Procedural vs. Declarative Control of Decision Making

Cher's project, but in (faster) Python code. An overview of this project can be found on this BioarXiv preprint. Instead of implementing models and running simulations in ACT-R, the two subsystems of declarative memory and procedural memory are directly modeled as Python code.

There are two major advantages to this strategy:

• The models are much (100x) faster to run and estimate, and we can use Nelder-Mead instead of discrete grid search;
• log likelihood is now calculated on a trial by trial bases, which gives significantly more data.

Log-Likelihood Implementation

Declarative and procedural models will be compared using log-likelihood. Instead of using aggregate data, we will use Nathaniel Daw's (2011) original approach and calculate the probability that a given model generates the series of decisions made across two runs.

The likelihood of a model $m$ given that a participant $p$ has made choice $c_{i,r}$ at the $i$-th trial of run $r$ is simply the probability of that choice being made by model $m$:

$\mathcal{L}(m|c_{p,r,i}) = P(c_{p,r,i}|m)$

The likelihood of a model $m$ given all the choices made by participant $p$ across all runs is simply the likelihood of a model making all of the participant's choices:

$\mathcal{L}(m|p) = \mathcal{L}(m|c_{p,1,1}, \dots, c_{p,r,i})$

In turn, this is just the product of the probability that the model would make every choice in the sequence:

$\mathcal{L}(m|p) = \prod_r \prod_i P(c_{p,r,i}|m)$

Finally, as it is convenient, we will take the log-likelihood, thus transforming all products of probabilities into sums of log-probabilities:

$\log \mathcal{L} = \sum_r \sum_i \log P(c_{p,r,i|m})$