# 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).↩︎