What Is The Expected Value (outside Of Frequentism)?

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Introduction

The concept of expected value is a fundamental idea in probability theory, used to quantify the average outcome of a random experiment. In the context of frequentism, the expected value is calculated as the sum of the product of each possible outcome and its probability of occurrence, over an infinite number of trials. However, this perspective is not the only way to understand the expected value, and in this article, we will explore alternative interpretations.

Frequentism vs. Non-Frequentism

Frequentism is a philosophical approach to probability that views probability as a long-run frequency of an event. In this framework, the expected value is calculated as the average outcome of a random experiment, repeated an infinite number of times. However, there are alternative perspectives on probability, such as Bayesianism, which views probability as a degree of belief or a measure of uncertainty.

Bayesian Interpretation of Expected Value

From a Bayesian perspective, the expected value is not just a mathematical concept, but a representation of our degree of belief about the outcome of a random experiment. In the context of the game mentioned earlier, where we win $4 if a coin lands heads and $2 if it lands tails, the expected value of our winnings is $3. This means that, according to our degree of belief, we expect to win an average of $3 per game.

Subjective Expected Value

The subjective expected value (SEV) is a concept that is closely related to the expected value, but with a twist. Instead of calculating the expected value based on the probability of each outcome, we calculate it based on our degree of belief about the outcome. In other words, the SEV is a measure of how much we expect to win, based on our subjective assessment of the probability of each outcome.

Imprecise Probability

Imprecise probability is a concept that is used to represent uncertainty when the probability of an event is not precisely known. In this context, the expected value is not a single number, but a range of values, representing the uncertainty about the outcome. This approach is particularly useful when dealing with complex systems or uncertain data.

Von Mises' Frequency Theory

Von Mises' frequency theory is a philosophical approach to probability that views probability as a long-run frequency of an event. However, unlike frequentism, von Mises' theory does not require an infinite number of trials to calculate the probability. Instead, it uses a finite number of trials, and the probability is calculated based on the frequency of the event in those trials.

De Finetti's Subjective Probability

De Finetti's subjective probability is a philosophical approach to probability that views probability as a degree of belief or a measure of uncertainty. In this context, the expected value is not just a mathematical concept, but a representation of our degree of belief about the outcome of a random experiment.

Expected Utility

Expected utility is a concept that is used to evaluate the desirability of a decision, based on the expected value of the outcome. In this context, the expected value is not just a mathematical concept, but a representation of the desirability of the outcomeConclusion

In conclusion, the expected value is a concept that has been interpreted in various ways, depending on the philosophical approach to probability. While frequentism views the expected value as a long-run frequency of an event, Bayesianism views it as a degree of belief or a measure of uncertainty. The subjective expected value, imprecise probability, von Mises' frequency theory, de Finetti's subjective probability, and expected utility are all alternative interpretations of the expected value, each with its own strengths and weaknesses.

References

  • De Finetti, B. (1937). "Foresight: Its Logical Laws, Its Subjective Sources." In Studies in Subjective Probability (pp. 53-118).
  • von Mises, R. (1928). Probability, Statistics and Truth. Dover Publications.
  • Savage, L. J. (1954). The Foundations of Statistics. John Wiley & Sons.
  • Walley, P. (1991). Statistical Reasoning with Imprecise Probabilities. Chapman and Hall.

Further Reading

  • Probability and Statistics: The Science of Uncertainty by David J. Hand
  • The Foundations of Statistics by Leonard J. Savage
  • Statistical Reasoning with Imprecise Probabilities by Peter Walley
  • Probability, Statistics and Truth by Richard von Mises
    Q&A: Expected Value (Outside of Frequentism) =====================================================

Q: What is the expected value, and how is it calculated?

A: The expected value is a measure of the average outcome of a random experiment. It is calculated by multiplying each possible outcome by its probability of occurrence and summing the results. However, the way we calculate the expected value can vary depending on the philosophical approach to probability.

Q: What is the difference between frequentism and non-frequentism?

A: Frequentism views probability as a long-run frequency of an event, while non-frequentism views probability as a degree of belief or a measure of uncertainty. Non-frequentism includes approaches such as Bayesianism, subjective expected value, imprecise probability, von Mises' frequency theory, and de Finetti's subjective probability.

Q: What is Bayesianism, and how does it relate to the expected value?

A: Bayesianism is a philosophical approach to probability that views probability as a degree of belief or a measure of uncertainty. From a Bayesian perspective, the expected value is not just a mathematical concept, but a representation of our degree of belief about the outcome of a random experiment.

Q: What is the subjective expected value (SEV), and how is it different from the expected value?

A: The subjective expected value (SEV) is a concept that is closely related to the expected value, but with a twist. Instead of calculating the expected value based on the probability of each outcome, we calculate it based on our degree of belief about the outcome. In other words, the SEV is a measure of how much we expect to win, based on our subjective assessment of the probability of each outcome.

Q: What is imprecise probability, and how does it relate to the expected value?

A: Imprecise probability is a concept that is used to represent uncertainty when the probability of an event is not precisely known. In this context, the expected value is not a single number, but a range of values, representing the uncertainty about the outcome.

Q: What is von Mises' frequency theory, and how does it relate to the expected value?

A: Von Mises' frequency theory is a philosophical approach to probability that views probability as a long-run frequency of an event. However, unlike frequentism, von Mises' theory does not require an infinite number of trials to calculate the probability. Instead, it uses a finite number of trials, and the probability is calculated based on the frequency of the event in those trials.

Q: What is de Finetti's subjective probability, and how does it relate to the expected value?

A: De Finetti's subjective probability is a philosophical approach to probability that views probability as a degree of belief or a measure of uncertainty. In this context, the expected value is not just a mathematical concept, but a representation of our degree of belief about the outcome of a random experiment.

Q: What is expected utility, and how does it relate to the expected value?

A: Expected utility is a concept that is used to evaluate the desirability a decision, based on the expected value of the outcome. In this context, the expected value is not just a mathematical concept, but a representation of the desirability of the outcome.

Q: Can you provide some examples of how the expected value is used in real-world applications?

A: Yes, the expected value is used in a wide range of real-world applications, including finance, insurance, and decision-making under uncertainty. For example, in finance, the expected value is used to calculate the expected return on investment, while in insurance, it is used to calculate the expected payout for a policy.

Q: What are some common misconceptions about the expected value?

A: One common misconception about the expected value is that it is always a single number. However, as we have seen, the expected value can be a range of values, representing uncertainty. Another misconception is that the expected value is always a good indicator of the outcome of a random experiment. However, the expected value can be influenced by a wide range of factors, including the probability of each outcome and the magnitude of each outcome.

Q: What are some best practices for using the expected value in decision-making under uncertainty?

A: Some best practices for using the expected value in decision-making under uncertainty include:

  • Using a range of values to represent uncertainty, rather than a single number
  • Considering multiple scenarios and outcomes
  • Using sensitivity analysis to evaluate the impact of different assumptions
  • Using decision-making tools, such as decision trees and influence diagrams, to evaluate the expected value of different decisions.

References

  • De Finetti, B. (1937). "Foresight: Its Logical Laws, Its Subjective Sources." In Studies in Subjective Probability (pp. 53-118).
  • von Mises, R. (1928). Probability, Statistics and Truth. Dover Publications.
  • Savage, L. J. (1954). The Foundations of Statistics. John Wiley & Sons.
  • Walley, P. (1991). Statistical Reasoning with Imprecise Probabilities. Chapman and Hall.

Further Reading

  • Probability and Statistics: The Science of Uncertainty by David J. Hand
  • The Foundations of Statistics by Leonard J. Savage
  • Statistical Reasoning with Imprecise Probabilities by Peter Walley
  • Probability, Statistics and Truth by Richard von Mises