Mathematical Psychology
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SIMPLE Model

The SIMPLE model (Scale-Invariant Memory, Perception, and Learning) explains memory retrieval as a discrimination problem in which items compete based on their temporal distinctiveness, with no separate short-term memory store required.

P(i) = [1 / Σⱼ e^(−c·|log(Tᵢ)−log(Tⱼ)|)] / [1/Σⱼ e^(−D(i,j)) + θ]

SIMPLE, developed by Gordon Brown, Ian Neath, and Nick Chater (2007), is a model that reconceptualizes memory retrieval as a problem of temporal discrimination. Rather than positing separate short-term and long-term memory stores, SIMPLE proposes that all memory phenomena arise from the distinctiveness of items along a temporal (or other) dimension, following principles borrowed from the psychophysics of perception.

Temporal Distinctiveness

In SIMPLE, each memory is located at a point along a temporal dimension (its time of encoding). The psychological distance between two memories is computed using a logarithmic transformation of their temporal distances from the present moment. Items that are temporally isolated from their neighbors are more distinctive and therefore more retrievable:

Psychological Distance D(i,j) = c · |log(Tᵢ) − log(Tⱼ)|

where Tᵢ and Tⱼ are the temporal distances of items i and j from the time of retrieval, and c is a sensitivity parameter. The log transform ensures that the model is scale-invariant: multiplying all temporal distances by a constant does not change the relative distinctiveness of items.

Discriminability and Retrieval

The probability of correctly retrieving item i is determined by its discriminability from all other items in the relevant memory set. Discriminability is computed using a similarity-based choice rule analogous to Shepard's (1987) universal law of generalization:

Retrieval Probability P(i) = [1 / Σⱼ e^(−D(i,j))] / [1 / Σⱼ e^(−D(i,j)) + θ]

where the sum is over all items in the memory set and θ is a threshold parameter representing the possibility of recall failure. Items that are very similar to many neighbors are hard to discriminate and are therefore poorly recalled.

Accounting for Classic Phenomena

SIMPLE explains recency effects as a natural consequence of temporal distinctiveness: recent items are spaced further apart on the log-time scale and are therefore more discriminable. The ratio rule (Bjork & Whitten, 1974) follows directly from the scale-invariant property: what matters is the ratio of inter-item spacing to retention interval, not the absolute durations. Primacy effects arise when a distinctiveness bonus for the first item or an encoding advantage is included.

Beyond Time: Multidimensional SIMPLE

While the basic model uses only a temporal dimension, SIMPLE can incorporate additional dimensions such as positional, semantic, or physical similarity. When multiple dimensions are included, the psychological distance is computed as a weighted Euclidean distance across all relevant dimensions. This multidimensional version accounts for phenomena in absolute identification, relative judgment, and category learning, unifying memory and perception under common principles.

Memory as Discrimination

SIMPLE's core insight is that memory retrieval and perceptual discrimination obey the same mathematical principles. Just as you have difficulty identifying a tone that is surrounded by similar tones, you have difficulty retrieving a memory that is surrounded by similar memories. This elegant unification dissolves the distinction between perception and memory at the mathematical level.

Related Topics

References

  1. Brown, G. D. A., Neath, I., & Chater, N. (2007). A temporal ratio model of memory. Psychological Review, 114, 539-576.
  2. Neath, I., & Brown, G. D. A. (2006). SIMPLE: Further applications of a local distinctiveness model of memory. In B. H. Ross (Ed.), Psychology of Learning and Motivation (Vol. 46).

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