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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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Interpreting by dividing up results of AB test

at my company we are doing AB tests (with 95% confidence) for features of our game (mobile app, hyper-casual game, Global scale). After the tests had ran its course, we have a practice of dividing up ...
Thinh Vuong's user avatar
4 votes
1 answer
75 views

Sample mean of Bernoulli trials is admissible under squared loss

Let $X_1,\ldots,X_n$ be i.i.d. Bernoulli trials with probability $\theta\in(0,1)$, and let $L:(0,1)\times[0,1]\to\mathbb{R}$ be the squared loss function, i.e. $L(\theta,a)=(\theta-a)^2$. I am trying ...
Anon's user avatar
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What time window to use for independent variables?

I have 10000 listings on various days. My y variable is % of transactions that happened 7 days before listing with a loss > 0 is 30% or more then 1 else 0. Now losses can happen before listing also....
Mnag's user avatar
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3 votes
1 answer
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Difference between generalisation error (Vapnik risk) and frequentist (statistical) risk

I'm reading these lecture notes: http://www.iro.umontreal.ca/~slacoste/teaching/ift6269/A19/notes/lecture5.pdf I always learned: "risk is the expected loss". In these lecture notes I see two ...
Tchaikovski's user avatar
2 votes
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62 views

Decision theory when distributions don't have a first moment? [closed]

Lets say that we are presented with two gambling opportunities and would like to decide between them in a decision-theoretic framework. For gamble 1, the cost is $1$ and the payoff is $X_1$ where $X_1 ...
QMath's user avatar
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fuzzy decission making

I'm working on decision-making using Double Hierarchy Fuzzy Linguistic terms. I want to ask if can we compare two Double Hierarchy Fuzzy Linguistic terms. I am trying to implement the TODIM algorithm ...
Mehwish Tahir Mathematics's user avatar
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General Lower Bound of Power in Neyman-Pearson

Let $X$ be an $\sigma$-finite space $(\mathcal{X}, F_{\mathcal{X}}, \nu)$ valued absolutely continuous random variable whose distribution is one of $P_0 = f_0(x)d\nu(x)$ or $P_1 = f_1(x)d\nu(x)$. We ...
温泽海's user avatar
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Is there a good review on complete class theorems?

I'm trying to get an overview of the various results called "complete class theorems" and their relatives, especially the ones that say things along the lines of "every admissible ...
N. Virgo's user avatar
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How to Visualize Integrated Risk in a GMM with the Defined Risk Function

In my Bayesian decision theory research within a Gaussian Mixture Model (GMM) framework, I've come across the need to visualize the integrated risk function, especially in the context of two ...
Alireza Ghazavi's user avatar
3 votes
2 answers
100 views

What is a scoring rule for binary classification that is not dependent on the "difficulty" of classification?

Consider a model that predicts the probability of some binary event $Y$ (potentially given some features $X$). Denote the estimated probability of $Y$ occurring as $\hat{p}$. One possible choice for a ...
ischmidt20's user avatar
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Neyman-Pearson Testing: Swapping the main and alternative hypotheses to ensure P(Type I) < P(Type II)

I have been reading up on hypothesis testing, and realized I misunderstood something, which made me mix Fisher's p-values with Neyman-Pearson's critical regions. I am going to amend that situation, so ...
W_vH's user avatar
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Fitting probabilities for decision trees

I perform an accident analysis with decision trees. Some cases are clear (all info exists), other ones haven’t it. By my previous experience, I can add missing data (restoration costs). But I don’t ...
manro's user avatar
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3 votes
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Find the formal bayes rule

Consider a decision rule as an interval, $\delta(x) = [a_1,a_2], \ a_1 \leq a_2, \ a_1,a_2 \in \mathbb{R}$ and the loss function $$L(y,a) = L(y,[a_1,a_2]) = (a_2-a_1) + c(1-1_{y \in [a_1,a_2]}).$$ ...
Oskar's user avatar
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1 answer
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What studied statistical model (if any) fits this application?

I'm having trouble identifying what statistical model or methodology is suited for my application. My situation is as follows: I want to create a stock trading agent that trades a single stock-cash ...
QMath's user avatar
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Outlier detection on a measurement stream. Decision theoretic, Bayesian approaches?

I have a stream of real valued measurements $x_1, x_2, \dotsc$ that I expect to be, for the most part, normal distributed with some unknown mean $\mu > 0$ and unknown variance $\sigma^2$. However, ...
ummg's user avatar
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Predicting willingness-to-pay for a risk-averse person who can 'select' lotteries

I'm studying how the willingness-to-pay differs for individuals who can 'select' lotteries. Individuals are presented with L1 first and can pay some amount to get lottery L2. Assume these are my ...
ordering-lotteries-help's user avatar
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Long-term frequentist decision strategy. How to "unreject" H0 when false positive occurs?

TLDR: How to quantify the H0/H1 plausibility, when there are multiple statistics available on the matter? What is a good source to learn more about frequentist decision making strategy? Once in a ...
kroszczek's user avatar
2 votes
1 answer
48 views

Do Bernoulli bandits need a different treatment if the rewards are sparse?

I have a problem where, effectively, my slot machines have very low payout probability (on the order of 1% for the "best" slot machines) and my goal is to minimize the number of actions to ...
Alexander Soare's user avatar
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Where does cross-validation fit into a model selection workflow with inference?

Say we have some model, $f(x) = \hat{Y}$, such as linear regression, that estimates an output dependent value for some set of input data. We want to do inference on the coefficients of the model to ...
Estimate the estimators's user avatar
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Is there any bias introduced by evaluating a model and decisions based on this model on the same data set?

As an example, let's say we have some financial time series such as closing prices of some stock and we would like to evaluate the ability of different models to forecast future closing prices as well ...
QMath's user avatar
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Why is the statement "the bayes estimator is bayes optimal" profound?

I'm trying to understand why people make a big deal about the optimality of a Bayes estimator. Certainly, if I have a Bayes estimator, then my expected loss is minimized, almost by definition. So, $$ \...
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Which method should be used to determine the class ID of multiple SVM models?

I'm using Support Vector Machine(SVM) with image classification. Each SVM model results a linear model $$y = wx + b$$ Where $w$ and $b$ is the SVM parameters. If I have multiple SVM models, I will get ...
euraad's user avatar
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3 votes
1 answer
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Modeling the probability distributions for the Bayes classifier

According to the Wikipedia, the Bayes classifier assumes knowledge of the distributions of $X | Y$, where $X$ and $Y$ are the random variables of the features and the classes, respectively. Let's ...
synack's user avatar
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3 votes
1 answer
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Reconciling Nondeterministic and Probabilistic Decision Rules

I've been getting a bit stuck recently on how to reconcile the two seemingly-competing ideas of nondeterministic and probabilistic decision rules. As an example: Let $t=0$ denote the current time and ...
QMath's user avatar
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1 answer
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Is there an equivalent for Yates' correction for a confusion matrix-derived metrics?

Given the following table of predictions vs. actual states: ...
Bryan's user avatar
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4 votes
1 answer
108 views

Minimizing the expected loss (PRML)

In Bishop's PRML in section 1.5.2, the author introduces a loss function for classification, which is the expected loss, $$ E[L]=\sum_k \sum_j \int_{R_j} L_{kj}p(\textbf{x},C_k)d\text{x} $$ where Lkj ...
Bruce Murdock's user avatar
11 votes
2 answers
520 views

How much would you wager for a Cauchy distributed return?

Suppose there's a casino that has a game where you sample from a Cauchy (100, 1) distribution (mode is 100). If the sample is positive, then the casino pays you that amount, otherwise you'd have to ...
George Chang's user avatar
2 votes
0 answers
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Which concept applies here?

Suppose I want to learn about a random variable $X$ (e.g., whether the subject is infected with Covid). I have already accessed a random variable $Y$ (e.g., an antigen test), which can help me ...
Ypbor's user avatar
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2 answers
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Equivalent definition of stochastic dominance

Note that a distribution function (cadlag etc) $F$ is said to be stochastically dominated by a distribution function $G$ if $F(x)\geq G(x)$ for all $x \in \mathbb{R}$. The following result ...
Yashaswi Mohanty's user avatar
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1 answer
41 views

Alternate definitions of risk in decision theory?

In decision theory, given a family of distributions $(P_{\theta})_{\theta \in \Theta}$ on the data space $\mathcal{X}$, the risk $R_L(\theta, \delta)$ of an estimator $\delta$ for a given loss ...
gordta_chichrron's user avatar
1 vote
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54 views

Clarification Question from Berger's Statistical Decision Theory (Chapter 1, Exercise 12 a))

I have a CS background with intro stats courses, and currently, I am working through Berger's Statistical Decision Theory (the theoretical questions). I'd like to ask a clarification question: this is ...
gordon's user avatar
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1 answer
20 views

Population bias in survey leading to inaction

This isn’t exactly an academic statistics question, but it is a real problem that I’m trying to understand with regards to bias in survey statistics leading to issues in real-world decision making. I’...
Concerned Sampling Person's user avatar
1 vote
1 answer
86 views

Questions regarding power of test and type II error

I´m preparing for a lecture in decision theory and I´m a little bit confused by the notation used by my prof. On the first slide under remark 3.2 point v) its written, that $\beta(\varphi)$ is equal ...
this_is_not_easy's user avatar
1 vote
0 answers
28 views

Modeling multiple-choice data

In my experiment, participants had to make a series of decisions between different options. On each trial, they were presented with a different number of options to choose from, and each option varied ...
TanZor's user avatar
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1 vote
1 answer
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N-ary decision tree with categorical features

I want to build an n-ary decision tree with categorical features. I am using ordinary ID3 algorithm to build a tree. Lets take the next dataset as a training dataset for building a decision tree: ...
dzi's user avatar
  • 113
6 votes
1 answer
705 views

Understanding what defines a Bayes optimal classifier in classification tasks

Say we are given a data-distribution $D$ over $\mathcal{X} \times \mathcal{Y}$, where $\mathcal{X}$ is our input/feature space and $\mathcal{Y}$ the set of (discrete) labels. Hence, $D$ is a joint-...
saper0's user avatar
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1 vote
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How to derive a decision rule as derived from Bayes rule?

I have a dataset consisting of 5 classes and the prior probability is $p(\omega_c)=\frac{|D_c|}{\sum_{i = 1}^{5}|D_i|}$. Suppose each class $c$ associated with the likelihood $p(x|ω_c)\,=\,\text{N}(\...
Furkan Mola's user avatar
5 votes
1 answer
155 views

How to estimate when an event of interest is overdue?

I'm looking for a principled way to estimate when an event of interest is overdue (a binary decision/alert), not just predicting when it is supposed to happen. In the survival analysis literature I ...
Georg M. Goerg's user avatar
2 votes
0 answers
24 views

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!
Abderrahmen Hamdi's user avatar
3 votes
0 answers
315 views

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 ...
qp212223's user avatar
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1 vote
0 answers
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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 ...
Milap Jhumkhawala's user avatar
0 votes
1 answer
70 views

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 ...
szm's user avatar
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1 vote
0 answers
99 views

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 ...
Christian Chapman's user avatar
3 votes
1 answer
80 views

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 ...
user avatar
2 votes
1 answer
229 views

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 ...
DancingIceCream's user avatar
1 vote
1 answer
107 views

At what value of p are you indifferent between action A and action B?

Problem statement: Suppose you are deciding between two actions, A, and B, and are testing between two mutually exclusive hypotheses, H1 and H2. If you choose action A, you receive 1 dollar if H1 is ...
Gioi Hoc Sinh's user avatar
1 vote
0 answers
75 views

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 ...
Eric Johnson's user avatar
4 votes
1 answer
379 views

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 ...
user551504's user avatar
0 votes
1 answer
15 views

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 ...
user346541's user avatar
2 votes
0 answers
78 views

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 ...
Mark's user avatar
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