Suppose I look at the registry of residents of my town and
pick a person at random.
What is the probability that this person is
under 18 years of age?
The answer is about 25%.
Suppose now that I tell you that this person is married.
Will you give the same answer?
Of course not.
The probability of being less than 18 years
old is now much smaller.
What happened here?
We started with some initial probabilities that reflect
what we know or believe about the world.
But we then acquired some additional
knowledge, some new evidence--
for example, about this person's family situation.
This new knowledge should cause our beliefs to change,
and the original probabilities must be replaced with new
probabilities that take into account the new information.
These revised probabilities are what we call conditional
probabilities.
And this is the subject of this lecture.
We will start with a formal definition of conditional
probabilities together with the motivation behind this
particular definition.
We will then proceed to develop three tools that rely
on conditional probabilities, including the Bayes rule,
which provides a systematic way for incorporating new
evidence into a probability model.
The three tools that we introduce in this lecture
involve very simple and elementary mathematical
formulas, yet they encapsulate some very powerful ideas.
It is not an exaggeration to say that much of this class
will revolve around the repeated application of
variations of these three tools to increasingly
complicated situations.
In particular, the Bayes rule is the foundation for the
field of inference.
It is a guide on how to process data and make
inferences about unobserved quantities or phenomena.
As such, it is a tool that is used all the time, all over
science and engineering.