Thesis Proposal

version 1 10/09/04

Praveen Paritosh

Title: The sense of the quantitative

Abstract: Quantities are ubiquitous and an important part of our understanding about the world, e.g., cars have engine horsepower, size, mileage, price; countries have GDP, population, area; birds have wingspan, weight, surface area, and so on. Current models of similarity, retrieval and generalization do not take quantities.

This thesis presents a theory of quantity – cognitively plausible representations and principles for generating those representations.The theory is implemented in a computational model, CARVE, that builds symbolic representations of quantity by exposure to examples.

Details: Analysis of Range Frequency Theory

Ri' = (Si – Smin,i) / (Smax,i – Smin,i)

Fi' = (Ranki,i – 1) / (i – 1)

When doing incremental category building, there are some issues. Smin, Smax, and Ranki are global properties of a stimulus which can be evaluated (efficiently) for that stimulus when all the data is available. But thats an odd assumption for CARVE to make, that it can make inferences only when all the data is available. Plus, what happens when new data comes along – how does CARVE classify the new points, and does it change any of the earlier representations?

Design and architecture for CARVE

CARVE

starts with dimensions, or cases?

Adds the symbolic representations generated to the KB?

What dimensions to generalize on? [Alignable]

Theoretical questions:

How does quantity effect similarity judgments? Even though I am not directly setting myself for answering that questions, I am indirectly claiming something along those lines when I say that quantity is missing in current theories of similarity and retrieval.

My current answer is that the qualitative representations account for the quantity effects. But is that all? Is it qualitative all the way down? Sexy if I can claim/prove that. Or is there a part of quantitative similarity that will still remain?

How is one ever going to test such a claim?

In considering the effects of quantity on similarity, we can imagine two kinds of effects – for example, consider the example of a quantity q with a limit point p. The value of p is 10 units. And lets say, in four cases the quantity takes on values q1 (11), q2(9), q3(14), and q4(15).

Qualitative similarity: that part of the similarity because of this quantity thats captured by saying (> q1 p), (< q2 p) and (> q3 p). This makes q1 and q3 more similar than q1 and q2. This is based on the result by Goldman 1972.

sim(q1,q3) < sim (q1, q2) --1

Quantitative similarity: In the above case for q4, we also get (> q4 p). So according to purely qualitative sense of similarity

sim(q1, q3) = sim (q1, q4) -- 2

However, we know that numerically, q1 is closer to q3 than q4. So, if there is a quantitative similarity effect, we will find that people think that

sim(q1, q3) > sim (q1, q4) -- 3

It seems (almost obvious) that if the difference between q3 and q4 is greater than the just noticeable difference than people should notice that and agree with eqn 3. But wait, is that obvious. With the same subject it seems eqn 3 is right. What about across subjects? The similarity ratings across subjects might be closer to 2 than 3. With the same subject, many things are happening. They are possibly creating finer distinctions and newer landmarks based on more anchor points (there is a Cech and Shoben study that supports this idea, pull the cite out). This is the strong claim – eqn 1 and 2 accurately capture the role of quantity in similarity and 3, in fact, is not true. The trick is to do the 2/3 experiment across subjects and argue that 3 happens just because the subject is introducing newer limit points to the mix, or the dynamic nature of the notion of similarity. As methodology goes, what does comparing similarity across subjects means?

Analogical Estimator

Strategy library and kinds of strategies

2004-10-09T16:49:30