Questions tagged [decision-theory]

Decision theory is the science of making optimal decisions in the face of uncertainty. Statistical decision theory is concerned with the making of decisions when in the presence of statistical knowledge (data) which sheds light on some of the uncertainties involved in the decision problem.

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Is there a distribution where the data is generated from a decision process?

Sometimes my data is the result of the decisions each individuum of the observed population made: how much invest, how much gambled etc. The outcome would be a metric variable: money spent in total by ...
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admissibility under weighted error loss implies admissibility under square error loss

Show that admissibility of a decision rule under weighted square error loss implies it's admissibility under square error loss. Weighted Square error loss = $\frac {(d(x) - \theta)^2}{\theta*(1-\...
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admissibility of bayes rule

How to show that for a binomial(n, p) distribution, the MLE X/n is admissible under square error loss? The Bayes rule undr square error loss with beta($\alpha, \beta$) prior is X+$\alpha$/ (n +$\...
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Bayes estimator under LINEX loss?

Suppose we are trying to estimate a real valued parameter under the linear exponential (LINEX) loss. Suppose we have some prior $g$ that has finite mean, variance and a moment generating function. ...
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64 views

Bayesian calculation of parameter multiplying normal

I am interested in a model like: $y_{i} = \sum_{k\in K}{\beta_{k} z_{k}}$, with $z_{k} \tilde{} N(\mu_{k}, \sigma_{k})$. where $\beta \equiv(\beta_{k})_{k\in K}$ is not known, but all else is. I ...
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Detecting a consistent pattern in a dataset via Decision Trees and cross-validation

Assume a classification problem where there are two classes and the aim is to detect a consistent pattern which successfully separates the input dataset regardless of how we divide it into training / ...
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Is 'fair statistics' a thing?

Given that statistics can often be abused to deliberately present 'facts' to support a pre-existing viewpoint. (Lies, damned lies and statistics). And given confirmation bias. Is there an ...
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Optimal Number of Entries in Contest of Skill

Objective: I'm looking for the optimal number of unique entries in a contest of skill with monetary prizes. Description: The contests vary from as little as 20 entries up to 10,000+ entries. You ...
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Intuition behind the Stein's paradox [duplicate]

I had read wiki and some sources. jmanton's blog, Wasserman's blog the background is that: You have Xi ∼ N(θi, 1), and we want to estimate the each θi. Where Xi are ...
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Drawing numbered balls from an urn

PROBLEM There is an urn with a set of balls where each ball is labeled with a different integer. The numbers on the balls are known and are not a range of integers. For example the set of balls could ...
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Bayes decision theory: Classification error probability

In Bayesian decision theory: Given $\omega_1$ and $\omega_2$ as two classes for classification, $P\left( \omega_1 \right)$ and $P\left( \omega_2\right)$ their prior probabilities, $x$ the feature ...
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What is the difference between a loss function and decision function?

I see that both functions are part of data mining methods such as Gradient Boosting Regressors. I see that those are separate objects too. How is the relationship between both in general?
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Bayesian Learning with Dynamic Programming

This question is more general than a regular programming question. I am hoping for a reference type of answer. I am working on a model that involves bayesian learning in a dynamic programming problem....
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393 views

Loss function that relates ROPE with HDI?

In Doing Bayesian Data Analysis (link to the book) and Bayesian Estimation Supersedes the t-Test, J. Kruschke proposes using the following criterion to reject or accept the null hypothesis in a ...
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174 views

Decision function problem based on the logistic function

Suppose we have a bunch of a sampled pairs $(x_1,y_1)...(x_n,y_n)$ with the $y_i =\pm1$. Then consider the decision function $h(x) = -1$ if $p(x)=\frac{1}{1+e^{-x}}\leq0.5$, and $h(x) = 1$ if $p(x) ...
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Given 3 discriminant functions I was able to classify 2D patterns, but how do I plot decision boundaries via matplotlib?

I have implemented the discriminant function and was able to classify the 2D patterns (via Python), but I have troubles thinking about an approach to plot the decision boundaries. Hope anyone has an ...
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Formal justification of Bayesian inference as a model for belief

I remember a proof that Bayesian probability theory is the only valid method for representing beliefs, it went something like we represent belief by some non-negative function over some domain of ...
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Admissibility and domination for estimators [closed]

Watching a video by the "mathematicalmonk" on the web, I was wondering how to answer this kind of questions: Given $X_1,\ldots,X_n\sim \mathcal{N}\left(\mu,\sigma^2\right)$. Assume that $\mu$ is ...
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What is a robust way to find the max of $n$ independent, non-identical random variates?

Suppose I observe $n$ random variates along with their variance (but not mean) and I'd like to select the one with the largest mean as frequently as possible. The procedure must be memoryless--you ...
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Given $n$ different univariate non-normal sample sets calculate for a new sample, $x$, which it most likely belongs to [duplicate]

Say you have $n$ different, non-normal, potentially overlapping data sets of samples. Maybe their densities look something like: and you are given a new sample $x$, how would you decide to which of ...
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197 views

Maximizing returns - A Bayesian approach

I want to design a Bayesian model for a simple asset allocation problem. Say I can buy $a_i$ amounts of $N$ assets. The return values of these assets are given by random variables $r_i$ with known ...
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2k views

How to incorporate constraints in random forest output

Suppose I am doing random forest classification of labels $A$,$B$,$C$,$D$. There is some theoretical ordering to this output such that when $A$ is more likely than $B$, $B$ is also more likely than $C$...
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Computing Confidence in A/B testing

First of all, I'm fairly new to Decision Theory and my question might be rather too simple for the experts. I am having some issues in computing confidence for A/B Testing. I will first explain the ...
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determining the number of variables in the kernel of a U-stat

I'm working through some problems on deriving U-statistics, and I am unsure about determining the number of samples (random variables?) that are needed for the kernel of a given U-stat. For example: ...
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651 views

Showing that a statistic is ancillary for a parameter

Working through a HW problem, and a hint is that for a decision rule $$T(X) = \frac{X_{(1)} + X_{(n)}}{2}$$ Then $$T - \bar{X} $$ is ancillary. Intuitively this makes complete sense, but I am ...
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Selection of sites for wind power generation using time series from 600 candidate sites

I have 3 hourly power generation data for around 600 locations for a year. (i.e. 8 data per day for 365 days for each location.) I want to find out a way where out of this 600 locations, I can say ...
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How do I combine multiple prior components and a likelihood?

Lets imagine I am comparing two groups of animals (treatment/control). There is previous data from cell cultures indicating the treatment should have a positive effect. This gives me "prior component ...
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684 views

Markov Decision Process and its generality

My major is CS and I have a question about Markov decision process. I have been reading a book, planning with markov decision process an AI perspective. While reading it, I have a question regarding ...
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596 views

Prove/counter example: A minimax decision rule is always Bayes wrt some proper prior

Not sure whether the claim is true or false. If claim is true, intuitively, it might have something to do with "least favorable priors", but am not able to figure out the connection. If claim is ...
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What is a loss function in decision theory?

My notes define a loss function as the 'cost' incurred when the true value of $\theta$ is estimated by $\hat\theta$. What kind of cost is it talking about? monetary cost? or is it something related to ...
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156 views

Using decision trees with very infrequent outcomes

I am working on decision tree model and a value in dependent variable (churn) is very less. we have 1.5 lac records and only 1700 records have churn = 1. While using decision tree model, tree is not ...
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Decision boundary plot for a perceptron

I am trying to plot the decision boundary of a perceptron algorithm and I am really confused about a few things. My input instances are in the form $[(x_{1},x_{2}), y]$, basically a 2D input instance (...
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Example Of Strict von Neumann Inequality

Let $r(\pi, \delta)$ denote the Bayes risk of an estimator $\delta$ with respect to a prior $\pi$, let $\Pi$ denote the set of all priors on the parameter space $\Theta$, and let $\Delta$ denote the ...
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Is the expected value a valid decision-making method in a very short term?

This might be related to game theory more than statistics, but I decided to ask this question here. Let's assume you're offered a lottery. There are a hundred balls in a bowl: 99 white balls and one ...
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Optimizing “twenty questions”-like decision tree queries

I'm looking for an algorithm which can optimize the selection of queries used to build a decision tree. However, unlike most decision trees, I am constrained to ask the same set of questions for every ...
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Value of Information for a simple investment problem

Assume the following problem: You're deciding whether to invest into an opportunity with uncertain cost $c$ and value $v$. The cost has been estimated to be normally distributed with 90% CI between 1 ...
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327 views

The meaning of translation completeness w.r.t a random variable

I stumbled upon this term in McFadden - Analysis of qualitative choice behavior (page 111). It is said that "A random Variable $X$ is translation complete if for a function h of bounded ...
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Question about proof for luce choice axiom w.r.t. conditional probability

In Luce (1959) the choice axiom is definied, that for a finite subset $T$ of $U$ such that, for every $S\subset T$, $P_S$ is defined. If $P(x,y)\ne 0,1$ for all $x,y\in T$, then for $R\subset S\...
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Under which conditions do Bayesian and frequentist point estimators coincide?

With a flat prior, the ML (frequentist -- maximum likelihood) and the MAP (Bayesian -- maximum a posteriori) estimators coincide. More generally, however, I'm talking about point estimators derived as ...
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What technical language to describe the degree to which probabilities are likely to be modified by future data?

I'm trying to reason about something I call "estimate stability," and I'm hoping you can tell me whether there’s some relevant technical language, so that I can learn about it and then write a ...
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1answer
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Maximizing number of positive targets in first decile in logistic regression

When We estimate logistic regression using MLE we try to minimize $-2ln(likelihood)$, which is equivalent to minimization of sum of squared deviance residual. Is it possible to fit logistic ...
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Sampling to maximize model accuracy

Suppose you have a relatively small random sample and have a corresponding model $\ Y$ ~ $\operatorname{Bernoulli}(p_i) $ $\ \operatorname{logit}( \hat{p_i} )=\hat{\beta}*X$ and now want to draw a ...
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Coin flipping, decision processes and value of information

Imagine the following setup: You have 2 coins, coin A which is guaranteed to be fair, and coin B which may or may not be fair. You are asked to do 100 coin flips, and your objective is to maximize the ...
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Bandits without exploitation: finding the best items with incomplete information

I'm trying to analyze a general game. This is probably well-known, in which case pointers to relevant literature would suffice (but explanation would not be declined!). If it's not standard, of course ...
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1answer
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Decision tree model evaluation for “training set ” vs “testing set ” in R

So I got my training set with 70% of my data called "train" / 30% "test" I use ctree to get my decision tree model with something like this code below : ...
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682 views

A constant as an admissible estimator

This is a homework question so I would appreciate hints. I believe I have the first part correct, but I fail to see how the second part is different. Assume square error loss, $L(\theta ,a)=(\theta -...
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SVM decision function

our decision function e.g. in SVMs for binary classification (where the response is labeld by $y_i \in \{-1,1\}$) has the form: $f(\mathbf{x}) = \text{sgn}(\mathbf{w}^\top \mathbf{x} + b)$ where $\...
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Decision theory - reject option

In decision theory, we define a reject option ($\theta$) so that when making decision is difficult, the case will be ignored. Suppose $1/k \leq \theta \leq1$: If $\theta=1/k$ no cases will be ...
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175 views

Classification optimal decisions considering a loss function

Suppose we're given data from three different classes which are normally distributed with the following means and variances: $C_1: \mu_1=(1,2)^T, \Sigma_1^{-1}=( \begin{array}{ccc}2 & 1 \\1 &...
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1answer
109 views

A logistic problem about decision theory

Belows is the question and the solution for part c which is the part that i don't understand. Can someone explains to me? I don't quite get how it gets $3\over 7$ and why does it needs to?Any hint ...