Bayes’ Theorem

Bayes’ Theorem: Grasping the Basics

Mathematical concepts are used to explain and solve a wide variety of problems involving virtually every aspect of life. For those who easily understand mathematics, theorems are logical and easy to comprehend. However, anyone who has some difficulties with math will often find the constructs difficult to grasp. Bayes’ Theorem is an excellent example of a concept people have problems fully understanding.

The central idea is that it’s possible to predict an event based on existing knowledge. In fact, Bayes’ Theorem is actually not long or convoluted—it’s a single equation, not a long, drawn out series of equations. The conclusions resulting can be used to hypothesize future outcomes, which makes it easier to plan for those outcomes.

Insurance companies, for example, may use historical data related to heart attacks and strokes to estimate their potential liabilities in the future. Researching historical data allows insurers to determine what portion of the general population is likely to suffer a heart attack or stroke and at what age those events are most likely to occur.

Those statistical probabilities make it possible for insurers to set rates that reflect their true risk exposures quite accurately. Of course, statistical probabilities enable relatively accurate estimates that assist planners in other areas of research as well. The trick, as with all types of research, is to start with accurate historical data to ensure the results are not skewed.

Exploring the Issues When Applying Bayes’ Theorem

First, it’s important to understand how data is derived. Tests, for example, are not events, so data derived only from test results rather than true events will likely prove to be somewhat inaccurate.

False positives resulting from testing will skew final results. All test formats tend to generate a certain percentage of false positives, which means the final results won’t necessarily provide the level of accuracy needed to reach constructive conclusions. Defining the probabilities for false results and applying those probabilities will provide more accurate end results.

Science itself isn’t always accurate. In any experiment, the potential for errors always exists. A basic premise can be in error, test parameters may not be properly defined, and equipment used can also be flawed, and any of those potential issues will lead to erroneous conclusions.

However, if researchers have access to accurate data to begin with, it’s possible to correct measurement errors. In the example cited above, there is sufficient historical data available related to the incidence of heart attacks and strokes. That means researchers should be easily able to chart the potential future heart attack and stroke patterns. However, there are other factors impacting the conclusions.

Diet, exercise, and other lifestyle choices impact the odds of heart attacks and strokes. That means, to be truly accurate, tests or other measurements must be devised that reflect the physiological changes created by cultural mores. If the evolution of lifestyle patterns is not factored in, the end result will accurately reflect the totals to date, but not consider how those changes will likely alter future statistics.

Discovering Other Applications

Bayes’ Theorem can be applied to other research topics as well. Education is another area where existing data is available to assist in making future predictions. If a specific number of students from well-defined demographic groups have successfully completed or failed an area of study in the past, is it possible to predict how many will successfully complete the coursework in the future? In most cases, it would be possible to do so but, as with so many examples, the data used must reflect current conditions if the conditions are changing.

In the education example, comparing different test populations and the trends defining the results for each population using Bayes’ Theorem can assist educators to develop strategies to improve outcomes in the future. However, applying data from a specific population subset to a larger population would likely not be effective. That’s one of the problems with most standardized testing formats.

When looking at the heart attack and stroke example, the results derived after applying the theorem provide immediately usable data for the insurance companies, but that data can also be used to help medical researchers develop plans and strategies that target specific demographics to remediate the causes of heart attacks and strokes.

Is Bayes’ Theorem Practical?

Bayes’ Theorem allows users to employ the basic equation to take test results and correct for skewing occurring due to false positives and other issues. At that point, the theorem allows users to quite accurately predict the likelihood of a specific event occurring. Assuming the data used is, indeed, accurate, the equation should provide a reliable prediction of a particular outcome.