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.
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Ranking methods for [1-X] voters and N candidates
Situation
I must rank N options (N = 54 here, but could be lower or higher) according to X voters (X = 1 here, though I am also ...
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Negation of an inequality
Working on semi orders, for a binary relation $R$ on a set $A$ we have that it is a semi order if the following holds
$$
aRb\longleftrightarrow u(a)\geq u(b) + q
$$
For $q\geq 0$ and $u : A \to\mathbb{...
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Unbiased decision rule.
The question is Problem 12 (p97, pdf p97) in Section 1.7 in Mathematical Statistics: Basic Ideas and Selected Topics. It can be calculated that
$$
\begin{aligned}
& E_{\theta} l (\...
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Complete Directed Graph and Decision Theory
In decision theory, condition $ \beta $ is defined as follows: If $a,b \in A \subset B, a, b \in C(A)$, and $b \in C(B)$, then $a \in C(B) $. $C(.)$ here is the choice correspondence of a decision ...
- 141
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Which branch of math theory could solve the task?
Imagine that we have a value $s_i = f(s_{i-1}, x_{i-1})$, reccurent formula $s_i$ with parameter $x_i$. $x_i$ values depends on $x_0$ and each $x_i$ is calculated in a diffenrent way. I guess it is ...
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Optimal strategy to get maximum element of $n$ sequentially i.i.d uniform distribution $X_1, \cdots, X_n \sim \text{Uniform}(0,1)$?
This is a homework for probability. To warm-up let's consider the case of $3$.
Consider three random variables $X_1, X_2, X_3$ that are independently and identically distributed according to the ...
- 1,149
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Evaluating a minimum mean square error explicitly
I have a minimum mean squared error (MMSE) expression that I would like to evaluate explicitly:
$$
\mathbb{E}[(X+\mathbb{E}[X|CX+G])^2],
$$
where:
$X\sim\text{Bernoulli}(\nu)$ where $\nu$ is some ...
- 730
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MCDA, Fuzzy analytic hierarchy process(FAHP), how to integrate user input data based on criteria into AHP calculation
i am developing a website that gives the best recommendation of school for all student using Fuzzy AHP(analytic hierarchy process). and i have a problem on how to integrate student data into my AHP ...
- 1
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Does there exist an MDP policy with this property?
Consider a discrete-time MDP with finite states and actions. For any policy $\pi$ and state $s$, let $u_t^{\pi}(s)$ be the expected total reward for using $\pi$ at times $t, t+1, ..., N$ if the ...
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How to create a prediction/decision model when decisions can impact future observations?
Apologies if this is not the correct topic for this question.
I am looking for a general approach/potential references/terms to search for regarding the following situation or similar situations as it ...
- 156
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How to model an if statement as a linear transformation
While working with the rectified linear activation function or ReLU, which can be mathematically expressed as ReLU(X) = max(0,X) where X
$$
X = \begin{pmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{pmatrix}...
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To prove non-decidability between two sets is it required to prove non-computability between all elements of two power sets
I am constructing a set of information $Y = \{y_{1}, \dots, y_{n}\}$ and from another set $X = \{x_{1}, \dots, x_{n}\}$, some publicly known information and some secrets.
The construction should ...
- 323
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Expected Utility, decision theory
I’m slightly confused how to assign utility if there are 2 pieces of information given about its value. For example, there are 2 decisions: to go to the movies or to go fishing. Provided that you get ...
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two persons roll dice and bid game: optimal strategy
Two persons $A, B$ roll a fair $n$-face dice separately and get $1 \le x,y \le n$ points. Then the third party will put $x + y$ dollars in a black box. $A$ and $B$ only know the point they roll and ...
- 1,149
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Minimizing the expected loss when there is a general loss matrix
This question is related to question 1.23 of "Pattern Recognition and Machine Learning" by Bishop.
The question asks "Derive the criterion for minimizing the expected loss when there is ...
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Recommendations on Intermediate Level Probability/Applied Statistics Book
So I'm an Internal Medicine Resident with an interest in mathematics and I have a BS in physics and MS in math. Lately I've been getting more into the statistical interpretation of diagnostic test, ...
- 31
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Will the duality gap be zero if the constraint is satisfied with equality?
Consider the following problem that aims to find an optimal policy $\pi$ mapping a state $s\in\mathcal{S}$ to an action $a\in\mathcal{A}$:
\begin{array}{cl}\tag{1}
\displaystyle \underset{\pi:\mathcal{...
- 3,365
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votes
1
answer
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What to do when in Coombs voting method there two equal weights for candidates to be elimenated?
I've read about Coombs method on Wikipedia.
I understand that we eliminate candidate with the most last-place votes. But what do we do when, for example, two candidates A and B have equal number of ...
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Minimizing average depth of a decision tree: when is greedy optimal?
Fix parameters $m$ and $n$. Consider a finite set $A$ and a set of attributes $f_1,\ldots,f_m$ where $f_i: A \rightarrow [n]$. We want to construct a decision tree $T$ with minimum average depth.
We ...
1
vote
1
answer
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Proof in a clique decision problem (karp reduction)
Considering the following decision problems:
E_CLIQUE(G, k), where G = (V, E) is a simple graph and k >= 1 an integer. Does
G have a clique of size 2 · k?
and
CLIQUE(G, l), where G = (V, E) is a ...
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answers
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Why does the uniform law of large numbers hold with non-i.i.d. random variables in Bayesian experimental design?
This paper, Asymptotic theory of information-theoretic experimental design, studies Bayesian experimental design where in each round $n$, the experimenter selects a stimuli $X_n$ that maximizes mutual ...
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An interesting game "The Truel"
There is an interesting paper called The Truel. It is about 3 players A , B and C shooting under some rules.The two snippets of the pages relevant to my Question are given below, the full paper is ...
- 3,872
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Prove that the preferences follow the Von Neumann and Morgensten's axioms
I'm studying Decision Theory from the book 'An introduction to decision theory' by Martin Peterson, and there is a problem that I don't understand how to solve. The problem is:
You prefer a fifty-...
- 3
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Example of computing value of risk function of decision rule
Risk function is defined as
$$R(\theta, \delta) := \int L(\theta, \delta(x)) P_{\theta}(dx)$$
where $x = (x_1, ... x_n)$ is an observation and $\delta(x)$ is a decision rule.
$P_{\theta}(x) = (\frac{1}...
- 241
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1
answer
207
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Finding formal bayes rule for a binary classification problem using zero-one loss function
I am currently practicing decision theory and bayes rule. I want to find the optimal decision given the problem below.
Consider a binary classification problem where we have a pair
$(X,Y)$ with $X \in ...
- 11
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votes
1
answer
37
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How lower than P(S|R) P(S|~R) must be in order for the expected value of R to be higher than the expected value of ~R
I was asking myself how lower than P(S|R) P(S|~R) must be in order for the expected value of option R to be (strictly) higher than the expected value of ~R, given the following value assignments to ...
- 235
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Secretary problem
I have a problem with the secretary problem, I wanted to prove that maximum value of the probability function of choosing the best applicant is decreasing as n gets bigger. So, in other words:
$$F(n):=...
- 31
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votes
1
answer
190
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Finding stable sets from a graph
I am trying to understand what a stable set is and have the following graph:
What are examples of a stable set from this graph?
If possible, what is the maximum stable set of this graph?
My current ...
user960518
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95
views
Does an ergodic Markov Decision Process have a unique optimal gain?
It is known from chapter 5 of Dynamic Programming and Optimal Control Vol II that a uni-chain Markov Decision Process (MDP) has a unique gain-bias solution $(J,\vec{h})$ to the following infinite-...
- 315
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An efficient stopping rule to determine the sign of the mean of an i.i.d. sequence of random variables.
Do there exist a family of measurable functions $(f_t^\delta)_{t \in \mathbb{N}, \delta \in (0,1)}$ and constants $C,c>0$ such that, for each $t \in \mathbb{N}$ and $\delta \in (0,1)$ we have that $...
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Von Neumann–Morgenstern: compare coefficients in Archimedean axiom
Now we have:
Axiom1: Completeness of $\succeq$.
Axiom2: Transitivity of $\succeq$.
Axiom3: Independence: For any $N$ and $p\in (0,1]$, if $L\succ M$, then $pL+(1-p)N\succ pM+(1-p)N$.
Axiom4: ...
- 193
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Model or algorithm for a balanced graph
I have a graph which each nodes has the following features:
A node can produce some "energy" (or something like that);
A node has to satisfy the need energy and so use the energy produced ...
- 91
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31
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How would you design an algorithm to solve this decision problem?
Suppose I want to design an algorithm that, for an arbitrary polynomial $p$, returns YES iff there are two roots $z_1$ and $z_2$ of $p$ such that $\left|z_1 - z_2\right| = 1$.
How do I design such an ...
- 524
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answers
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Which voting algorithm to use to assign N number of people to G groups based on their ranked choice preference
I've been looking through social choice theory textbooks and videos trying to find the right sort of algorithm for this, but struggling. Basically I have N (say 21) people that I need to assign into G ...
- 31
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answer
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Equivalence of Leaky Integrator and Low-Pass Filter Decision Models
I'm working with a decision making model from the cognitive neuroscience literature (the Urgency Gating Model) and have found two different implementations. My concern is that these two ...
- 131
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589
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Non-Optimality of First-Fit-Decreasing Algorithm for Bin Packing
The First-Fit-Decreasing algorithm solves the bin packing decision problem for given weights $w_1,\dotsc,w_n\in [0,1]$ and number of bins $k$ in quadratic time. This would mean that the bin packing ...
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Question about Bayes risk and best rule Bayes
I'm start to learn Decision Theory and I'm trying to solve (analytically) the exemple 2 from Berger, pag. 5-6 (James O. Berger - Statistical Decision Theory - 1980).
I can't understand the result (how ...
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Is a fixed welfare function that outputs the same answer regardless of the inputs independence of irrelevant alternative?
I'm taking this course to learn game theory and I'm confused about a question in Unit 1.5.
Background. In game theory, independence of irrelevant alternatives (IIA) says the social welfare function $W$...
- 426
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Getting a first-order condition of Risk Adverse selection problem
I'm struggling to find out some basic maximization problem associated to the first order condition of a problem. The problem is an insurance example, where there's an strictly risk-adverse decision ...
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vote
1
answer
67
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Minimizing uncertainty in a POMDP
Consider a partially observable Markov decision process (POMDP), see here for a complete definition.
The general definition allows for a reward function to be defined in terms of (pairs of) states and ...
- 383
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Learn this decision problem
Problem statement: Here's a single-player probabilistic game. In front of you are $L$ urns, each containing bills of various values. You get $N$ chances to draw a bill from any urn you like, check its ...
- 1,016
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Proof: if the preference relation $\succsim$ is continuous, then the upper contour set is closed
I am having some trouble with this demonstration.
I know that the definition of continuous preference relation is: $$\succsim \text{is continuous iff } \forall \text{ pair of sequences } \{x^n\}_{n>...
- 83
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Von-Neumann-Morgenstern Axioms Clarification Question
I want to solidify my understanding of the von-Neumann-Morgenstern axioms. How would you analyze these scenarios using the lens of these axioms?
Suppose that Person A owns a potentially expensive ...
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Bayes estimate of upper limit of uniform distribution with exponential prior
Let $X_{1}, . . . , X_{n} > 0$ be a random sample from $U(0, \theta)$. Suppose $\theta$ has the prior $\pi(θ) = e^{-\theta} ; \theta > 0$. Find the Bayes estimate of $\frac{1}{\theta}$ with ...
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votes
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answer
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Consistency in Statistical Decision Theory
Let $(\mathcal X,\mathcal F,\mathcal P)$ be a statistical model with $\mathcal P = \{P_\theta : \theta\in\Theta\}$. A decision rule is a measurable function $\delta:(\mathcal X, \mathcal F)\rightarrow(...
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Probability of Error: Binary Communication Channel?
I have a binary communication channel shown below:
with the following parameters:
$\pi_0 = 0.3$
$\pi_1 = 0.7$
$\epsilon_0 = 0.2$
$\epsilon_1 = 0.1$
Goal: compute the probability of error and ...
- 504
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0
answers
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Decision based on decision tree
Based on a text I created a decision tree. This decision tree shows options A, B and C. I calculated the value that could be expected for each decision and the highest one would be that of option C ...
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Use Jensen's inequality to show $\underset {\theta}{\operatorname {max}} \mathbb E[L(\theta, \delta(X)]\ge ...$
Consider binary random vector $X \in \{0, 1\}^n$. Consider the most general model for such a random vector
$\Omega = \{ \theta=(\theta_x)_{x \in\{0,1\}^n} | \theta_x\ge0, \forall x \in \{0, 1\}^n, \...
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Expected Utility
I am confused by something regarding expected utility calculation.
When we are calculating the expected utility where 2 positive outcomes can happen at the same time such as the example below:
We win ...
- 15
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1
answer
111
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Is it true that every closed formula is decidable? Why?
I was wondering if every closed formula is decidable (in a complete system). Clearly a non closed formula is not decidable, since it can take some values for which it is true, and other for which it ...
