Example 1: Imagine a billiard table, and suppose that the billiard track bounces off the table many times, so we don't know where the ball will end up. Let's say the billiard table is L, and the probability that the ball ends up x feet from the edge is x/L.

The inverse probability problem: Suppose we observe that the ball's final stop is a foot away from the table, but we don't know the length of the table, L.

Bayes' main work is to break this cognitive asymmetry and propose a method to estimate the inverse probability.


Inverse-probability: break cognitive asymmetry 

Inverse probability, as supposed to forward probability, is a conditional probability that a hypothesis (cause), H, is true given that the evidence (effect), E is true.  Inverse probabiity is harder to estimate than estimating a forward probability.  

It can be expressed as:

	 $$ P(H | E) $$


University of Michigan - Ann Arbor

Bayes Theorem is about predicting the probability of an event A occurring given an event B has already occurred. We can use the inverse probability of this statment and the probability of each event to find this value. 
P(A) is called prior probability, P(B|A) is Likelihood, P(B) is evidence, and P(A|B) is posterior probability.


Bayes Theorem Overview

Pearl, J., & Mackenzie, D. (2018). $$\mathit{The \ Book \ of \ Why: \ The \ New \ Science \ of \ Cause \ and \ Effect.}$$ Basic Books.

The Book of Why

A method of supervised classification that takes many features, assumed to be independent, and uses Bayes' Theorem to determine given those independent variables how likely the likelihood of another event occurring.

It's called "Naive" because it assumes that features are conditionally independent, given the class, i.e., for all instances of a given class, the features have little or no correlation with each other.

(Naive) Bayes Classifier

According to Bayes' theorem, we have
$$
P\left(Y=k | X=x\right)=\frac{P\left(Y=k\right) P\left(X=x | Y=k\right)}{\sum_{l=1}^{K} P\left(Y=l\right) P\left(X=x | Y=l\right)}\\
=\frac{\pi_{k} f_{k}(x)}{\sum_{l=1}^{K} \pi_{l} f_{l}(x)}.
$$
In LDA with the assumption of Gaussian distribution in every class, we have the Gaussian density 
$$
f_k(x)=\frac{1}{(2 \pi_k)^{p / 2}|\boldsymbol{\Sigma}|^{1 / 2}} \exp \left(-\frac{1}{2}(x-\mu_k)^{T} \boldsymbol{\Sigma}^{-1}(x-\mu_k)\right)
$$
for a $p$-dimensional random vector $X$ with the distribution $N(\mu_k, \mathbf{\Sigma})$. All we need to do is to plug $f_k(x)$ into Bayes' theorem above and find out the $k$ which maximize $P\left(Y=k | X=x\right)$. LDA is a Bayes classifier with the assumption of Gaussian distribution in every class.

Using Bayes' theorem in LDA

Bayesian networks were developed to understand reasoning under conditions of uncertainty. It is based on Bayes Theorem and is a probabilistic model that represents a set of variables and their conditional dependencies by being able to find the probability of facts being true/false given already defined facts. A Bayesian network is represented by a directed acyclic graph which helps to visualize the probabilistic model for reviewing relationships between random variables and reasonings for causal probabilities given certain facts



Bayesian Networks

This example in the book is a discussion on how we can update our beliefs using Bayes's rule.  The question the example tries to address is, "What is the probability that she has breast cancer, given that the test came out positive?", which can be expressed as P(D | T), where D is the hypothesis, disease, and T is the evidence, test.

$$
        $$(Updated probability of D) = P(D | T) = (likelihood ratio)) × (prior probability of D)$$
      $$

(Note) The "likelihood ratio" is given by P(T | D)/P(T). 



Mammogram (Breast Cancer Screening Example)

There are three out of four binary classification metrics mentioned in the Book of Why; true positive, false positive, and false negative.  The fourth one that is not mentioned in the book is true negative.  The following are the two primary building blocks of such four metrics: 

1. True vs. False: These are about prediction's correctness whether a prediction outcome was made accurately/ correctly.  The correct prediction is True, and the incorrect prediction is False.
2. Positive vs. Negative: These are about a binary classification -(e.g.) In the mammogram example in the book, "positive" is positive test results (breast cancer), whereas "negative" is negative test results (not breast cancer).  

The combinations of the above concepts result in True positive, True negative, false positive, and false negative, each of which is explained in the subsequent nodes to follow.  


Binary Classification Metrics

Inverse Probability

Forward probability, as supposed to inverse probability, is the conditional probability that an evidence (effect), E, is true given that the hypothesis (cause), H is true.  For the definition of inverse probability, see the node, "Inverse Probability Defined." 

It can be expressed as:

  $$P(E | H)$$

Estimating the forward probability is easier than estimating the inverse probability because, as long as we know the cause, it is easy to estimate the probability of the effect.  

Pearl calls forward probability, calculating conditional probability in the direction, in which "our judgment is more reliable."


Forward Probability

Bayes' Theorem can also be used as the following :

$$ P(A | B) = \frac{P(A \cap B)}{P(B)}$$
The way to think about this is that given two events A and B, the probability that A happens, given B occurs, is the probability of their intersection occuring i.e. both events occuring, over the probability that B occurs.
The easiest way to see why this works is for Independent events. If A and B are independent, the probability that they both occur is the product of their individual probabilities. Which then simplifies the above equation to :
$$ P(A|B)    =    \frac{P(A \cap B)}{P(B)}$$
$$ \implies P(A|B) = \frac{P(A).P(B)}{P(B)}$$
$$ \implies P(A|B) = P(A)$$
Which makes sense since the events are independent.

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