Questions tagged [decision-theory]

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3
votes
1answer
50 views

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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0answers
10 views

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 ...
2
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2answers
77 views

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 ...
6
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3answers
195 views

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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0answers
14 views

Is the admissible minimax decision rule ever a randomized action in frequentist statistics?

Are randomized action as opposed to pure action ever an admissible minimax rule in frequentist statistics,
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0answers
25 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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0answers
16 views

Statistical literature on task prioritisation problems

I am lookig for statistical papers on task prioritisation problems. In particular I am looking for solutions to the following problem or slight variations thereof: You have a set of tasks, each with a ...
0
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1answer
26 views

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 ...
0
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0answers
29 views

Explain Dempster Shafer Equation

I have a question about the Dempster Shafer theory application. I have four models where the output is of abstract level (crisp). I understand I have to use the confusion matrix (precision/recall) to ...
2
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1answer
26 views

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 ...
0
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1answer
25 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 ...
3
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1answer
76 views

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 ...
0
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0answers
8 views

Integrated AHP and Fuzzy logic for Supplier categorization

I am looking into a supplier classification problem. As I have a lot of vague and subjective criteria I am using Fuzzy Logic to classify suppliers on two dimensions. However not all criteria are ...
2
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1answer
39 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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0answers
51 views

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 ={...
0
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0answers
67 views

Why does AdaBoost use decision stumps instead of 0-depth trees?

Why is it that AdaBoost uses decision stumps for the weak learners? It seems simpler to me to just use the weighted majority of the data points for the classification. Why shouldn't we do this?
3
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1answer
54 views

The proper way to compute the posterior distribution of a distribution

Suppose I am a Bayesian working with multi-level data, $j$ and $t$. I run a model using $t$ that calculates the posterior distribution of a parameter $\theta_j$ for each $j$, which I then use to ...
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0answers
24 views

Bayesian decision making

I have a real world problem which I have reformulated into a simpler problem which hopefully you can help me solve. Picture this, I have the option to build a factory next to a conservation park. The ...
1
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0answers
12 views

Decision making with respect to utility function

I am currently working on a small project targeted towards predicting survival times (red, green functions) of certain engine parts. The ultimate goal is to decide what part would be the best choice ...
1
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0answers
36 views

Why would a Bayesian want to maximize expectation? [closed]

A Frequentist interprets probability as an estimate of how frequent an event is giving that we can repeat the experiment many times. It is natural for them to try to maximize the expected utility ...
0
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0answers
33 views

Aren't multi-armed bandits basically the same things as the Von Neumann-Morgenstern utility theorem?

I can't seem to find any material connecting the two ideas. How would one who is more knowledgeable about these topics relate them to one another? Is it that multi-armed bandits are just one way of ...
1
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0answers
21 views

Modeling and updating the reliability of two sources of information

I do not know the general framework this might fall under, apologies for the vague title. Assume that a decision maker's choice is dependent on two sources of information $f_1$ and $f_2$. Assume for ...
0
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0answers
27 views

Are the following terminologies error/risk/marmgin/regret bounds related?

I recently come across papers with titles resembling "Error/Risk/Margin/Regret Bounds" and I can't help but wondering if there is any fundamental (mathematical) difference between these terminologies? ...
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0answers
49 views

Optimal decisions based on frequentist estimators

Consider a decision problem aimed at minimizing the expected loss1 where the argument is a parameter estimate. In a Bayesian setting, given a posterior distribution of the parameter and the loss ...
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0answers
12 views

Detection Theory minimax with non differentiable interior

The minimax is used in detection theory and decision theory for minimizing the overall average risk for the worst case prior. $$ \min_{\delta} \max_{\pi_0} r(\pi_0,\delta) = \max_{\pi_0} \min_{\delta}...
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2answers
45 views

Book on decision theory

I am from physics background. I know basics about statistics (upto 'Statistical Inference - Casella'). I came across some articles talking about terms like 'reciever operating characteristics curve', '...
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0answers
50 views

Bayesian Decision Theory - Rejection & Bishop plot

Reading through Bishop, I stumbled upon this picture on p. 42 top left under the topic of Bayesian classification, but I am unclear on how this can be two posterior distributions, as they seemingly do ...
6
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1answer
139 views

Differentiable programming for general Bayesian decision theory

It is my understanding that differentiable programming and thus libraries like TensorFlow (e.g. TFP) and JAX can be used to solve Bayesian decision theory problems where e.g. we have a probabilistic ...
2
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0answers
132 views

How to choose operation point from precision recall curves for multi-label classification

Is there a commonly accepted method for selecting an operating point for a multilabel classifier to optimize for each of these aggregate metrics: micro averaged recall at some minimal acceptable ...
0
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1answer
57 views

Bayesian estimator $\theta(x)$

Given a training set of $(X, Y )$'s where the $X$'s are the source variables and the $Y$'s are the targets, derive an estimator that minimizes the mean squared error between target values and ...
1
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0answers
27 views

How to use the likelihood-ratio to compute the error probability? [closed]

In Bayesian decision theory, There is an analytical form of error rate, which is $$P(e)=\int P(e|\bf{x})p(\bf{x})d\bf{x}$$. For binary classification, we can compute the type I error probability with: ...
5
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1answer
190 views

Admissible Empirical Bayes Examples

I would like to hear about a few simple empirical bayes estimators that are admissible for high (i.e. at least 3) dimensional parameter space. What are some textbook lollipop examples to study for ...
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2answers
151 views

Derivation of Bayes classifier in Murphy's book

I am reading Kevin Murphy's Machine Learning book (MLAPP, 1st printing) and want to know how he got the expression for the Bayes classifier using minimization of the posterior expected loss. He wrote ...
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0answers
34 views

Is the shrinkage of subgroup analyses in meta-analysis an example of Stein's paradox?

This paper writes (edited for concision): Consider, a doctor in Germany confronted by a meta-analysis of long term‚ $\beta$ blockade after myocardial infarction. Although a robust beneficial effect ...
1
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0answers
46 views

Cannot understand a notation detail in ESL's Statistical Decision Theory EPE minimization

In The Elements of Statistical Learning, at page 18 the authors explain that, in order to minimize the EPE (Expected Prediction Error defined as the mean of the loss function: $\text{EPE}(f) = \mathbb{...
1
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1answer
65 views

Bayes Decision Theory With 3 Classes

I'm trying to create a Bayes classificator in 1 dimension with 3 classes. I have created the following graph, where you can see that from zero to $x_{bnd1}$ is the first area $R1$, then from $x_{bnd1}$...
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0answers
22 views

Understanding Randomized Estimators in Statistical Decision Theory

I'm reading through The Bayesian Choice by CP Robert with a particular focus on understanding randomized decision rules vs non-randomized statistical rules in the Bayesian context. In Section 2.3, he ...
0
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0answers
182 views

Compute the Risk function

Suppose we are given $(X_1,...,X_n)$ random variables which are iid. from $\mathcal{N}(\mu,\theta)$ and finite variance. Let $Y=\frac{1}{n}\sum_{i=1}^n(X_i-\overline X)^2$ and define a loss function $...
11
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4answers
193 views

Getting all answers correct by taking the same exam for fewest times

Rain never studies, so she is completely clueless during the midterm even though it consists of Yes/No questions only. Fortunately, Rain's professor allows her to re-take the same midterm as many ...
1
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1answer
73 views

Optimal classification rule given data, model and loss function

Setup Suppose I have a data set with a categorical variable $Y$ (with possible values $j=1,\dots,J$) and another variable $X$. I wish to classify $Y$ based on the information in $X$. For simplicity, ...
0
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0answers
29 views

Expected utility maximization when beliefs are inaccurate

In the framework of maximization of expected utility (MEU), is it somehow optimal or justifiable to make choices based on the subjective probability distribution when we know it may be inaccurate (...
4
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1answer
97 views

In what sense does interim monitoring of clinical trials “cost” a Bayesian?

I have read (and will seek a specific reference on the subject) that unlike Frequentist trials, Bayesians can continually monitor data as it accrues. A Frequentist tries to control, and thus ...
4
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2answers
561 views

Minimizing expected brier score and Brier score interpretation

For a probabilistic binary forecast, the BS (Brier score) is given by $$ \text{BS}= \begin{cases} (1-f_i)^2\\ f_i^2\\ \end{cases} $$ Where $f$ is the forecast. If the event occurs with probability $...
1
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1answer
45 views

Role of expected loss of the best forecast in decision theory

Suppose we have a random variable $Y$ with an unknown distribution $P$. We model it with a distribution $Q$. We are asked to make a point forecast under some type of loss $L$. We choose the loss ...
1
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1answer
47 views

Value of using a better normal distribution

I tried to derive this on my own, but my stats education proved too far back… (This is a problem in Bayesian decision theory – if that makes you uncomfortable, feel free to reformulate it) Let's say ...
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0answers
40 views

Does intended model use affect Bayesian parameter estimation?

Bayesian parameter estimation results in a posterior distribution for model parameters. The user may or may not be interested equally much in all properties of the distribution. Perhaps the user ...
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0answers
20 views

What does it mean when Risk function turns out to be a number?

I have a statistical decision making theory problem.I have to calculate the Risk Function for each of 4 decision rules.However,it turns out that the fourth Risk function is not a function of θ and it ...
2
votes
1answer
70 views

Why is the maximum risk of an estimator independent of a prior distribution over the parameter?

One way of choosing an estimator $\delta(x)$ for data $X$ distributed as $P_{\theta}(X)$, where $\theta \in \Theta$ is: $$minimize \sup_{\theta \in \Theta} Risk(\delta(x), \theta)$$ In this case why ...
0
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2answers
39 views

Bayesian (In)Decision

Let $A_j$ be the action of person $j$, $A_k$ be the action of person $k$, and $p(A)$ be the probability of an action. Using Bayes Rule, $$p(A_j=x|A_k=y)=\frac{p(A_k=y|A_j=x)p(A_j=x)}{p(A_k=y)}$$ If $...
0
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0answers
59 views

Response time of sequential probability ratio test for continuous-time observation process?

I hope to simulate the response time of a binary decision problem given continuous-time observation using sequential probability ratio test (SPRT). Traditionally with discrete-time SPRT, we calculate ...

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