Questions tagged [decision-theory]

For questions regarding formal decision problems. In contrast, questions involving strategic aspects (where the solution depends on the behavior of others) are discussed in game theory.

Filter by
Sorted by
Tagged with
1
vote
1answer
35 views

How to find a utility function

The choices are of the form $(x; y)$ where $x$ represents the amount of time you have left to live, say anywhere from $0$ to $50$ years, and $y$ represents the amount of time you have left to work, ...
1
vote
1answer
23 views

Creating a Majority Graph from multiple preference orders

I can't find much on voting theory on this exceptional site. I am trying to find a way of constructing a majority graph based on a few preference. When I try to construct one, I end up breaking the ...
0
votes
0answers
35 views

probability shuffling decks of cards?

You have the opportunity to invest $20 , with an uncertain return. To simulate uncertainty, we will use a deck of four playing cards consisting of two aces and two kings. The deck has been shuffled ...
2
votes
1answer
65 views

Understanding a part of the theorem from Ferguson's book

The following images are part of a proof of a theorem in the Ferguson's seminal book "Mathematical Statistics-A decision theoretic approach". I did not get some parts which uses certain concepts from ...
0
votes
0answers
15 views

Analytic hierarchy process

can someone help me with this question regarding AHP? I've uopladed a picture below, click the link: AHP Question Thank you! :-)
0
votes
1answer
34 views

Supporting hyperplane for Bayes boundary of a convex set

I was reading the book 'Optimal Statistical Decisions' by DeGroot. I came across the following claim, without proof: G is a convex set in $R^k$ that is the convex hull of a finite number of points. ...
0
votes
0answers
28 views

Showing the existence of a unique Bayes estimator for a parameter

Suppose $X \sim N(\theta, 1)$ and $\theta$ has a uniform prior on $(-1,1)$. Denote the density of $X$ as $f(x)$. We are interested in proving the Bayes estimator for a function of $\theta$, say $h(\...
3
votes
0answers
56 views

Loss of a randomized decision rule

I am looking into the Wikipedia article with the topic Randomised decision rule. In the "Definition and interpretation" section, I see the formula of randomized loss: $$L(\theta,d^*)=\int_{A\in\...
0
votes
1answer
29 views

Deconvolution with respect to a particular function

Let $\mathcal L, \mathcal L^*: \Theta \times \mathcal A \to \mathbb R$ be functions. When can $\mathcal L$ be expressed as the convolution of $\mathcal L^*$ with some third function $U$? That is, when ...
0
votes
0answers
32 views

Decision tree construction

I recently failed a university exam in Decision theory. The question is as follows: 2. A car manufacturer wishes to market its latest design. They have already performed the minimum mandatory level ...
0
votes
0answers
44 views

I want introductory book in Statistical decision theory for self-study

Of course it should be with exercises (and solutions to said exercises!), just reading theory isn't enough for me.
0
votes
0answers
14 views

Dynamic Multiple Variable Decision Support Probability Help

All, I'm faced with an unfamiliar, conceptual problem (at least to me). I'm sure there are a number of ways to execute, but I'm curious as to the best or most intuitive approach to achieve the ...
1
vote
1answer
47 views

NP-completeness of chromatic sum in list coloring problem with capacity constraints

I am trying to solve a problem that can be considered as minimizing the sum of colored numbers in a List Coloring Problem while satisfying some restricted constraints. In the List Coloring Problem (...
1
vote
0answers
33 views

Decision Analysis on game - Y vs L uncover pattern

I am trying to understand the logic of the community in choosing a specfic strategy to maximize profit in a game. The game can be described as follows: Every ticket has nine spaces, with a number ...
0
votes
1answer
32 views

Calculate decision boundary of two Gaussians with different missclassification costs

Assuming we have two classes $C_1$ and $C_2$ represented as two Gaussians with $(2\mu_2, \sigma)$ and $(\mu_2, \sigma)$. We know further that $\mu_2 > 0$ and $p(C_1) = p(C_2)$. We want now ...
0
votes
0answers
28 views

Combine multiple source sets to make a decision

Let's say I have multiple sources from which I have to make a decision, what element is my input X? I am getting the possible matches with elements: Each element ...
1
vote
0answers
34 views

Compute conditional probability for a decision analysis network

I have to resolve an exercise for decision analysis network. I have the following decision tree for that decision analysis network: $$\begin{array}{l} G&\to&Y&\to&D&\to&X&...
0
votes
0answers
32 views

Compute conditional probability

I have to solve this problem for a decision analysis network: "There is a genetic trait present in 20% of the population that makes that, for men over 65 years, the probability of suffering the ...
1
vote
1answer
136 views

Deriving of optimal decision boundary of two Gaussians

Given two Gaussians with the same variance $\sigma$ and means $\mu_1$ and $\mu_2$, where each Gaussian represents a class $C_1$ and $C_2$ with the same prior probabilities, i.e. $p(C_1) = p_(C_2)$, we ...
0
votes
0answers
16 views

Finding optimum time for change of equipment based on shift pattern available resources

I am trying to sort out a relatively simple problem, where I believe an algorithm or solving technique may already exist (Hungarian Algorithm?). I'd like to solve it using an methodology rather than ...
0
votes
2answers
49 views

Is this expected utility correct?

I'm learning Bayesian Networks and Decision Making but I'm not very good with notation. I have written down this expected utility (more info here): $$UE=\sum_s\max_b\sum_{s,r}\max_q\sum_tP(t)P(s|t)P(...
0
votes
0answers
21 views

How to calculate pi sub D, upper opt

I¡m learning Bayesian Networks and now I'm studying Decision Making. I have the following probability table (sorry about the format, I don't know how to do it better): ________| $\psi(y,t,d)$ $+y,+t,...
0
votes
0answers
26 views

Playing a normal-form game against another player

Suppose we have the following game: Game and we need to play the game twice against another player who we do not know. First I have to play as the row player and then next as the column player ...
0
votes
0answers
88 views

Difference and usefulness between performance measure and utility

I would like to know if my understanding is correct. Here is how I understand the difference between performance measure and utility in terms of agents. It seems that performance measure is a binary ...
0
votes
0answers
30 views

Choquet integral for negative function on a finite set

I've to calculate the Choquet integral (with respect to a capacity) for a $\textbf{negative}$ discrete value function on a finite set, the problem is that I know only these two formulas. $\textbf{1)...
0
votes
1answer
62 views

Unbiased Decision Rule

A decision rule $\delta$ is said to be unbiased if $\mathbb E_\theta[L(\theta^\prime,\delta]\geq\mathbb E_\theta[L(\theta,\delta]$ for all $\theta,\theta^\prime\in\Theta$. In the context of testing ...
0
votes
0answers
44 views

Bayes decision rule: Bayes Risk.

Suppose that we replace the deterministic decision function $\alpha(x)$ with a randomized rule, the probability $P(\alpha_i|x)$ of taking action $\alpha_i$ upon observing $x$. (a) Show that the ...
0
votes
1answer
28 views

Game theory - How was the table of decision analysis formulation constructed?

Could someone please explain how the table in the solution was constructed? Particularly in the first row, why $0$ and why $54$ millions? Why was $6$ and $600$ millions not included ? Solution ...
0
votes
1answer
32 views

is there a linear bounded automaton the decides $A_{nfa}$?

first post here :) I was wondering, since regular languages are context sensitive, and since linear bounded automatons can act as an acceptors for context sensitive language, is it possible or is ...
2
votes
1answer
74 views

A Maximize or Explore Problem over a Finite Time Series

I recently read "Reinforcement Learning" Bardo and Sutton which motivated me to come up with this problem (which I hope is well posed): The Problem Some sort of reward maximizing agent finds itself ...
0
votes
0answers
47 views

Dutch Book for VNM Axiom 4

I'm unconvinced that my refusal to accept the fourth of the von Neumann-Morgenstern axioms is irrational. Wikipedia claims that there is a Dutch book argument against me, but I do not see how that can ...
3
votes
1answer
173 views

Why is the stopping rule in the secretary problem 'optimal' when it can be shown not to be optimal for $n=2$?

Several authors have proved that the optimal strategy in the secretary problem is a stopping rule whereby the interviewer rejects the first $r-1$ applicants (let $M$ be the best applicant among these $...
2
votes
1answer
64 views

Algorithm for optimal assignment of tasks to a team of people

Is there an algorithm to get a team of people to complete a certain number of tasks the fastest, where the time taken to complete a certain task is different for different people? Each task must be ...
2
votes
2answers
144 views

Does the von Neumann-Morgenstern utility theorem work for infinitely many outcomes?

The von Neumann-Morgenstern utility theorem is easy to prove for a finite number of outcomes. Is it still true for an infinite number of outcomes? With infinite outcomes, a lottery can now be any ...
2
votes
1answer
54 views

Integration of probabilities - decision theory - minimizing misclassification rate

I have trouble understanding an equation in a book I'm reading. Basically, Consider a decision rule that divides the input space into regions $R_k$ called decision regions, one for each class, such ...
0
votes
0answers
31 views

How should I create a matrix of costs based on a set of variable?

I'm trying to make a matrix that depicts costs for a set of tasks. These tasks are the rows of the matrix. The cost matrix is to be based on a multi-attribute weightage of the tasks based on their ...
1
vote
0answers
25 views

Optimal number of experiments

There is a random variable and we know that it is either uniformly distributed on $(0, 1)$ or uniformly distributed on $(0, \frac{1}{2})$. Both cases are equally likely to be. We are to guess the ...
2
votes
1answer
886 views

Decision theory vs. Game theory?

Game theory is defined (here) as follows: "Game theory, branch of applied mathematics that provides tools for analyzing situations in which parties, called players, make decisions that are ...
1
vote
1answer
50 views

Secretary problem - Is there an equation that allows one to have $r = 0$?

Secretary problem's equation I found this equation on wikipedia to resolve the secretary problem. I understand it but I have a small problem. Theoretically, if I would not want to reject any ...
2
votes
0answers
21 views

Which act should be chosen according to the principle of maximizing expected monetary value ( EVM )?

Assume that the decision maker's utility u of money is linear. Consider ...
0
votes
1answer
27 views

Using posterior expected loss to make a decision between two pdfs

I am trying to solve a question from a past exam paper. Suppose you have a single observation $X$ from a continuous distribution for which the probability density function (pdf) is either $f_0$ ...
1
vote
1answer
198 views

Log utility function and the St. Petersburg paradox

In the log utility model the formula solving the St. Petersburg paradox $$\Delta E(U)=\sum_{k=1}^\infty \frac{1}{2^k}\left [\ln(w + 2^k - c) - \ln(w) \right ]$$ relates the wealth, $w,$ of the ...
2
votes
2answers
80 views

What is Meant by a Statistical Decision Problem?

I am reading Mathematical Statistics - A Decision Theoretic Approach by Ferguson (Academic Press 1967). A game in the book is defined as a triple $(\Theta, \mathcal A, L)$, where $L$ is a function ...
0
votes
0answers
22 views

Computing the error rate of rejecting the null hypothesis (from the decision theoretic perspective)

Consider the following passage from the bottom of pg. 23 of Statistical Decision Theory and Bayesian Analysis: Problem. I'm having trouble understanding the red highlighted; it seems that the ...
2
votes
0answers
66 views

Bayes decision theory - step in derivation

I am self studying Bayes Decision theory from these lecture notes page 30 / 31 and there is a step a struggle to understand mathematically Background context Given Bayes risk defined as: $$ r_B(\...
0
votes
1answer
36 views

What approaches can be used to calculating a fair weighted score?

I want to calculate a weighted score in a fair way, for a soccer substitution algorithm. There might not be a clear answer to this question, but I am looking for guidance on how to approach computing ...
1
vote
1answer
56 views

Likelihood ratio test between two hypothesis

I have two hypothesis as below: Under $H_1$, $f_x(x) = 3/2 * x^2$ where $x \epsilon (-1,1)$ Under $H_0$, $f_x(x) =$ Uniformly distributed between $x \epsilon (-1,1)$ What is the maximum likelihood ...
0
votes
2answers
327 views

Value of the game from payoff matrix

I am absolutely new to decision theory . I came across this following payoff matrix in the book.(Math. Stats : John E Freund). ...
0
votes
1answer
126 views

A characterization of the Choquet integral as the supremum of finite sums

The following question refers to Ehud Lehrer's and Roee Teper's paper The Concave Integral over Large Spaces, which was published August 2008 in Fuzzy Sets and Systems 159(16):2130-2144. The above ...
1
vote
2answers
45 views

A question on the relation of entropy to the secretary problem

I recently read about the Secretary problem and it made me wonder if there was any relation to the entropy within the quality distribution of secretaries (i.e. how much their quality varies, which we ...

1 2 3 4 5