Nature has rules. Universal and permanent rules
Whatever happens in the future is the result of applying the rules to the current state of the universe
\[\text{State}_{t+1} = F(\text{State}_t, \text{Parameters})\]
We just need to follow the logic consequences
If we launch a ball, and we know the angle and speed, then we can predict where it will fall
We can launch a rocket and land in the moon
We can put a satellite to explore the Earth, find our position using GPS, and watch TV from other countries
We can build a plane that can fly and carry us to other countries
If the world is deterministic, and we know
then we can predict everything that will happen
and everything that has happened before
We just need to use logic
We do not know all the rules
Among the rules that we know, some
have complex solutions. They are hard to calculate
depend on parameters that we do not know
give very different results when parameters change a little bit
Since we have imperfect knowledge, we must deal with degrees of certainty
How much we believe some predicate is true
We want to give a numeric value to the chances that our experiment is successful
We want to compare the chances of success versus failure
We will call experiment to any procedure generating a result which we do not knew before doing it
This include “natural experiments” and observations of the nature
An experiment produces a single outcome
We do not know the outcome until we perform the experiment
If we knew the outcome before doing the experiment, we would not be doing it
It is useful to give a name to the set of all possible outcomes
We will call it \(Ω\)
Exercise: What is \(Ω\) in each case?
An event is a yes-no question that will be answered by the experiment
Having fever is an event. Thermometer showing 38.2°C is an outcome.
An event will become either true or false after an experiment
For example, a dice can be either 4 or not
We want to give a value to our rational belief that the event will become true after the experiment
The numeric value is called Probability
Most people are familiar with the naive idea of Probability
\[ℙ(A)=\frac{\text{Number of cases where }A\text{ is true}}{\text{Total number of cases}}\]
This is a useful first approach, but it is easy to get confused
It is not obvious which are the cases
For example, if you throw a dice, what is the probability of getting a 6?
We have to be careful
new information may change our confidence
For example, if we learn that the dice outcome is an even number, what is the probability of getting a 6?
What if we learn that the outcome is an odd number?
They
They are subjective, because different subjects may have different knowledge
But they are not arbitrary.
We must use all the available information, and follow all the rules
We will use capital letters to represent events. For example
\(A\): The dice outcome is 6
\(B\): The dice outcome is even
The probability of \(A\), given that we know \(B\) is \[ℙ(A|B)\]
This is called conditional probability
We always evaluate probabilities based on what whe know
If the background knowledge is well known, and does not change, we sometimes write \[ℙ(A)\]
This is to simplify notation.
But do not forget that there is an implicit context.
The order is relevant \[ℙ(A|Z)≠ℙ(Z|A)\] There are two events, 𝐴 and 𝑍
The one written after |
is what we assume to be true
The one written before |
is what we are asking for
One we know, the other we do not
The set of all possible outcomes is often called Ω
An event 𝐴 can be seen as the subset of all outcomes that make the event true
For example, \[Fever=\{Temp>37.5°C\}\]
It is useful to think that the probability of an event is the area in the drawing
The total area of Ω is 1
Usually we do not know the shape of 𝐴
Our rational beliefs depend on our knowledge
If we represent our knowledge (or hypothesis) by 𝑍, the the probability of an event 𝐴 is written as \[ℙ(A|Z)\] We read “the probability of event 𝐴, given that we know 𝑍”
For example, “the probability that we get a 4, given that the dice is symmetrical”
Now outcomes are limited only to the 𝑍 region
We measure the area of \(ℙ(A|Z)\) with respect to the area of 𝑍 instead of Ω
The shape of 𝑍 is often unknown
It has been proven that probabilities must be like this
A probability is a number between 0 and 1 inclusive \[ℙ(A) ≥ 0\text{ and } ℙ(A)≤1\]
The probability of an sure event is 1 \[ℙ(\text{True}) = 1\]
The probability of an impossible event is 0 \[ℙ(\text{False}) = 0\]
We are interested in non-trivial events, that are usually combinations of smaller events
For example, we may ask “what is the probability that, in a group of 𝑛 people, at least two persons have the same birthday?”
Fortunately, any complex event can be decomposed into simpler events, combined with and, or and not connectors
Exercise: decompose the birthday event into simpler ones
If the event 𝐴 becomes more and more plausible, then the opposite event not 𝐴 becomes less and less plausible
It can be shown that we always have \[ℙ(\text{not } A) = 1-ℙ(A)\]
We have \[ℙ(\text{not } A) = 1-ℙ(A)\] therefore \[ℙ(A) + ℙ(\text{not } A) = 1\]
A coin is an experiment where \(Ω=\{"H","T"\}\)
Let’s take \(A\) to be the outcome is “H”
\[ℙ(X="H") + ℙ(X="T") = 1\]
Without more information (or hypothesis) we cannot know more
If we do not have any reason to believe that one side of the coin has more chance than the other, then we assume that both sides have equal chances
If all alternatives are symmetric, then the probabilities are equal \[ℙ(X="H")= ℙ(X="T")\] Therefore \[ℙ(X="H")= ℙ(X="T")=\frac 1 2\]
Lets consider two different events \(A\) and \(B\)
(for instance, if X is the result of a dice, “X>3” and “X is
even”)
\[\underbrace{B}_{m} = B\text{ and }(A\text{ or not }A) = \underbrace{(B\text{ and }A)}_{m_1}\text{ or }\underbrace{(B\text{ and not }A)}_{m_2}\] We see that \(m=m_1+m_2\) because, for an outcome where \(B\) is true, we have either “\((B\text{ and }A)\) is true” or “\((B\text{ and not }A)\) is true”, but never both
Show that for a fair dice the probability of each side is 1/6.