## Introduction

In this series of articles, We’ll dig deep into understanding **Bayesian inference**, starting from the basics.

The main idea behind Bayesian statistics is the Bayes theorem, we need to understand some concepts first.

## TL;DR

Let´s take the following poker cards as an example:

Our `experiment`

is to take two cards for the deck, one at a time, each card extraction is an `event`

. We define as outcomes that a card is of a defined

If we introduce the card in the deck again before extracting the next one, we’ll have `indepedent`

events, if we don’t the events will be `dependent`

, due to the fact that the fact of removing one card from the deck will alter the probabilities of the next event.

## Concepts

Let’s illustrate these concepts with poker cards and rolling dices:

- An
**experiment**is a well-defined action, that might result in a number of outcomes. **Outcome**: Possible result of an`experiment`

. Each outcome is unique, and different outcomes are`mutually exclusive`

.- The
**sample space**is the set of all possible outcomes. - An
**event**is a collection of outcomes. - A
**probability**is a number between 0 and 1, both included, that describes the degree of confidence that we have in a prediction, 0 is none or false and 1 is all or true.

$$ P ( hearts ) = 0.25 $$

The probability of an event $ E $ is $ P(E) $ is the number of possible outcomes of event $ E $ is $ n(E) $ divided by the number of outcomes of the sample space $ n(S) $. $$ \mathrm { P } ( \mathrm { E } ) = \frac { \mathrm { n } ( \mathrm { E } ) } { \mathrm { n } ( \mathrm { S } ) } $$

#### Mutually exclusive events

We call **mutually exclusive events**, to two or more events that cannot happen simultaneously. For example in Poker cards we cannot draw a card that is hearts and diamonds at the same time, but we can have a card that is 2 and diamonds.
$$
P ( hearts \cup diamonds ) = P ( hearts ) + P ( diamonds )
$$

#### Conditional probability

A **conditional probability** is a probability based on some premise. I know that in my city there is a 5% chance of rain, but given that is autumn the probability raises to 25%, the premise is that it is autumn, this premise gives us additional information that allows us to have a more accurate prediction.
$$
P ( A \cap B ) = P ( A ) * P ( B )
$$

We roll a six-sided dice, this is an experiment, we have 6 possible outcomes. If we get the number

3by rolling the dice, we’ll say that from this event of rolling the dice we observed the outcome3, that has a prior probability of 1/6.

#### Independent Events

We’ll say that 2 events are *independent* if one event doesn’t give any information about another event.

We have the following outcomes that we want to analyse:
A = Outcome 3 in the first dice.
B = Outcome 3 in the second dice.
$$
P ( A | B ) = P ( A )
$$
We’ll read “*The probability of observing 3 in the first dice given that we got 3 in the second dice is the observing of getting 3 in the first dice.*”

Rolling a second dice doesn’t give us any information about the first dice.

When two events are *independent* the probability of both of them occurring is:

$$ P ( A \cap B ) = P ( A ) * P ( B ) $$ We can generalize this to multiple events:

$$ P \left[ \bigcap _ { i = 1 } ^ { n } A _ { i } \right] = \prod _ { i = 1 } ^ { n } P \left( A _ { i } \right) $$ Here A means rolling a dice, it can be the same dice or different dices.

#### Dependent Events

When two events are *dependent* the probability of both of them occurring is:
$$
\mathrm { P } ( \mathrm { A } \text { and } \mathrm { B } ) = \mathrm { P } ( \mathrm { A } ) \times \mathrm { P } ( \mathrm { B } | \mathrm { A } )
$$

#### Prior and Posterior distributions

The **prior probability distribution** (prior), is ones beliefs about its quantity before some evidence is taken into account.

The **posterior probability distribution** (posterior), is the revised probability of an event occurring after some evidence has been taken into account.

We´ll see in the next article more about these probability distributions.