6.1 Probability

To make a good decision, an agent cannot simply assume what the world is like and act according to those assumptions. It must consider multiple possible contingencies and their likelihood. Consider the following example.

Example 6.1: Many people consider it sensible to wear a seat belt when traveling in a car because, in an accident, wearing a seat belt reduces the risk of serious injury. However, consider an agent that commits to assumptions and bases its decision on these assumptions. If the agent assumes it will not have an accident, it will not bother with the inconvenience of wearing a seat belt. If it assumes it will have an accident, it will not go out. In neither case would it wear a seat belt! A more intelligent agent may wear a seat belt because the inconvenience of wearing a seat belt is far outweighed by the increased risk of injury or death if it has an accident. It does not stay at home too worried about an accident to go out; the benefits of being mobile, even with the risk of an accident, outweigh the benefits of the extremely cautious approach of never going out. The decisions of whether to go out and whether to wear a seat belt depend on the likelihood of having an accident, how much a seat belt helps in an accident, the inconvenience of wearing a seat belt, and how important it is to go out. The various trade-offs may be different for different agents. Some people do not wear seat belts, and some people do not go out because of the risk of accident.

Reasoning under uncertainty has been studied in the fields of probability theory and decision theory. Probability is the calculus of gambling. When an agent makes decisions and uncertainties are involved about the outcomes of its action, it is gambling on the outcome. However, unlike a gambler at the casino, the agent cannot opt out and decide not to gamble; whatever it does - including doing nothing - involves uncertainty and risk. If it does not take the probabilities into account, it will eventually lose at gambling to an agent that does. This does not mean, however, that making the best decision guarantees a win.

Many of us learn probability as the theory of tossing coins and rolling dice. Although this may be a good way to present probability theory, probability is applicable to a much richer set of applications than coins and dice. In general, we want a calculus for belief that can be used for making decisions.

The view of probability as a measure of belief, as opposed to being a frequency, is known as Bayesian probability or subjective probability. The term subjective does not mean arbitrary, but rather it means "belonging to the subject." For example, suppose there are three agents, Alice, Bob, and Chris, and one die that has been tossed. Suppose Alice observes that the outcome is a "6" and tells Bob that the outcome is even, but Chris knows nothing about the outcome. In this case, Alice has a probability of 1 that the outcome is a "6," Bob has a probability of (1)/(3) that it is a "6" (assuming Bob believes Alice and treats all of the even outcomes with equal probability), and Chris may have probability of (1)/(6) that the outcome is a "6." They all have different probabilities because they all have different knowledge. The probability is about the outcome of this particular toss of the die, not of some generic event of tossing dice. These agents may have the same or different probabilities for the outcome of other coin tosses.

The alternative is the frequentist view, where the probabilities are long-run frequencies of repeatable events. The Bayesian view of probability is appropriate for intelligent agents because a measure of belief in particular situations is what is needed to make decisions. Agents do not encounter generic events but have to make a decision based on uncertainty about the particular circumstances they face.

Probability theory can be defined as the study of how knowledge affects belief. Belief in some proposition, α, can be measured in terms of a number between 0 and 1. The probability α is 0 means that α is believed to be definitely false (no new evidence will shift that belief), and a probability of 1 means that α is believed to be definitely true. Using 0 and 1 is purely a convention.

Adopting the belief view of probabilities does not mean that statistics are ignored. Statistics of what has happened in the past is knowledge that can be conditioned on and used to update belief. (See Chapter 7 for how to learn probabilities.)

We are assuming that the uncertainty is epistemological - pertaining to an agent's knowledge of the world - rather than ontological - how the world is. We are assuming that an agent's knowledge of the truth of propositions is uncertain, not that there are degrees of truth. For example, if you are told that someone is very tall, you know they have some height; you only have vague knowledge about the actual value of their height.

If an agent's probability of some α is greater than zero and less than one, this does not mean that α is true to some degree but rather that the agent is ignorant of whether α is true or false. The probability reflects the agent's ignorance.

For the rest of this chapter, we ignore the agent whose beliefs we are modeling and only talk about the probability.