Decision Theory & Bayesian Analysis

Section- A

 

1 - State and Prove Minimax Theorem.

 

The Minimax Theorem is a fundamental theorem in game theory. It states that in two-player zero-sum games, there exists a value V and a mixed strategy for each player, such that given Player 2's strategy, the best payoff possible for Player 1 is V, and given Player 1's strategy, the best payoff possible for Player 2 is -V. This theorem was proven by John von Neumann in 1928.

 



Let K and C be inhabited totally bounded convex subsets of normed spaces E and F, respectively, and let f : K × C → R be a uniformly continuous function such that f (·, y) : K → R is convex for each y ∈ C; f (x,·) : C → R is concave for each x ∈ K. Then sup y∈C inf x∈K f (x,y) = inf x∈K sup y∈C f (x,y).

 

Lemma A

 

 Let X and Y be inhabited totally bounded metric spaces, and let f : X ×Y →Rbeauniformly continuous function. Then supy∈Y infx∈X f (x,y) and infx∈X supy∈Y f (x,y) exist, and sup y∈Y

 

A proof of the minimax theorem Proof. By Lemma A

, it suffices to show that sup y∈C inf x∈K f (x,y) ≥ inf x∈K sup f (x,y). y∈C

 

 Let c = supy∈C infx∈K f(x,y), and suppose that c < inf x∈K sup y∈C f (x,y).

 

Then, by Lemma 21, there exist y1,...,yn ∈ C

 

 such that c < inf x∈K max{f (x,yi) | 1 ≤ i ≤ n}.

 

2 - State and Prove complete class Theorem.

 

The Complete Class Theorem is a fundamental theorem in game theory. It states that for any finite two-player game with perfect information, either (1) one player has a strategy that guarantees them a win no matter what the other player does, or (2) both players have strategies that guarantee them a draw no matter what the other player does.

 

To prove this theorem, we must first define some terms. A two-player game with perfect information is a game where each player has complete information on all of their opponent's moves. A strategy is a set of rules that a player follows in order to maximize their payoff.

 

Now, let G be a two-player game with perfect information and payoff matrix M, where Mij represents the payoff for Player 1 if Player 2 plays the strategy i and Player 1 plays the strategy j. Let X be the set of all possible strategies for Player 1, and let Y be the set of all possible strategies for Player 2.

 

To prove the Complete Class Theorem, we must show that either (1) there exists a strategy x in X such that Mxy is greater than or equal to Mxy' for all y in Y, or (2) there exists a strategy y in Y such that Mxy is less than or equal to Mxy' for all x in X.

 

Let p be the probability of Player 1 winning if both players play their optimal strategies. If p is greater than 0, then there exists a strategy x in X such that Mxy is greater than or equal to Mxy' for all y in Y, and the theorem is proven. If p is less than 0, then there exists a strategy y in Y such that Mxy is less than or equal to Mxy' for all x in X, and the theorem is also proven.

 

3 - State is the basic difference between Bayes and Minimax Principles.

 

The basic difference between the Bayes and Minimax principles is that the Bayes Principle is a method of statistical inference in which Bayes' theorem is used to update the probability for a hypothesis as more evidence or data is collected. The Bayesian approach is subjective and relies on prior beliefs, while the minimax approach is objective and seeks to maximize the expected utility of a decision.

The Bayes Principle is based on the concept of Bayesian updating, which is a process in which a prior belief is updated with new information or data in order to arrive at a posterior belief. The prior belief is expressed in terms of a probability distribution, which is then updated with new data to form a posterior belief. The Bayesian approach is based on the idea that probabilities should be assigned to beliefs based on the evidence available, and that the probabilities should be updated as more evidence is acquired.

In contrast, the Minimax Principle is a decision-making strategy that seeks to minimize the maximum possible loss. It is based on the concept of game theory, which is a branch of mathematics that deals with the analysis of strategies in situations of conflict or competition. The Minimax Principle is used to evaluate the potential outcomes of a decision and to determine the best possible course of action. This approach is considered to be more objective than the Bayesian approach, as it takes into account the likelihood of the worst-case scenario.

Overall, the Bayes Principle is a subjective approach to statistical inference and relies heavily on prior beliefs and data. The Minimax Principle, on the other hand, is an objective approach to decision-making that seeks to maximize expected utility and minimize the maximum possible loss. Both approaches have their own advantages and disadvantages, and it is important to understand the differences between them in order to make the best decisions.

 

Section - B

 

 

1 - Discuss about the Invariance and ordering.

 

Invariance is a property that a mathematical object holds which remains the same even after the object is subjected to a series of transformations. This property is important in many mathematical and scientific fields, such as game theory and machine learning. Ordering is a mathematical concept which describes the relationship between objects. It is used to compare objects and determine which is greater or lesser than the other.

In game theory, invariance and ordering are used to study decision problems. An ordering can be used to compare the outcomes of different decisions and determine which is more preferable. Invariance can be used to identify decision problems which remain the same even when subjected to a certain transformation. The minimax theorem is an example of an invariance theorem which states that in certain sequential decision problems, there exists a minimax procedure among the class of strategies.

In machine learning, invariance and ordering can be used to identify patterns and make predictions. Invariance can be used to detect patterns that remain the same even after being subjected to certain transformations, such as rotation or scaling. Ordering can be used to compare different inputs and determine which is more important or relevant for the task at hand.

In conclusion, invariance and ordering are important concepts in mathematics and science. They are used to compare and identify patterns in various fields, such as game theory and machine learning.

 

 

2 - Write short notes on (a) Admissibility (b) Completeness

 

(a) Admissibility: Admissibility refers to the process by which evidence is deemed to be valid and relevant to a particular case. In order to be considered admissible, evidence must meet certain requirements, such as relevance, materiality, and competency. Additionally, certain types of evidence, such as hearsay or evidence obtained in violation of a defendant’s constitutional rights, may be excluded from consideration.

(b) Completeness: Completeness refers to the process by which evidence is deemed to be valid and complete. In order for evidence to be considered complete, it must include all relevant facts and information. Additionally, evidence must be properly authenticated and corroborated in order to be considered complete. Completeness is an important factor in determining the admissibility of evidence in a court of law.

 

3 - Define extended Bayes rule.

 

Extended Bayes Rule, also known as Bayes Theorem, is a mathematical formula used to calculate a probability when there are multiple events occurring. It is named after Thomas Bayes who first developed the formula in 1763. The formula is used to calculate the probability of an event occurring, given that certain other events have already occurred.

The formula states the following:

P(A|B) = P(A) x P(B|A) / P(B)

Where P(A|B) is the probability of event A occurring, given that event B has already occurred. P(A) is the probability of event A occurring, and P(B|A) is the probability of event B occurring, given that event A has occurred. Finally, P(B) is the probability of event B occurring.

The Extended Bayes Rule formula is used in a variety of fields, including statistics, machine learning, and artificial intelligence. In statistics, this formula is used to calculate the likelihood of a hypothesis being true given the data that is available. In machine learning, the formula can be used to calculate the probability of a certain outcome given the data that has been collected. Finally, in artificial intelligence, the formula can be used to calculate the probability of certain decisions being made based on the data that is available.

The Extended Bayes Rule formula can be used to calculate the probability of any situation, given the data that is available. It is a powerful tool for making decisions and can be used in a variety of fields.

 

4 - Give examples of (i) an improper prior distribution and (ii) a proper prior distribution.

 

(i) An improper prior distribution is one that does not sum to a finite value and cannot be normalized to sum to a finite value. Examples include the uniform prior, which assigns each parameter an equal probability, and the Dirichlet prior, which assigns probabilities proportional to a parameter's prior probability.

 

(ii) A proper prior distribution is one that sums to a finite value and can be normalized to sum to a finite value. Examples include the Jeffreys-rule prior and the normal distribution prior. The Jeffreys-rule prior is a non-informative prior which assigns equal probability to all possible parameter values. The normal distribution prior is an informative prior which assigns higher probability to values close to the mean and lower probability to values far from the mean.

In general, a proper prior distribution should be chosen based on the data available and the desired inference goal. In some cases, an improper prior may be desirable, as it may lead to more accurate estimates of the parameters. In other cases, a proper prior may be more appropriate, as it will help to prevent overfitting and ensure that the parameters are accurately estimated.

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