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Shapes of Uncertainty [Bayesian Inference
Posted on June 9, 2013 @ 04:38:00 PM by Paul Meagher

In my last 2 blogs, I discussed the idea of a profit distribution. I argued that it is better to estimate profit using a profit distribution rather than a single most-likely value (e.g., we should make 100k next year). A distribution is more epistemically informative than a single most-likely value. I'll illustrate what I mean by this in today's blog on the shapes of uncertainty.

In this blog, I want to focus on what to look for in a profit distribution. A profit distribution can have many shapes and these shapes are quite informative about the type and level of uncertainty involved in an estimate.

To demonstrate why profit distribution shapes matter, I have prepared 3 new Google profit distributions for your consideration.

  1. A flat distibution.
  2. A peaked distribution.
  3. A distribution with a reduced x range, or a "shrunk" distribution.

It is useful to acquire the skill of reading and interpreting a profit distribution. That skill involves attending to significant aspects of the distribution shape and understanding what the shapes mean.

Flat Profit Distribution

If our profit distribution for Google was flat, this would mean that our level uncertainty was the same over all the profit intervals. In the graph below, the estimated profit could fall within the full range of values with the same probabiliy (i.e.,16.6%) of being in any interval. Some Bayesian textbooks advise that you start with a flat distribution if you have no strong convictions where an estimated parameter might lie.

Peaked Profit Distribution

In a peaked profit distribution one of the intervals has significantly more probability mass than other profit intervals. This refects an increased level of certaintly that the estimated profit will actually be within that interval. As we acquire more information about the company and its lines of business (e.g., second quarter financials), we might expect that our profit distribution estimate would begin to change shape in this manner first.

Shrunk Profit Distribution

As we learn even more about a company and their lines of business, then the range of possible profiit outcomes should be reduced so that instead of a Google profit range running from 10.0b to 12.4b, perhaps it only covers the range from 10.8b to 12.0b (see below). We show our confidence in our prediction by how narrow our profit distribution is. This does not necessarily change the shape of the profit distribution, it changes the x axis of the profit distirbution (both shapes might be peaked, but they would be on x axis with different ranges of possible values).

Conclusion

The shape of a profit distribution tells us alot about the nature of the uncertainty surrounding our estimate of profit. We have seen that our confidence in an estimate is reflected in how peaked our profit distribution is and how shrunk the range of possible profits are. This suggests strategies one might adopt to increase confidence in an estimate - gather information that helps you establish a more peaked profit distribution and that helps you reduce the range of the profit distribution.

In this article we have examined three ways in which a profit distibution can appear on a graph - flat, peaked, or shrunk. There are other aspects of shape that we have not examined, namely, the skew factor and the kurtosis factor (second and third moments of the distribution). Using these shape controls, we might be able to approximate the peaked distribution above as a normal distibution with a skew and kurtosis setting that would help match a theoretical normal distribution to the estimated profit distribution. A normal distribution is an example of a function that generates points on a probability curve (sums to 1) based upon the values fed into it (i.e., mean, standard devitation, skew, kurtosis, x-values). We might want to take this additional step of creating a profit distribution function if we thought it would simplify calculations (or thinking) or if we thought it was a better representation of the data than a discrete historgram of possible profit intervals. Step functions are potentially limited as a means of representing the actual shape of our uncertainty about a parameter.

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