Back of the Envelope (BoTE) Reasoning

 

 

A type of qualitative reasoning, where the goal is to come up with a qualitative (in the sense of being approximate, in the ballpark), yet numeric/quantified value(s) of the parameters in question.

 

One of the central claims of qualitative simulation is that it does not miss any (possible) behaviors of the system, because it does not (implicitly) make any assumption that quantification might warrant. The main sources of “qualitativeness” in a qualitative model are – 1) quantity value (represented as quantity spaces, landmark values and intervals, +/-/0, etc), and 2) functional dependence (modeled as monotonic functions, influences, etc). At no stage does one make an assumption that is not known, and the ambiguity/uncertainty is propagated to come up with a (usually) large envisionment.

 

As opposed to the above, BoTE usually is about narrowing down to one (most strongly) possible behavior, by making the most reasonable assumptions at all stages. Reasonable assumption on quantity value would mean assuming the most typical value of that quantity in that (or a similar) scenario. This is the first part of the problem that I am now working on, the parameter estimator. The parameter estimator will be able to churn out reasonable values for parameters in that domain. The domain taken here is transportation. When asked to come up with a value for something (the dimensions of  Ford Explorer, for example), the strategies that one might use are –

  1. Recall if we directly know the value of the parameter for that object.
  2. Identify the class of things to which the object belongs (here SUVs), and see if we know anything about the bounds/reasonable values for the class.
  3. Compare, with other objects (sedan < SUV < truck) in the class hierarchy, to determine bounds on the values that parameter can take.
  4. Although it is not necessary that the comparison be restricted to the class hierarchy – so, when asked how high is the ceiling, a typical way to proceed is: People are about 6ft tall, and the ceiling is higher than that, and seems to be less higher than two 6ft tall people put end to end, so about 10 ft.
  5. Guess, not arbitrarily, but using some very weak form of strategy applied in 4.

 

1, 2, and 3 are not very difficult to implement, and I think that 4 is quite important. Another example - when asked how much is the energy in dry cell, a lot of people found out  a place where the dry cell is used (a light torch, transistor radio) and then used to power consumption, etc to go back to energy calculation, instead of first principles (Linder, 1999). Comparing, placing objects in situations where there are lesser unknowns, or just placing it aside a known object are crucial in parameter estimation. People, when learning a new domain, are not necessarily always comfortable with the quantitative understanding – there is a phase of “calibration” when they create these representations.

 

Problems in handling numbers –

  1. The key problem is how to handle numbers without having a priori bounds and statistics.  The value density of water is a safe assumption for the density of human body, as it is mostly water – how much exactly we mean by “mostly”? When we say high/low, hot/cold, there aren’t crisp boundaries. When to say two things are similar – when they differ by +/- 10% or 12.6%?
  2. Need a numSME – how to use the numeric data to measure similarity (on top of structural similarity). Its not just Euclidean distance – Jaguars are similar to Mercedes, even though the dimensions can be quite different, here the key thing which determines similarity here is the price, and the engine power. Possible (but not at all cognitively grounded) solutions –  
    1. A decision tree algorithm (like ID4) which can learn the classification based on the nature of data.
    2. Statistical hypothesis testing methods.

 

Some data on cars –

 

Car Data          SUV Data        Braking Distances         An analogy between car and the human body

 

 

Praveen Paritosh, 07/12/00