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.
102
questions with no upvoted or accepted answers
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Approximate Dynamic Programing - Discount Factor for Very Long Horizons
I want an optimal strategy for a very long time horizon, say $K=100000$. I have dynamic decision making problem where next state $x_{k+1}$ is determined by the probability distribution $f(x_{k+1}|u_{k}...
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votes
0
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100
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Minimax Estimator for Normal Random Vector
Question. Suppose $Y_i \sim N(\mu_1, 1)$. Let $Y := (Y_1, Y_2)$, and $T_y = (Y_1, 0)$. Denote $\Theta$ as the space of all estimators $\mu := (\mu_1, \mu_2)$. Is it necessarily true that $\hat{\mu}$ ...
- 311
5
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214
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Find a functional property satisfied by union of choice functions
Consider $X$ a finite set and let $2^{X}-\emptyset$ denotes its power set (excluding the empty set).
Definition 1: A choice function is a function $c:2^{X}-\emptyset\mapsto X$ satisfying $c(A)\in ...
- 869
4
votes
0
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156
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Inadmissibility of Simpson's rule
Let $B_t$, $t\ge0$ be a standard Brownian motion and suppose $0<x_1<x_2<\cdots<x_n<1$. Then the conditional expectation
$$
\mathbb E\left(\int_0^1 B_t\,dt \,\middle\vert\, B_0, B_{x_1},...
3
votes
1
answer
134
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Recommendations on Intermediate Level Probability/Applied Statistics Book
So I'm an Internal Medicine Resident with an interest in mathematics and I have a BS in physics and MS in math. Lately I've been getting more into the statistical interpretation of diagnostic test, ...
- 31
3
votes
0
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281
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Which voting algorithm to use to assign N number of people to G groups based on their ranked choice preference
I've been looking through social choice theory textbooks and videos trying to find the right sort of algorithm for this, but struggling. Basically I have N (say 21) people that I need to assign into G ...
- 31
3
votes
0
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109
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Loss of a randomized decision rule
I am looking into the Wikipedia article with the topic Randomised decision rule.
In the "Definition and interpretation" section, I see the formula of randomized loss:
$$L(\theta,d^*)=\int_{A\in\...
- 31
3
votes
1
answer
82
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Applying Markov Decision Processes to an arrival forecasting problem
I have the following problem and I'd like to know if it's something that was already studied in the literature or not. I'm not sure about the naming conventions either.
I have a system $S$ that can ...
- 141
2
votes
0
answers
55
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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 ...
- 121
2
votes
1
answer
61
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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 ...
- 73
2
votes
0
answers
31
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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 ...
- 21
2
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0
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41
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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
...
- 121
2
votes
0
answers
99
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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(\...
2
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0
answers
51
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Relation between two Uniform Upper Probabilities on $\wp\omega$
There are two candidates I know of for uniform upper probabilities on $\wp\omega$. The first is the usual relative frequency and the second is a function which I do not know the name of but I read ...
- 577
2
votes
0
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165
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Stochastic decision problem with normal distribution
Suppose the decision maker receives a piece of information (signal) $s=\theta+e$, where the true parameter $\theta$ and error $r$ are normally distributed, and makes decision $d\ge 0$ in order to ...
- 4,105
2
votes
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42
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Proving that a specific Bayes rule is least favourable
Suppose $\pi$ is a prior distribution on $\Theta$ such that the Bayes risk of the Bayes rule equals $\sup_{\theta\in \Theta}R(\delta_\pi,\theta)$, where $R(\delta,\theta)$ is the risk function ...
- 387
2
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0
answers
55
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Characterize joint distribution from marginals
Let $Z$ be an arbitrary set. Let $X=\prod_{i=1}^n Z_i$ and $Z_i=Z$ for each $i=1,\ldots,n$, $n$ fixed. Consider the $n$-tuple $(\mu_1,\ldots,\mu_n)$ with $\mu_i\in\Delta_s(Z)$ for each $i=1,\ldots,n$, ...
- 464
2
votes
0
answers
47
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Matrix (geometric sum) orbit problem
Is the following algorithmic problem known to be decidable/undecidable?
Input: an element $\mathbf{v} \in \mathbb{Z}^n$, a matrix $\mathbf{A} \in GL_n(\mathbb{Z})$, and a subgroup $H \leqslant \...
- 697
2
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0
answers
561
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Classification problem: admissible rule is a Bayes rule for some prior $\pi$
I have a classification problem where I want to place an observation $X$ into a population described by a pdf equal to either $f_1$ or $f_2$.
Given $P_{f_i}(\frac{f_1(X)}{f_2(X)}=j)=0$ for all $j\in [...
- 21
2
votes
0
answers
146
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Lower bound on uncertainty reduction
Let $T$ be a set of tuples such that each score tuple $s(t_i)$, $t_i \in T$ is uncertain (i.e., not known deterministically). The score $s(t_i)$ can be represented as a uniform probability density ...
- 363
2
votes
1
answer
532
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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 ...
- 1,261
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vote
0
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61
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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
1
vote
1
answer
148
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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 ...
- 41
1
vote
0
answers
75
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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 ...
- 49
1
vote
0
answers
161
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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
1
vote
0
answers
51
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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 $...
- 5,803
1
vote
0
answers
38
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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
1
vote
0
answers
40
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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
1
vote
0
answers
211
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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, \...
- 11.1k
1
vote
0
answers
35
views
Is it possible to get no solution from an optimal stopping problem
I recently read about the 37-percent rule as the solution to the secretary problem.
It says
To have the highest chance of getting the best applicant from a pool of applicants, you should interview ...
- 11
1
vote
1
answer
126
views
Total Utility Value Composition of Different Utility Functions
Let's suppose we have a variable $x$ with a domain $X \in [0,1000]$ and two utility functions $uf_1(x)$ and $uf_2(x)$ that describe the utility of $x$ with respect to two different properties. We ...
- 11
1
vote
1
answer
142
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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, ...
- 147
1
vote
0
answers
37
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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 ...
- 73
1
vote
0
answers
44
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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&...
- 378
1
vote
0
answers
27
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 ...
- 141
1
vote
1
answer
308
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Decision rule that minmize the probability of error
Given
We consider a real-valued, discrete-time communication system with a channel gain $h$ and additive white Laplacian noise of unit scale with two possible signals $s \in (-\mu,+\mu)$ that are ...
1
vote
0
answers
61
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Generating cycles on a strongly connected graph
I have been thinking about the problem of generating cycles from a given node, on a strongly connected graph. The goal is to generate cycles that are good, with respect to an objective function $f$.
...
- 9,424
1
vote
0
answers
261
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What methods exist to prove a best strategy?
In decision or game theory, what methods exist to prove a "best" strategy? For example: Consider a game where a standard deck is shuffled and one card drawn face-down. The single player wins if they ...
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1
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0
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108
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How do you solve the Absent Minded Gambler problem?
Introduction
The following is a decision problem I created today, and which I am unable to resolve. I would state the problem in the general form, and not assign specific payoffs to any of the ...
- 1,217
1
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0
answers
305
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Far-too-simple proof of (limited) Debreu representation theorem; where's the error?
Debreu's theorem: Let $X$ be a topological space that is connected separable or second countable. A binary relation relation $\succsim$ on $X$ is complete, transtitive, and continuous if and only if ...
- 439
1
vote
0
answers
138
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Fixed point to maximum probability measure in decision problem
I am not a mathematician, so I am amply challenged by this issue. It may be very easy, or it may be impossible, indeed I have found some references but I have trouble with the jargon. Perhaps, if you ...
- 111
1
vote
0
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207
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Proving that the language is in class P
I was asked to prove that the next language is in class P (polynomial):
L={ $a$ | $a$ is a 3CNF and we can split the formula into two 3CNF formulas such that there exists an interpretation that ...
- 51
1
vote
0
answers
21
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Prove that all Bayes' solutions are admissable
I want to prove that all Bayes' solutions (those of maximum expected utility) are admissible, meaning that there's no other decision that dominates it.
By definition, a decision $d_a$ dominates $d_b$ ...
- 555
1
vote
0
answers
506
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MPE versus MAP estimates
In my class notes both MPE (Minimum Probability of Error) and MAP (Maximum Aposteriori Probability) estimators are shown as:
$$h_{MPE}(\mathbf x) = posterior\ mode = arg\ max_{y\in Y}p(y|\mathbf x)$$
...
- 690
1
vote
2
answers
57
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Need some clarification on what "decidability" means.
[I am relatively new to computability theory, so please try to avoid complicated jargon except as absolutely necessary. Thank you for your time!]
So I get that a decision problem is decidable iff ...
- 524
1
vote
0
answers
39
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Is the question whether the value of a given definite integral has a closed-form decideable?
Suppose, we have a definite convergent integral (possibly improper) with an elementary function as an integrand.
Is there an algorithm deciding whether the value of the integral has a closed-form ...
- 85.5k
1
vote
0
answers
261
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What are some techniques of constructing a good utility matrix?
A utility matrix is considered to be subjective and arbitrarily defined. Therefore, we run the risk of over-emphasizing or under-emphasizing the possible alternatives. Are there ways to design an ...
- 555
1
vote
0
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160
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Bayes risk and Bayes decision
We are considering a sample of size $n$ from an exponential distribution, with parameter $w >0$. We wish to produce an estimate for $d$, for $w$ , with loss function: $L(w, d)=w(w-d)^2$
The prior ...
- 713
1
vote
0
answers
574
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Undecidability of first-order satisfiability problem?
I need some clarification on understanding the "undecidability of" First-Order Logic (onwards, FOL). I understand that it means that the set of FOL theorems is undecidable (i.e. there is no effective ...
1
vote
0
answers
79
views
Expected utility of action, given probability model
We record measurements of an appartus every day. If apparatus doesn't break (it has probability equal to $1-p_2$), it will measure zero with probability $p_1$. If apparatus breaks (probability $p2$), ...
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