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 @@ -914,7 +915,7 @@

System configuration
-
Last updated: Tue Sep 10 2024
+
Last updated: Wed Sep 11 2024
 
 Python implementation: CPython
 Python version       : 3.12.5
@@ -924,10 +925,10 @@ 
System configuration

diff --git a/dev/notebooks/0.2-Creating_networks.html b/dev/notebooks/0.2-Creating_networks.html
index d2f411a62..e188827a0 100644
--- a/dev/notebooks/0.2-Creating_networks.html
+++ b/dev/notebooks/0.2-Creating_networks.html
@@ -444,6 +444,7 @@
 
  • Application to reinforcement learning
    • 
 
 
+
    @@ -752,7 +753,7 @@
    Continuous value coupling -../_images/3bd8b196e0d8eed3711c18708279a838ac059ec6c3de50ce92b3d58832268e33.png +../_images/a08fba24e7383f6639e82a4025421e77679caec1f5e6f7c675169beca6dca405.png
    @@ -965,7 +966,7 @@

    Value coupling -
    /tmp/ipykernel_2217/1724106460.py:6: UserWarning: The figure layout has changed to tight
+
    /tmp/ipykernel_2221/1724106460.py:6: UserWarning: The figure layout has changed to tight
   plt.tight_layout()
 
    
 
    
@@ -1087,7 +1088,7 @@ 
    System configuration
    -
    Last updated: Tue Sep 10 2024
+
    Last updated: Wed Sep 11 2024
 
 Python implementation: CPython
 Python version       : 3.12.5
@@ -1097,10 +1098,10 @@ 
    System configuration
    
diff --git a/dev/notebooks/0.3-Generalised_filtering.html b/dev/notebooks/0.3-Generalised_filtering.html
index 2d3297577..b7b792b61 100644
--- a/dev/notebooks/0.3-Generalised_filtering.html
+++ b/dev/notebooks/0.3-Generalised_filtering.html
@@ -444,6 +444,7 @@
 
  • Application to reinforcement learning
    • 
 
 
+
    @@ -965,7 +966,7 @@

    System configuration

    -
    Last updated: Tue Sep 10 2024
+
    Last updated: Wed Sep 11 2024
 
 Python implementation: CPython
 Python version       : 3.12.5
@@ -976,8 +977,8 @@ 
    System configurationvar togglebuttonSelector = '.toggle, .admonition.dropdown';
     
     
-    
+    
     
     
     
@@ -447,6 +447,7 @@
 
  • Application to reinforcement learning
    • 
 
 
+
 
 
    
 
    
@@ -803,10 +804,10 @@ 
    Sampling
 
    NUTS: [tonic_volatility_2]
 
    
 
    
-
    
-
    
-
    
-
    
+
    
+
    
+
    
+
    
 
    Sampling 4 chains for 1_000 tune and 1_000 draw iterations (4_000 + 4_000 draws total) took 11 seconds.
 
    
 
    
@@ -819,7 +820,7 @@ 
    Sampling
 
    
 
    
 
    
-../_images/da88306f4d8f40a66e9000eb4358f207af1dcde397c8be14a473dbe5866b8428.png
+../_images/847cac0fe8c5b4c566330729302d641e30ba7543d68f973fe30434788001414a.png
 
    
 
    
 
@@ -856,7 +857,7 @@ 
    Using the learned parameters
-../_images/aba79dfa12a5f821cfb649fabf4562f7594d13a12d01ce741211bbcbb9718720.png
+../_images/65ebb718ef626c907be9668eb0c71e4ec56d42d74b7e7b74a3d613c5e6795727.png
 
    
 
    
 
    
@@ -866,7 +867,7 @@ 
    Using the learned parameters
-
    Array(202.5296, dtype=float32)
+
    Array(202.52988, dtype=float32)
 
    
 
    
 
    
@@ -918,7 +919,7 @@ 
    Visualizing the model
-../_images/e3093df40c606a365fce51328bb90c832e9b204e89fed0d0688e3e2fdb1677dd.svg
+../_images/e23b939a310486df3afa5ada7aac1a37266da7c16855b27f94e2ecf820ea7441.svg
 
    
 
    
 
@@ -944,11 +945,11 @@ 
    Sampling#
 
    NUTS: [tonic_volatility_2, tonic_volatility_3]
 
    
 
    
-
    
-
    
-
    
-
    
-
    Sampling 4 chains for 1_000 tune and 1_000 draw iterations (4_000 + 4_000 draws total) took 18 seconds.
+
    
+
    
+
    
+
    
+
    Sampling 4 chains for 1_000 tune and 1_000 draw iterations (4_000 + 4_000 draws total) took 17 seconds.
 
    
 
    
 
    
@@ -962,7 +963,7 @@ 
    Sampling#
 
    
 
    
 
    
-../_images/dd25a7607fa3908363ee62a91e7f78eb315f29a1e80c1e60e3da2c9c84218ea0.png
+../_images/646be6aed89af26c7d6879a60cf3d4ae988f935a42c5ac46830ce33a23a8a1ec.png
 
    
 
    
 
@@ -1000,7 +1001,7 @@ 
    Using the learned parameters
-../_images/5919f66dd9a658c7ad732706526144ce8ba088671f973e6512653962160b16f5.png
+../_images/c96d5341c765949f9f12b3866315d97fe265d54af89e98c2390816535895d067.png
 
 
 
    
@@ -1010,7 +1011,7 @@ 
    Using the learned parameters
-
    Array(203.01239, dtype=float32)
+
    Array(203.01517, dtype=float32)
 
    
 
    
 
    
@@ -1028,7 +1029,7 @@ 
    System configuration
 
    
 
    
-
    Last updated: Tue Sep 10 2024
+
    Last updated: Wed Sep 11 2024
 
 Python implementation: CPython
 Python version       : 3.12.5
@@ -1038,10 +1039,10 @@ 
    System configurationApplication to reinforcement learning
 
 
+
 
 
    
 
    
@@ -582,7 +583,7 @@ 
    Simulating a dataset
 
    
 
    
-../_images/f46a030957cd4f6254936411e7bcb6efd0fa799f0deeaa9e0f680959ddcd1c64.png
+../_images/04800461fa8e48e3576559542f4111c2207d663e6fce49ae0f4925582ce02bf2.png
 
    
 
    
 
    
@@ -696,7 +697,7 @@ 
    Fitting the model forwards
-../_images/ced14a7c9e2161eae4fe9600ffe3ada73cf69d6a51719fd3e8f787b66a0d7718.png
+../_images/801246fe9ea7b5afccdf6aed41c1d1818b31b3060546e38084c3ea92240aed48.png
 
    
 
    
 
@@ -844,7 +845,7 @@ 
    System configuration
 
    
 
    
-
    Last updated: Tue Sep 10 2024
+
    Last updated: Wed Sep 11 2024
 
 Python implementation: CPython
 Python version       : 3.12.5
@@ -854,12 +855,12 @@ 
    System configuration
    
diff --git a/dev/notebooks/1.3-Continuous_HGF.html b/dev/notebooks/1.3-Continuous_HGF.html
index 7c7d86abc..58483c4ea 100644
--- a/dev/notebooks/1.3-Continuous_HGF.html
+++ b/dev/notebooks/1.3-Continuous_HGF.html
@@ -52,7 +52,7 @@
     
     
     
-    
+    
     
     
     
@@ -447,6 +447,7 @@
 
  • Application to reinforcement learning
    • 
 
 
+
 
 
    
 
    
@@ -831,7 +832,7 @@ 
    Visualizing the model
 
    
 
    
-../_images/276f245ebd9fd4bcae5e926f51a69ac0c4a8b7994776641b7ca80207aba0e073.svg
+../_images/0747ca4d2db1d73075db0036fa2d7adb40e86a06d3c587c73103c1d2a5698bb6.svg
 
    
 
    
 
@@ -857,10 +858,10 @@ 
    Sampling
 
    NUTS: [tonic_volatility_1]
 
    
 
    
-
    
-
    
-
    
-
    
+
    
+
    
+
    
+
    
 
    Sampling 4 chains for 1_000 tune and 1_000 draw iterations (4_000 + 4_000 draws total) took 13 seconds.
 
    
 
    
@@ -874,11 +875,11 @@ 
    Sampling
 
    
 
 
    
-
    /tmp/ipykernel_3157/2516081684.py:2: UserWarning: The figure layout has changed to tight
+
    /tmp/ipykernel_3229/2516081684.py:2: UserWarning: The figure layout has changed to tight
   plt.tight_layout()
 
    
 
    
-../_images/aeec0f3ac9853bb87b43b6cebd9d0b763ebcca20fcddcc84ed2a22ffb6cc4f8b.png
+../_images/2ea0005916d19b8c03e2441b06262f48e55d68ef057e911a246c3216add17974.png
 
    
 
    
 
@@ -913,7 +914,7 @@ 
    Using the learned parameters
-../_images/11eaeef0b67d76fc55e9b97cc6ec6a8d6294c37676b49365e87dfc794782fe9e.png
+../_images/32e4f31c6b48129c9795c05e3205c1f49aa52e20238b5ce47abe26f4d6653456.png
 
    
 
 
    
@@ -923,7 +924,7 @@ 
    Using the learned parameters
-
    Array(-1106.1132, dtype=float32)
+
    Array(-1106.0792, dtype=float32)
 
    
 
    
 
    
@@ -973,7 +974,7 @@ 
    Visualizing the model
-../_images/276f245ebd9fd4bcae5e926f51a69ac0c4a8b7994776641b7ca80207aba0e073.svg
+../_images/0747ca4d2db1d73075db0036fa2d7adb40e86a06d3c587c73103c1d2a5698bb6.svg
 
    
 
    
 
@@ -999,11 +1000,14 @@ 
    Sampling#
 
    NUTS: [tonic_volatility_1]
 
    
 
    
-
    
-
    
-
    
-
    
-
    Sampling 4 chains for 1_000 tune and 1_000 draw iterations (4_000 + 4_000 draws total) took 17 seconds.
+
    
+
    
+
    
+
    
+
    Sampling 4 chains for 1_000 tune and 1_000 draw iterations (4_000 + 4_000 draws total) took 16 seconds.
+
    
+
    
+
    There were 1 divergences after tuning. Increase `target_accept` or reparameterize.
 
    
 
    
 
    
@@ -1015,7 +1019,7 @@ 
    Sampling#
 
    
 
 
    
-../_images/d408bfa7c4c2762f497a96a7670caccc77e80dda4a315576f77e2aee0f7373a6.png
+../_images/da6f314215d30d0b43ca2d0f3edddc1436f8b448b4c108e35cf59ddde58d75f6.png
 
    
 
 
@@ -1049,7 +1053,7 @@ 
    Using the learned parameters
-../_images/876be502e67c7863c51246fae257d17e4f70499651c6c368d737987f4feba756.png
+../_images/412470ec90c19f83d1990416d922775c0ff02b86acd0c0b9804ef9074e1baa2a.png
 
 
 
    
@@ -1059,7 +1063,7 @@ 
    Using the learned parameters
-
    Array(-1118.0286, dtype=float32)
+
    Array(-1118.0323, dtype=float32)
 
    
 
    
 
    
@@ -1077,7 +1081,7 @@ 
    System configuration
 
    
 
    
-
    Last updated: Tue Sep 10 2024
+
    Last updated: Wed Sep 11 2024
 
 Python implementation: CPython
 Python version       : 3.12.5
@@ -1087,11 +1091,11 @@ 
    System configuration
    
diff --git a/dev/notebooks/2-Using_custom_response_functions.html b/dev/notebooks/2-Using_custom_response_functions.html
index 49eaba2a6..978e27a89 100644
--- a/dev/notebooks/2-Using_custom_response_functions.html
+++ b/dev/notebooks/2-Using_custom_response_functions.html
@@ -52,7 +52,7 @@
     
     
     
-    
+    
     
     
     
@@ -447,6 +447,7 @@
 
  • Application to reinforcement learning
    • 
 
 
+
 
 
    
 
    
@@ -901,10 +902,10 @@ 
    Recovering HGF parameters from the observed behaviors
    NUTS: [tonic_volatility_2]
 
    
 
    
-
    
-
    
-
    
-
    
+
    
+
    
+
    
+
    
 
    Sampling 4 chains for 1_000 tune and 1_000 draw iterations (4_000 + 4_000 draws total) took 10 seconds.
 
    
 
    
@@ -950,19 +951,19 @@ 
    Recovering HGF parameters from the observed behaviors
     
       tonic_volatility_2
-      -3.944
-      0.493
-      -4.882
-      -3.03
-      0.011
-      0.008
-      1841.0
-      2789.0
+      -3.951
+      0.502
+      -4.888
+      -3.021
+      0.012
+      0.009
+      1732.0
+      2071.0
       1.0
     
   
 
-
    ../_images/54ca9936893f35687d224af885ce5ab69d44ce29b0c27b54c1c599ac4b3e458b.png
+
    ../_images/ddaed825bd1224c2dae259d386e13934e8296bdc789e47fbd039bcaaf2308775.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.
    
@@ -990,7 +991,7 @@ 
    System configuration
 
 
    
-
    Last updated: Tue Sep 10 2024
+
    Last updated: Wed Sep 11 2024
 
 Python implementation: CPython
 Python version       : 3.12.5
@@ -1000,12 +1001,12 @@ 
    System configuration
    
diff --git a/dev/notebooks/3-Multilevel_HGF.html b/dev/notebooks/3-Multilevel_HGF.html
index 99b25fed7..d8b4a31da 100644
--- a/dev/notebooks/3-Multilevel_HGF.html
+++ b/dev/notebooks/3-Multilevel_HGF.html
@@ -52,7 +52,7 @@
     
     
     
-    
+    
     
     
     
@@ -447,6 +447,7 @@
 
  • Application to reinforcement learning
    • 
 
 
+
 
 
    
 
    
@@ -786,7 +787,7 @@ 
    Plot the computational graph
-../_images/1d9dbfc9e1b93c190d6b7a5113ad84c343c300bdcf7182f8ddd8c861a08f1dd0.svg
+../_images/c2923450c77f6c6ef719766a4bc10533412be15579ef49cc29d0f0a3e35eb331.svg
 
    
 
    
 
@@ -812,9 +813,9 @@ 
    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 50 seconds.
+
    
+
    
+
    Sampling 2 chains for 1_000 tune and 1_000 draw iterations (2_000 + 2_000 draws total) took 48 seconds.
 
    
 
    
 
    There were 1 divergences after tuning. Increase `target_accept` or reparameterize.
@@ -849,7 +850,7 @@ 
    Visualization of the posterior distributions
-../_images/57d351bf3f18ba97c164e391d305ea74c2f52c82e7d39586127eee744c316d8e.png
+../_images/273a1aabcc14f043f102eb2cea8cdb4c19de3758731ccd9e0e21a5c681c2ec23.png
 
    
 
    
 
    The reference values on both posterior distributions indicate the mean of the distribution used for simulation.
    
@@ -876,17 +877,17 @@ 
    Model comparison
    Computed from 2000 posterior samples and 3200 observations log-likelihood matrix.
 
          Estimate       SE
-elpd_loo -1684.29    25.62
-p_loo       17.98        -
+elpd_loo -1684.41    25.65
+p_loo       18.18        -
 
 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)     3186   99.6%
    (0.70, 1]   (bad)         1    0.0%
-   (1, Inf)   (very bad)   12    0.4%
+   (1, Inf)   (very bad)   13    0.4%
 
    
 
    
 
    
@@ -903,7 +904,7 @@ 
    System configuration
 
 
    
-
    Last updated: Tue Sep 10 2024
+
    Last updated: Wed Sep 11 2024
 
 Python implementation: CPython
 Python version       : 3.12.5
@@ -913,14 +914,14 @@ 
    System configuration
    
diff --git a/dev/notebooks/4-Parameter_recovery.html b/dev/notebooks/4-Parameter_recovery.html
index 59d427873..ce6730557 100644
--- a/dev/notebooks/4-Parameter_recovery.html
+++ b/dev/notebooks/4-Parameter_recovery.html
@@ -50,7 +50,7 @@
     
     
     
-    
+    
     
     
     
@@ -445,6 +445,7 @@
 
  • Application to reinforcement learning
    • 
 
 
+
 
 
    
 
    
@@ -670,12 +671,12 @@ 
    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 57 seconds.
+
    
+
    
+
    Sampling 2 chains for 1_000 tune and 1_000 draw iterations (2_000 + 2_000 draws total) took 60 seconds.
 
    
 
    
-
    There were 2000 divergences after tuning. Increase `target_accept` or reparameterize.
+
    There were 1998 divergences after tuning. Increase `target_accept` or reparameterize.
 
    
 
    
 
    We recommend running at least 4 chains for robust computation of convergence diagnostics
@@ -745,7 +746,7 @@ 
    Visualizing parameters recovery
-../_images/0af1e2194558be695fd881780dfb668909d97cd8738ce94874a9d181911b5fc7.png
+../_images/7da46b4ab110367e829e77adc2db4f2de202a52d8ea9a0ab01caefe6f18def86.png
 
    
 
    
 
@@ -760,7 +761,7 @@ 
    System configuration
 
    
 
    
-
    Last updated: Tue Sep 10 2024
+
    Last updated: Wed Sep 11 2024
 
 Python implementation: CPython
 Python version       : 3.12.5
@@ -770,13 +771,13 @@ 
    System configuration
    
diff --git a/dev/notebooks/5-Non_linear_value_coupling.html b/dev/notebooks/5-Non_linear_value_coupling.html
index 8d16491ea..564cfb8bd 100644
--- a/dev/notebooks/5-Non_linear_value_coupling.html
+++ b/dev/notebooks/5-Non_linear_value_coupling.html
@@ -921,7 +921,7 @@ 
    System configuration
 
    
 
    
-
    Last updated: Tue Sep 10 2024
+
    Last updated: Wed Sep 11 2024
 
 Python implementation: CPython
 Python version       : 3.12.5
@@ -931,11 +931,11 @@ 
    System configuration
    
diff --git a/dev/notebooks/Example_1_Heart_rate_variability.html b/dev/notebooks/Example_1_Heart_rate_variability.html
index 187bd1743..818eb675b 100644
--- a/dev/notebooks/Example_1_Heart_rate_variability.html
+++ b/dev/notebooks/Example_1_Heart_rate_variability.html
@@ -50,7 +50,7 @@
     
     
     
-    
+    
     
     
     
@@ -445,6 +445,7 @@
 
  • Application to reinforcement learning
    • 
 
 
+
 
 
    
 
    
@@ -567,16 +568,16 @@ 
    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,  1.79it/s]
+
    Downloading ECG channel:  50%|█████     | 1/2 [00:00<00:00,  1.95it/s]
 
    
 
    
-
    Downloading Respiration channel:  50%|█████     | 1/2 [00:00<00:00,  1.79it/s]
+
    Downloading Respiration channel:  50%|█████     | 1/2 [00:00<00:00,  1.95it/s]
 
    
 
    
-
    Downloading Respiration channel: 100%|██████████| 2/2 [00:01<00:00,  1.65it/s]
+
    Downloading Respiration channel: 100%|██████████| 2/2 [00:01<00:00,  1.78it/s]
 
    
 
    
-
    Downloading Respiration channel: 100%|██████████| 2/2 [00:01<00:00,  1.67it/s]
+
    Downloading Respiration channel: 100%|██████████| 2/2 [00:01<00:00,  1.80it/s]
 
    
 
    
 
    
@@ -673,8 +674,8 @@ 
    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.
 
    
 
    
@@ -690,7 +691,7 @@ 
    Model#<
 
    
 
    
 
    
-../_images/3456114bf6a49bc6b1dcc7e5826517fce089b3fb83fd171adb7e00fb06d62e76.png
+../_images/498e9dac4c6e972b4e607090053c41c2627d095020624bfbe6054340645b7eeb.png
 
    
 
    
 
    
@@ -722,7 +723,7 @@ 
    Model#<
 
    
 
    
 
    
-../_images/9f8e055d1baf6d37c2c0e4abfc014d5b66b8a595806cbaa85d631c8ab72717bc.png
+../_images/bc36edcaef7d49fcaf90a31db56c7146586ff64ad013b043663b09a239db803e.png
 
    
 
    
 
@@ -737,7 +738,7 @@ 
    System configuration
 
    
 
    
-
    Last updated: Tue Sep 10 2024
+
    Last updated: Wed Sep 11 2024
 
 Python implementation: CPython
 Python version       : 3.12.5
@@ -747,11 +748,11 @@ 
    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 358558516..b23085613 100644
--- a/dev/notebooks/Example_2_Input_node_volatility_coupling.html
+++ b/dev/notebooks/Example_2_Input_node_volatility_coupling.html
@@ -442,6 +442,7 @@
 
  • Application to reinforcement learning
    • 
 
 
+
 
 
    
 
    
@@ -685,7 +686,7 @@ 
    System configuration
 
    
 
    
-
    Last updated: Tue Sep 10 2024
+
    Last updated: Wed Sep 11 2024
 
 Python implementation: CPython
 Python version       : 3.12.5
@@ -695,10 +696,10 @@ 
    System configuration
    
diff --git a/dev/notebooks/Example_3_Multi_armed_bandit.html b/dev/notebooks/Example_3_Multi_armed_bandit.html
index 8a9a154b9..1b98ecc4d 100644
--- a/dev/notebooks/Example_3_Multi_armed_bandit.html
+++ b/dev/notebooks/Example_3_Multi_armed_bandit.html
@@ -52,7 +52,7 @@
     
     
     
-    
+    
     
     
     
@@ -447,6 +447,7 @@
 
  • Application to reinforcement learning
    • 
 
 
+
 
 
    
 
    
@@ -1086,12 +1087,12 @@ 
    Bayesian inference
    NUTS: [omega]
 
    
 
    
-
    
-
    
+
    
+
    
 
    Sampling 2 chains for 1_000 tune and 1_000 draw iterations (2_000 + 2_000 draws total) took 8 seconds.
 
    
 
    
-
    There were 7 divergences after tuning. Increase `target_accept` or reparameterize.
+
    There were 3 divergences after tuning. Increase `target_accept` or reparameterize.
 
    
 
    
 
    We recommend running at least 4 chains for robust computation of convergence diagnostics
@@ -1109,7 +1110,7 @@ 
    Bayesian inference
-../_images/1babfc1859e33ae6f3818b65c624f7a59e0de7e8fc84f1c2c9da8ac5564bcfe2.png
+../_images/04464c9720c3b245104d7ab82301be47e42110ba2b0dca2c7de4991f6054b95c.png
 
    
 
    
 
@@ -1124,7 +1125,7 @@ 
    System configuration
 
    
 
    
-
    Last updated: Tue Sep 10 2024
+
    Last updated: Wed Sep 11 2024
 
 Python implementation: CPython
 Python version       : 3.12.5
@@ -1134,14 +1135,14 @@ 
    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 fe38ea9e2..bf7ee9a3a 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
@@ -444,6 +444,7 @@
 
  • Application to reinforcement learning
    • 
 
 
+
 
 
    
 
    
@@ -735,29 +736,6 @@ 
    Model inversion: the generalized Hierarchical Gaussian Filtermean, precision and tonic_volatility that can be provided when creating the network. Using the function above, try to change these values. How does this influence the belief trajectories?
    
 
 
    
-
    
-
    
-
    (
-    Network()
-    .add_nodes(kind="continuous-input")
-    .add_nodes(
-        kind="continuous-state",
-        value_children=0,
-        mean=1.0,
-        precision=1e2,
-        tonic_volatility=-2.0,
-    )
-    .add_nodes(kind="continuous-state", volatility_children=1)
-    .input_data(input_data=timeseries)
-    .plot_trajectories()
-);
-
    
-
    
-
    
-
    
-../_images/d6c46049226920d87843a3a297ddbd0c63c1e23cb0dab919f6596fbeed367051.png
-
    
-
    
 
 
    
 
    Practice: Filtering the worlds weather#
    
@@ -1047,7 +1025,7 @@ 
    System configuration
 
    
 
    
-
    Last updated: Tue Sep 10 2024
+
    Last updated: Wed Sep 11 2024
 
 Python implementation: CPython
 Python version       : 3.12.5
@@ -1058,13 +1036,13 @@ 
    System configuration
    
diff --git a/dev/notebooks/Exercise_2_Bayesian_reinforcement_learning.html b/dev/notebooks/Exercise_2_Bayesian_reinforcement_learning.html
index 946a1bb79..3711a57ff 100644
--- a/dev/notebooks/Exercise_2_Bayesian_reinforcement_learning.html
+++ b/dev/notebooks/Exercise_2_Bayesian_reinforcement_learning.html
@@ -52,7 +52,7 @@
     
     
     
-    
+    
     
     
     
@@ -447,6 +447,7 @@
 
  • Application to reinforcement learning
    • 
 
 
+
 
 
    
 
    
@@ -554,8 +555,8 @@
 
    
 
 
    
-
    
-
    Reinforcement learning with the binary HGF#
    
+
    
+
    The binary HGF#
    
 
    In the first part of the tutorial, we introduced neural networks for predictive coding, a flexible and modular way to define hierarchically structured systems of predictive nodes. We discussed how value coupling and volatility coupling are two kinds of interactions between nodes that can influence how prediction and prediction errors are propagated through the network.
    
 
    However, several experiments, especially when considering reinforcement learning, require working with variables that have discrete states. This is the case for example when the agent has to learn the association between one stimulus and a reward. Such experiments, like the one-armed bandit task that we will be using below, generally produce two types of information at each trial:
    
 
      
@@ -674,7 +675,7 @@ 
      Parameters optimization
 
    
 
    
-../_images/ecd192b619fcda3b40cdb5f737d2b936691486677eb7a4f82b5cbc7f55367e9e.svg
+../_images/7d29572ae423fb4c8d4ea736f01dab651bf0ee09a3d53b5838bcf639ba2eaf01.svg
 
    
 
    
 
    
@@ -697,10 +698,10 @@ 
    Parameters optimization
    NUTS: [tonic_volatility_2]
 
    
 
    
-
    
-
    
-
    
-
    
+
    
+
    
+
    
+
    
 
    Sampling 4 chains for 1_000 tune and 1_000 draw iterations (4_000 + 4_000 draws total) took 10 seconds.
 
    
 
    
@@ -713,7 +714,7 @@ 
    Parameters optimization
 
    
 
    
-../_images/652b901809491a58f53fd3df351d4d6093868ede9348d2a0fd662ca1e9f8c393.png
+../_images/23a947492740a4387ade6fb6412ab3c996fff3f3ee9c4626ded56992baa4024a.png
 
    
 
    
 
    
@@ -755,14 +756,14 @@ 
    Parameters optimization
     
       tonic_volatility_2
-      -2.567
-      0.385
-      -3.34
-      -1.906
+      -2.591
+      0.393
+      -3.363
+      -1.904
       0.011
       0.008
-      1328.0
-      1097.0
+      1254.0
+      1078.0
       1.0
     
   
@@ -806,10 +807,10 @@ 
    Biased random
    NUTS: [bias]
 
    
 
    
-
    
-
    
-
    
-
    
+
    
+
    
+
    
+
    
 
    Sampling 4 chains for 1_000 tune and 1_000 draw iterations (4_000 + 4_000 draws total) took 4 seconds.
 
    
 
    
@@ -822,7 +823,7 @@ 
    Biased random
-../_images/51ace5e0df230f26a825e40316b57289c523e35a03567f4edd3b0dbaebf5f96b.png
+../_images/7684f781393fddcb5b12fb0266f447c0d53da287bfc7f6bcdd77f1ce716e86b0.png
 
    
 
    
 
    Assess model fitting, here using leave-one-out cross-validation from the Arviz library.
    
@@ -894,7 +895,7 @@ 
    Rescorla-Wagner{"version_major": 2, "version_minor": 0, "model_id": "eb7d63d8c4024c6ca6a346386f5c74fc"}
    /opt/hostedtoolcache/Python/3.12.5/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/3.12.5/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()
 
    
 
    
@@ -911,7 +912,7 @@ 
    Rescorla-Wagner
-../_images/71a10420090755270a8de124f690cf28abc2ac8da75a2bdc544159c8b6bbe7b0.png
+../_images/d976b93af7bf12b1517046b96de7e43ee0ef0473e3e7888654e502f60d986e90.png
 
    
 
    
 
@@ -1002,10 +1003,10 @@ 
    Two-level HGF
    NUTS: [tonic_volatility_2]
 
    
 
    
-
    
-
    
-
    
-
    
+
    
+
    
+
    
+
    
 
    Sampling 4 chains for 1_000 tune and 1_000 draw iterations (4_000 + 4_000 draws total) took 10 seconds.
 
    
 
    
@@ -1018,7 +1019,7 @@ 
    Two-level HGF
-../_images/febf29c2036063f463af158cf478b4b53583b7ff186b032dd0ea3f04cb95aa0b.png
+../_images/c0548c1593129791eadcdeaab49a72208a0b1c53afaded766502a4f257440da9.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.
    
@@ -1112,14 +1113,14 @@ 
    Three-level HGF
    NUTS: [tonic_volatility_2]
 
    
 
    
-
    
-
    
-
    
-
    
+
    
+
    
+
    
+
    
 
    Sampling 4 chains for 1_000 tune and 1_000 draw iterations (4_000 + 4_000 draws total) took 12 seconds.
 
    
 
    
-
    There were 111 divergences after tuning. Increase `target_accept` or reparameterize.
+
    There were 126 divergences after tuning. Increase `target_accept` or reparameterize.
 
    
 
    
 
    
@@ -1131,7 +1132,7 @@ 
    Three-level HGF
-../_images/fdd38c86511a7c09898bc9d85dd7380eb66e1a4ec7193d30537e76517fc94963.png
+../_images/7bc8e8923009bbd35cfaa57a4245bdbe40e2c5a87d9305e27c7fdb715dd920f8.png
 
    
 
    
 
    Move pointwise estimate into the likelihood field.
    
@@ -1204,11 +1205,11 @@ 
    Model comparison
-../_images/8eecb320911c9c326f2b7e7904084bfcd78435ed0eca866075f495f5922861d7.png
+../_images/fcc839c923456dc5194b1282e849ceb22dde9cfbd8a11b43edff24a8bec505e8.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.
    
@@ -1377,13 +1378,27 @@ 
    Beliefs trajectories
 
 
    
-../_images/db0bddabbf65c408dea4166716d91670658e01f204e51eb573b570181ae9a2f4.png
+../_images/1e60d6a08ed4ed956bafa287e21493d4bd9aafef53ec5cd3e8eec134b49b0a5b.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.
    
 
 
 
+
    
+
    Where to go next?#
    
+
    Thank you for completing this tutorial on the generalized Hierarchical Gaussian Filter. Let’s briefly recap what we have covered so far:
    
+
      
+
    • In the first notebook you learned how to simulate the generative model on which the gHGF is built and how to create probabilistic networks that can invert this model with PyHGF. We applied these principles to real data and tried different parametrisations that could improve model performances.
      • 
+
    • In the second notebook we covered the binary HGF, an extension of this model to binary inputs for reinforcement learning. We then used MCMC sampling to infer the probability distribution of key parameters and compared the performances of our model to other RL classics.
      • 
+
    
+
    All these sections should give you a solid understanding of the model and how to use it in context. If you want to apply this to your dataset, we recommend exploring the tutorial section of the documentation:
    
+
      
+
    • Using custom response models will explore the creation of custom models for behaviours that can match your experimental design and theory.
      • 
+
    • Multilevel modelling will discuss modelling at the group/condition level in multilevel Bayesian networks.
      • 
+
    • Parameter recovery explains how to simulate a dataset and perform parameter recovery, as a prior validation of your models.
      • 
+
    
+
    
 
    
 
    Solutions#
    
 
    
@@ -1397,7 +1412,7 @@ 
    System configuration
 
    
 
    
-
    Last updated: Tue Sep 10 2024
+
    Last updated: Wed Sep 11 2024
 
 Python implementation: CPython
 Python version       : 3.12.5
@@ -1409,8 +1424,8 @@ 
    System configurationSystem configuration
     
      
 
    • Application to reinforcement learning
diff --git a/dev/searchindex.js b/dev/searchindex.js
index 607357b88..1a131bd61 100644
--- a/dev/searchindex.js
+++ b/dev/searchindex.js
@@ -1 +1 @@
-Search.setIndex({"alltitles": {"": [[76, "exercise1"], [76, "exercise2"], [76, "exercise3"], [76, "exercise4"], [76, "exercise5"]], "API": [[0, "api"]], "Acknowledgments": [[61, "acknowledgments"]], "Add data": [[66, "add-data"], [66, "id4"], [68, "add-data"], [68, "id3"]], "Adding a drift to the random walk": [[63, "adding-a-drift-to-the-random-walk"]], "Application to reinforcement learning": [[77, null]], "Autoregressive processes": [[63, "autoregressive-processes"]], "Bayesian inference": [[75, "bayesian-inference"]], "Beliefs trajectories": [[77, "beliefs-trajectories"]], "Biased random": [[77, "biased-random"]], "Binary inputs": [[0, "binary-inputs"]], "Binary nodes": [[0, "binary-nodes"], [0, "id1"]], "Binary state nodes": [[0, "binary-state-nodes"]], "Bivariate normal distribution": [[65, "bivariate-normal-distribution"]], "Categorical nodes": [[0, "categorical-nodes"]], "Continuous inputs": [[0, "continuous-inputs"]], "Continuous nodes": [[0, "continuous-nodes"], [0, "id2"]], 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"How to cite?": [[1, null]], "Imports": [[66, "imports"]], "Inference from the simulated behaviours": [[71, "inference-from-the-simulated-behaviours"]], "Inference using MCMC sampling": [[67, "inference-using-mcmc-sampling"]], "Inputs": [[0, "inputs"]], "Installation": [[61, "installation"]], "Introduction to the Generalised Hierarchical Gaussian Filter": [[63, null]], "Introduction to the generalised Hierarchical Gaussian Filter": [[76, null]], "Kown mean, unknown precision": [[74, "kown-mean-unknown-precision"]], "Learn": [[62, null]], "Learning parameters with MCMC sampling": [[66, "learning-parameters-with-mcmc-sampling"], [68, "learning-parameters-with-mcmc-sampling"]], "Loading and preprocessing physiological recording": [[73, "loading-and-preprocessing-physiological-recording"]], "Math": [[0, "math"]], "Model": [[0, "model"], [73, "model"]], "Model comparison": [[70, "model-comparison"], [77, "model-comparison"]], "Model fitting": [[61, "model-fitting"]], "Model inversion: the generalized Hierarchical Gaussian Filter": [[76, "model-inversion-the-generalized-hierarchical-gaussian-filter"]], "Modifying the attributes": [[64, "modifying-the-attributes"]], "Modifying the edges": [[64, "modifying-the-edges"]], "Multivariate coupling": [[64, "multivariate-coupling"]], "Non-linear predictions": [[72, "non-linear-predictions"]], "Non-linear value coupling": [[72, null]], "Non-parametric predictive coding": [[62, "non-parametric-predictive-coding"]], "Parameter recovery": [[75, "parameter-recovery"]], "Parameters optimization": [[77, "parameters-optimization"]], "Plot correlation": [[68, "plot-correlation"]], "Plot the computational graph": [[70, "plot-the-computational-graph"]], "Plot the signal with instantaneous heart rate derivations": [[73, "plot-the-signal-with-instantaneous-heart-rate-derivations"]], "Plot trajectories": [[66, "plot-trajectories"], [66, "id5"], [68, "plot-trajectories"], [68, "id4"]], "Plots": [[0, "plots"]], "Posterior predictive sampling": 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