The theory stems from Roger Shepard, who in 1987 published a paper in
*Science* on generalisation (Toward a universal law of generalization
for psychological science, *Science*, **237**, 1317-1323). Applied
to spatial generalisation, and in my words, the argument runs as follows.

You have found something good to eat in a container at some location, called S+ in trade jargon. A while later, you wander in the region again in search of sustenance, and come across another container. It looks and smells like the one you've encountered before, the one from which you got the goodies, but this one is at a different location x. Do you bet that this container also contains food? In formulating the problem this way, it is assumed that you can tell the two locations apart. What you are concerned with is the consequence of interest (food in container): do the two places have the same consequence?

Recasting the problem, you have learned that a container at S+ has food. The world being what it is, you assume that some 'hot region' around S+ has the same consequence of interest. S+ is in the hot region. But is x? The problem now becomes: given that S+ is in the hot region, what is the probability that x is also in the hot region? Of course, this formulation applies to any dimension on which S+ and x might differ, not just the spatial location.

The answer is surprisingly simple. Under a wide range of assumptions about the hot region, Shepard found that the probability is given by an exponential function:

**y = exp (-kx)**

y is the probability of x being in the hot region

k is a scaling parameter

x is the appropriately scaled distance of x from S+ (I will unpack the scaling business soon)

Then the argument takes a functional form. This exponential function characterises the way the world is. Animals' brains should have evolved to reflect the structure of the world. Hence, in generalisation, animals should follow the exponential function.

experiments on bees

I tested spatial generalisation on bees using these paradigms:

Spatial location was defined with respect to a landmark, a bottle on a table. Two series of experiments tested for generalisation over different distances from the landmark (A) or different directions from the landmark (B). Bees were trained in each experiment to find food (sugar water) at one place (S+). To make sure that the place was defined by the landmark, the setup was moved around the table from trial to trial. The lab has plenty of cues to let the bee tell which direction is which. The sugar water was in a cap, the cap sat on a piece of blue cardboard paper (so that the container was very visible). A strip of yellow cardboard running the length of the table (A) or a ring of yellow cardboard (B) gave us defined regions to count the presence of the bee on video. The demarcation lines on the yellow cardboard were drawn in pencil; the bees would not be able to see them.

The procedure created the problem scenario above for the bees. They got trained to find food at one place. Then they got asked to 'bet' on a bunch of places, including the S+, on unrewarded tests. On a test, tap water (yuk!) replaced sugar water in a cap. The time a bee searched over the area of the proferred tap water measured how much it betted on the location. The locations used on tests are indicated by the regions. Durations of search were made relative to amount of searching at S+ to derive the y values for testing Shepard's law.

Crucial in testing Shepard's law is scaling the x axis. The scales, for distance and direction from a landmark, have to reflect how the bees scale distance and direction. We have theory on how bees measure distance from a landmark: they use retinal size and motion parallax.

Retinal size means the angle subtended by a landmark on the eye. Motion parallax means how much an object appears to move when the perceiver is moving: the closer the object, the more it appears to move. I derived scales based on both retinal size and motion parallax.

On these scales, but not on a linear scale, the generalisation gradients follow Shepard's law:

Corresponding datapoints in A do have the same value. For example, the second point from the left on linear, euclidean and city-block scales have the same value. They appear different because of a visual illusion. Euclidean and city-block scales are different ways of combining the width and height dimensions of the landmark. It doesn't matter how width and height are weighted; the fits were all good.

Given that scales based on retinal size and parallax are similar, it doesn't matter how the bee combines the two dimensions. The resulting exponential fit would still be good.

For generalisation over directions, the data delivered the appropriate scale for testing Shepard's law. All the gradients obtained, with S+ at different locations, looked similar. The only way that could happen is if the scale were uniformly circular. That is to say, directional difference increases for the bee from 0 to 180 degrees as x's direction to the landmark differs from S+'s, and the change is uniform. It means that the x axis, from 0 to 180 degrees, can be left untransformed. Measured this way, the generalisation gradients (all averaged into one) supported Shepard's law again in having an exponential form:

And just by the way, spatial generalisation in humans also obey Shepard's law:

For the human story see story on spatial cognition in humans.

a universal law

Few laws in psychology are claimed to be universal. Shepard's law aspires to the claim of universality because it is based on good functional arguments. The structure of the world is analysed with regard to generalisation. Then the argument is that animals should have evolved to generalise in a way that reflects this structure. The law had been confirmed only in humans and pigeons. Finding support in the invertebrate honeybee adds confidence to the generality of the law, and suggests that the structure of the world might have shaped the evolution of cognition in diverse animals.

back to Ken's home page