What are the basic components of decision-making under risk?
The components include available actions/alternatives, events or states of the world, outcomes, and preferences.
What is the difference between a certain and an uncertain state in decision-making?
Under certainty, alternatives are directly associated with deterministic outcomes, while under uncertainty, the outcome depends on an external event or state of the world.
How is uncertainty modeled under risk in decision theory?
Uncertainty is fully specified by assigning probabilities to possible outcomes or states of the world. Alternatives can be represented by a "lottery."
What is a lottery in decision theory?
A lottery is a representation of alternatives, where each alternative is associated with a certain probability and an outcome, e.g., L=(p1,...,pj,...,pn,x1,...,xj,...,xn)L=(p1,...,pj,...,pn,x1,...,xj,...,xn).
What do the axioms of probability theory according to Kolmogoroff state?
The three axioms are:
p(E)∈[0,1]p(E)∈[0,1] for every event EE.
p(Θ)=1p(Θ)=1 for the entire sample space ΘΘ.
The probability of the union of disjoint events is the sum of their individual probabilities, i.e., p(E1∪E2)=p(E1)+p(E2)p(E1∪E2)=p(E1)+p(E2) if E1∩E2=∅E1∩E2=∅.
What is meant by conditional probability?
Conditional probability is the probability of an event occurring given that another event has already occurred. It is denoted as p(E1∣E2) and calculated as p(E1∩E2)/p(E2).
What is the concept of "Bayes' Rule"?
Bayes' Rule is used to update the probability of an event based on new information. It is given by:
p(E∣F)=p(E)⋅p(F∣E)/p(F)
What is the purpose of using a decision matrix?
A decision matrix is used to organize and analyze the outcomes of different actions under various states of the world, facilitating structured decision-making.
What are sensitivity and specificity in the context of medical tests?
Sensitivity measures the test's ability to correctly identify individuals with the condition (true positive rate). Specificity measures the test's ability to correctly identify individuals without the condition (true negative rate).
What does stochastic independence between two events mean?
Two events E1E1 and E2E2 are stochastically independent if p(E1∣E2)=p(E1), or equivalently, p(E1∩E2)=p(E1)⋅p(E2)
What is the interpretation of probabilities in decision theory?
Probabilities can be interpreted in different ways, such as the frequentist interpretation (based on relative frequencies of an event occurring) or the subjective interpretation (based on an individual's belief about an event's likelihood).
What is the frequentist interpretation of probability?
It views probabilities as limits of relative frequencies over a large number of repetitions. For example, if an event E occurs in m out of n repetitions, the probability is m/n
What is the subjective interpretation of probability?
The subjective interpretation assigns probabilities based on an individual's personal belief or degree of confidence in the occurrence of an event, given their knowledge and information.
What is meant by utility in decision-making?
Utility refers to the numerical measure of the value of an outcome, as perceived by the decision-maker. It allows for the ranking of outcomes from worst to best based on preferences.
Why is assigning probabilities to states of the world important in decision-making under risk?
Assigning probabilities helps quantify uncertainty and allows decision-makers to compare and evaluate alternatives based on their expected outcomes and risks.
What is the principle of "expected utility" in decision-making?
Expected utility is a concept in decision theory that helps decision-makers evaluate risky alternatives. It is calculated as the sum of the utilities of all possible outcomes, each weighted by its probability. Decision-makers choose the alternative with the highest expected utility.
What is the difference between risk and uncertainty in decision-making?
Risk refers to situations where the probabilities of different outcomes are known, whereas uncertainty refers to situations where these probabilities are unknown or cannot be assigned.
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