# 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.

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