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 = 1087.99 seconds
Compute duration = 1607.0 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] -0.0772 0.1360 0.0179 487.3965 49.2938 2.2348 0.3033
learning_rate.β[2] 0.1384 0.2277 0.0677 27.1742 88.4432 2.2461 0.0169
learning_rate.β[3] -0.2957 0.3313 0.1921 3.2763 2.1370 2.2399 0.0020
learning_rate.ranef_1.σ[1] 0.7915 0.2666 0.0303 89.3771 92.1977 2.2334 0.0556
learning_rate.ranef_1.r[1] -2.5884 1.2142 0.3600 89.8795 129.6627 2.2395 0.0559
learning_rate.ranef_1.r[2] 0.3158 0.5824 0.0952 253.2225 63.9834 2.2538 0.1576
learning_rate.ranef_1.r[3] -0.4122 0.5462 0.1593 9.6980 46.1141 2.2327 0.0060
learning_rate.ranef_1.r[4] 0.0266 0.8071 0.3003 8.4308 112.9638 2.2336 0.0052
learning_rate.ranef_1.r[5] -0.5968 0.8017 0.1527 14.9271 113.9311 2.2606 0.0093
learning_rate.ranef_1.r[6] 0.5159 0.9456 0.4666 4.0995 147.9296 2.2560 0.0026
reward_sensitivity.β[1] -0.3721 0.1417 0.0062 454.0315 171.0156 2.2324 0.2825
reward_sensitivity.β[2] -0.2280 0.1980 0.0084 784.1333 186.5376 2.2410 0.4879
reward_sensitivity.β[3] -0.1269 0.2567 0.1102 7.1871 73.9149 2.2373 0.0045
reward_sensitivity.ranef_1.σ[1] 0.3080 0.3458 0.2031 2.5866 2.0325 2.2411 0.0016
reward_sensitivity.ranef_1.r[1] -0.0281 0.5865 0.0434 133.8821 64.2951 2.2494 0.0833
reward_sensitivity.ranef_1.r[2] -0.1471 0.4197 0.1410 12.6425 102.8753 2.2323 0.0079
reward_sensitivity.ranef_1.r[3] -0.3805 0.6003 0.2973 4.3431 190.7738 2.2324 0.0027
reward_sensitivity.ranef_1.r[4] -0.2724 0.5417 0.1801 15.5057 95.5693 2.2349 0.0096
reward_sensitivity.ranef_1.r[5] -0.3043 0.5930 0.2429 7.0046 169.8609 2.2336 0.0044
reward_sensitivity.ranef_1.r[6] -0.3014 0.5066 0.1829 13.1553 68.7551 2.2457 0.0082
loss_aversion.β[1] -0.2295 0.1325 0.0240 93.2163 62.4714 2.2355 0.0580
loss_aversion.β[2] -0.5151 0.2477 0.1015 8.8090 128.3562 2.2343 0.0055
loss_aversion.β[3] -0.1075 0.2213 0.0068 1021.8084 190.9917 2.2451 0.6358
loss_aversion.ranef_1.σ[1] 0.3433 0.1558 0.0509 13.5332 48.3537 2.2489 0.0084
loss_aversion.ranef_1.r[1] 0.0845 0.2773 0.1465 3.4900 55.3386 2.2434 0.0022
loss_aversion.ranef_1.r[2] -0.4048 0.3150 0.0720 13.5790 241.7611 2.2362 0.0084
loss_aversion.ranef_1.r[3] -0.0612 0.2270 0.0280 21.3886 43.6145 2.2566 0.0133
loss_aversion.ranef_1.r[4] 0.1964 0.3587 0.2081 3.0690 213.5619 2.2427 0.0019
loss_aversion.ranef_1.r[5] -0.3142 0.3231 0.1390 4.8800 2.3495 2.2407 0.0030
loss_aversion.ranef_1.r[6] -0.0454 0.2132 0.0086 647.8554 40.0776 2.2729 0.4031
action_noise.β[1] 0.4864 0.2020 0.0983 5.1146 182.6105 2.2332 0.0032
action_noise.β[2] 0.3061 0.2145 0.0086 833.6584 58.4463 2.2369 0.5188
action_noise.β[3] 0.0251 0.3679 0.2245 2.8257 2.0637 2.2348 0.0018
action_noise.ranef_1.σ[1] 0.8973 0.2060 0.0764 12.6386 65.1240 2.2329 0.0079
action_noise.ranef_1.r[1] 0.0396 0.6841 0.3466 4.3725 2.2788 2.2326 0.0027
action_noise.ranef_1.r[2] 0.3654 0.3704 0.0147 680.1321 75.5211 2.2518 0.4232
action_noise.ranef_1.r[3] 1.0046 0.4627 0.1700 12.5407 70.6206 2.2328 0.0078
action_noise.ranef_1.r[4] 2.0815 0.9658 0.6018 2.7899 165.1563 2.2351 0.0017
action_noise.ranef_1.r[5] 1.3146 0.6855 0.2200 14.1099 127.2067 2.2324 0.0088
action_noise.ranef_1.r[6] 1.7732 0.9773 0.6145 3.0343 329.0925 2.2337 0.0019
Quantiles
parameters 2.5% 25.0% 50.0% 75.0% 97.5%
Symbol Float64 Float64 Float64 Float64 Float64
learning_rate.β[1] -0.4074 -0.1173 -0.0434 -0.0434 0.1852
learning_rate.β[2] -0.4053 0.0263 0.2354 0.2354 0.4982
learning_rate.β[3] -0.5593 -0.5593 -0.5593 -0.0211 0.4548
learning_rate.ranef_1.σ[1] 0.1538 0.8148 0.8305 0.8305 1.3092
learning_rate.ranef_1.r[1] -4.2693 -3.1422 -3.1422 -2.2011 0.0920
learning_rate.ranef_1.r[2] -0.6299 0.1503 0.1503 0.3207 1.8329
learning_rate.ranef_1.r[3] -1.3694 -0.6314 -0.6314 -0.1268 0.8657
learning_rate.ranef_1.r[4] -2.3442 -0.1559 0.4395 0.4395 0.9060
learning_rate.ranef_1.r[5] -2.4989 -0.8487 -0.8487 -0.1064 1.1653
learning_rate.ranef_1.r[6] -1.7711 -0.0519 1.1499 1.1499 1.2904
reward_sensitivity.β[1] -0.6891 -0.3840 -0.3840 -0.3590 -0.0410
reward_sensitivity.β[2] -0.7182 -0.2274 -0.2036 -0.2036 0.1915
reward_sensitivity.β[3] -0.7447 -0.2807 0.0229 0.0229 0.2032
reward_sensitivity.ranef_1.σ[1] 0.0270 0.0270 0.0452 0.5964 1.0571
reward_sensitivity.ranef_1.r[1] -1.2953 -0.1284 0.0141 0.0141 1.4797
reward_sensitivity.ranef_1.r[2] -1.3006 -0.2604 0.0455 0.0455 0.3986
reward_sensitivity.ranef_1.r[3] -1.8838 -0.7646 0.0282 0.0282 0.0696
reward_sensitivity.ranef_1.r[4] -1.7846 -0.3824 -0.0145 -0.0145 0.3254
reward_sensitivity.ranef_1.r[5] -1.9911 -0.4873 0.0264 0.0264 0.2105
reward_sensitivity.ranef_1.r[6] -1.7051 -0.4443 -0.0452 -0.0452 0.1989
loss_aversion.β[1] -0.5610 -0.2702 -0.1925 -0.1925 0.0337
loss_aversion.β[2] -0.8350 -0.6539 -0.6539 -0.3910 0.1362
loss_aversion.β[3] -0.5636 -0.1210 -0.1210 -0.1056 0.4021
loss_aversion.ranef_1.σ[1] 0.0343 0.2382 0.4046 0.4046 0.7093
loss_aversion.ranef_1.r[1] -0.6530 -0.0540 0.2837 0.2837 0.2837
loss_aversion.ranef_1.r[2] -1.2822 -0.5041 -0.5041 -0.1663 0.1338
loss_aversion.ranef_1.r[3] -0.5646 -0.1184 -0.1184 -0.0061 0.5145
loss_aversion.ranef_1.r[4] -0.6620 -0.0378 0.4822 0.4822 0.4822
loss_aversion.ranef_1.r[5] -0.7576 -0.5087 -0.5087 -0.0540 0.3040
loss_aversion.ranef_1.r[6] -0.6355 -0.0375 -0.0375 -0.0145 0.3657
action_noise.β[1] 0.0141 0.3540 0.6209 0.6209 0.7048
action_noise.β[2] -0.2105 0.3098 0.3134 0.3134 0.7585
action_noise.β[3] -0.2856 -0.2856 -0.2856 0.3352 0.8054
action_noise.ranef_1.σ[1] 0.3496 0.7904 1.0078 1.0078 1.1864
action_noise.ranef_1.r[1] -0.5347 -0.4402 -0.4402 0.5198 1.5679
action_noise.ranef_1.r[2] -0.3802 0.3142 0.3142 0.3987 1.3214
action_noise.ranef_1.r[3] -0.0569 0.7098 1.2434 1.2434 1.6849
action_noise.ranef_1.r[4] 0.1318 1.2417 2.9216 2.9216 2.9216
action_noise.ranef_1.r[5] 0.2197 0.9941 0.9941 1.6238 2.9385
action_noise.ranef_1.r[6] -0.0495 0.8779 2.6332 2.6332 2.6332
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 1.19806 0.0397095 1.09555 0.408584
2 │ 117 6.45123 0.337413 0.732784 0.411977
3 │ 105 3.48517 0.584723 0.259128 0.367704
4 │ 136 6.87877 0.341462 0.257926 0.363276
5 │ 130 25.9703 0.459288 1.18377 0.407202
6 │ 149 19.4623 0.633474 0.703993 0.399811We 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 0.196366
3 │ 103 2 0.0 0.0 0.40012
4 │ 103 3 0.0 0.0 0.272774
5 │ 103 4 0.0 0.0 0.272774 ⋯
6 │ 103 5 0.0 0.0 0.466106
7 │ 103 6 0.260652 0.0 0.466106
8 │ 103 7 0.531113 0.0 0.466106
⋮ │ ⋮ ⋮ ⋮ ⋮ ⋮ ⋱
600 │ 149 94 -3.41555 6.66582 4.15543 ⋯
601 │ 149 95 -3.41555 6.66582 3.39351
602 │ 149 96 -3.41555 6.66582 3.39351
603 │ 149 97 -3.41555 6.66582 3.39351
604 │ 149 98 -3.41555 6.66582 2.83433 ⋯
605 │ 149 99 -3.41555 6.73865 2.83433
606 │ 149 100 2.8967 6.73865 2.83433
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 = 15.38 seconds
Compute duration = 15.38 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] 0.2127 0.0437 0.0020 499.7635 264.2679 1.0202 32.4838
action_probabilities.session[2, 1] 0.2225 0.0420 0.0018 566.9277 258.8146 1.0044 36.8494
action_probabilities.session[3, 1] 0.1339 0.0345 0.0015 530.6587 323.6614 0.9988 34.4920
action_probabilities.session[4, 1] 0.4309 0.0498 0.0022 509.9878 362.1328 1.0011 33.1484
action_probabilities.session[1, 2] 0.1076 0.0303 0.0011 836.8070 318.1140 1.0023 54.3911
action_probabilities.session[2, 2] 0.2874 0.0445 0.0017 713.0993 441.4632 1.0126 46.3503
action_probabilities.session[3, 2] 0.3563 0.0456 0.0018 612.0289 408.3394 0.9993 39.7809
action_probabilities.session[4, 2] 0.2487 0.0410 0.0018 500.2803 400.5173 1.0012 32.5174
action_probabilities.session[1, 3] 0.1825 0.0356 0.0015 534.1960 408.3394 1.0001 34.7219
action_probabilities.session[2, 3] 0.3469 0.0445 0.0017 646.2085 366.2060 0.9995 42.0025
action_probabilities.session[3, 3] 0.1239 0.0335 0.0013 610.8795 283.5594 0.9991 39.7062
action_probabilities.session[4, 3] 0.3466 0.0477 0.0021 549.8255 274.6839 0.9998 35.7378
action_probabilities.session[1, 4] 0.3339 0.0496 0.0023 447.1155 285.2636 0.9992 29.0618
action_probabilities.session[2, 4] 0.2102 0.0395 0.0017 535.5974 297.4440 1.0018 34.8130
action_probabilities.session[3, 4] 0.2034 0.0425 0.0018 607.4729 377.4493 1.0036 39.4848
action_probabilities.session[4, 4] 0.2526 0.0410 0.0017 597.4313 229.7026 0.9995 38.8321
action_probabilities.session[1, 5] 0.2181 0.0428 0.0018 525.8366 298.1951 1.0073 34.1785
action_probabilities.session[2, 5] 0.2728 0.0449 0.0016 766.2143 400.7875 0.9996 49.8027
action_probabilities.session[3, 5] 0.3544 0.0502 0.0023 474.3856 470.5002 1.0031 30.8343
action_probabilities.session[4, 5] 0.1547 0.0371 0.0016 543.4503 308.5591 1.0010 35.3234
action_probabilities.session[1, 6] 0.1613 0.0367 0.0015 615.0265 367.4394 1.0003 39.9757
action_probabilities.session[2, 6] 0.2803 0.0420 0.0019 510.9267 425.4347 0.9988 33.2094
action_probabilities.session[3, 6] 0.2588 0.0431 0.0018 549.6302 414.4278 1.0068 35.7251
action_probabilities.session[4, 6] 0.2997 0.0445 0.0019 521.1288 314.9595 0.9987 33.8725
Quantiles
parameters 2.5% 25.0% 50.0% 75.0% 97.5%
Symbol Float64 Float64 Float64 Float64 Float64
action_probabilities.session[1, 1] 0.1334 0.1822 0.2070 0.2417 0.3036
action_probabilities.session[2, 1] 0.1477 0.1956 0.2200 0.2502 0.3105
action_probabilities.session[3, 1] 0.0732 0.1100 0.1313 0.1524 0.2107
action_probabilities.session[4, 1] 0.3273 0.3993 0.4334 0.4654 0.5225
action_probabilities.session[1, 2] 0.0558 0.0863 0.1053 0.1270 0.1702
action_probabilities.session[2, 2] 0.2003 0.2574 0.2842 0.3162 0.3784
action_probabilities.session[3, 2] 0.2728 0.3246 0.3531 0.3854 0.4552
action_probabilities.session[4, 2] 0.1739 0.2183 0.2459 0.2778 0.3276
action_probabilities.session[1, 3] 0.1192 0.1583 0.1796 0.2055 0.2582
action_probabilities.session[2, 3] 0.2645 0.3153 0.3456 0.3796 0.4290
action_probabilities.session[3, 3] 0.0640 0.0996 0.1214 0.1425 0.1949
action_probabilities.session[4, 3] 0.2640 0.3130 0.3428 0.3787 0.4435
action_probabilities.session[1, 4] 0.2437 0.2983 0.3327 0.3659 0.4304
action_probabilities.session[2, 4] 0.1370 0.1841 0.2074 0.2353 0.2901
action_probabilities.session[3, 4] 0.1253 0.1757 0.1997 0.2296 0.2974
action_probabilities.session[4, 4] 0.1801 0.2235 0.2523 0.2781 0.3388
action_probabilities.session[1, 5] 0.1414 0.1899 0.2159 0.2452 0.3125
action_probabilities.session[2, 5] 0.1881 0.2389 0.2732 0.3035 0.3591
action_probabilities.session[3, 5] 0.2518 0.3214 0.3543 0.3849 0.4532
action_probabilities.session[4, 5] 0.0890 0.1303 0.1534 0.1766 0.2413
action_probabilities.session[1, 6] 0.0995 0.1350 0.1584 0.1858 0.2361
action_probabilities.session[2, 6] 0.2021 0.2494 0.2794 0.3071 0.3661
action_probabilities.session[3, 6] 0.1818 0.2295 0.2581 0.2866 0.3446
action_probabilities.session[4, 6] 0.2124 0.2693 0.2996 0.3295 0.3903
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.