Bayes’ Theorem, the Khaleesi of all theorems that will change your life for the better!
If I ask you what the scientists searching for the Covid-19 vaccine and the nerds trying to code their AI (Artificial Intelligence) have in common with the way you see the world, you would think absolutely nothing. Or just that we are all humans, stick in 2020.
Yet we are all, in some shape or form, using Bayes’ Theorem.
The only difference resides in the fact that scientists or AI specialists are aware of how they implement Bayes’ theorem in their workflow. While most of us are not aware of how we are using consciously or not Bayes’ theorem in our everyday life. Most importantly, we are not seeing how powerful it can be to willingly understand and apply the formula in our own lives and the huge positive impacts we can reap from it.
Bayes’ theorem is a formula about probabilities. And a little disclaimer before we go further. I am not at all good at probabilities. I am quite the opposite. When I was in engineering studies, I was the valedictorian for our probability and statistics test, but from the end. So just bear that in mind while I try to explain to you my understanding of this theorem and how I apply it to my own life to update my worldview, based on new evidence I receive from the universe.
Before we go further, I recommend watching this video, explaining Bayes’ theorem better than I will ever do:
BAYES’ THEOREM — Now let me explain it t you with my own words and visuals. Let’s assume that the whole universe of possibilities can be represented as below and let’s call it Omega (𝝮):
Now let’s imagine 2 events A and B that intersect as below:
Probability is just a world of proportions.
The blue circle represents all the events in the Omega universe where A is true. You can see it as the map of a country, for example. Let’s call A Americana, B as Britannica.
A represents all the citizens of Americana, B of all Britannica.
Now if we want to know the proportion of Americanans in the whole Omega universe, we make the ratio between the circle representing A and the whole universe.
So: P(A) = (Area of A)/(Area of Omega) (1)
Let assume that the area of Omega is 1, for simplicity.
P(A) represents the probability of one individual, picked randomly, being a citizen of Americana.
So is P(B) for Britannica.
Now let’s imagine that we want to know the proportions of citizens of Americana who also happen to be Britannicans. Let’s call if P(A if B).
The dual national individuals are represented by the purple area, where A and B intersect.
In probability, it is represented by an intersection sign, ∩.
P(A∩B) represents the proportion of dual nationals.
So the proposition of Americana citizens having also a Britannica passport can be evaluated as follows:
P(A if B) = P(A∩B) / P(B) (2)
Mathematicians love symbols. And the “if” in probability is represented by the pipe sign, I. So P(A if B) is represented by P(A I B), where you read the probability of A while B is true.
So the formula become:
P(A I B) = P(A∩B) / P(B) (3)
The formula above represents the proportion of Americanans having a Britannican citizenship.
In the same way, we can derive the proportion of Britannicans having an Amricanan’s passport:
P(B I A) = P(A∩B) / P(A) (4)
Now we know that the number of dual nationals is the same in equations (3) and (4).
P(A I B) = P(A∩B) / P(B) (3) ⇒ P(A∩B) = P(A I B) x P(B) (5)
P(B I A) = P(A∩B) / P(A) (4) ⇒ P(A∩B) = P(B I A) x P(A) (6)
Equations (5) and (6) are 2 ways to evaluate the proportion of dual nationals.
By writing (5) is equal to (6), we derive the famous Bayes’ formula:
(5)=(6) ⇒ P(A∩B) = P(A I B) x P(B) = P(B I A) x P(A) (7)
Then let’s arrange (7) like this, by dividing each side by P(B):
P(A I B) x P(B) = P(B I A) x P(A) ⇒ P(A I B) = P(A)x (P(B I A) / P(B) ) (8)
There you have it, the famous Bayes’ theorem of formulae.
Let’s explain each term in plain text:
P(A I B) is the probability that A is true if B. It is a probability a posteriori, after the appearance of B.
P(A) is the probability of A, on its own. It is a probability a priori.
P(B I A) / P(B) is the correction factor, based on the new evidence, by taking into account the appearance of B in the universe of A.
BAYES’ THEOREM IN REAL LIFE — Now let’s give an example. Let’s assume that A represents your vision or hypothesis on a certain subject. Let’s call your world of hypothesis H.
B represents new events coming into your life, the whole universe of events coming after you have formed your opinions. Let’s call that universe, E as the Evidence universe.
From (8) by replacing A by H and B by E in the equation (8) we have the new equation:
P(H I E) = P(H)x (P(E I H) / P(E)) (9)
P(H) represents the probability that my hypothesis of this world without any evidence. It is a representation of the world I have, based on no evidence.
P(E I H) / P(E) represents the new evidence coming into my life. It is the correction factor. Or simply put, this term is the universe providing me with new data.
P(H I E) means the world where both my hypothesis and the new event coexist.
Bayes’ formula says that the only way for my hypothesis to be true with new evidence is to update it with the correction factor.
Yet for most of us, in our everyday life, we reduce that correction factor to zero in the equation (9) above.
In mathematics, it translates into raising the correction factor to the power of zero:
(P(E I H) / P(E) )⁰ = 1 (10)
Basically, we have annihilated any possibility of the correction factor.
Most of us are dogmatic in our life and have this reduced incomplete equation tattooed in our mind:
P(H I E) = P(H)x (P(E I H) / P(E)) (9) where (P(E I H) / P(E) )⁰ = 1
This is the dark equation of our life:
P(H I E) = P(H) (11)
With (11), no matter how things change around us, no matter how much new evidence we are provided with, we just hold on to our worldview a priori.
We become rigid, blind, dogmatic, and deaf to the music of the universe. We create clans and we clash with others who do not share our worldview.
By adopting the mindset of Bayes’ theorem in our life, we are open to new evidence. And this does not mean that we have to get rid of our worldview. What it means is that we need to update constantly our vision of the world based on new inputs.
I apply it to my life every day by always listening to what the universe is trying to show me. I also surround myself with people who have the same mindset and different views of the world so that we can grow together.
By incorporating Baye’s formula into my life, I am filled with humility and the curiosity to learn and grow.
Yes, I have my worldview, not always based on evidence, and it is OK. However, I know that new evidence will always come. And if I listen and learn, I will be able to update my worldview and be more in alignment with the universe around me.
There is my take on Bayes’ theorem, from the valedictorian from the bottom, in probability and statistics in my class of 2006.
How do you update your worldview based on new evidence?
How would you rate your love of probabilities and proportions after reading this article?
Leave a comment below.
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You can read my previous article on 3 secrets to be the best DP of your own life!