Iowa Gambling Task
In this tutorial, we will fit the PVL-Delta model to data from the Iowa Gambling Task (IGT) using the ActionModels package. In the IGT, participants choose cards from four decks, each with different reward and loss probabilities, and must learn over time which decks are advantageous. We will use data from Ahn et al. (2014), which includes healthy controls and participants with heroin or amphetamine addictions. There are more details about the collected data in the docs/example_data/ahn_et_al_2014/ReadMe.txt
Loading data
First, we load the ActionModels package. We also load CSV and Dataframes for loading the data, and StatsPlots for plotting the results.
using ActionModels
using CSV, DataFrames
using StatsPlotsThen we load the ahn et al. (2014) data, which is available in the docs/example_data/ahn_et_al_2014 folder.
#Import data
data_healthy = CSV.read(
joinpath(docs_path, "example_data", "ahn_et_al_2014", "IGTdata_healthy_control.txt"),
DataFrame,
)
data_healthy[!, :clinical_group] .= "1_control"
data_heroin = CSV.read(
joinpath(docs_path, "example_data", "ahn_et_al_2014", "IGTdata_heroin.txt"),
DataFrame,
)
data_heroin[!, :clinical_group] .= "2_heroin"
data_amphetamine = CSV.read(
joinpath(docs_path, "example_data", "ahn_et_al_2014", "IGTdata_amphetamine.txt"),
DataFrame,
)
data_amphetamine[!, :clinical_group] .= "3_amphetamine"
#Combine into one dataframe
ahn_data = vcat(data_healthy, data_heroin, data_amphetamine)
ahn_data[!, :subjID] = string.(ahn_data[!, :subjID])
#Make column with total reward
ahn_data[!, :reward] = Float64.(ahn_data[!, :gain] + ahn_data[!, :loss]);
show(ahn_data)12883×7 DataFrame
Row │ trial deck gain loss subjID clinical_group reward
│ Int64 Int64 Int64 Int64 String String Float64
───────┼─────────────────────────────────────────────────────────────
1 │ 1 3 50 0 103 1_control 50.0
2 │ 2 3 60 0 103 1_control 60.0
3 │ 3 3 40 -50 103 1_control -10.0
4 │ 4 4 50 0 103 1_control 50.0
5 │ 5 3 55 0 103 1_control 55.0
6 │ 6 1 100 0 103 1_control 100.0
7 │ 7 1 120 0 103 1_control 120.0
8 │ 8 2 100 0 103 1_control 100.0
⋮ │ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮
12877 │ 94 4 70 0 344 3_amphetamine 70.0
12878 │ 95 2 150 0 344 3_amphetamine 150.0
12879 │ 96 4 50 0 344 3_amphetamine 50.0
12880 │ 97 2 110 0 344 3_amphetamine 110.0
12881 │ 98 4 70 -275 344 3_amphetamine -205.0
12882 │ 99 2 150 0 344 3_amphetamine 150.0
12883 │ 100 3 70 -25 344 3_amphetamine 45.0
12868 rows omittedFor this example, we will subset the data to only include two subjects from each clinical group. This makes the runtime much shorter. Simply skip this step if you want to use the full dataset.
ahn_data =
filter(row -> row[:subjID] in ["103", "117", "105", "136", "130", "149"], ahn_data);Creating the model
Then we construct the model to be fitted to the data. We use the PVL-Delta action model, which is a classic model for the IGT. The PVL-Delta is a type of reinfrocement learning model that learns the expected value for each of the decks in the IGT. First, the observed reward is transformed with a prospect theory-based utlity curve. This means that the subjective value of a reward increses sub-linearly with reward magnitute, and that losses are weighted more heavily than gains. The expected value of each deck is then updated using a Rescorla-Wagner-like update rule. Finally, the probability of selecting each deck is calculated using a softmax function over the expected values of the decks, scaled by an inverse action noise parameter. In summary, the PVL-Delta has four parameters: the learning rate $\alpha$, the reward sensitivity $A$, the loss aversion $w$, and the action noise $\beta$. See the section on the PVL-Delta premade model in the documentation for more details.
We create the PVL-Delta using the premade model from ActionModels. We specify the number of decks, and also that actions are selected before the expected values are updated. This is because in the IGT, at least as structured in this dataset, participants select a deck before they receive the reward and update expectations.
action_model = ActionModel(PVLDelta(n_options = 4, act_before_update = true))-- ActionModel --
Action model function: pvl_delta_act_before_update
Number of parameters: 5
Number of states: 1
Number of observations: 2
Number of actions: 1
We then specify which column in the data corresponds to the action (deck choice) and which columns correspond to the observations (deck and reward). We also specify the columns that uniquely identify each session.
action_cols = :deck
observation_cols = (chosen_option = :deck, reward = :reward)
session_cols = :subjID;Finally, we create the full model. We use a hierarchical regression model to predict the parameters of the PVL-Delta model based on the clinical group (healthy, heroin, or amphetamine). First, we will set appropriate priors for the regression coefficients. For the action noise and the loss aversion, the outcome of the regression will be exponentiated before it is used in the model, so pre-transformed outcomes around 2 (exp(2) ≈ 7) are among the most extreme values to be expected. For the learning rate and reward sensitivity, we will use a logistic transformation, so pre-transformed outcomes around around 5 (logistic(5) ≈ 0.993) are among the most extreme values to be expected.
regression_prior = RegressionPrior(
β = [Normal(0, 0.2), Normal(0, 0.3), Normal(0,0.3)],
σ = truncated(Normal(0, 0.3), lower = 0),
)
plot(regression_prior.β[1], label = "Intercept")
plot!(regression_prior.β[2], label = "Effect of clinical group")
plot!(regression_prior.σ, label = "Random intercept std")
title!("Regression priors for the Rescorla-Wagner model")
xlabel!("Regression coefficient")
ylabel!("Density")We can now create the popluation model, and finally create the full model object.
population_model = [
Regression(
@formula(learning_rate ~ clinical_group + (1 | subjID)),
logistic,
regression_prior,
),
Regression(
@formula(reward_sensitivity ~ clinical_group + (1 | subjID)),
logistic,
regression_prior,
),
Regression(
@formula(loss_aversion ~ clinical_group + (1 | subjID)),
exp,
regression_prior,
),
Regression(
@formula(action_noise ~ clinical_group + (1 | subjID)),
exp,
regression_prior,
),
]
model = create_model(
action_model,
population_model,
ahn_data,
action_cols = action_cols,
observation_cols = observation_cols,
session_cols = session_cols,
)-- ModelFit object --
Action model: pvl_delta_act_before_update
Linear regression population model
4 estimated action model parameters, 6 sessions
Posterior not sampled
Prior not sampled
Fitting the model
We are now ready to fit the model to the data. For this model, we will use the Enzyme automatic differentiation backend, which is a high-performance automatic differentiation library. Additionally, to keep the runtime of this tutorial short, we will only fit two chains with 500 samples. We will pass MCMCThreas() in order to parallelize the sampling across the two chains. This should take up to 10-15 minutes on a standard laptop. Switch to only a single thread for a learer progress bar.
#Set AD backend
using ADTypes: AutoEnzyme
import Enzyme: set_runtime_activity, Reverse
ad_type = AutoEnzyme(; mode = set_runtime_activity(Reverse, true));
#Fit model
chns = sample_posterior!(model, MCMCThreads(), init_params = :MAP, n_chains = 2, n_samples = 500, ad_type = ad_type)Chains MCMC chain (500×52×2 Array{Float64, 3}):
Iterations = 251:1:750
Number of chains = 2
Samples per chain = 500
Wall duration = 3183.22 seconds
Compute duration = 3988.36 seconds
parameters = learning_rate.β[1], learning_rate.β[2], learning_rate.β[3], learning_rate.ranef_1.σ[1], learning_rate.ranef_1.r[1], learning_rate.ranef_1.r[2], learning_rate.ranef_1.r[3], learning_rate.ranef_1.r[4], learning_rate.ranef_1.r[5], learning_rate.ranef_1.r[6], reward_sensitivity.β[1], reward_sensitivity.β[2], reward_sensitivity.β[3], reward_sensitivity.ranef_1.σ[1], reward_sensitivity.ranef_1.r[1], reward_sensitivity.ranef_1.r[2], reward_sensitivity.ranef_1.r[3], reward_sensitivity.ranef_1.r[4], reward_sensitivity.ranef_1.r[5], reward_sensitivity.ranef_1.r[6], loss_aversion.β[1], loss_aversion.β[2], loss_aversion.β[3], loss_aversion.ranef_1.σ[1], loss_aversion.ranef_1.r[1], loss_aversion.ranef_1.r[2], loss_aversion.ranef_1.r[3], loss_aversion.ranef_1.r[4], loss_aversion.ranef_1.r[5], loss_aversion.ranef_1.r[6], action_noise.β[1], action_noise.β[2], action_noise.β[3], action_noise.ranef_1.σ[1], action_noise.ranef_1.r[1], action_noise.ranef_1.r[2], action_noise.ranef_1.r[3], action_noise.ranef_1.r[4], action_noise.ranef_1.r[5], action_noise.ranef_1.r[6]
internals = lp, n_steps, is_accept, acceptance_rate, log_density, hamiltonian_energy, hamiltonian_energy_error, max_hamiltonian_energy_error, tree_depth, numerical_error, step_size, nom_step_size
Summary Statistics
parameters mean std mcse ess_bulk ess_tail rhat ess_per_sec
Symbol Float64 Float64 Float64 Float64 Float64 Float64 Float64
learning_rate.β[1] -0.1458 0.2100 0.0075 788.0090 684.2905 1.0026 0.1976
learning_rate.β[2] 0.0258 0.2978 0.0128 555.0488 473.9848 1.0016 0.1392
learning_rate.β[3] -0.0388 0.3037 0.0092 1084.7973 681.9421 1.0017 0.2720
learning_rate.ranef_1.σ[1] 0.8265 0.3166 0.0446 55.5097 149.7110 1.0257 0.0139
learning_rate.ranef_1.r[1] -2.2781 1.3366 0.1821 63.9628 287.6264 1.0194 0.0160
learning_rate.ranef_1.r[2] 0.6344 0.7523 0.0642 129.4021 298.4270 1.0246 0.0324
learning_rate.ranef_1.r[3] -0.2557 0.8156 0.0812 69.7595 110.2648 1.0572 0.0175
learning_rate.ranef_1.r[4] -0.3105 0.9908 0.0575 319.8078 264.4876 1.0030 0.0802
learning_rate.ranef_1.r[5] -0.3655 1.1238 0.1418 84.8017 100.9919 1.0166 0.0213
learning_rate.ranef_1.r[6] -0.0655 0.8830 0.0473 324.6192 337.1385 1.0027 0.0814
reward_sensitivity.β[1] -0.3467 0.2017 0.0075 731.9576 750.4011 1.0016 0.1835
reward_sensitivity.β[2] -0.2743 0.2754 0.0084 1085.5429 501.6289 0.9991 0.2722
reward_sensitivity.β[3] -0.2852 0.2698 0.0085 998.7901 746.8395 0.9991 0.2504
reward_sensitivity.ranef_1.σ[1] 0.5639 0.2822 0.0216 166.6972 274.8063 1.0244 0.0418
reward_sensitivity.ranef_1.r[1] -0.0097 0.7782 0.0542 199.3430 499.4656 1.0139 0.0500
reward_sensitivity.ranef_1.r[2] -0.3440 0.4772 0.0242 386.4615 567.3087 1.0097 0.0969
reward_sensitivity.ranef_1.r[3] -0.7425 0.6589 0.0391 264.5037 659.1312 1.0169 0.0663
reward_sensitivity.ranef_1.r[4] -0.5864 0.6479 0.0341 360.9692 746.4718 1.0096 0.0905
reward_sensitivity.ranef_1.r[5] -0.6366 0.6637 0.0330 404.4595 845.9577 1.0081 0.1014
reward_sensitivity.ranef_1.r[6] -0.5395 0.5876 0.0315 346.4858 803.3882 1.0121 0.0869
loss_aversion.β[1] -0.2400 0.1774 0.0067 681.1574 631.0527 0.9997 0.1708
loss_aversion.β[2] -0.3553 0.2788 0.0086 1043.7526 835.4924 1.0068 0.2617
loss_aversion.β[3] -0.1182 0.2846 0.0085 1127.7070 677.8353 1.0058 0.2827
loss_aversion.ranef_1.σ[1] 0.3315 0.2254 0.0210 103.8532 175.4735 1.0099 0.0260
loss_aversion.ranef_1.r[1] -0.1244 0.3454 0.0261 276.6112 156.6776 1.0114 0.0694
loss_aversion.ranef_1.r[2] -0.3756 0.4912 0.0338 229.5728 463.3372 1.0058 0.0576
loss_aversion.ranef_1.r[3] 0.0172 0.4176 0.0262 331.8085 200.5155 1.0096 0.0832
loss_aversion.ranef_1.r[4] -0.1442 0.3947 0.0206 585.8721 266.3164 1.0066 0.1469
loss_aversion.ranef_1.r[5] -0.1656 0.4171 0.0195 595.0356 310.7070 1.0026 0.1492
loss_aversion.ranef_1.r[6] -0.0540 0.3672 0.0134 834.8783 493.4550 1.0009 0.2093
action_noise.β[1] 0.3594 0.1950 0.0075 671.6886 702.4170 1.0031 0.1684
action_noise.β[2] 0.2839 0.2819 0.0104 752.5935 650.2428 1.0049 0.1887
action_noise.β[3] 0.3276 0.2775 0.0101 767.9811 747.2489 1.0014 0.1926
action_noise.ranef_1.σ[1] 0.7755 0.2636 0.0297 85.8663 117.7332 1.0320 0.0215
action_noise.ranef_1.r[1] 0.4668 0.6362 0.0557 137.7539 422.5892 1.0107 0.0345
action_noise.ranef_1.r[2] 0.4040 0.4937 0.0206 586.2093 813.7125 1.0067 0.1470
action_noise.ranef_1.r[3] 0.7979 0.5807 0.0414 188.6492 432.9618 1.0235 0.0473
action_noise.ranef_1.r[4] 1.2002 0.7684 0.0654 125.4608 428.7655 1.0207 0.0315
action_noise.ranef_1.r[5] 1.5845 0.8334 0.0901 77.8296 122.1488 1.0309 0.0195
action_noise.ranef_1.r[6] 0.9108 0.6571 0.0546 127.1097 475.1995 1.0204 0.0319
Quantiles
parameters 2.5% 25.0% 50.0% 75.0% 97.5%
Symbol Float64 Float64 Float64 Float64 Float64
learning_rate.β[1] -0.5417 -0.2902 -0.1459 0.0056 0.2452
learning_rate.β[2] -0.5368 -0.1872 0.0277 0.2393 0.5740
learning_rate.β[3] -0.6108 -0.2470 -0.0356 0.1748 0.5579
learning_rate.ranef_1.σ[1] 0.1831 0.6430 0.8718 1.0481 1.3719
learning_rate.ranef_1.r[1] -4.4500 -3.2627 -2.5507 -1.3297 0.1445
learning_rate.ranef_1.r[2] -0.7627 0.1135 0.6070 1.1111 2.1959
learning_rate.ranef_1.r[3] -2.1956 -0.6280 -0.1933 0.1858 1.3038
learning_rate.ranef_1.r[4] -2.9022 -0.7483 -0.2089 0.2736 1.4440
learning_rate.ranef_1.r[5] -3.2458 -0.8141 -0.1636 0.3024 1.4904
learning_rate.ranef_1.r[6] -1.8845 -0.5593 -0.0443 0.3578 1.8946
reward_sensitivity.β[1] -0.7254 -0.4893 -0.3399 -0.2093 0.0400
reward_sensitivity.β[2] -0.8266 -0.4668 -0.2792 -0.0873 0.2650
reward_sensitivity.β[3] -0.8187 -0.4540 -0.2895 -0.1030 0.2495
reward_sensitivity.ranef_1.σ[1] 0.0836 0.3542 0.5673 0.7629 1.1182
reward_sensitivity.ranef_1.r[1] -1.6819 -0.4480 -0.0311 0.3670 1.6934
reward_sensitivity.ranef_1.r[2] -1.4332 -0.6060 -0.2561 -0.0235 0.4284
reward_sensitivity.ranef_1.r[3] -2.1907 -1.1728 -0.6198 -0.2149 0.1553
reward_sensitivity.ranef_1.r[4] -2.1377 -0.9732 -0.4718 -0.0986 0.3562
reward_sensitivity.ranef_1.r[5] -2.2113 -1.0776 -0.4997 -0.1037 0.3214
reward_sensitivity.ranef_1.r[6] -1.9096 -0.8795 -0.4236 -0.1083 0.3144
loss_aversion.β[1] -0.6091 -0.3505 -0.2379 -0.1268 0.1131
loss_aversion.β[2] -0.8753 -0.5557 -0.3521 -0.1715 0.1882
loss_aversion.β[3] -0.6753 -0.3068 -0.1143 0.0771 0.4047
loss_aversion.ranef_1.σ[1] 0.0402 0.1435 0.2940 0.4755 0.8554
loss_aversion.ranef_1.r[1] -0.9862 -0.2421 -0.0540 0.0640 0.4161
loss_aversion.ranef_1.r[2] -1.5874 -0.6068 -0.2265 -0.0200 0.1943
loss_aversion.ranef_1.r[3] -0.9246 -0.1311 0.0057 0.1855 0.9497
loss_aversion.ranef_1.r[4] -1.1359 -0.2926 -0.0542 0.0565 0.4857
loss_aversion.ranef_1.r[5] -1.2414 -0.3027 -0.0696 0.0484 0.4907
loss_aversion.ranef_1.r[6] -0.9308 -0.1926 -0.0231 0.1012 0.6751
action_noise.β[1] -0.0249 0.2262 0.3689 0.4897 0.7407
action_noise.β[2] -0.2989 0.1041 0.2818 0.4732 0.8300
action_noise.β[3] -0.1970 0.1313 0.3349 0.5195 0.8583
action_noise.ranef_1.σ[1] 0.2123 0.6081 0.7742 0.9622 1.2752
action_noise.ranef_1.r[1] -0.7337 0.0262 0.3956 0.8898 1.7589
action_noise.ranef_1.r[2] -0.5064 0.0545 0.3613 0.7266 1.4281
action_noise.ranef_1.r[3] -0.1885 0.3539 0.7901 1.1799 1.9866
action_noise.ranef_1.r[4] -0.0710 0.6157 1.1314 1.7229 2.8707
action_noise.ranef_1.r[5] 0.0771 1.0067 1.5401 2.1184 3.2687
action_noise.ranef_1.r[6] -0.1384 0.4052 0.8403 1.3284 2.2996
We can now inspect the results of the fitting process. We can plot the posterior distribution over the beta parameters of the regression model. We can see indications of lower reward sensitivity and lower loss aversion, as well as higher action noise, in the heroin and amphetamine groups compared to the healthy controls. Note that the posteriors would be more if we ha used the full dataset.
plot(
plot(
plot(
title = "Learning rate",
grid = false,
showaxis = false,
bottom_margin = -30Plots.px,
),
density(
chns[Symbol("learning_rate.β[2]")],
title = "Heroin",
label = nothing,
),
density(
chns[Symbol("learning_rate.β[3]")],
title = "Amphetamine",
label = nothing,
),
layout = @layout([A{0.01h}; [B C]])
),
plot(
plot(
title = "Reward sensitivity",
grid = false,
showaxis = false,
bottom_margin = -50Plots.px,
),
density(
chns[Symbol("reward_sensitivity.β[2]")],
label = nothing,
),
density(
chns[Symbol("reward_sensitivity.β[3]")],
label = nothing,
),
layout = @layout([A{0.01h}; [B C]])
),
plot(
plot(
title = "Loss aversion",
grid = false,
showaxis = false,
bottom_margin = -50Plots.px,
),
density(
chns[Symbol("loss_aversion.β[2]")],
label = nothing,
),
density(
chns[Symbol("loss_aversion.β[3]")],
label = nothing,
),
layout = @layout([A{0.01h}; [B C]])
),
plot(
plot(
title = "Action noise",
grid = false,
showaxis = false,
bottom_margin = -50Plots.px,
),
density(
chns[Symbol("action_noise.β[2]")],
label = nothing,
),
density(
chns[Symbol("action_noise.β[3]")],
label = nothing,
),
layout = @layout([A{0.01h}; [B C]])
),
layout = (4,1), size = (800, 1000),
)We can also extract the session parameters from the model.
session_parameters = get_session_parameters!(model);
#TODO: plot the session parametersAnd we can extract a dataframe with the median of the posterior for each session parameter
parameters_df = summarize(session_parameters, median)
show(parameters_df)6×5 DataFrame
Row │ subjID action_noise learning_rate loss_aversion reward_sensitivity
│ String Float64 Float64 Float64 Float64
─────┼────────────────────────────────────────────────────────────────────────
1 │ 103 2.17007 0.0614588 0.731244 0.395373
2 │ 105 2.84362 0.620093 0.406255 0.280489
3 │ 117 3.13918 0.416547 0.77889 0.269566
4 │ 130 6.23599 0.414338 0.619374 0.240845
5 │ 136 9.02988 0.427001 0.47848 0.233851
6 │ 149 4.65828 0.440308 0.678762 0.248081We can also look at the implied state trajectories, in this case the expected value.
state_trajectories = get_state_trajectories!(model, :expected_value);
#TODO: plot the state trajectoriesThese can also be summarized in a dataframe, for downstream analysis.
states_df = summarize(state_trajectories, median)
show(states_df)606×6 DataFrame
Row │ subjID timestep expected_value[1] expected_value[2] expected_value[ ⋯
│ String Int64 Any Any Any ⋯
─────┼──────────────────────────────────────────────────────────────────────────
1 │ 103 0 0.0 0.0 0.0 ⋯
2 │ 103 1 0.0 0.0 0.341685
3 │ 103 2 0.0 0.0 0.685106
4 │ 103 3 0.0 0.0 0.450515
5 │ 103 4 0.0 0.0 0.450515 ⋯
6 │ 103 5 0.0 0.0 0.806783
7 │ 103 6 0.457852 0.0 0.806783
8 │ 103 7 0.927495 0.0 0.806783
⋮ │ ⋮ ⋮ ⋮ ⋮ ⋮ ⋱
600 │ 149 94 -1.0957 2.99291 2.17897 ⋯
601 │ 149 95 -1.0957 2.99291 2.04027
602 │ 149 96 -1.0957 2.99291 2.04027
603 │ 149 97 -1.0957 2.99291 2.04027
604 │ 149 98 -1.0957 2.99291 1.89753 ⋯
605 │ 149 99 -1.0957 3.05706 1.89753
606 │ 149 100 0.720223 3.05706 1.89753
2 columns and 591 rows omittedComparing to a simple random model
We can also compare the PVL-Delta to a simple random model, which randomly samples actions from a fixed Categorical distribution.
function categorical_random(
attributes::ModelAttributes,
chosen_option::Int64,
reward::Float64,
)
action_probabilities = load_parameters(attributes).action_probabilities
return Categorical(action_probabilities)
end
action_model = ActionModel(
categorical_random,
observations = (chosen_option = Observation(Int64), reward = Observation()),
actions = (; deck = Action(Categorical)),
parameters = (; action_probabilities = Parameter([0.3, 0.3, 0.3, 0.1])),
)-- ActionModel --
Action model function: categorical_random
Number of parameters: 1
Number of states: 0
Number of observations: 2
Number of actions: 1
Fitting the model
For this model, we use an independent session population model. We set the prior for the action probabilities to be a Dirichlet distribution, which is a common prior for categorical distributions.
population_model = (; action_probabilities = Dirichlet([1, 1, 1, 1]),)
simple_model = create_model(
action_model,
population_model,
ahn_data,
action_cols = action_cols,
observation_cols = observation_cols,
session_cols = session_cols,
)
#Set AD backend
using ADTypes: AutoEnzyme
import Enzyme: set_runtime_activity, Reverse
ad_type = AutoEnzyme(; mode = set_runtime_activity(Reverse, true));
#Fit model
chns = sample_posterior!(simple_model, n_chains = 1, n_samples = 500, ad_type = ad_type)
#TODO: plot the resultsChains MCMC chain (500×36×1 Array{Float64, 3}):
Iterations = 251:1:750
Number of chains = 1
Samples per chain = 500
Wall duration = 10.18 seconds
Compute duration = 10.18 seconds
parameters = action_probabilities.session[1, 1], action_probabilities.session[2, 1], action_probabilities.session[3, 1], action_probabilities.session[4, 1], action_probabilities.session[1, 2], action_probabilities.session[2, 2], action_probabilities.session[3, 2], action_probabilities.session[4, 2], action_probabilities.session[1, 3], action_probabilities.session[2, 3], action_probabilities.session[3, 3], action_probabilities.session[4, 3], action_probabilities.session[1, 4], action_probabilities.session[2, 4], action_probabilities.session[3, 4], action_probabilities.session[4, 4], action_probabilities.session[1, 5], action_probabilities.session[2, 5], action_probabilities.session[3, 5], action_probabilities.session[4, 5], action_probabilities.session[1, 6], action_probabilities.session[2, 6], action_probabilities.session[3, 6], action_probabilities.session[4, 6]
internals = lp, n_steps, is_accept, acceptance_rate, log_density, hamiltonian_energy, hamiltonian_energy_error, max_hamiltonian_energy_error, tree_depth, numerical_error, step_size, nom_step_size
Summary Statistics
parameters mean std mcse ess_bulk ess_tail rhat ess_per_sec
Symbol Float64 Float64 Float64 Float64 Float64 Float64 Float64
action_probabilities.session[1, 1] 0.2105 0.0401 0.0011 1330.2875 349.8661 0.9981 130.6894
action_probabilities.session[2, 1] 0.2210 0.0401 0.0011 1273.2539 384.7125 1.0001 125.0863
action_probabilities.session[3, 1] 0.1347 0.0329 0.0011 917.2124 306.8386 0.9987 90.1083
action_probabilities.session[4, 1] 0.4338 0.0481 0.0013 1257.3936 425.5089 0.9992 123.5282
action_probabilities.session[1, 2] 0.1825 0.0422 0.0013 1023.3875 311.1505 1.0016 100.5391
action_probabilities.session[2, 2] 0.3462 0.0495 0.0018 763.2401 243.8135 1.0011 74.9818
action_probabilities.session[3, 2] 0.1244 0.0309 0.0011 716.8152 353.1196 0.9992 70.4210
action_probabilities.session[4, 2] 0.3469 0.0439 0.0014 1010.2876 364.0013 1.0002 99.2521
action_probabilities.session[1, 3] 0.1068 0.0309 0.0008 1174.0060 387.6724 0.9991 115.3361
action_probabilities.session[2, 3] 0.2904 0.0448 0.0016 906.6387 310.2592 1.0017 89.0695
action_probabilities.session[3, 3] 0.3553 0.0497 0.0015 1004.3480 320.3494 1.0011 98.6686
action_probabilities.session[4, 3] 0.2475 0.0415 0.0014 914.4503 199.4877 1.0015 89.8369
action_probabilities.session[1, 4] 0.2197 0.0400 0.0012 1087.2613 461.4608 0.9984 106.8142
action_probabilities.session[2, 4] 0.2702 0.0389 0.0011 1190.6484 399.3389 1.0031 116.9711
action_probabilities.session[3, 4] 0.3560 0.0442 0.0014 954.4264 398.1367 1.0127 93.7643
action_probabilities.session[4, 4] 0.1540 0.0357 0.0013 765.7528 332.4821 1.0088 75.2287
action_probabilities.session[1, 5] 0.3367 0.0479 0.0018 754.7404 392.5808 1.0032 74.1468
action_probabilities.session[2, 5] 0.2119 0.0401 0.0015 732.0202 371.1017 0.9989 71.9147
action_probabilities.session[3, 5] 0.2018 0.0387 0.0013 854.1399 448.9903 1.0071 83.9120
action_probabilities.session[4, 5] 0.2496 0.0429 0.0012 1217.3508 491.6032 1.0035 119.5943
action_probabilities.session[1, 6] 0.1661 0.0389 0.0012 1094.3367 353.1196 0.9981 107.5093
action_probabilities.session[2, 6] 0.2780 0.0418 0.0015 878.3651 359.4191 1.0017 86.2919
action_probabilities.session[3, 6] 0.2586 0.0412 0.0014 835.3493 362.4674 0.9995 82.0659
action_probabilities.session[4, 6] 0.2973 0.0430 0.0015 842.8835 279.0662 1.0086 82.8061
Quantiles
parameters 2.5% 25.0% 50.0% 75.0% 97.5%
Symbol Float64 Float64 Float64 Float64 Float64
action_probabilities.session[1, 1] 0.1438 0.1817 0.2080 0.2357 0.2913
action_probabilities.session[2, 1] 0.1510 0.1918 0.2181 0.2478 0.3041
action_probabilities.session[3, 1] 0.0756 0.1120 0.1335 0.1532 0.2039
action_probabilities.session[4, 1] 0.3435 0.4013 0.4308 0.4645 0.5285
action_probabilities.session[1, 2] 0.1096 0.1516 0.1788 0.2094 0.2715
action_probabilities.session[2, 2] 0.2547 0.3119 0.3442 0.3790 0.4447
action_probabilities.session[3, 2] 0.0711 0.1025 0.1231 0.1435 0.1894
action_probabilities.session[4, 2] 0.2514 0.3197 0.3484 0.3737 0.4335
action_probabilities.session[1, 3] 0.0561 0.0836 0.1036 0.1290 0.1697
action_probabilities.session[2, 3] 0.2077 0.2580 0.2860 0.3201 0.3786
action_probabilities.session[3, 3] 0.2627 0.3199 0.3528 0.3937 0.4485
action_probabilities.session[4, 3] 0.1709 0.2177 0.2488 0.2755 0.3284
action_probabilities.session[1, 4] 0.1518 0.1897 0.2170 0.2491 0.2983
action_probabilities.session[2, 4] 0.2032 0.2397 0.2704 0.2964 0.3418
action_probabilities.session[3, 4] 0.2719 0.3261 0.3559 0.3861 0.4372
action_probabilities.session[4, 4] 0.0927 0.1285 0.1505 0.1772 0.2315
action_probabilities.session[1, 5] 0.2460 0.3021 0.3348 0.3693 0.4360
action_probabilities.session[2, 5] 0.1421 0.1819 0.2084 0.2381 0.2925
action_probabilities.session[3, 5] 0.1338 0.1740 0.1990 0.2280 0.2869
action_probabilities.session[4, 5] 0.1727 0.2190 0.2479 0.2787 0.3344
action_probabilities.session[1, 6] 0.0979 0.1363 0.1650 0.1908 0.2490
action_probabilities.session[2, 6] 0.2017 0.2470 0.2760 0.3056 0.3611
action_probabilities.session[3, 6] 0.1873 0.2292 0.2560 0.2861 0.3401
action_probabilities.session[4, 6] 0.2148 0.2725 0.2977 0.3226 0.3893
We can also compare how well the PVL-Delta model fits the data compared to the simple random model.
#TODO: model comparisonThis page was generated using Literate.jl.