Uncertainty

Consider a situation where an action $A_t$ is leave for airport $t$ minutes before flight.

Problem: Will $A_{25}$ get me to the airport on time?

Since there a lot of uncertainties, we can make assumptions

• Assume my car doesn’t have a flat tire
• Assume $A_{25}$ works unless contradicted by evidence

Issues: what assumptions are reasonable and how should we handle contradictions?

Probability can help

• Given the available evidence, $A_{25}$ will get me there on time with probability of 0.04

Probability

Probabilistic assertions summarize effects of

laziness: failure to enumerate exceptions, qualifications, etc.

ignorance: lack of relevant facts, initial conditions, etc

Subjective or Bayesian probability:

Probabilities relate propositions to one’s own state of knowledge

e.g $P(A_{25}\; |\; \text{no reported accident}) = 0.06$

Probabilities of propositions change with new evidence

e.g. $P(A_{25}\; |\; \text{no reported accident, 5a.m.}) = 0.15$

Making Decisions under uncertainty

Suppose I believe the following:

$P(A_{25}\text{ gets me there on time}\; |\; ...) = 0.04$

$P(A_{90}\text{ gets me there on time}\; |\; ...) = 0.70$

$P(A_{120}\text{ gets me there on time}\; |\; ...) = 0.95$

$P(A_{1440}\text{ gets me there on time}\; |\; ...) = 0.99999$

Which action should we choose?

• Depends on preference for missing flight vs. airport cuisine, etc.
• Utility theory says that every state has a degree of usefulness, or utility, to an agent and that the agent will prefer states with higher utility