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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Expected vs empirical loss with data augmentation
In a typical supervised learning setup, we assume our data $X$ with labels $Y$ comes from a data distribution $(X,Y) \sim P(X,Y)$.
The expected loss for some loss function, $L$, is:
$R_{L,P,f} = \...
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Representation of a particular functional form [closed]
Does anyone know of any papers or references which have representation theorems for functions of (or generalisations of) the form:
$f(x_1, x_2, \ldots, x_n) = \left( \sum_{i=1}^{n} a_i x_i^{\rho} \...
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Estimating the value of information for determining when to stop an experiment
Suppose that there are two parameters $p_1, p_2 \in [0, 1]$, and you begin with an uninformative prior $\textrm{Beta}(1, 1)$ for both of them. An experiment has been running for $d$ days. On each day ...
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Optimal strategy in a number-picking game against a perfect logician?
I'm thinking about a game scenario involving three players: myself, an opponent, and a referee. Each player picks a real number between 0 and 1, and the referee will select a number randomly between 0 ...
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How to solve for Nash Equilibrium?
Image:
I am currently studying for a college exam next week in Games Theorie. Unfortunately the example questions are very different to the course material and im stuck on this one. I would solve the ...
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Proving a (Representing Utility) Function is Continuous
I sincerely apologize for posting such a long question. The question involves a complicated proof of a theorem in mathematical economics. I feel it will be better for me to state my question first. I ...
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Calculation of normalized values for cost-type criteria in the weighted sum model
According to pymcdm WSM is calculated as follows:
$$ A_i^{\text{score}} =\sum_{j=1}^n \bar{x}_{i j} w_j \quad \text{for } i=1,2,3,\ldots ,m$$
Where:
$m$ is the number of alternatives
$n$ is the ...
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Minimax Estimation: What's the difference between Minimax, Sharp Minimax, First-Order Sharp Minimax and Second-Order Sharp Minimax Estimator?
I am currently working on my dissertation on Biased Data, and the second chapter focuses on distribution function estimation. Efromovich's work appears to be an outstanding reference on the topic, but ...
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Every cut is the union of (edge-disjoint) minimal cuts
I am tasked with proving the following statement:
"Every cut is the union of edge-disjoint minimal cuts"
The only information given, is the existance of the cut-set subspace $W_S(G)$. It ...
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Infinite sum $\sum_{t=c}^{n-1}(\frac{1}{t^2-1})$ as part of cardinal payoff variant
What does this sum $\sum_{t=c}^{n-1}(\frac{1}{t^2-1})$ equal?
For context, I am trying to digest the cardinal payoff variant of the secretary problem. There is an interview process with $n$ candidates ...
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Example where r.v. $X_2$ stochastically dominates $X_1$ but $P(X_1 > X_2) \geq 0.95$
The problem is from a textbook I'm reading, but even with the hint, I'm not being able to come up with a solution.
Let $X_1$ and $X_2$ be two random variables with CDFs $F_1$ and $F_2$. We say $X_2$ ...
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Determining Perfect vs. Imperfect Information in Calculating Expected Value
In this scenario, you are presented with an opportunity to engage in a game for a fee of $50. On a table, there are two boxes: a large box and a small box. The large box contains a total of 40 balls, ...
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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
$$
\begin{aligned}
& E_{\theta} l (\...
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Complete Directed Graph and Decision Theory
In decision theory, condition $ \beta $ is defined as follows: If $a,b \in A \subset B, a, b \in C(A)$, and $b \in C(B)$, then $a \in C(B) $. $C(.)$ here is the choice correspondence of a decision ...
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Which branch of math theory could solve the task?
Imagine that we have a value $s_i = f(s_{i-1}, x_{i-1})$, reccurent formula $s_i$ with parameter $x_i$. $x_i$ values depends on $x_0$ and each $x_i$ is calculated in a diffenrent way. I guess it is ...
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Optimal strategy to get maximum element of $n$ sequentially i.i.d uniform distribution $X_1, \cdots, X_n \sim \text{Uniform}(0,1)$?
This is a homework for probability. To warm-up let's consider the case of $3$.
Consider three random variables $X_1, X_2, X_3$ that are independently and identically distributed according to the ...
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How to create a prediction/decision model when decisions can impact future observations?
Apologies if this is not the correct topic for this question.
I am looking for a general approach/potential references/terms to search for regarding the following situation or similar situations as it ...
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How to model an if statement as a linear transformation
While working with the rectified linear activation function or ReLU, which can be mathematically expressed as ReLU(X) = max(0,X) where X
$$
X = \begin{pmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{pmatrix}...
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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 ...
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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, ...
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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{...
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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 ...
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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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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 ...
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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-...
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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}...
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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 ...
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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 ...
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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):=...
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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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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-...
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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: ...
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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 ...
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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 ...
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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 ...
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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 ...
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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$...
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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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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 ...
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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 ...
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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>...
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