Bayesian methods





"...why, therefore, if I am half blind, must I take for my guide one that cannot see at all?"
Thomas Bayes (1702-1761), in a letter of 1736.

Although the reverend and statistician Thomas Bayes wrote his seminal paper "Essay toward solving a problem in the doctrine of chances" in the 18th Century, only in the last fifty years has it been adopted, at least by some, in the Oil Industry (Newendorp, 1971). Philosophically, the theorem is about cause and effect. While one normally predicts an effect from a cause, Bayes turns the problem around and asks which is the most likely cause, out of several possibilities, that produced the observed effect?

This approach, independently rediscovered by the famous Laplace in France a few years after the Bayes publication, was perfected to a workable method by Laplace. However it led to a heated discussion amongst statisticians, partly because of the element of subjectivism in estimating prior probabilities. For an interesting account of the history of rejection by the "frequentists" and acceptance of the bayesian method by the "bayesians" I recommend the book by Bertsch (2011).

The Bayesian application seeks to optimally combine information from two sources:

  1. "Prior" notions of what is likely to occur in general, and,
  2. "New information" that will change our prior notions, according to probabilistic rules.
The result is denoted as "posterior" and this forms the "update" of the prior information.

The prior information can be completely subjective, but it is better to have some database to tabulate and count experience. Then we can better specify prior probabilities or distributions. Exploration is typically a process where we slowly gather information. In a virgin area, it will be difficult to know if source rocks are present. Now we could, in despair, say that we have no idea at all about source rocks in sedimentary basins (the completely blind man in the above quote), or we could use our world-wide experience that tells us that basins are more likely to contain at least one sourcerock than not ("half-blind", because this data is very general and may not really be representative for our particular virgin basin).
New information may come in the form of geological age of this virgin area, say Jurassic to Cretaceous. Undoubtedly, this will affect our prior in a positive way.
It is this process of adjusting our views and probabilities at every step when new information becomes available. In the process a prior becomes a posterior, which in the next phase of updating becomes a new prior, and so forth.

Newendorp (1971) suggested the use of the bayesian method. In the appraisal systems that I developed for prospects and basins/plays, I used bayesian logic as far as practical. The applications are the following:

Application to Play and Prospect POS assessment

When evaluating prospects in an play, a fair amount of drilling may have taken place. From the results it is possible to obtain an average success rate (sometimes called "base rate"). In case this seems not to be constant in time, a creaming analysis has to be made, giving a best estimates for the near future of this average POS value. (Note that this POS is not the so-called "Play risk". This is the chance that at least on discovery will be made in the play.) We take this play average as the prior distribution. This means that if we had no geological information about the prospect, such as charge, trap, seal, etc. then we would assume the prior as the POS of the prospect.

If we may consider that the geological parameters of a prospect are assessed quite independently from the total play knowledge, the total POS of the prospect will usually differ from the play average POS value. It forms "New information".With a set of prospects in the play, it may occur that the mean POS of the prospects is, for instance, much higher than the play POS. Are the prospect POS's overestimated? If it is believed that this the case, a bayesian update of the prospect risk may be advised. This works as follows:

The Bayes rule can be written in words as follows:

In Bayes parlance we want a "posterior probability" p" from the combination of the "prior probability" p' and the "prospect probability" p. The easy way is to use "odds" (a typical bookmakers' term) as in the formula in words above, where the odds associated with a probability p is p divided by (1 - p). Here the odds of the prospect POS is interpreted as the Likelihood ratio. This ratio is normally the ratio of two probability densities, see the bayesian discriminant analysis. As we have only point estimates of POS for the prospect, the probability density function of POS is formed by two spikes at 0 and at 1.

The we derive the formula for p" as a more familiar form as:

Note that the result is invariant under an exchange of p with p'.

Example:
Assume the play p'= 0.420 and the prospect POS p = 0.700, the p" = 0.628, so the prospect POS is brought nearer to the play average POS. In the example given by Milkov (2017) with some 25 prospects conclusively drilled, the correspondence between the actual and predictive is much better after the above-mentioned update of POS. However, the question remains if we can assume that the geologists making their prospect POS assessment have not alraedy in some subjective way used the Play prior information, in which case the "update" process would inadvertently cause a bias up or down.

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