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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How did the probability change to integral in bishop's 1.78 formula

In the equation R is the decision region and C is the classification. I don't understand how did he go from probability to integral!
2 votes
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kNN Classifier Asymptotic Error Rate versus Bayes Error Rate

Suppose we are in the realm of $M$ class classification, $M \in \mathbb{N}$. I have seen the following result stated many times, but only proven for the case $k = 1$. I would like to prove it for ...
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How do I build a decision tree model with a dataset that only has categorical values

kI'm trying to build a decision tree model on a dataset that only has categorical values, an example fragment of the dataset is below. My training dataset consists of 40 observations ...
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How to combine information from expected losses for a multiparameter model?

I have Y~N(mean, variance), then the expected loss for each parameter, mean, E[L(mean,d*)], and variance E[L(variance,d*)]. I want to have a measure that integrates both expected losses.
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Can we use drift diffusion model for three-alternative forced choices?

It is well known that the classic drift-diffusion model (DDM) was proposed for two alternative forced choices. However, is it possible to use the DDM for an experiment with three alternative forced ...
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1 answer
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Randomly choose between options with multiple criteria

Here's the problem: I have some options. Each is represented with three attributes or say criteria (with normalized values between 0 and 1). I want to randomly choose one of these options based on ...
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Is this decision rule well-known/optimal in some setting?

First, you'll have to forgive me if my exposition of this is not the best, I am a computer scientist, not statistician. I have a certain classification task where I am given two (say discrete for ...
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Are there any (exponential) families without a minimal sufficient statistic?

Bahadur's theorem says that if a minimal sufficient statistic exists, then a complete sufficient statistic is also minimal sufficient. Are there any (homogenous, identifiable) families with a complete ...
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How would you compare competing Bayesian hierarchical models that are estimating the same time-series?

Suppose we have two models $\mathcal{M}_1$ and $\mathcal{M}_2$ that attempt to estimate a time-series $\boldsymbol{\theta}$. Let's assume that I actually know the true values for $\boldsymbol{\theta}$....
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How to eliminate constant to derive the decision rule in terms of the sufficient statistic $\bar{X}$ for normal distribution means hypothesis test?

Suppose that we have a random sample, of size $n$, from a population that is normally-distributed. Both the mean, $\mu$, and the standard deviation, $\sigma$, of the population are unknown. We want to ...
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Interpretation of decision curve analysis

I have read but apparently not internalized many of the resources at decisioncurveanalysis.org I would like to report DCA for a predictive model. The results of my DCA are visually similar to this ...
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Does scoring rules really only apply to categorical outcomes?

The wikipedia article on scoring rule says that It is applicable to tasks in which predictions must assign probabilities to a set of mutually exclusive outcomes or classes. The set of possible ...
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Drift-diffusion model: Can the accumulated evidence be expressed as probability?

For a reaction-time model, I am considering whether I can compare 1) a probabilistic classifier or survival model and 2) a drift-diffusion model (DDM). I am interested in predicting reaction ...
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In Bayesian Parameter Estimation - How Is The Parameter's Priori PDF Found? [duplicate]

I wish to explain my confusion through an example, so I understand all the contextuals aswell: Say I want to predict the chance of someone being a European man $C_M$ based on their height $x_i$. I.e I ...
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A Proper Conjugate Model for A/B Test for Revenue per Click (RPC)

What would be a proper Conjugate Posterior model for Earning / Revenue per Click in A/B test? The data is the total number of visitors and the total revenue per day per variant (A and B). What are the ...
4 votes
1 answer
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James-Stein estimator with multiple samples

Let $X_1, \dots, X_n \in \mathbb{R}^p$ be i.i.d. samples from the $p$-normal distribution $N(\theta, \tau^2 I)$. Suppose we are interested in estimating $\theta$ with known variance $\tau^2$. Take the ...
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1 answer
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Does a high-listed attribute in a Decision Tree represent a major cause for the target class?

I am wondering about two questions: Let us assume we have a Decision Tree, which wants to predict health. If the attribute "smoking (yes/no/occasionally)" is listed relatively high in the ...
2 votes
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How to Build a Model with Correlation / Statistical Dependency for Bayesian A / B Testing

I use the Beta Binomial model for A/B testing. I wonder if there a way to build a model in PyMC which models correlation between the conversion rate of group A with ...
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Which predictive models output the posterior distribution?

In a supervised learning context, the posterior distribution of the target given the predictors is often discussed in foundational treatments of the subject. One way this comes up is in decision ...
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Why do we have a Neyman-Pearson lemma for type one error but not an analogous one for precision and recall of a classifier?

We fix alpha to .05 in statistical testing. We never fix the precision or recall of a test. There's no decision theory lemma that guarantees we can find a decision region with fixed precision or ...
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1 answer
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Please help me understand a Figure in Bishop's "Pattern Recognition and Machine Learning", Sec 1.5.1 Minimizing the misclassification rate

The figure is Figure 1.24 on page 40 of Bishop's "Pattern Recognition and Machine Learning", Sec 1.5.1 Minimizing the misclassification rate: I don't understand this figure, starting from &...
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Please help me understand a sentence in Bishop's "Pattern Recognition and Machine Learning"

The sentence is in section 1.5.1 "Minimizing the misclassification rate" (page 39), underlined in red: The author thinks this statement is "clear", but I just can't understand. ...
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A Frequentist approach to modeling uncertainty around decision optimization

I'm curious about how a Frequentist would approach an optimization problem, where said problem is constructed using inferred parameters. As an example, I'll use price optimization given a demand curve....
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2 votes
1 answer
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Computing the Bayesian Estimator with Jeffreys prior for the Gamma distribution

Question: Let $X_1, · · · , X_n$ be a random sample from $Gamma(1, θ)$. The population mean is $θ$. Assume that the Jeffreys prior is used. Find the generalized Bayesian estimator of θ under the SEL (...
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Generalized Bayesian estimator (rule) of θ

Question: Let $X_1, · · · , X_n$ be a random sample from $Poisson(θ)$. The prior for θ is $G(α, β)$ Find the Bayesian estimator (rule) of θ under the SEL(squared error loss). Find the generalized ...
1 vote
1 answer
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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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2 votes
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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 ...
2 votes
1 answer
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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 ...
6 votes
1 answer
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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 ...
1 vote
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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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2 votes
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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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1 answer
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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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3 votes
1 answer
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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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1 answer
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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:...
6 votes
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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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1 answer
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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 ...
2 votes
1 answer
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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 ...
0 votes
1 answer
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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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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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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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1 answer
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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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1 vote
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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 | ...
2 votes
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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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1 answer
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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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1 answer
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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 ...
4 votes
1 answer
365 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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0 votes
1 answer
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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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2 votes
1 answer
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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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