Uncertainty

Agents may not always have a complete understanding of their current state or tomorrow's outcomes after their actions.
Agents maintain a "belief state" which is a representation of all possible states of the environment.

Large Belief State: Requires considering every possible states and
related actions, even highly unlikely ones.
Complex Contingency Plans: Plans may become very large, accounting for unlikely scenarios.

Sometimes, no guaranteed plans can lead to achieving the goal, necessitating agents to act despite uncertainty.

Goal: Automated taxi must deliver a passenger to
the airport on time.
Plan: Leave 90 minutes before the flight and
drive at a reasonable speed.
What would be the challenges that agent is not
considering? Challenges in Certainty: A logical agent cannot guarantee that the selected plan will succeed due to numerous uncontrollable
factors: Car Breakdown, Accidents, road closures, Weather and other rare occurences
Conclusion: Plan is only conditionally valid - success depends on the absence of various unpredictable failures

Not all toothaches indicate cavities; several other causes could exist like gum disease or abscess.

Laziness: Enumerating all possible causes and outcomes for strict rules is impractical.
Theoretical Ignorance: Medicine doesn't have a complete theory for all conditions and relationships.
Practical Ignorance: Uncertainties can stem from incomplete patient information due to unperformed tests.

Probability theory offers a framework for representing uncertainty stemming from laziness and ignorance.
Although certain about a patient's condition is not always feasible, statistical estimates like 80% can be inferred from prior data regarding toothaches and cavities.

Probabilities refer to inferred conditions rather than absolute truths. E.g., "The probability that a patient has a cavity given they have a toothache is 0.8."
The probability may differ when incorporating additional data, leading to contrasting evaluations (0.4, 0).

The complete set of potential scenarios is termed the sample space, wherein:
- Mutually Exclusive: No two scenarios can exist simultaneously.
- Exhaustive: Every possible scenario must be represented.

All possible worlds have probabilities ranging between 0 and 1 (0 <= P(ω) <= 1).
The total probability encompassed by all possible worlds should equal 1 (ΣωΩ P(ω) = 1).

Rolling a die has distinct probabilities (1/6) of landing on each number due to equal possibility.
Rolling two dice leads to varied probabilities for sums (e.g., P(sum to 12) = 1/36 and P(sum to 7) = 6/36).

Unconditional probabilities (e.g. P(sum to 12), P(sum to 7)) convey beliefs without further conditions.
In contrast, conditional probabilities focus on probabilities under specific evidence or conditions, crucial for intelligent decision-making systems.

Initial Information: If a patient has a toothache, P(cavity|toothache) = 0.6 based on previous data.
New Evidence: If further examination finds no cavities, one should update their probability assessments accordingly.

Independence between events suggests the occurrence of one event has no impact on another; in this case, dental issues and weather are independent.
Conditional independence can signify that two variables do not affect each other when a third variable is fixed.

A Bayesian Network captures dependencies among various random variables in a structured, graphical formulation; this representation facilitates easier data interpretation.
Nodes symbolize random variables, and arrows depict dependency relationships, aiding in conditional probability calculations.