian Filter was inspired by the original Matlab HGF Toolbox. A Julia implementation with similar aims is also available here.

      +

      This implementation of the Hierarchical Gaussian Filter was inspired by the original Matlab HGF Toolbox. A Julia implementation is also available here.

References#

diff --git a/dev/notebooks/0.1-Theory.html b/dev/notebooks/0.1-Theory.html index aad1b507b..2cb92db02 100644 --- a/dev/notebooks/0.1-Theory.html +++ b/dev/notebooks/0.1-Theory.html @@ -931,11 +931,11 @@

System configuration diff --git a/dev/notebooks/0.2-Creating_networks.html b/dev/notebooks/0.2-Creating_networks.html index 045a6d393..4c3dd2394 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/116ba179f38fe997ec8dd5339a7b7b45dfa4cde1ba660bc3e17fb7650e0e8251.png +../_images/24a47a30e61963b01e9fe86fc49f9457db4b4cba70774597cb0b25bd4df90686.png
@@ -1135,12 +1135,12 @@

System configuration

diff --git a/dev/notebooks/0.3-Generalised_filtering.html b/dev/notebooks/0.3-Generalised_filtering.html index 4016155b6..c41f10816 100644 --- a/dev/notebooks/0.3-Generalised_filtering.html +++ b/dev/notebooks/0.3-Generalised_filtering.html @@ -1000,13 +1000,13 @@

System configuration diff --git a/dev/notebooks/1.1-Binary_HGF.html b/dev/notebooks/1.1-Binary_HGF.html index 6ae0b0095..cf54f1124 100644 --- a/dev/notebooks/1.1-Binary_HGF.html +++ b/dev/notebooks/1.1-Binary_HGF.html @@ -52,7 +52,7 @@ - + @@ -803,7 +803,7 @@

Visualizing the model
-../_images/ecd192b619fcda3b40cdb5f737d2b936691486677eb7a4f82b5cbc7f55367e9e.svg +../_images/7d29572ae423fb4c8d4ea736f01dab651bf0ee09a3d53b5838bcf639ba2eaf01.svg

@@ -829,17 +829,14 @@

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.
 

 

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

 

-
The rhat statistic is larger than 1.01 for some parameters. This indicates problems during sampling. See https://arxiv.org/abs/1903.08008 for details
-

-

 

 
 

@@ -849,7 +846,7 @@ 
Sampling
 

 
 

-../_images/5a99396fb2358fd3bde2ff8f77e2f78d0f7e3ccfc4c3dfae1a701d179cc8db48.png
+../_images/444a5b2d70e469c857d0b6f9ff4444b2da08c87eb1a82cf600adef5ec0674682.png
 

 
 
@@ -886,7 +883,7 @@ 
Using the learned parameters
-../_images/6354f5db8b71a5abba6da48fe89d82f65317368c8531aed7fb8329a7425e3bec.png
+../_images/a736e444ca28d861f03c5b1dbdbf7a5534c2e37289ddb786f22fbed2cf8b8537.png
 
 
 

@@ -896,7 +893,7 @@ 
Using the learned parameters
-
Array(202.53006, dtype=float32)
+
Array(202.52985, dtype=float32)
 

 

 

@@ -948,7 +945,7 @@ 
Visualizing the model
-../_images/c220766790f0ac4c23d169d2a6b8286c1b66ca2bf50f19e9b1c2b87730c5efa1.svg
+../_images/e3093df40c606a365fce51328bb90c832e9b204e89fed0d0688e3e2fdb1677dd.svg
 

 

 
@@ -974,9 +971,9 @@ 
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.
+

+

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

 

 
We recommend running at least 4 chains for robust computation of convergence diagnostics
@@ -993,7 +990,7 @@ 
Sampling#
 

 

 

-../_images/8bf90f4f5d0ef6dadf9792093612016fa74bc479b8d59f26cec6598902af92a1.png
+../_images/b590b5cd35858b7bc5468f4ec04fb54d78f94f07ab100ccb8c0654aa00868bac.png
 

 

 
@@ -1031,7 +1028,7 @@ 
Using the learned parameters
-../_images/73d8ac4564bd7c20f638b42d402ff89f69cb42565445a575c3ae7a7eaed6265a.png
+../_images/233a6798deb950bae4abbce540aa2dfeecb8eed2dff2588e1274a89d750fa37b.png
 

 
 

@@ -1069,13 +1066,13 @@ 
System configuration

diff --git a/dev/notebooks/1.2-Categorical_HGF.html b/dev/notebooks/1.2-Categorical_HGF.html
index 6e3f820ac..4c532ef87 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/03fb89787c0643accb8e98ac41a9d68876aa156045737432f9b16d2298ee1722.png
+../_images/c816db92f3ac56f0832f5324c2ee4379cc3e7a74400c2bd6c298ccb92ab42712.png
 

 
 

@@ -701,7 +701,7 @@ 
Fitting the model forwards
-../_images/97166c2bbfb27aa4121c75c0556324aeed771b61ae8c83290ba1f5e3145b71ff.png
+../_images/29295e06c1eb5848e8e93428b9578e19846e291904b3a10dac6b94bf6757c447.png
 

 
 
@@ -859,14 +859,14 @@ 
System configuration
diff --git a/dev/notebooks/1.3-Continuous_HGF.html b/dev/notebooks/1.3-Continuous_HGF.html
index a5a2800c1..356127e10 100644
--- a/dev/notebooks/1.3-Continuous_HGF.html
+++ b/dev/notebooks/1.3-Continuous_HGF.html
@@ -52,7 +52,7 @@
     
     
     
-    
+    
     
     
     
@@ -880,8 +880,8 @@ 
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.
 

 

@@ -902,7 +902,7 @@ 
Sampling
   plt.tight_layout()
 

 
-../_images/1e72cff820cbd98e5bda7efefd8f287bfbe7f4e7ace42d66120f8d70f0b6cf7b.png
+../_images/c2323feef5f512fa702bf0200886fedd958eff5909a06f494c20993ed655e808.png
 
 
 
@@ -937,7 +937,7 @@ 
Using the learned parameters
-../_images/0654015973a91d306adb272fbcc4682092de2e20d14e9438c2a0d62578a1f8bd.png
+../_images/c411ee3d5073f99c48bd56f340be14983ec30e0e6d7608ae6da3ae1f9897a83f.png
 
 
 

@@ -947,7 +947,7 @@ 
Using the learned parameters
-
Array(-1106.1259, dtype=float32)
+
Array(-1106.1246, dtype=float32)
 

 

 

@@ -1023,8 +1023,8 @@ 
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.
 

 

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

 

 

-../_images/f7d82d5f5d4e76e152d144cab2ec19d7306e6744f7beffa04d8bbefb09e155cb.png
+../_images/c2addc030e95a741b204403278a335639b22c9f55b68c4cc241636030def7471.png
 

 

 
@@ -1074,7 +1074,7 @@ 
Using the learned parameters
-../_images/dfd5447e2f78b946df72f1f7169ec6d98555857bcc7e2db1bfdc805be5fcb4fc.png
+../_images/76b5d8f3b2720d8b557c96ba18307540d719cc34c421c5c0daea3276bc5e79cc.png
 
 
 

@@ -1084,7 +1084,7 @@ 
Using the learned parameters
-
Array(-1117.9681, dtype=float32)
+
Array(-1118.0105, dtype=float32)
 

 

 

@@ -1112,13 +1112,13 @@ 
System configuration

diff --git a/dev/notebooks/2-Using_custom_response_functions.html b/dev/notebooks/2-Using_custom_response_functions.html
index 68120897d..34bd62c50 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
-      -3.943
-      0.482
-      -4.893
-      -3.081
-      0.016
-      0.011
-      946.0
-      1153.0
+      -3.952
+      0.5
+      -4.842
+      -2.981
+      0.017
+      0.012
+      889.0
+      1261.0
       1.0
     
   
 
-
../_images/43e79bcefd6590e092d3b59bc4dd131a58b705fd6ad7a83cba69ab5ea61c6a9d.png
+../_images/153b556a97f25d970d6291b1327cb3fcb130c3b279d03dc7b8cde6ca0282360c.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.

@@ -990,12 +990,12 @@ 
System configurationvar togglebuttonSelector = '.toggle, .admonition.dropdown';
     
     
-    
+    
     
     
     
@@ -806,7 +806,7 @@ 
Plot the computational graph
-../_images/cd388337c92dcb3193299b1087136df0ab25680b6d2e7bbaa43435e0a2db95b6.svg
+../_images/a98848f23487345169ba0443080bc719c8abc65be7ebe18b918c4a829074b564.svg
 
 
 
@@ -832,17 +832,23 @@ 
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 45 seconds.
+

+

+
Sampling 2 chains for 1_000 tune and 1_000 draw iterations (2_000 + 2_000 draws total) took 27 seconds.
 

 

-
There were 1 divergences after tuning. Increase `target_accept` or reparameterize.
+
There were 1000 divergences after tuning. Increase `target_accept` or reparameterize.
 

 

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

 

+
The rhat statistic is larger than 1.01 for some parameters. This indicates problems during sampling. See https://arxiv.org/abs/1903.08008 for details
+

+

+
The effective sample size per chain is smaller than 100 for some parameters.  A higher number is needed for reliable rhat and ess computation. See https://arxiv.org/abs/1903.08008 for details
+

+

 

 

 

@@ -869,7 +875,7 @@ 
Visualization of the posterior distributions
-../_images/663965b834f90357f811ded970ac0de9a39a11aa030eb6b9a3ba2792c2a0fa96.png
+../_images/b2380c69c6d5faafdf3900c8af8e443b44078eca2d972dd859ee992193092346.png
 

 

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

@@ -896,17 +902,17 @@ 
Model comparison
Computed from 2000 posterior samples and 3200 observations log-likelihood matrix.
 
          Estimate       SE
-elpd_loo -1684.48    25.64
-p_loo       18.25        -
+elpd_loo -2511.06    76.83
+p_loo      739.62        -
 
 There has been a warning during the calculation. Please check the results.
 ------
 
 Pareto k diagnostic values:
                          Count   Pct.
-(-Inf, 0.70]   (good)     3187   99.6%
+(-Inf, 0.70]   (good)     2167   67.7%
    (0.70, 1]   (bad)         1    0.0%
-   (1, Inf)   (very bad)   12    0.4%
+   (1, Inf)   (very bad) 1032   32.2%
 

 

 
@@ -933,15 +939,15 @@ 
System configuration
diff --git a/dev/notebooks/4-Parameter_recovery.html b/dev/notebooks/4-Parameter_recovery.html
index 1eb0930ea..536fda6a1 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 59 seconds.
+

+

+
Sampling 2 chains for 1_000 tune and 1_000 draw iterations (2_000 + 2_000 draws total) took 49 seconds.
 

 

 
There were 1999 divergences after tuning. Increase `target_accept` or reparameterize.
@@ -751,7 +751,7 @@ 
Visualizing parameters recovery
-../_images/e471bdfe6e97fd35d7668e61f2864bc512c2551617b3837477c2925b4d3a7abc.png
+../_images/75ba141784c329581c8af75cb93813932383738b8ca02503f189e93e19a34736.png
 

 

 
@@ -776,15 +776,15 @@ 
System configuration

diff --git a/dev/notebooks/5-Non_linear_value_coupling.html b/dev/notebooks/5-Non_linear_value_coupling.html
index f700d2291..45b1e93ec 100644
--- a/dev/notebooks/5-Non_linear_value_coupling.html
+++ b/dev/notebooks/5-Non_linear_value_coupling.html
@@ -989,12 +989,12 @@ 
System configuration

diff --git a/dev/notebooks/Example_1_Heart_rate_variability.html b/dev/notebooks/Example_1_Heart_rate_variability.html
index 0af555f52..c026e5bbc 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,  1.56it/s]
+
Downloading ECG channel:  50%|█████     | 1/2 [00:00<00:00,  2.35it/s]
 

 

-
Downloading Respiration channel:  50%|█████     | 1/2 [00:00<00:00,  1.56it/s]
+
Downloading Respiration channel:  50%|█████     | 1/2 [00:00<00:00,  2.35it/s]
 

 

-
Downloading Respiration channel: 100%|██████████| 2/2 [00:01<00:00,  1.44it/s]
+
Downloading Respiration channel: 100%|██████████| 2/2 [00:00<00:00,  2.19it/s]
 

 

-
Downloading Respiration channel: 100%|██████████| 2/2 [00:01<00:00,  1.45it/s]
+
Downloading Respiration channel: 100%|██████████| 2/2 [00:00<00:00,  2.22it/s]
 

 

 

@@ -680,8 +680,8 @@ 
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.
 

 

@@ -697,7 +697,7 @@ 
Model#<
 

 

 

-../_images/959562fb3a66897575e221465c228219570f40a9822de7a34232571fb9f96c69.png
+../_images/845bbb99a5a68380c8f37c9a41b24da15cdbf5600b51de63e556a08bea0cebf0.png
 

 

 

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

 

 

-../_images/543db542df16a7681d3b72a722653b6598b2d1e06fc5c4d3cf7a9b8da13ebbcc.png
+../_images/fcaf294a75fafe9c0dabc7e2e2643f61e1c78b78bf8d7b7c74f489d637b37f5e.png
 

 

 
@@ -755,13 +755,13 @@ 
System configuration

diff --git a/dev/notebooks/Example_2_Input_node_volatility_coupling.html b/dev/notebooks/Example_2_Input_node_volatility_coupling.html
index 6fe11d0ed..4097345c0 100644
--- a/dev/notebooks/Example_2_Input_node_volatility_coupling.html
+++ b/dev/notebooks/Example_2_Input_node_volatility_coupling.html
@@ -704,12 +704,12 @@ 
System configuration

diff --git a/dev/notebooks/Example_3_Multi_armed_bandit.html b/dev/notebooks/Example_3_Multi_armed_bandit.html
index ac0414608..a6cc9ef5c 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/8eaaca2275dc895d6c8d7af5dd218d2046fa14b35405a262a1944435e63f3cb2.png
+../_images/484b00abaaf4ac30b26ae8ca84aea19f5e0a7f9f5a6630a5ff919f28ff2e7177.png
 

 

 
@@ -1134,15 +1134,15 @@ 
System configuration

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 24e3f73fa..4073c574c 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
@@ -1044,13 +1044,13 @@ 
System configuration

diff --git a/dev/notebooks/Exercise_2_Bayesian_reinforcement_learning.html b/dev/notebooks/Exercise_2_Bayesian_reinforcement_learning.html
index 3621b24e0..736b9578b 100644
--- a/dev/notebooks/Exercise_2_Bayesian_reinforcement_learning.html
+++ b/dev/notebooks/Exercise_2_Bayesian_reinforcement_learning.html
@@ -52,7 +52,7 @@
     
     
     
-    
+    
     
     
     
@@ -703,7 +703,7 @@ 
Parameters optimization
 
 

-../_images/7d29572ae423fb4c8d4ea736f01dab651bf0ee09a3d53b5838bcf639ba2eaf01.svg
+../_images/ecd192b619fcda3b40cdb5f737d2b936691486677eb7a4f82b5cbc7f55367e9e.svg
 

 
 

@@ -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/26758b7f65167119e7f137514a3bb762c860517b1390d4fcfe48baf410db71e9.png
+../_images/ee706bd40c79698e726fe0acdf05a8694dc4031d023476e5da2318161cd5cf03.png
 

 
 

@@ -785,15 +785,15 @@ 
Parameters optimization
     
       tonic_volatility_2
-      -2.593
-      0.402
-      -3.38
-      -1.94
-      0.019
-      0.013
-      479.0
-      568.0
-      1.01
+      -2.599
+      0.401
+      -3.361
+      -1.872
+      0.015
+      0.011
+      645.0
+      562.0
+      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/3c2af49a091ef1221eda36afd570a2dddbb7a0b9d97e8db6202753549f2caf95.png
+../_images/7e3bd8562916a392a1a27ed3e938adba5309476da2be4f67e8bcd1e4f37b375e.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": "ae7ba9f4c882455e81f93c84712ed58a"}
/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/e4a0ac1e1d1a1c73a4ea74ca9c2684ff9ab30c93926efc4d3621dd368b9cab39.png
+../_images/9d3c39a39f31badddb0ffc16c50176f00817b02bd365f7370cd035c9e158ccab.png
 

 

 
@@ -1036,8 +1036,8 @@ 
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.
 

 

@@ -1053,7 +1053,7 @@ 
Two-level HGF
-../_images/b166c9e058ffa8deed6689419f3e89e8e9b3cc5fb37d3b9af4f796813f732b5d.png
+../_images/0114064ed85fc215380fbdabb181c31d981c52cd56374cf008fa5b330f9abcd4.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.

@@ -1147,12 +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 76 divergences after tuning. Increase `target_accept` or reparameterize.
+
There were 44 divergences after tuning. Increase `target_accept` or reparameterize.
 

 

 
We recommend running at least 4 chains for robust computation of convergence diagnostics
@@ -1167,7 +1167,7 @@ 
Three-level HGF
-../_images/8fea7168fa17fc6b3d40923278b3870df3af8a2ff131b79212752f9137921a7d.png
+../_images/acbf6fbb2ca082f097ee859d4d468b99bced871ccca24ef9f1f162c2d526acf5.png
 

 

 
Move pointwise estimate into the likelihood field.

@@ -1240,11 +1240,11 @@ 
Model comparison
-../_images/e17b5b4df71d585ba8be22e3032e1d2cdd32f7dce1d4f014dbaa4b34790e3a4c.png
+../_images/5d435b3fb0dce7a7d45dea210135000420dfc59cdf7bc28380708a90f0e7acd2.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.

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

-../_images/caa79567721dd81873a5b2b664d2e17b8b5d6b2ea1e074167f759160207a973d.png
+../_images/d795c16f780b3629d7892222d25649d814af2b4887a94bfe8a42519157ee1c95.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.

@@ -1486,15 +1486,15 @@ 
System configuration
diff --git a/dev/searchindex.js b/dev/searchindex.js
index 8715207ab..f8d49274c 100644
--- a/dev/searchindex.js
+++ b/dev/searchindex.js
@@ -1 +1 @@
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