A Visual Approach to Bayesianism for beginners
September 2022
Summary
“To see is to believe”, that famous quote is essential to delve into one of the most
fascinating actual domains of Mathematics and Statistics, known as Bayesianism. The theory
about it is known since the 18th century, but it is only now that this rule in probability is
gaining success among scientists as an alternative to machine learning. Even the finder of
this rule did not find it “exciting” and just considered it as just another mathematical
equation. But the truth is very balanced as some rare scientists have been trying out to
understand that equation. Anyway, it is still very difficult for the newbie to follow up
because of the complexity of this rule that even embraces the philosophy of life. Through
this note, we build a comprehensive and unique visual way to give critical clues for the
scholar to begin the long march through Bayesianism.
,Contents
1 The Marlon Case................................................................................................................. 4
2 The Frequentist vs the Bayesian......................................................................................... 5
2.1 Three questions to change the game .......................................................................... 6
2.2 The Bayesian way......................................................................................................... 8
3 Why is the Bayesian approach so powerful? ..................................................................... 9
3.1 Do not reject any hypothesis ....................................................................................... 9
3.2 No more calculation complexity .................................................................................. 9
3.3 Subjectivity is part of scientific process..................................................................... 10
3.4 Remember your priors............................................................................................... 10
3.5 Who chooses the budgets? ....................................................................................... 11
3.6 Imagine your theory is wrong. What would the world look like? ............................. 11
3.7 Updating in Small Steps ............................................................................................. 14
4 Conclusion ........................................................................................................................ 16
, INTRODUCTION
Bayesianism gets its name from Reverend
Thomas Bayes's discovery of the Bayes formula,
which was interested in probabilities in risk
assessment. For him, it was just a mathematical
theorem, a somewhat trivial and very interesting
relationship, he did not even publish it. This formula
was later found in his writings, posthumously.
That formula seems very simple. Do not be impressed by the annotation. It just says that if in a certain
sample there is a certain proportion of elements that have the property A and another proportion of
elements that have the property B, we can calculate the proportion of elements that have the 2 properties
at a time (a). Thus, we take the percentage of the elements that are A and multiply it by the percentage
of those who are B among those who are already at A. And then we can also do the opposite and take
the percentage of the elements which are B and we multiply by the percentage of those who are A among
those who are already at B (b).
So, we have this equality (c), that Pierre Simon Laplace, a few years later, understood its true interest.
(a) (b) (c)
Indeed, by rewriting it rather, he notices that it can be used to return the conditional probabilities, that is
to say, find the probability of B and knowing A from the probability of knowing B. Before him, one could
only calculate the probabilities of events if the cause was known.
Nowadays, Bayes' rule has become more and more influential in scientific thinking. Still, however, many
people, even among the scientific community can't explain how the Bayes rule works or what it is, so this
study's note is a contribution to remedying this incongruity because using the formula seems quite
complicated, but it is quite easy to visualize.
September 2022
Summary
“To see is to believe”, that famous quote is essential to delve into one of the most
fascinating actual domains of Mathematics and Statistics, known as Bayesianism. The theory
about it is known since the 18th century, but it is only now that this rule in probability is
gaining success among scientists as an alternative to machine learning. Even the finder of
this rule did not find it “exciting” and just considered it as just another mathematical
equation. But the truth is very balanced as some rare scientists have been trying out to
understand that equation. Anyway, it is still very difficult for the newbie to follow up
because of the complexity of this rule that even embraces the philosophy of life. Through
this note, we build a comprehensive and unique visual way to give critical clues for the
scholar to begin the long march through Bayesianism.
,Contents
1 The Marlon Case................................................................................................................. 4
2 The Frequentist vs the Bayesian......................................................................................... 5
2.1 Three questions to change the game .......................................................................... 6
2.2 The Bayesian way......................................................................................................... 8
3 Why is the Bayesian approach so powerful? ..................................................................... 9
3.1 Do not reject any hypothesis ....................................................................................... 9
3.2 No more calculation complexity .................................................................................. 9
3.3 Subjectivity is part of scientific process..................................................................... 10
3.4 Remember your priors............................................................................................... 10
3.5 Who chooses the budgets? ....................................................................................... 11
3.6 Imagine your theory is wrong. What would the world look like? ............................. 11
3.7 Updating in Small Steps ............................................................................................. 14
4 Conclusion ........................................................................................................................ 16
, INTRODUCTION
Bayesianism gets its name from Reverend
Thomas Bayes's discovery of the Bayes formula,
which was interested in probabilities in risk
assessment. For him, it was just a mathematical
theorem, a somewhat trivial and very interesting
relationship, he did not even publish it. This formula
was later found in his writings, posthumously.
That formula seems very simple. Do not be impressed by the annotation. It just says that if in a certain
sample there is a certain proportion of elements that have the property A and another proportion of
elements that have the property B, we can calculate the proportion of elements that have the 2 properties
at a time (a). Thus, we take the percentage of the elements that are A and multiply it by the percentage
of those who are B among those who are already at A. And then we can also do the opposite and take
the percentage of the elements which are B and we multiply by the percentage of those who are A among
those who are already at B (b).
So, we have this equality (c), that Pierre Simon Laplace, a few years later, understood its true interest.
(a) (b) (c)
Indeed, by rewriting it rather, he notices that it can be used to return the conditional probabilities, that is
to say, find the probability of B and knowing A from the probability of knowing B. Before him, one could
only calculate the probabilities of events if the cause was known.
Nowadays, Bayes' rule has become more and more influential in scientific thinking. Still, however, many
people, even among the scientific community can't explain how the Bayes rule works or what it is, so this
study's note is a contribution to remedying this incongruity because using the formula seems quite
complicated, but it is quite easy to visualize.
