Introduction

Up till now our agents all operated under the assumption that everything is true, false, or unknown
Would like agents that can operate given some probability that something is true.
For example, we would like to make plans and say a plan will succeed provided certain conditions are met at time of execution. Then estimate probability those conditions will be met to do reasoning.
As a particular instance of this: We might reason that "the having fun in park plan" will succeed if it is not raining. We then might estimate this probability to try to figure out if we want to execute plan at a given time in future.
The right thing to do in a given situation -- the rational decision --depends both on the relative importance of various goals and the likelihood that, and degree to which, these goals can be achieve.
We will now spend some time developing the probability needed to talk about rational decision making and later about statistical learning.

How to make a logical model of uncertainty.

Consider:
`forall p, Sym\p\t\om(p, \T\o\othache) => Disease(p, Cavity)`
This is not always true.
For example, gum disease might have been the cause of the symptom.
To make a valid statement we might try to enumerate the possibilities:
`forall p Sym\p\t\om(p, \T\o\othache) => Disease(p, Cavity) vv Disease(p, GumDisease) vv ...`
This kind of enumerations is quite hard: We are often too lazy or don't know all the possibilities.
The point of above is modeling uncertainty using just first-order logic or propositional logic is both awkward and potentially hard to do -- We want to use a formalism more geared to modeling these things.
We will use probability theory to do this.
We will keep a notion of propositional and 1st order sentence, but now add a notion of degree of believe in that sentence.

Uncertainty and Rational Decisions

Before we go into probability let's briefly look at how we would like to modify our agents to work with them.
First to make a decision as to what to do next the agent need to have preferences between it possible goals. This is called a utility theory. For example I prefer a cup of coffee twice as much as a cup of tea. This is relative to the agent.
A Decision Theory takes this utility and adds to it some calculation based on the probability of achieving each of the possible goals.
The agent should choose an action which yields the highest expected payoff among the available choices. This is called the principle of maximum expected utility (MEU).

Below is some pseudo-code for an agent acting in this way:

function DT-Agent(percept) returns an action
    persistent belief-state, probabilistic beliefs about 
                             the current state of the world
               action, the agent's action

    update belief-state based on action and percept
    calculate outcome probabilities for actions
       given action descriptions and current belief-state
    select amounst these action  the one with highest expected utility
        given probabilities of outcomes and utility information
    return action

What probabilities are about

In probability, we will have a notion of possible worlds, which we will call the sample space
These are the sets of things that could be true. Elements/possible worlds of this space are mutually exclusive and they exhaust the space.
For example, for a pair of dice, the sample space might be all possible ordered pairs of the numbers 1-6: (1,1), (1,2), ... (2,1) ..., (6,6).
Often `Omega` is used to refer to a sample space and `omega` to an element of this space.
A probability model associates a numerical number `P(omega)` between `0` and `1` with element.
We require that `sum_(omega in Omega) P(omega) = 1`.
Probabilistic assertions and queries are not usually about particular worlds, but about sets of them. For example, what's the odd that two dice add up to 11.
In probability theory, these sets of possible worlds are called events and these events are almost always described by propositions in a formal language.
For any proposition `\phi`, `P(phi) = sum_(omega in phi) P(omega)`. For example P(Total = 11) = P((5,6)) + P((6,5)) = 1/36 + 1/36 = 1/18, assuming the die are fair.

Conditional versus Unconditional Probabilities

Probabilities such as P(Total = 11) and P(doubles) are called unconditional or prior probabilities; they refer to degrees of belief in propositions in the absence of any other information.
Most times we have some information, usually called evidence.
For example, we might ask what's the odd of getting 11 given that we know one dice has already landed as a 5?
This is called a conditional probability. We would write this probability as `P(\T\o\tal = 11 | Die_1 = 5)`
Mathematically, conditional probabilities can be computed using the formula:
`P(a|b) = frac(P(a^^b))(P(b))`
which holds whenever `P(b) > 0`.
For example,
`P(doubl\es | Die_1 =5) = frac(P(doubl\es ^^ Die_1 =5))(P(Die_1=5)) = frac(1/36)(1/6) = 1/6`

A Language for Probability Assertions

In propositional logic or first-order logic we build up statements with variables.
We would like a similar notion for probability...
For us, a random variable is a function from a set of possible states into probabilities (values between 0 and 1).
For example, the domain of Total is `{2, ... 12}`. Its range could be the probability of the dice summing to the given value.
There are a few different common kinds of random variables:
- A boolean random variable has domain {true, false}. For example, one might have variable Cavity with domain {truth, false} where `Cavity (true) = p in [0,1]` and `Cavity (false) = 1 - p`.
- A discrete random variable has domain some fixed finite set of values. For example, Weather has domain {Sunny, rainy, cloudy, snowy} with Weather(Sunny) = .4 Weather(rainy) = .2 Weather(cloudy) = .38 Weather(snowy) = .02. We call the vector (0.4, 0.2, 0.38, 0.02) a probability distribution.
- A continuous random variable which has domain usually consisting of an infinite number of states and where the function is continuous on this domain (pre-image of open set is open). For example, the real interval [18 to 26] might be the domain. The function might be the odds of getting that temperature in some city at noon on some day. Such a continuous random variable is often called a probability density function.
Given two domains it is often interesting to take their Cartesian product to make a new domain. We can often create a new probability distribution on the resulting set called a joint probability distribution. For example, P(Weather, Cavity).
This start to allow us to talk about probability for sentences because we can write things like:
`P(Weather = su\n\n\y ^^ Cavity = true)`
So joint distributions give us a notion of probabilities for `^^`.

Probabilities for Propositional Formulas

Given an event `a`. The probability that `a` did not happen is the sum over all the situations where `a` did not occur. By our earlier definitions, this and the situation where `a` did occur must sum to 1. Hence. `P(neg a) + P(a) = 1` and so `P(neg a) = 1 - P(a)`.
We can use the principle of inclusion exclusion to get a formula for the probability of the OR of two things as:
`P(a vv b) = P(a) + P(b) - P(a ^^ b)`.
Next day, we will talk about how to use probabilities to do inference.

Quantifying Uncertainty

Outline

Introduction

How to make a logical model of uncertainty.

Uncertainty and Rational Decisions

What probabilities are about

Conditional versus Unconditional Probabilities

A Language for Probability Assertions

Probabilities for Propositional Formulas