RW1972"
In calmr: Canonical Associative Learning Models and their Representations

The mathematics behind RW1972

The most influential associative learning model, RW1972 [@rescorla_theory_1972], learns from global error and posits no changes in stimulus associability.

1 - Generating expectations

Let $v_{k,j}$ denote the associative strength from stimulus $k$ to stimulus $j$. On any given trial, the expectation of stimulus $j$, $e_j$, is given by:

$$ \tag{Eq.1} e_j = \sum_{k}^{K}x_k v_{k,j} $$

$x_k$ denotes the presence (1) or absence (0) of stimulus $k$, and the set $K$ represents all stimuli in the design.

2 - Learning associations

Changes to the association from stimulus $i$ to $j$, $v_{i,j}$, are given by:

$$ \tag{Eq.2} \Delta v_{i,j} = \alpha_i \beta_j (\lambda_j - e_j) $$

where $\alpha_i$ is the associability of stimulus $i$, $\beta_j$ is a learning rate parameter determined by the properties of $j$^note1, and $\lambda_j$ is a the maximum association strength supported by $j$ (the asymptote).

3 - Generating responses

There is no specification of response-generating mechanisms in RW1972. However, the simplest response function that can be adopted is the identity function on stimulus expectations. If so, the responses reflecting the nature of $j$, $r_j$, are given by:

$$ \tag{Eq.3} r_j = e_j $$

The implementation of RW1972 allows the specification of independent $\beta$ values for present and absent stimuli (beta_on and beta_off, respectively).