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Motivating example

We introduce the concept of maximum entropy through a simple example. Suppose we wish to model an expert translator's decisions concerning the proper French rendering of the English word in. Our model tex2html_wrap_inline1484 of the expert's decisions assigns to each French word or phrase f an estimate, tex2html_wrap_inline1488 , of the probability that the expert would choose f as a translation of in. To guide us in developing tex2html_wrap_inline1492 , we collect a large sample of instances of the expert's decisions. Our goal is to extract a set of facts about the decision-making process from the sample (the first task of modeling) that will aid us in constructing a model of this process (the second task).

One obvious clue we might glean from the sample is the list of allowed translations. For example, we might discover that the expert translator always chooses among the following five French phrases: {dans, en, à, au cours de, pendant}. With this information in hand, we can impose our first constraint on our model p:


This equation represents our first statistic of the process; we can now proceed to search for a suitable model which obeys this equation. Of course, there are an infinite number of models tex2html_wrap_inline1496 for which this identity holds. One model which satisfies the above equation is tex2html_wrap_inline1498 ; in other words, the model always predicts dans. Another model which obeys this constraint predicts pendant with a probability of tex2html_wrap_inline1500 , and à with a probability of tex2html_wrap_inline1502 . But both of these models offend our sensibilities: knowing only that the expert always chose from among these five French phrases, how can we justify either of these probability distributions? Each seems to be making rather bold assumptions, with no empirical justification. Knowing only that the expert chose exclusively from among these five French phrases, the most intuitively appealing model is


This model, which allocates the total probability evenly among the five possible phrases, is the most uniform model subject to our knowledge. (It is not, however, the most uniform overall; that model would grant an equal probability to every possible French phrase.)

We might hope to glean more clues about the expert's decisions from our sample. Suppose we notice that the expert chose either dans or en 30% of the time. We could apply this knowledge to update our model of the translation process by requiring that tex2html_wrap_inline1504 satisfy two constraints:


Once again there are many probability distributions consistent with these two constraints. In the absence of any other knowledge, a reasonable choice for tex2html_wrap_inline1510 is again the most uniform--that is, the distribution which allocates its probability as evenly as possible, subject to the constraints:


Say we inspect the data once more, and this time notice another interesting fact: in half the cases, the expert chose either dans or à. We can incorporate this information into our model as a third constraint:


We can once again look for the most uniform tex2html_wrap_inline1518 satisfying these constraints, but now the choice is not as obvious. As we have added complexity, we have encountered two problems. First, what exactly is meant by ``uniform,'' and how can one measure the uniformity of a model? Second, having determined a suitable answer to these questions, how does one find the most uniform model subject to a set of constraints like those we have described?

The maximum entropy method answers both these questions. Intuitively, the principle is simple: model all that is known and assume nothing about that which is unknown. In other words, given a collection of facts, choose a model which is consistent with all the facts, but otherwise as uniform as possible. This is precisely the approach we took in selecting our model tex2html_wrap_inline1520 at each step in the above example.

next up previous
Next: Maxent Modeling Up: Overview Previous: Overview

Adam Berger
Fri Jul 5 11:43:50 EDT 1996