What Bayes’ Theorem Has to Do With Anxiety and Screening in Primary Healthcare.

After entire weeks of mutism I was just about to write a post about how I couldn’t write at all. Of course, I danced for a while with some skepticism texts to justify my own disability, and that lead me to consume audiovisual material about epistemology, Sanders Peirce’s semiotics, induction, abduction, deduction, logic and argumentation, statistics, probability, and, finally, Bayes theorem.

This last encounter was so interesting to me that I just chose to abandon the initial concept (writing about my own writer’s block) and decided to organize everything I’d learned so far about said theorem.

Although what I’m writing now are not entirely conclusion of my own merit merit, I believe it is worthwhile to share what I’ve learned from what others have said before me — especially because I found the theorem to be a great contribution to learning how to live a good life, which is what I’m trying to verbalize today.

The theorem.

Bayes’ theorem is an equation that allows us to induce the probability of a hypothesis given evidence-based data. Therefore, it is an equation that allows us to estimate conditional probability — i.e., that a hypothesis (H) occurs given the evidence (E). The theorem looks like this:

Whereas the “|” symbol denotes condition (in English we’d translate that symbol to “given” or “given that”), H is the hypothesis, and E the evidence that would or would not back up said hypothesis.

The uses of this theorem in statistics and probability are multiple, particularly in the field of sequential analysis. I do not pretend to explain each element of the theorem or discuss the validity of the equation, but rather to explain why I believe that it is useful and important to know Bayes’ theorem in the first place.

Once you understand the overall purpose and uses of the theorem, it becomes very clear what it is concretely made for. However, its utility in everyday life goes a little further. Bayes theorem essentially distinguishes rational beliefs (or propositions), and separates them from irrational ones. If you contrast evidence-based data with any proposition that you might be potentially upholding, you’ll have a pretty clear idea of how irrational or rational your thoughts about it are.

Bayes’ Theorem applies to any observation, new or previously established, about any particular circumstance. From how likely it is that it will rain today all the way up to your friend replying to your Whatsapp message. The theorem appeals to inductive argumentation, introducing new data based on previous and current events in order to estimate how likely it is for a given event to happen or repeat itself.

For example, if your town has the tendency to have rainy weather from March all the way through May and today is April’s 10th, the “today is going to rain” hypothesis is highly probable. The same applies if your friend never replies to any of your Whatsapp texts. If he hasn’t replied in the last two weeks, the probability of him doing it today is quite low.

Bayesian inference.

Bayes’ theorem is an open theorem, at least in the sense that it allows data to update itself as new things happen.

If it rains in your town from March to May, but there hasn’t been a drop of rain in the last three years, the value of P(R|E) changes.

For this particular example, R is a rainy April’s 10th and E is the evidence that would or wouldn’t back up the hypothesis that rain is going to happen that day.

The value of P(R|E) was higher three years ago because E was different. The weather seemed to have a steady, cyclical and uninterrupted pattern. But what happens today is different, so evidence gets an update and, with it, the value of the probability also updates itself.

Keep in mind that Bayes’ theorem is not used to deduce. Nobody is saying here that a proposition is true or false without anything in the middle. The induction that the theorem appeals to assigns a measure of credibility to a hypothesis based on its certainty probability given the evidence.

The higher the probability, the more credible the hypothesis is.

This credibility approach for a hypothesis implies that if the evidence supporting a hypothesis fluctuates, its occurrence probability will also fluctuate and, naturally, so will its credibility. That’s why three years ago it was credible that April’s 10th would be a rainy day, but today it isn’t. In 2014 rain in that particular day was more probable than it is in 2017, so the hypothesis’ credibility adjusts itself to this pattern change, and adapts its value.

If your friend starts texting you via Whatsapp with some frequency, for instance, the value of P(W|E) changes. Again, in this case W is the “your friend texting you via Whatsapp” hypothesis, and E is the evidence that backs it up. Six months ago, W had a different value and, therefore, P(W|E) was also different.

From the theorem’s approach and its ability to admit data update for each of its elements (this is what Bayesian inference is all about), if your friend has texted you very frequently these past three weeks, it is highly probable that he will text you today.

It is not false that the value of P(W|E) was low a month ago, nor is it false that P(W|E) is high today. Probability changed given the new evidence and the theorem is still useful and pertinent in both cases.

Induction, the theorem and is uses in everyday life.

Inductive thinking does have its limitations, and that would be material for a whole other article. The past isn’t necessarily a representation of the future, and many real and measureable observations have been unpredictable. But induction still has an important use, and we all appeal to inductive thinking in everyday life in order to predict or make decisions about something.

Of course, as the imperfect machines that we all are, we frequently ignore the conclusions that inductive thinking leads us to. And this is not because we choose to deduct or abduct, but because we, as humans, have that tendency to the irrational that seems as old as humanity itself.

For instance, and this is incredibly common, many people turn to Dr. Google, M.D., when they get a headache. It’s not unusual for a person to close their Chrome tabs believing that they have an ateriovenous malformation with an imminent aneurism, a brain tumor, or an incredibly rare auto-immune disease. If we’re friends, the next step is to Whatsapp me about your serious brain condition. If we are not, and if you are not close to someone who is a doctor, the panorama is a lot more obscure.

Given that serious causes of headache are rare (all of them together represent less than 10% of all causes of headaches, according to the World Health Organization) thinking that your particular headache is caused by a brain tumor is irrational.

The most likely scenario is for you to have a headache because you’re having a migraine episode, because you didn’t have enough sleep last night, or because your coworker just made you angry. Given that these are the most probable causes, the most rational belief is that your headache is being caused by a benign cause that will not lead you to serious complications. You do not have a brain tumor, an aneurism is not going to burst and, no, you do not have Lupus.

Some anonymous YouTube hero applied the Bayes theorem to breast cancer screening, aiming his example at the particular case of a woman in her between 40 and 50 years old. If you’re a woman on this age-range, you will find the video very interesting. But if you’re a doctor or a health practitioner, you will not only find it interesting — it will also change a lot the way you behave and make decisions in your professional practice.

Instead of keeping up with screening tests of which you know nothing more than the fact that they are used for screening, it would be amazing to take the Bayes theorem into account, the specificity and sensibility of the methods you’re utilizing, the probabilities of a given patient to be ill with the disease given the evidence of previous years, among many other factors that I will not explain in much detail.

Using conditional probability and applying Bayes’ theorem to medical practice can induce major changes in the way patients are treated and costs are allocated.

Bayes theorem is useful for Medicine, especially for those of us in primary care as well for prevention and early diagnosis protocols. But for its utility to translate into wellbeing, health improvement, and the design of cost-effective health planning, it is imperative for doctors to know about it.

On the other hand, if you’re a person suffering from anxiety, the theorem might also be very useful. Let’s assume that you have a crippling anxiety episode that’s keeping you from going out with your group of friends because you don’t want them to see you eating and dropping your food, or you don’t want them to consider you a boring person. You’ve gone out with them before and nobody seems to be looking at you while you eat, you’ve only dropped your food once, and if they’re still inviting you out it’s highly probable that they do not think you’re boring or useless.

The probability of P(D|G), where G areprevious gatherings and D your inner drama, seems pretty low — at least given the evidence (G) provided by past experiences.

So, if you want to go out and the only thing that is keeping you from doing it is this anxiety issue, you might want to consider that D is quite an irrational belief given G. If you start looking at things this way, it seems like going out today is a good idea, and staying home feeling frustrated is kinda foolish.

And just like in these particular cases I just mentioned, I have to state that this theorem is also useful for logic and argumentation-related purposes. Bayes theorem is, to certain degree, very useful to overcome our everyday cognitive biases — especially the Base Rate Bias (BRB), consisting of a bias where, given a large amount of general information about a subject and a small amount about a particular derivation of that same subject, our mind focuses on the latter instead of the former.

While the Base Rate Bias is taking place, the evidence that would back up a hypothesis is abandoned because prior considerations (priors) are not taken into account. Using the same elements of the theorem to explain this bias, we’d say the BRB would be represented by this assumption:

P(H|E) = P(E|H)

As you see, the elements on each side of the equation are different and have different conditional probabilities and values, but if this bias is present, people might see them and evaluate them as equal and identical, thus giving more weight to specific details about an issue rather than to big, more probable general information about that same issue.

So, I’m not saying that everyone should just walk around emptying numeric data on the theorem in order to know if something is just about to happen, or if it is rational to think that it would.

What I am trying to say is that knowing how this theorem works helps us make better choices and have better life quality thanks to its usefulness in opposing rational beliefs versus irrational ones.

Bayes theorem is particularly useful for anxiety issues and medical screening protocols, as well as many other stuff, like we just revised before.

On the next article I will discuss how us humans are not Bayesian networks and why knowing that is so important if you want to understand human behavior in a better, broader way.