Blending Expert Estimates

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Q: You have a diamond but are not sure whether its a real or a fake one. You show the diamond to two experts who independently estimate the probability that it is real at \(p_1\) and \(p_2\). What is your estimate of what the true probability is?

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A: Surprisingly, there is no clear answer or strategy on how to approach and blend such subjective estimates together. Most people run into similar situations all the time wherein they get estimates from different sources on a fixed decision they need to make.

A naive way to approach this would be to average the two estimates. This would yield \(\frac{p_1 + p_2}{2}\) in the present case. But this doesn't address all the cases. For example if \(p_1 = p_2 = 0.6\), i.e. the experts agree independently the overall estimate has to be greater than 0.6. This is because the fact that they agree is worth something from an information perspective. A good approach then is to resort to a Bayesian perspective.

Let \(p\) be the sort estimate. The new estimate can be computed as
$$
\frac{p}{1 - p} = \frac{p_0}{1 - p_0}\times \frac{p_1}{1- p_1} \times \frac{p_2}{1 - p_2}
$$

In the above formulation, \(p_0\) is our prior estimate. As we typically don't have a prior to go by, we can assume that to be \(\frac{1}{2}\) and the term \(\frac{p_0}{1 - p_0}\) becomes 1.

Watch what happens when we plug in \(0.6\) into the above formula,
$$
\frac{p}{1-p} = \frac{0.6}{1 - 0.6}\times \frac{0.6}{1 - 0.6} = \frac{9}{13} = 69\%
$$

It works out to be greater than\(60\%\) as expected. Also notice another elegant property, if both experts give an uncertain \(50\%\) each, the final estimate remains \(50\%\) as it intuitively should.

If you are looking to buy some books in probability here are some of the best books to learn the art of Probability

Book	Notes/Comments
Fifty Challenging Problems in Probability with Solutions (Dover Books on Mathematics)	This book is a great compilation that covers quite a bit of puzzles. What I like about these puzzles are that they are all tractable and don't require too much advanced mathematics to solve.
Introduction to Algorithms	This is a book on algorithms, some of them are probabilistic. But the book is a must have for students, job candidates even full time engineers & data scientists
Introduction to Probability Theory	Overall an excellent book to learn probability, well recommended for undergrads and graduate students
An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd Edition	This is a two volume book and the first volume is what will likely interest a beginner because it covers discrete probability. The book tends to treat probability as a theory on its own
The Probability Tutoring Book: An Intuitive Course for Engineers and Scientists (and Everyone Else!)	A good book for graduate level classes: has some practice problems in them which is a good thing. But that doesn't make this book any less of buy for the beginner.
Introduction to Probability, 2nd Edition	A good book to own. Does not require prior knowledge of other areas, but the book is a bit low on worked out examples.
Bundle of Algorithms in Java, Third Edition, Parts 1-5: Fundamentals, Data Structures, Sorting, Searching, and Graph Algorithms (3rd Edition) (Pts. 1-5)	An excellent resource (students, engineers and even entrepreneurs) if you are looking for some code that you can take and implement directly on the job
Understanding Probability: Chance Rules in Everyday Life	This is a great book to own. The second half of the book may require some knowledge of calculus. It appears to be the right mix for someone who wants to learn but doesn't want to be scared with the "lemmas"
Data Mining: Practical Machine Learning Tools and Techniques, Third Edition (The Morgan Kaufmann Series in Data Management Systems)	This one is a must have if you want to learn machine learning. The book is beautifully written and ideal for the engineer/student who doesn't want to get too much into the details of a machine learned approach but wants a working knowledge of it. There are some great examples and test data in the text book too.
Discovering Statistics Using R	This is a good book if you are new to statistics & probability while simultaneously getting started with a programming language. The book supports R and is written in a casual humorous way making it an easy read. Great for beginners. Some of the data on the companion website could be missing.
A Course in Probability Theory, Third Edition	Covered in this book are the central limit theorem and other graduate topics in probability. You will need to brush up on some mathematics before you dive in but most of that can be done online
Probability and Statistics (4th Edition)	This book has been yellow-flagged with some issues: including sequencing of content that could be an issue. But otherwise its good