Bayesian Reasoning: Discrete Inference with Sequential Data

Or, One Way I Learned to Quit Believing My Prejudices

In my last article on this topic, I showed that considering background information can play a significant role in helping us make less biased judgments. What I hope to show now is that while we learn by updating the information we have through experience, limited experiences can often lead to prejudices about the way we interpret the world; but even broad and deep experience should rarely lead us to certain conclusions.

To get started, imagine playing a game in which someone asks you to infer the number of sides of a die based on the face numbers that show up in repeated throws of the die. The only information you are given beforehand is that the actual die will be selected from a set of seven die having these number of faces: (4, 6, 8, 10, 12, 15, 18). Assuming you can trust the person who reports the outcome on each throw, after how many rolls of the die will you be willing to specify which die was chosen?

Let's use the R programming language to help us think through the problem. Start by specifying the set of the die possibilities such that each number represents the number of sides of a given die. (You might also want to refer to my previous article on Bayesian analysis to familiarize yourself with some of the terminology that follows.)

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Bayesian Reasoning: Gender Inference from a Specimen Measurement

Imagine that we have a population of something composed of two subset populations that, while distinct from each other, share a common characteristic that can be measured along some kind of scale. Furthermore, let’s assume that each subset population expresses this characteristic with a frequency distribution unique to each. In other words, along the scale of measurement for the characteristic, each subset displays varying levels of the characteristic among its members. Now, we choose a specimen from the larger population in an unbiased manner and measure this characteristic for this specific individual. Are we justified in inferring the subset membership of the specimen based on this measurement alone? Baye’s rule (or theorem), something you may have heard about in this age of exploding data analytics, tells us that we can be so justified as long as we assign a probability (or degree of belief) to our inference. The following discussion provides an interesting way of understanding the process to do this. More importantly, I present how Baye’s theorem helps us overcome a common thinking failure associated with making inferences from an incomplete treatment of all the information we should use. I’ll use a bit of a fanciful example to convey this understanding along with showing the associated calculations in the R programming language.

Suppose we are aliens from another planet conducting scientific research on this strange group of bipedal organisms called humans...



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