Things I don’t understand: Bayes’ theorem

TODO:
* clean up the discussion about conditional probability
* put in footnotes
* make Fig. 1

In which I lay down foundations

Probability is interesting to me for two reasons:

  • It’s useful.
  • It’s a generalisation of logic1.

The problem is, I have never had any formal training in it. Not even in high school. So that leaves me in a bit of a limbo because I see understanding probability as one of the cornerstones of Good Thinking.

This is my attempt at curing that.

What I shall do is start from the axioms then prove theorems as I tackle classic problems. Which means three things: a) this page will be a work-in-progress indefinitely2, b) the vocabulary I will build here may not correspond to the standard one, and c) I’ll probably get things wrong a lot of times so this may not be the best page to cite in support of your Internet argument.

Before we start, I’d like to introduce a rule and a notion. I won’t start from the very foundations of mathematics because that will just waste everyone’s time. But I really admire the rigor of the Bourbaki school (in great part because I’m still in my “rigorous phase”3). So I shall follow a simple rule: every assertion must have a proof or a reference to one. And to do that more efficiently I’ll borrow a notion from programming and “import” complete mathematical objects in this manner:

IPT 1: (the algebra of sets)
see Wikipedia

Eventually, those external links should become internal ones.

Right. Let’s get to work. Here are some definitions:

DEF: (data point)
a value (e.g., “red”, “42.4 seconds”)
DEF: (data set)
a set of data points
DEF: (universe, the Universal Data Set)
\Omega, or the data set containing all data sets
DEF: (probability)
the probability P(E) of a data set E is a real number associated with E

These are the basic building blocks of probability theory.

As far as I know, probability theory is then completely axiomatised by the following three axioms (which come from A. Kolmogorov, according to Wikipedia):

AXM 1: (“All probabilities are non-zero.”)
(\forall S \subseteq \Omega)(P(S) \geq 0)
AXM 2: (“The probability of the Universal Data Set is 1.”)
P(\Omega) = 1
AXM 3: (“The probabilities of disjoint data sets are additive.”)
(\forall E_1, E_2 \subseteq \Omega)( (E_1 \cap E_2 = \emptyset) => (P(E_1 \cup E_2) = P(E_1) + P(E_2)) )

Using these we can already say a few basic facts about probabilities.

THM 1: “The probability of the empty set is 0.”
(P(\emptyset) = 0)
\begin{aligned} &\rightsquigarrow \emptyset \cap \Omega = \emptyset \\ &\implies P(\emptyset \cup \Omega) = P(\Omega) = P(\emptyset) + P(\Omega) \\ &\implies P(\emptyset) = 0\\ &\Box \\ \end{aligned}

 

THM 2: “The probability of the complement of a data set is one (1) minus the probability of the original.”
(P(A^c) = 1 - P(A))
\begin{aligned} &\rightsquigarrow A \cap A^c = \emptyset \\ &\implies P(A \cup A^c) = P(\Omega) = 1 = P(A) + P(A^c)\\ &\implies P(A^c) = 1 - P(A) \\ &\Box \\ \end{aligned}

(Can I just say how awful and tedious it is to write multi-line \LaTeX in default WordPress?)

What we have though is still too bare. It lacks flavour. So let’s define a few more things:

DEF: (reduced universe)
the reduced universe \Omega_E of a data set E is the subset of the universe \Omega where E is true
DEF: (joint probability)
the probability P(A \cap B), or the probability of both A and B being true
DEF: (conditional probability)
the probability P(A|B) = \frac{P(A \cap B)}{P(B)}, or the probability of A being true given that B is true
DEF: (independence)
two data sets are independent if P(A \cap B) = P(A)P(B)

What are these definitions for?

  • We defined the reduced universe as such because we want to be able to say, “In the universe where data set A is true…”.
  • What do we mean by a data set being true in the first place? Say A = \{\text{Alice is a big mouse.}\}. Then in a particular universe, A is true if Alice is a mouse and if she is big and not otherwise. The truthiness of A is the truthiness of all its conditions.
  • Why the definition of conditional probability? We want to have a way of saying, “The truth of A depends on the truth of B by this much.” […]
  • In this vein, saying that two data sets are independent is saying that whether or not B is true does not affect whether or not A is true: P(A \cap B) = P(A)P(B) \implies P(A|B) = \frac{P(A \cap B)}{P(B)} = \frac{P(A)P(B)}{P(B)} = P(A).

[…]

Fig. 1: A versus \not A


Definitions

Good Thinking
following what works; see the Twelve Virtues of Rationality, particularly the twelfth virtue

1. E. T. Jaynes. “Probability: The Logic of Science.” 1995. Print.

2. See the About section of gwern.net.

3. See Terry Tao’s post.

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