MAC1975

The mathematics behind MAC1975

A grand departure from global error term models such as RW1972 (Rescorla & Wagner, 1972), the MAC1975 model (Mackintosh, 1975) uses local error terms and changes stimulus associability ( $\alpha$ ) via an error comparison mechanism that promotes learning about uncertain stimuli:

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:

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

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

3 - Learning to attend

The parameter $\alpha_i$ changes as a function of learning, proportionally to the difference between the absolute errors conveyed by $i$ and all the other predictors¹, via:

$\tag{Eq.3} \Delta \alpha_{i} = x_i\theta_i \sum_{j}^{K}\gamma_j(|\lambda_j - \sum_{k \ne i}^{K}v_{k,j}|-|\lambda_j - v_{i,j}|)$ where $\theta_i$ is an attentional learning rate parameter for stimulus $i$ (usually fixed across all stimuli). Although Mackintosh (1975) did not extend their model to account for the predictive power of within-compound associations, the implementation of the model in this package does. This can sometimes result in unexpected behavior, and as such, Eq. 3 above includes an extra parameter $\gamma_j$ (defaulting to 1/K) that denotes whether the expectation of stimulus $j$ contributes to attentional learning. As such, the user can set these parameters manually to reflect the contribution of the different experimental stimuli. For example, in a simple “AB>(US)” design, setting $\gamma_{US}$ = 1 and $\gamma_{A} = \gamma_{B} = 0$ leads to the behavior of the original model.

4 - Generating responses

There is no specification of response-generating mechanisms in MAC1975. 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:

$\tag{Eq.4} r_j = e_j$

References

Le Pelley, M. E., Mitchell, C. J., Beesley, T., George, D. N., & Wills, A. J. (2016). Attention and associative learning in humans: An integrative review. Psychological Bulletin, 142, 1111–1140. https://doi.org/10.1037/bul0000064

Mackintosh, N. J. (1975). A theory of attention: Variations in the associability of stimuli with reinforcement. Psychological Review, 82, 276–298. https://doi.org/10.1037/h0076778