AdaptiveDesignOptimization

This package is a grid-based approach for performing Bayesian adaptive design optimization. After each observation, the optimizer chooses an experimental design that maximizes mutual information between model parameters and design parameters. In so doing, the optimizer selects designs that minimize the variance in the posterior distribution of model parameters.

Example

In this example, we will optimize a decision making experiment for the model called Transfer of Attention Exchange (TAX; Birnbaum, 2008). Additional examples can be found in the folder titled Examples.

julia using AdaptiveDesignOptimization, Random, UtilityModels, Distributions include("TAX_Model.jl") Random.seed!(25974)

Define Model

The model object contains a log likelihood function and optional prior distributions. Unless an array of distribution objects is passed as prior, uniform distributions will be used by default. Arguments in the log likelihood function must be ordered as follows:

• loglike(modelparameters..., designparameters..., data..., args...; kwargs...)

args... and kwargs... are optional arguments that may be preloaded through the Model constructor. Enter

julia ? Model for additional details. The likelihood function for the TAX model (see TAX_Model.jl) is defined as:

julia function loglike(δ, β, γ, θ, pa, va, pb, vb, data) eua,eub = expected_utilities(δ, β, γ, θ, pa, va, pb, vb) p = choice_prob(eua, eub, θ) p = max(p, eps()) return logpdf(Bernoulli(p), data) end

The model object is contructed with default uniform prior distributions.

```julia

model with default uniform prior

model = Model(;loglike) ```

Define Parameters

Define a NamedTuple of parameter value ranges. Note that the parameters listed in the same order that they appear in loglike.

julia parm_list = ( δ = range(-2, 2, length=10), β = range(.5, 1.5, length=10), γ = range(.5, 1.2, length=10), θ = range(.5, 3, length=10) )

The experiment will consist of two gambles with three outcomes each. The number of dimensions in the design space is large (2X2X3X3 = 36). In this case, we will sample random gambles and select a subset of 100 with high distributional overlap. In this case, the design_list will be a Tuple of design names and design values.

```julia

outcome distribution

dist = Normal(0,10) nvals = 3 nchoices = 2 designvals = map(x->randomdesign(dist, nvals, nchoices), 1:1000)

select gambles with overlapping distributions

filter!(x->abszscore(x) ≤ .4, designvals) designnames = (:p1,:v1,:p2,:v2) designlist = (designnames,designvals[1:100]) ```

Lastly, we will define a list for the data. The value true indicates gamble A was choosen and false indicates gamble B was chosen.

julia data_list = (choice=[true, false],)

Optimize Exeriment

In the following code blocks, we will run an optimized experiment and a random experiment. The first step is to generate the optimizer with the contructor Optimizer. Next, we specify true parameters for generating data from the model and initialize a DataFrame to collect the results on each simulated trial. In the experiment loop, data are generated with simulate. The data are passed to update in order to optimize the experiment for the next trial. Finally, the mean and standard deviation are added to the DataFrame for each parameter. A similar process is used to perform the random experiment.

```julia using DataFrames trueparms = (δ=-1.0, β=1.0, γ=.7, θ=1.5) ntrials = 100 optimizer = Optimizer(;designlist, parmlist, datalist, model) design = optimizer.bestdesign df = DataFrame(design=Symbol[], trial=Int[], meanδ=Float64[], meanβ=Float64[], meanγ=Float64[], meanθ=Float64[], stdδ=Float64[], stdβ=Float64[], stdγ=Float64[], stdθ=Float64[]) newdata = [:optimal, 0, meanpost(optimizer)..., stdpost(optimizer)...] push!(df, newdata)

for trial in 1:ntrials data = simulate(trueparms..., design...) design = update!(optimizer, data) newdata = [:optimal, trial, meanpost(optimizer)..., stdpost(optimizer)...] push!(df, newdata) end ```

Random Experiment

```julia randomizer = Optimizer(;designlist, parmlist, datalist, model, designtype=Randomize); design = randomizer.bestdesign newdata = [:random, 0, meanpost(randomizer)..., stdpost(randomizer)...] push!(df, new_data)

for trial in 1:ntrials data = simulate(trueparms..., design...) design = update!(randomizer, data) newdata = [:random, trial, meanpost(randomizer)..., stdpost(randomizer)...] push!(df, newdata) end ```

Results

As expected, in the figure below, the posterior standard deviation of δ is smaller for the optimal experiment compared to the random experiment.

julia using StatsPlots @df df plot(:trial, :std_δ, xlabel="trial", ylabel="σ of δ", grid=false, group=:design, linewidth=2, ylims=(0,1.5), size=(600,400))

References

• Birnbaum, M. H., & Chavez, A. (1997). Tests of theories of decision making: Violations of branch independence and distribution independence. Organizational Behavior and human decision Processes, 71(2), 161-194. Birnbaum, M. H. (2008). New paradoxes of risky decision making. Psychological review, 115(2), 463.

• Myung, J. I., Cavagnaro, D. R., and Pitt, M. A. (2013). A tutorial on adaptive design optimization. Journal of Mathematical Psychology, 57, 53–67.

• Yang, J., Pitt, M. A., Ahn, W., & Myung, J. I. (2020). ADOpy: A Python Package for Adaptive Design Optimization. Behavior Research Methods, 1--24. https://doi.org/10.3758/s13428-020-01386-4