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-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 deleted file mode 100644 index 7c431c9a2..000000000 Binary files a/dev/_images/acbf6fbb2ca082f097ee859d4d468b99bced871ccca24ef9f1f162c2d526acf5.png and /dev/null differ diff --git a/dev/_images/af4d4741e5bc71839707ead7a8738ea417a80edb7eb6eeb7458d70423dd9d49b.png 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a/dev/_images/fdd6d877846c97d412099272c3cda0b105e770024d9b6e23f6b5263597b4fc60.png b/dev/_images/fdd6d877846c97d412099272c3cda0b105e770024d9b6e23f6b5263597b4fc60.png new file mode 100644 index 000000000..f3ff43a4e Binary files /dev/null and b/dev/_images/fdd6d877846c97d412099272c3cda0b105e770024d9b6e23f6b5263597b4fc60.png differ diff --git a/dev/_images/ff6ae2c0ae1851f198ac15a979cfa4ebb5d5bfc268b50bc68ad883e32a14a80f.svg b/dev/_images/ff6ae2c0ae1851f198ac15a979cfa4ebb5d5bfc268b50bc68ad883e32a14a80f.svg new file mode 100644 index 000000000..b34093df5 --- /dev/null +++ b/dev/_images/ff6ae2c0ae1851f198ac15a979cfa4ebb5d5bfc268b50bc68ad883e32a14a80f.svg @@ -0,0 +1,135 @@ + + + + + + +%3 + + +cluster10 + +10 + + +cluster10 x 320 + +10 x 320 + + + +mu_temperature + +mu_temperature +~ +Normal + + + +inverse_temperature + +inverse_temperature +~ +LogNormal + + + +mu_temperature->inverse_temperature + + + + + +mu_volatility + +mu_volatility +~ +Normal + + + +volatility + +volatility +~ +Normal + + + +mu_volatility->volatility + + + + + +sigma_temperature + +sigma_temperature +~ +HalfNormal + + + +sigma_temperature->inverse_temperature + + + + + +sigma_volatility + +sigma_volatility +~ +HalfNormal + + + +sigma_volatility->volatility + + + + + +log_likelihood + +log_likelihood +~ +CustomDist_log_likelihood + + + +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/_sources/index.md.txt b/dev/_sources/index.md.txt index ec85757d1..4372a8e35 100644 --- a/dev/_sources/index.md.txt +++ b/dev/_sources/index.md.txt @@ -2,9 +2,10 @@ # PyHGF: A Neural Network Library for Predictive Coding -hgf +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. + +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. * 📖 [API Documentation](https://ilabcode.github.io/pyhgf/api.html) * ✏️ [Tutorials and examples](https://ilabcode.github.io/pyhgf/learn.html) @@ -34,7 +35,7 @@ Dynamic networks can be defined as a tuple containing the following variables: * A set of update functions. An update function receive a network tuple and returns an updated network tuple. * An update sequence (tuple) that defines the order and target of the update functions. -![png](https://raw.githubusercontent.com/ilabcode/pyhgf/master/docs/source/images/graph_network.svg) +networks You can find a deeper introduction to how to create and manipulate networks under the following link: diff --git a/dev/index.html b/dev/index.html index 9bd5459b5..9281200fe 100644 --- a/dev/index.html +++ b/dev/index.html @@ -426,8 +426,8 @@

pre-commit license codecov black mypy Imports: isort pip

PyHGF: A Neural Network Library for Predictive Coding#

-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.

+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.

@@ -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/444a5b2d70e469c857d0b6f9ff4444b2da08c87eb1a82cf600adef5ec0674682.png
+../_images/685fda99af028c72672e3206ea8ee16c1a52789d27948be8eaf87fa8d7d98181.png
 

 
 
@@ -883,7 +883,7 @@ 
Using the learned parameters
-../_images/a736e444ca28d861f03c5b1dbdbf7a5534c2e37289ddb786f22fbed2cf8b8537.png
+../_images/15dc40f4e6e445ad3c9d895fa3a76748cacf076c24ee80be754b237051cdc03c.png
 
 
 

@@ -893,7 +893,7 @@ 
Using the learned parameters
-
Array(202.52985, dtype=float32)
+
Array(202.52992, dtype=float32)
 

 

 

@@ -945,7 +945,7 @@ 
Visualizing the model
-../_images/e3093df40c606a365fce51328bb90c832e9b204e89fed0d0688e3e2fdb1677dd.svg
+../_images/906f003ac3c62510821da40b51513f921c716a616f423184ce543353e719929c.svg
 

 

 
@@ -971,9 +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 9 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
@@ -990,7 +990,7 @@ 
Sampling#
 

 

 

-../_images/b590b5cd35858b7bc5468f4ec04fb54d78f94f07ab100ccb8c0654aa00868bac.png
+../_images/621bd4bc41eb3bb1c3ba37af682abf523053170fe477709fab35f9e596212895.png
 

 

 
@@ -1028,7 +1028,7 @@ 
Using the learned parameters
-../_images/233a6798deb950bae4abbce540aa2dfeecb8eed2dff2588e1274a89d750fa37b.png
+../_images/3f9c9d096d1b45be79de680296b186aa4ef181d7e4390ab20a6046f038f7a98c.png
 

 
 

@@ -1038,7 +1038,7 @@ 
Using the learned parameters
-
Array(203.00745, dtype=float32)
+
Array(203.00859, dtype=float32)
 

 

 

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

diff --git a/dev/notebooks/1.2-Categorical_HGF.html b/dev/notebooks/1.2-Categorical_HGF.html
index 4c532ef87..8135a8d2f 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/c816db92f3ac56f0832f5324c2ee4379cc3e7a74400c2bd6c298ccb92ab42712.png
+../_images/4535da753e514502a56155b3635236909cb08ff6c3118a75b2b5c95c8c290167.png
 

 
 

@@ -701,7 +701,7 @@ 
Fitting the model forwards
-../_images/29295e06c1eb5848e8e93428b9578e19846e291904b3a10dac6b94bf6757c447.png
+../_images/8b6be87835fa80f2997c25a27b88d7803528429bdf38de7af260c18a3c7aec03.png
 

 
 
@@ -860,12 +860,12 @@ 
System configurationvar togglebuttonSelector = '.toggle, .admonition.dropdown';
     
     
-    
+    
     
     
     
@@ -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.
 

 

@@ -898,11 +898,11 @@ 
Sampling
 

 
 

-
/tmp/ipykernel_3226/2516081684.py:2: UserWarning: The figure layout has changed to tight
+
/tmp/ipykernel_3070/2516081684.py:2: UserWarning: The figure layout has changed to tight
   plt.tight_layout()
 

 

-../_images/c2323feef5f512fa702bf0200886fedd958eff5909a06f494c20993ed655e808.png
+../_images/346c7d30a296bd356a4761f846cd627fa166eed929d24c1963d7e4fa286b9638.png
 

 

 
@@ -937,7 +937,7 @@ 
Using the learned parameters
-../_images/c411ee3d5073f99c48bd56f340be14983ec30e0e6d7608ae6da3ae1f9897a83f.png
+../_images/c33edbdb345ad850c93c219b866a261642054de85b00a5c8ce9f5886f0db4526.png
 

 
 

@@ -947,7 +947,7 @@ 
Using the learned parameters
-
Array(-1106.1246, dtype=float32)
+
Array(-1106.1111, dtype=float32)
 

 

 

@@ -1023,9 +1023,9 @@ 
Sampling#
 
NUTS: [tonic_volatility_1]
 

 

-

-

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

+

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

 

 
We recommend running at least 4 chains for robust computation of convergence diagnostics
@@ -1040,7 +1040,7 @@ 
Sampling#
 

 

 

-../_images/c2addc030e95a741b204403278a335639b22c9f55b68c4cc241636030def7471.png
+../_images/fdd6d877846c97d412099272c3cda0b105e770024d9b6e23f6b5263597b4fc60.png
 

 

 
@@ -1074,7 +1074,7 @@ 
Using the learned parameters
-../_images/76b5d8f3b2720d8b557c96ba18307540d719cc34c421c5c0daea3276bc5e79cc.png
+../_images/fb695e13af21b24d9a38ac92d4775b743612ac26e9b6366320449a27e8706e8a.png
 

 

 

@@ -1084,7 +1084,7 @@ 
Using the learned parameters
-
Array(-1118.0105, dtype=float32)
+
Array(-1118.0354, 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 34bd62c50..a54a337a2 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
-      -3.952
-      0.5
-      -4.842
-      -2.981
-      0.017
+      -3.98
+      0.487
+      -4.858
+      -3.059
+      0.016
       0.012
-      889.0
-      1261.0
+      906.0
+      1453.0
       1.0
     
   
 
-
../_images/153b556a97f25d970d6291b1327cb3fcb130c3b279d03dc7b8cde6ca0282360c.png
+../_images/df92baea67d169287416455df87f373ce1d18462d4cca6db3be6d2ff9d46f2e6.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.

@@ -990,14 +990,14 @@ 
System configuration
diff --git a/dev/notebooks/3-Multilevel_HGF.html b/dev/notebooks/3-Multilevel_HGF.html
index 11f2364e5..d3d46f280 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/a98848f23487345169ba0443080bc719c8abc65be7ebe18b918c4a829074b564.svg
+../_images/ff6ae2c0ae1851f198ac15a979cfa4ebb5d5bfc268b50bc68ad883e32a14a80f.svg
 
 
 
@@ -832,8 +832,8 @@ 
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 27 seconds.
 

 

@@ -875,7 +875,7 @@ 
Visualization of the posterior distributions
-../_images/b2380c69c6d5faafdf3900c8af8e443b44078eca2d972dd859ee992193092346.png
+../_images/ed02c4af08d89f5d7ab0b508f60c80b173f5b42ea7caa307bb171a51dd6ca7a9.png
 

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

@@ -902,17 +902,17 @@ 
Model comparison
Computed from 2000 posterior samples and 3200 observations log-likelihood matrix.
 
          Estimate       SE
-elpd_loo -2511.06    76.83
-p_loo      739.62        -
+elpd_loo -2554.73    78.30
+p_loo      794.16        -
 
 There has been a warning during the calculation. Please check the results.
 ------
 
 Pareto k diagnostic values:
                          Count   Pct.
-(-Inf, 0.70]   (good)     2167   67.7%
+(-Inf, 0.70]   (good)     2281   71.3%
    (0.70, 1]   (bad)         1    0.0%
-   (1, Inf)   (very bad) 1032   32.2%
+   (1, Inf)   (very bad)  918   28.7%
 

 
 
@@ -941,13 +941,13 @@ 
System configuration
diff --git a/dev/notebooks/4-Parameter_recovery.html b/dev/notebooks/4-Parameter_recovery.html
index 536fda6a1..3369cec97 100644
--- a/dev/notebooks/4-Parameter_recovery.html
+++ b/dev/notebooks/4-Parameter_recovery.html
@@ -50,7 +50,7 @@
     
     
     
-    
+    
     
     
     
@@ -676,12 +676,12 @@ 
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 49 seconds.
+

+

+
Sampling 2 chains for 1_000 tune and 1_000 draw iterations (2_000 + 2_000 draws total) took 63 seconds.
 

 

-
There were 1999 divergences after tuning. Increase `target_accept` or reparameterize.
+
There were 1997 divergences after tuning. Increase `target_accept` or reparameterize.
 

 

 
We recommend running at least 4 chains for robust computation of convergence diagnostics
@@ -751,7 +751,7 @@ 
Visualizing parameters recovery
-../_images/75ba141784c329581c8af75cb93813932383738b8ca02503f189e93e19a34736.png
+../_images/777aa544ac4dcbf5f106862e13b673416c3e48aed23b0ed7616079e97a3e5e32.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 45b1e93ec..89d9de3d1 100644
--- a/dev/notebooks/5-Non_linear_value_coupling.html
+++ b/dev/notebooks/5-Non_linear_value_coupling.html
@@ -988,13 +988,13 @@ 
System configuration

diff --git a/dev/notebooks/Example_1_Heart_rate_variability.html b/dev/notebooks/Example_1_Heart_rate_variability.html
index c026e5bbc..3ccf70aa7 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,  2.35it/s]
+
Downloading ECG channel:  50%|█████     | 1/2 [00:01<00:01,  1.14s/it]
 

 

-
Downloading Respiration channel:  50%|█████     | 1/2 [00:00<00:00,  2.35it/s]
+
Downloading Respiration channel:  50%|█████     | 1/2 [00:01<00:01,  1.14s/it]
 

 

-
Downloading Respiration channel: 100%|██████████| 2/2 [00:00<00:00,  2.19it/s]
+
Downloading Respiration channel: 100%|██████████| 2/2 [00:02<00:00,  1.09s/it]
 

 

-
Downloading Respiration channel: 100%|██████████| 2/2 [00:00<00:00,  2.22it/s]
+
Downloading Respiration channel: 100%|██████████| 2/2 [00:02<00:00,  1.10s/it]
 

 

 

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

 

 

-../_images/8f02834ccc45a1fbb4bd4aabfac2d75a89a86b3ef90438ded772ac7bf43de688.svg
+../_images/8fc9c210ab62f5dd9a3be3ce533d3c8c7ee05351b66a0f94da4f8a26be10ae39.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/845bbb99a5a68380c8f37c9a41b24da15cdbf5600b51de63e556a08bea0cebf0.png
+../_images/1b2cd3602e88f0369f1e7f6ddc2996c4ff2d4d8f6adad0df74624b86e44acae7.png
 

 

 

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

 

 

-../_images/fcaf294a75fafe9c0dabc7e2e2643f61e1c78b78bf8d7b7c74f489d637b37f5e.png
+../_images/42ece0c760ca3462d1a64a909fdbb445dec3a2a13903bb304382033cc9c81657.png
 

 

 
@@ -754,14 +754,14 @@ 
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 4097345c0..8fbb38e1f 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 a6cc9ef5c..ddb261107 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/484b00abaaf4ac30b26ae8ca84aea19f5e0a7f9f5a6630a5ff919f28ff2e7177.png
+../_images/d5c5a97398656d0686b9767717490d1e07025f0da2e68fbe4d7a5a702cfc5da9.png
 

 

 
@@ -1133,16 +1133,16 @@ 
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 4073c574c..0b1f34205 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 736b9578b..5dd8d5fc8 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/ecd192b619fcda3b40cdb5f737d2b936691486677eb7a4f82b5cbc7f55367e9e.svg
+../_images/7d29572ae423fb4c8d4ea736f01dab651bf0ee09a3d53b5838bcf639ba2eaf01.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/ee706bd40c79698e726fe0acdf05a8694dc4031d023476e5da2318161cd5cf03.png
+../_images/d26c403c9e83b0b3dff7c5485a76b7732915a7254507dcead6ade3245ca07266.png
 

 
 

@@ -785,15 +785,15 @@ 
Parameters optimization
     
       tonic_volatility_2
-      -2.599
-      0.401
-      -3.361
-      -1.872
+      -2.583
+      0.379
+      -3.286
+      -1.916
       0.015
       0.011
-      645.0
-      562.0
-      1.0
+      610.0
+      590.0
+      1.01
     
   
 
@@ -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/7e3bd8562916a392a1a27ed3e938adba5309476da2be4f67e8bcd1e4f37b375e.png
+../_images/0471bfc7f5b457136d95ba644a9bfe011b2f621280ffbb07488b7d5a793e6a04.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": "185316b474b0480b92e62fdbbe031b10"}
/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 23 seconds.
+
Sampling 2 chains for 1_000 tune and 1_000 draw iterations (2_000 + 2_000 draws total) took 22 seconds.
 

 

 
We recommend running at least 4 chains for robust computation of convergence diagnostics
@@ -945,7 +945,7 @@ 
Rescorla-Wagner
-../_images/9d3c39a39f31badddb0ffc16c50176f00817b02bd365f7370cd035c9e158ccab.png
+../_images/27c79f989dc37a089cf770c31b5b78ec48ceb2afa29bf7edf7045201c97ee7e8.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/0114064ed85fc215380fbdabb181c31d981c52cd56374cf008fa5b330f9abcd4.png
+../_images/552486a3c2daff615e891fe08898274f80724ae60307d06194a91ea136381d07.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 44 divergences after tuning. Increase `target_accept` or reparameterize.
+
There were 52 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/acbf6fbb2ca082f097ee859d4d468b99bced871ccca24ef9f1f162c2d526acf5.png
+../_images/5cd5e876923055f6e9a080fad3af8d7656f54bdf27cfed81f3dd2225946b8e1f.png
 

 

 
Move pointwise estimate into the likelihood field.

@@ -1240,11 +1240,11 @@ 
Model comparison
-../_images/5d435b3fb0dce7a7d45dea210135000420dfc59cdf7bc28380708a90f0e7acd2.png
+../_images/368f7131ff797a25bddc793b46fefba095de3636305043a4322297ba6f4b8b45.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/d795c16f780b3629d7892222d25649d814af2b4887a94bfe8a42519157ee1c95.png
+../_images/beb370f4030370a25b65ecc7562478b3edc6cf905fd72a41d5096c626c408c88.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 f8d49274c..833b54ec2 100644
--- a/dev/searchindex.js
+++ b/dev/searchindex.js
@@ -1 +1 @@
-Search.setIndex({"alltitles": {"": [[72, "exercise1.1"], [72, "exercise1.2"], [72, "exercise1.3"], [72, "exercise1.4"], [72, "exercise1.5"], [72, "exercise1.6"], [73, "exercise2.1"], [73, "exercise2.2"]], "API": [[0, "api"]], "Acknowledgments": [[57, "acknowledgments"]], "Add data": [[62, "add-data"], [62, "id4"], [64, "add-data"], [64, "id3"]], "Adding a drift to the random walk": [[59, "adding-a-drift-to-the-random-walk"]], "Autoregressive processes": [[59, "autoregressive-processes"]], "Bayesian inference": [[71, "bayesian-inference"]], "Beliefs trajectories": [[73, "beliefs-trajectories"]], "Biased random": [[73, "biased-random"]], "Binary nodes": [[0, "binary-nodes"]], "Bivariate normal distribution": [[61, "bivariate-normal-distribution"]], "Categorical nodes": [[0, "categorical-nodes"]], "Continuous nodes": [[0, "continuous-nodes"], [0, "id1"]], "Continuous value coupling": [[60, "continuous-value-coupling"]], "Continuous volatility coupling": [[60, "continuous-volatility-coupling"]], "Coupling with binary nodes": [[60, "coupling-with-binary-nodes"]], "Create the model": [[62, "create-the-model"], [62, "id3"], [64, "create-the-model"], [64, "id2"]], "Creating a new response function": [[65, "creating-a-new-response-function"]], "Creating a new response function: the binary surprise": [[65, "creating-a-new-response-function-the-binary-surprise"]], "Creating and manipulating networks of probabilistic nodes": [[60, null]], "Creating custom update functions": [[60, "creating-custom-update-functions"]], "Creating custom update sequences": [[60, "creating-custom-update-sequences"]], "Creating probabilistic nodes": [[60, "creating-probabilistic-nodes"]], "Creating the decision rule": [[65, "creating-the-decision-rule"]], "Creating the model": [[62, "creating-the-model"], [62, "id7"], [64, "creating-the-model"], [64, "id5"]], "Creating the probabilistic network": [[63, "creating-the-probabilistic-network"]], "Decision rule": [[71, "decision-rule"]], "Dirichlet processes": [[0, "dirichlet-processes"]], "Distribution": [[0, "distribution"]], "Dynamic assignation of update sequences": [[60, "dynamic-assignation-of-update-sequences"]], "Dynamic beliefs updating": [[59, "dynamic-beliefs-updating"]], "Example 1: Bayesian filtering of cardiac volatility": [[69, null]], "Example 2: Estimating the mean and precision of a time-varying Gaussian distributions": [[70, null]], "Example 3: A multi-armed bandit task with independent rewards and punishments": [[71, null]], "Exercises": [[58, "exercises"]], "Exponential family": [[0, "exponential-family"]], "Filtering the Sufficient Statistics of a Non-Stationary Distribution": [[61, "filtering-the-sufficient-statistics-of-a-non-stationary-distribution"]], "Filtering the Sufficient Statistics of a Stationary Distribution": [[61, "filtering-the-sufficient-statistics-of-a-stationary-distribution"]], "Fitting behaviours to different RL models": [[73, "fitting-behaviours-to-different-rl-models"]], "Fitting the binary HGF with fixed parameters": [[62, "fitting-the-binary-hgf-with-fixed-parameters"]], "Fitting the continuous HGF with fixed parameters": [[64, "fitting-the-continuous-hgf-with-fixed-parameters"]], "Fitting the model forwards": [[63, "fitting-the-model-forwards"]], "Frequency tracking": [[68, "frequency-tracking"]], "From Reinforcement Learning to Generalised Bayesian Filtering": [[61, null]], "Gaussian Random Walks": [[59, "gaussian-random-walks"], [72, "gaussian-random-walks"]], "Getting started": [[57, "getting-started"]], "Glossary": [[59, "glossary"], [65, "glossary"]], "Group-level inference": [[66, "group-level-inference"]], "Hierarchical Bayesian modelling with probabilistic neural networks": [[66, null]], "How does it work?": [[57, "how-does-it-work"]], "How to cite?": [[1, null]], "Imports": [[62, "imports"]], "Inference from the simulated behaviours": [[67, "inference-from-the-simulated-behaviours"]], "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": [[60, "multivariate-coupling"]], "Non-linear predictions": [[68, "non-linear-predictions"]], "Non-linear value coupling between continuous state nodes": [[68, null]], "Parameter recovery": [[71, "parameter-recovery"]], "Parameters optimization": [[73, "parameters-optimization"]], "Plot correlation": [[64, "plot-correlation"]], "Plot the computational graph": [[66, "plot-the-computational-graph"]], "Plot the signal with instantaneous heart rate derivations": [[69, "plot-the-signal-with-instantaneous-heart-rate-derivations"]], "Plot trajectories": [[62, "plot-trajectories"], [62, "id5"], [64, "plot-trajectories"], [64, "id4"]], "Plots": [[0, "plots"]], "Posterior predictive sampling": [[73, "posterior-predictive-sampling"]], "Posterior updates": [[0, "posterior-updates"]], "Practice: Filtering the worlds weather": [[72, "practice-filtering-the-worlds-weather"]], "Prediction error steps": [[0, "prediction-error-steps"]], "Prediction steps": [[0, "prediction-steps"]], "Preprocessing": [[69, "preprocessing"]], "Probabilistic coupling between nodes": [[72, "probabilistic-coupling-between-nodes"]], "PyHGF: A Neural Network Library for Predictive Coding": [[57, null]], "ReLU (rectified linear unit) activation function": [[68, "relu-rectified-linear-unit-activation-function"]], "Real-time decision and belief updating": [[71, "real-time-decision-and-belief-updating"]], "Recovering HGF parameters from the observed behaviors": [[65, "recovering-hgf-parameters-from-the-observed-behaviors"]], "Recovering computational parameters from observed behaviours": [[67, null]], "References": [[57, "references"], [74, null]], "Rescorla-Wagner": [[73, "rescorla-wagner"]], "Response": [[0, "response"]], "Sampling": [[62, "sampling"], [62, "id9"], [64, "sampling"], [64, "id7"], [66, "sampling"]], "Simulate a dataset": [[66, "simulate-a-dataset"], [71, "simulate-a-dataset"]], "Simulate behaviours from a one-armed bandit task": [[67, "simulate-behaviours-from-a-one-armed-bandit-task"]], "Simulate responses from a participant": [[71, "simulate-responses-from-a-participant"]], "Simulating a dataset": [[63, "simulating-a-dataset"]], "Solution to Exercise 1": [[72, "solution-exercise1.1"]], "Solution to Exercise 2": [[72, "solution-exercise1.2"]], "Solution to Exercise 3": [[72, "solution-exercise1.3"]], "Solution to Exercise 4": [[72, "solution-exercise1.4"]], "Solution to Exercise 5": [[72, "solution-exercise1.5"]], "Solution to Exercise 7": [[73, "solution-exercise2.1"]], "Solution to Exercise 8": [[73, "solution-exercise2.2"]], "Solutions": [[72, "solutions"], [73, "solutions"]], "Static assignation of update sequences": [[60, "static-assignation-of-update-sequences"]], "Surprise": [[62, "surprise"], [62, "id6"], [64, "surprise"]], "System configuration": [[59, "system-configuration"], [60, "system-configuration"], [61, "system-configuration"], [62, "system-configuration"], [63, "system-configuration"], [64, "system-configuration"], [65, "system-configuration"], [66, 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