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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48 views

Select the best random variable [closed]

Let's assume I am offered to keep one of three slot machines, each with unknown and unique reward distributions. Each machine can output a -1, 0 or a 1 after each try. Given the following collected ...
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Why do we need the concept of Risk in Bayesian Decision theory?

I'm studying Bayesian decision theory as introduction to machine learning and I see the concept of Risk in a lot of places. In the course I read, they define risk as: Risk is the expected error ...
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How to interpret the results of the DCA curve?

I have a large sample of data, but only a small number of people have an event. I want to use a certain indicator to predict the occurrence of the event, but when I draw the DCA curve, I found a ...
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Summarization and resources for Bayesian decision theory

Looking for textbooks and/or resources to get familiar with Bayesian decision making. I have the book, Statistical Rethinking, by Richard McElreath and I've found this to be a really great resource ...
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51 views

Bayes Risk Not Connected to Observed Data

It puzzles me that the Bayes risk seems not connected to the observed data. Let me illustrate this with an example. Let a coin toss follow a Bernoulli distribution with a hidden parameter $\theta$ and ...
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Loss function in Supervised Learning vs Statistical Decision Theory

I am confused by the different definitions of Loss Function in statistical decision theory vs machine learning. In statistical decision theory, a loss function is typically defined as $L(\theta, \...
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There is no decision theory that isn’t Bayesian... or is there?

David Manheim says in a comment under a blog post: If you’re not making decisions, there’s no need for Bayes. If you are, you’re Bayesian whether you like it or not – there is no decision theory that ...
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James-Stein-style estimator when we place greater importance on some components

The James-Stein estimator allows us to get a better overall estimate of a mean vector (length $\ge 3$) than we would be able to get by estimating the components independently. My intuition is that, ...
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sufficient Condition for having same prediction in multiclassification

In statistical learning theory, the classification error is defined by $err(f)=P(f(X)\neq Y)$ where $f$ is a classifier $f:\mathcal{X}-->\mathcal{Y}$, $\mathcal{X}$ the instance space, $\mathcal{...
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Multiple hypothesis testing: lower bound for sample complexity of finding the different one

We have $m$ distributions $D_1,\dots,D_m$. We know that $m-1$ of them are $\mathcal{N}(\epsilon,\sigma^2)$ ($\epsilon>0$) and one of them is $\mathcal{N}(0,\sigma^2)$, but we don't know which one ...
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Is a constant ever inadmissible?

For now, assume square loss. Let's estimate some parameter $\theta$, such as $\theta = \mu$ in $N(\mu, 1)$. Is there ever a case where there is no such $c$ to make $\hat{\theta} = c$ an admissible ...
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What is an example of a loss function that is not minimized by the conditional expectation?

From statistical decision theory we know that if we want to minimize EPE (Expected prediction error) it is sufficient to minimize the conditional expectation of the loss function. $f(x) = argmin_{c} ...
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Cold Start and First Price Auctions

I have the following contrived scenario... I've participated on various auction platforms where I bid on widgets. Assume that win/loss outcomes on individual platforms are well-separated such that for ...
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Regression tree

Self-study question Given $(y_i, x_i)$, $i = 1, . . . , n$, where $y_i \in \mathbb{R}$ and $x_i ∈ R ⊂ \mathbb{R}^p$. Show that $\displaystyle \sum_{i:x_i \in R_1}(y_i − \hat{y}_{R_{1}})^2 +\sum _{i:...
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Decision Theory: Why is it called a "least favorable prior"?

I'm currently reading the chapter on Statistical Decision Theory in Larry Wasserman's "All of Statistics". Reading the section 13.4 about Minimax Rules he introduces the so called Least ...
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How is the threshold parameter practically selected for Scikit learn's decision tree algorithm and how to determine depth of tree?

I am referring to the so-called optimized CART algorithm that is explained on Scikit learn's website: https://scikit-learn.org/stable/modules/tree.html#mathematical-formulation I would appreciate if ...
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Random forest that aggregates by taking the maximum over the trees instead of taking the average

I want to make a Random forest that aggregates by taking the maximum over the decision trees instead of taking the average. By default Sklearn is taking the average, and I couldn't find how to change ...
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Better methodologies to make causal recommendations from correlated data?

I work as a data scientist at a SAAS company. We have an outcome variable, Y, that we consider "success" for our customers. We have a bunch of additional outcome variables X1, X2, X3 that ...
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284 views

Is $\frac1{n+1}\sum_{i=1}^n(X_i-\overline X)^2$ an admissible estimator for $\sigma^2$?

Consider a sample $X_1,X_2,\ldots,X_n$ from a univariate $N(\mu,\sigma^2)$ distribution where $\mu,\sigma^2$ are both unknown. Then it is known that under squared error loss, the sample variance $s^2=\...
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MDP optimal policy inverse problem

Given a map $\pi: S \to A$, is there an MDP with state ans action spaces $S,A$ such that it has $\pi$ as an optimal policy if we suppose the MDP is over an infinite time horizon and the optimality ...
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Uncertainty reduction with additional samples

I'm trying to estimate the uncertainty of a premium when we have $n$ observations of losses. My goal is to estimate the benefit (e.g., uncertainty reduction) when we add an additional sample to our ...
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MSE of randomized decision in Normal distribution

Suppose a sample $\bf{X}$$=(X_1,...,X_n)$ is from $X\sim N(\theta,1)$. The sample mean $T(\bf{X}$$)=\bar{X}$ is sufficient to the population mean $\theta$. For $\delta(\bf{X}$$)=X_1$, the decision $\...
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How to show that $d(X)$ is unique Bayes with respect to any prior whose mean is $\frac12$ and variance is $\frac18$?

Consider estimation of $\theta$ where $X\sim \text{Bernoulli}(\theta)$. Under squared error loss, I am asked to show that $d(X)=\frac{2X+1}{4}$ is unique Bayes with respect to any prior for which $\...
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Decision Theory and the 0-1 Loss

On Page 182 of Murphy's Probabilistic ML book (http://noiselab.ucsd.edu/ECE228/Murphy_Machine_Learning.pdf) he says that to pick class 1 this should be done iff $p(\hat{y} =0 | x) > p(\hat{y} =1 | ...
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Testing hypothesis in 0-1 loss function

Let's say we are testing $H_0=\theta\in\Theta$ vs $H_1=\theta\in\Theta_1$. Set the 0-1 loss function as $l(\theta,a)=I(r(\theta)\neq a)$ where $a\in\{0,1\}$ and $r(\theta)=I(\theta \in \Theta_1)$. I ...
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Standard deviation of X compared to 1-X, a problem related to utilities and statistical decisions

In health economics, a utility $U$ is defined as 1 = perfect health and 0 = death, though it is possible to have utilities $<0$ for conditions worse than death. Theoretically, $U$ is therefore in ...
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29 views

General Strategy to show an estimator is admissible?

I am getting into decision theory and I was wondering if there was a general way to check if a an estimator is admissible. (PS This question might have already been asked, sorry if that is the case I ...
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156 views

How to show that $X_{(1)}-\frac1n$ is the unique minimax estimator of $\theta$?

Let $X_1,\ldots,X_n$ be i.i.d shifted exponential with pdf $f_{\theta}(x)=e^{-(x-\theta)}\mathbf1_{x> \theta}$, where $\theta\in \mathbb R$. I have to show that $X_{(1)}-\frac1n$ is the unique ...
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Why the decision boundaries are linear in an input space?

In the section 4.2.1 in "Pattern Recognition and Machine Learning, Bishop", the author considered a 2-class problem and assumed class-conditional densities $p(x | C_k)$ are Gaussian, and all ...
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Explain equation 1.80 in Pattern Recognition and Machine Learning, Bishop

$$E[L] = \sum_k \sum_j \int_{R_j} L_{k,j} p(x, C_k)$$ L is a loss function that returns a real value given a pair (i,j), with i as the index of true class, and j as the index of the predicted class of ...
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Find a lower bound for the minimax risk $\displaystyle \min_{\delta}\max_{\theta\in\Omega}EL(\theta,\delta(X))\ge1-\frac{1}{2^n}$

Consider a random binary vector $X\in\{0,1\}^n. $Let $\theta\in\Omega$ be a probability vector in $\mathbb R^{2^n}$ with $X\sim\theta$. Consider the loss function $\displaystyle L(\theta,a)=\max_{x\in\...
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Is this case possible for Decision Tree?

I am studying decision tree and I would like to know if this case is possible: We have 2 features, each does not decrease the Gini of the previous node (=> not choose), but their combination (two ...
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192 views

Minimax and Bayes rule

Could anyone provide some examples that a decision rule which is minimax but not Bayes, and a decision rule which is Bayes but not minimax? Thanks!
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Decision boundary in a Bayesian classification problem

We are given a classification problem within Bayesian decision theory, with 3 equi-probable classes $w_i,\,i=1,2,3$ and 2 features (or dimensions). The average of the three classes are known to be $\...
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How can I determine the overall best algorithm from a set of algorithms given pairwise probabilities?

I am working on an evaluation of algorithms from a specific family of feature-selection algorithms. Using the Bayesian hierarchical correlated t-test, I evaluated each pair of algorithms to obtain the ...
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Maximising the utility when I have a large number of discrete decisions mixed with continuous decisions

I have an optimisation problem which is potentially quite tricky. Consider the problem of allocating a discrete number of resources (in my case, they are spare mechanical components for a large ...
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Repeated Multi-armed bandit trails of pure exploration: Bernoulli arms

I'm interested in analyzing a variant of the multi-armed bandit problem with pure exploration. In this variant, in each round we receive samples from two distributions and we need to estimate which ...
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202 views

Bayes classifier expected classification error for multiclass case

Assume a feature $x \in [a,b]$ and two classes $\omega_1, \omega_2$ with prior probabilities $P(\omega_1), P(\omega_2)$ and likelihood functions $p(x | \omega_1), p(x | \omega_2)$. Then, the expected ...
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Checking whether Brier score is a strictly proper scoring rule

I want to check whether Brier Score is a strictly proper scoring rule based on some definition I found here. Since the paper is behind a paywall, I provide the definition here: A scoring rule assigns ...
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Bayesian Decision making with a mixed effects model

Background A company runs an AB test in which the unit of randomization (the customer) can interact with the variant several times throughout the experiment. The outcome is a binomial random variable ...
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Decision tree: how you would expect the next split based on a set of variables?

I'm trying to understand the logic behind a question I was given during a mock test. Can somebody help me please? I am not sure I can understand the concept, hence be able to make it right in a ...
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Loss functions in statistical decision theory vs. machine learning?

I'm quite familiar with loss functions in machine learning, but am struggling to connect them to loss functions in statistical decision theory [1]. In machine learning, a loss function is usually only ...
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124 views

Why we use squared probabilities in the gini impurity

Why we are using squared probabilities instead of normal probabilities in gini impurity . Probabilities will always be positive , so why to square those ? Any leads would be highly apriciated , ...
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How do decision trees in random forests handle conflicts?

Let's say our input elements (training data) are 6 people with three attributes, Height, Weight, and Gender, and we are predicting if that person will have cancer or not (boolean 0 or 1). Let's say we ...
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235 views

Measuring the confidence of a softmax classification outcome

Suppose I have a softmax distribution produced by a classifier. There are four labels, and so the sum of the softmax probabilities over the four labels will be 1.0. I am looking for a measurement of ...
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How to quantify intangible costs for decision making

In many situations, decision-making requires weighing multiple losses. For example, you might determine the optimal threshold for a churn classification problem by comparing the cost of offering a ...
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27 views

Ordering list of items by two criteria

I have a list of items with two scores: scoreA and scoreB. To be more specific they represent the average of a list of accuracy scores and their maximum. Both of the scores range from 0 to 100%. I'm ...
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What is the best strategy for the simplified version of the multi-armed bandit?

Consider a simplified version of the multi-armed bandit problem, where: like in the standard multi-armed bandit: when you pull the lever of 1 bandit you win/lose some amount from that bandit ...
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225 views

How does Random Forest split?

Random forests or random decision forests are an ensemble learning method for classification, regression, and other tasks that operate by constructing a multitude of decision trees at training time ...
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Stochastic dominance and mean preserving spread

I need someones help on understanding the concepts of stochastic dominance and mean preserving spread. I have an exercise which could be used for explanation. Consider the following lotteries: L1 ={...

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