Probability for Machine Learning

Motivation

Given that many computer scientists and software engineers work in a relatively clean and certain environment, it can be surprising that machine learning makes heavy use of probability theory

Nearly all activities require some ability to reason in the presence of uncertainty

Sources of Uncertainty

1. Inherent stochasticity in the system being modeled 

   such as a hypothetical card game where we assume that the cards are truly shuffled into a random order.

2. Incomplete observability: Even deterministic systems can appear stochastic when we cannot observe all the variables that drive the behavior of the system e.g Monty Hall problem
3. Incomplete modelling:  When we use a model that must discard some of the information we have observed, the discarded information results in uncertainty in the model’s predictions.

Quiz: Monty Hall Problem

Behind one door is a car; behind the others, goats.

You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3,

which has a goat.

He then says to you, "Do you want to pick door No. 2?"

Is it to your advantage to switch your choice?

Suppose you're on a game show, and you're given the choice of three doors

Random Variables

A random variable is a variable that can take on different values randomly. They may be continous or discrete

A discrete random variable is one that has a finite or countably infinite number of states

A continuous random variable is associated with a real value.

Denote the random variable itself with a lowercase letter 

Probability Distributions

Is a description of how likely a random variable is to take on each of its possible states

Discrete variables & Probability Mass Functions (PMF)

Continous variables & Probability Density Functions (PDF)

Probability Distribution: PMF

• Denote PMF with capital P
• Properties
  • P must be the set of all possible states of x
  • Between 0 and 1
  •