RW1972

Victor Navarro

The mathematics behind RW1972

The most influential associative learning model, RW1972 (Rescorla & Wagner, 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 $j$¹, 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:

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

References

Rescorla, R. A., & Wagner, A. R. (1972). A theory of Pavlovian conditioning: Variations in the effectiveness of reinforcement and nonreinforcement. In A. H. Black & W. F. Prokasy (Eds.), Classical conditioning II: Current research and theory. (pp. 64–69). Appleton-Century-Crofts.