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📚 Statistical Inference for Everyone
Chapters ▾
1 Introduction to Probability2 Applications of Probability3 Random Sequences and Visualization4 Introduction to Model Comparison5 Applications of Model Comparison6 Introduction to Parameter Estimation7 Priors, Likelihoods, and Posteriors8 Common Statistical Significance Tests9 Applications of Parameter Estimation and Inference10 Multi-parameter Models11 Introduction to MCMC12 Concluding Thoughts13 Appendix A: Computational Analysis14 Appendix B: Notation and Standards15 Appendix C: Common Distributions and Their Properties16 Appendix D: Tables

1.1 Overview

Life's most important questions are, for the most part, nothing but probability problems. - Laplace

In 1968 a jury found defendant Malcolm Ricardo Collins and his wife defendant Janet Louise Collins guilty of second degree robbery1. The decision hinged on the testimony of bystanders, which stated that the perpetrators had been “black male, with a beard and moustache, and a caucasian female with blonde hair tied in a ponytail,” and that they escaped in a “yellow motor car.” A mathematician testified that the odds against this couple being innocent were one in twelve million, and this was enough for the jury to convict. Later, in an appeal, the California Supreme Court reversed the decision primarily because of lack of evidence, and faulty inference.

In another case, Sally Clark was convicted in 1999 of the murder of her two young sons2. Again, the testimony hinged on a statistical argument - the chances of one baby dying in their bed 1 in 8500, so therefore the chances of two of them dying in the same way is the square of this, or 1 in 73 million. Several years later, and a public statement from the Royal Statistical Society highlighting the erroneous logic, Sally Clark was released - although she never overcame the resulting damage to her life that the conviction had caused.

We will cover these cases in more detail later, and why the inference was faulty, but I introduce the stories here for two reasons. First, is to point out that there are cases in which proper statistical inference can be a life and death matter. Second, it is to highlight the fact that such inference can run counter to one's intuition. Part of the purpose of this book is to retrain your intuitions and your habits of intuition to avoid such failures.

We have to make decisions nearly every second of our lives, and those decisions are based on our state of knowledge. Unfortunately, we are never 100% sure of any information in our lives, so we are constantly forced to make decisions in the face of uncertainty. In many cases our common sense is enough to make sophisticated decisions, taking into account the uncertain nature of the situation. However, there are many times where our common sense is not enough to quantitatively resolve the level of uncertainty, and make valid inferences. It is in these cases that statistical inference is most useful.

Statistical inference refers to a field of study where we try to infer unknown properties of the world, given our observed data, in the face of uncertainty. It is a mathematical framework to quantify what our common sense says in many situations, but allows us to exceed our common sense in cases where common sense is not enough. Ignorance of proper statistical inference leads to poor decisions and wasted money. As with ignorance in any other field, ignorance of statistical inference can also allow others to manipulate you, convincing you of the truth of something that is false.

For example, in 1978 a Russian satellite deviated from its orbit and became increasingly erratic, and was going to crash into the Earth.3 This sort of event occurs from time to time, even including a recent crash of a US spy satellite in 2008.4 There was a local news broadcast about the impending Russian satellite crash which said something like, “the scientists had studied the trajectory of the satellite, and determined that there was only a 25% chance of it striking land, and even a much smaller chance striking a populated area.” The report was clearly designed to calm the public, and convince them that the scientists had a good handle on the situation. Unfortunately, given a little thought, one realizes that the Earth's surface consists of about 25% land and 75% water, so if you knew nothing about the trajectory of the satellite, you would simply state that it had a 25% chance of striking land. Instead of communicating knowledge of the situation, the news broadcast communicated (to those who knew basic statistical inference) that either the scientists were in complete ignorance of the trajectory or the reporter had misinterpreted a casual statement about probabilities and didn't realize what was implied. Either way, the intent of the message and the content of the message (to those who understood basic probability) were in direct conflict.

Adapted from Statistical Inference for Everyone, by Brian Blais (Bryant University), licensed under CC BY-SA 4.0 (dual-licensed under the GNU FDL 1.2 or later; this adaptation uses the CC BY-SA grant). Changes were made; this adaptation is distributed under the same license. License: CC-BY-SA-4.0.