George Rebane

[Trigger warning – this post is primarily for our technically oriented readers.]

We encounter ongoing minimally known processes (MKPs) continually in our daily round, and we often need to make decisions based on the likelihood of their terminating during a specific future time interval, or surviving it.  Yet few are aware of this perspective of the daily round, and fewer yet know how to obtain quantitative information from such processes, information that is important for decisions in such fields as finance, medicine, environmental risk, and other areas of science and technology.

The ongoing MKPs considered here are 1) a process about which we know only its age, or when the process started (label these MKPA); and 2) a process about which we know only its lifetime, or the length of time before such a process terminates (label these MKPL).  Both kinds of such observed and observable processes are of interest in that we want to know the probability the process will end during or survive through a specified future time interval.  A simple example of a MKPA is, say, a spring which started flowing five years ago, and today we want to know how likely it is stop sometime during the next nine months (or its complement, how likely is it to continue through the next nine months).  An example of a MKPL query is, given that such an ongoing process is known to last ten days, what is the probability that it will end or continue through the next 90 minutes.

The answers to such and even more complex questions can be calculated quickly through the use of formulas whose development is based on an ingenious application of the Copernican Principle first demonstrated on an MKPA problem by physicist J Richard Gott in 1993.  For reasons unknown, his work was discussed briefly and then quietly laid to rest.  My own involvement began with its serendipitous discover while researching approaches to solving another problem.  Since then, I have reformulated Gott’s work to expand and obviate its applicability, and expanded it to the class of MKPLs.  An interesting use of MKPLs is in allocating resources in the observation of periodic processes about which we only know their periodicity.

I have found these results useful in my own analysis of securities’ performance, and in making of healthcare decisions.  A little thinking on what processes qualify with the Copernican approach to minimally known (understood) processes continues to yield new uses.  As an important footnote to the application of this theory is in Bayesian decision making that require a reasonable prior probability to kick off the calculations as new evidence comes in.

We looked at MKPAs in a previous RR post here.  To update the elements of this theory and its resultant formulas, I have written a short technical note which you can download here – Download TN1902-1_lifetimeMKP , the progenitor of which you can download here – Download TN0708-1_GottExtensionConfirmation190120A .