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.

Filter by
Sorted by
Tagged with
0
votes
0answers
7 views

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 ...
0
votes
0answers
7 views

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 ...
0
votes
0answers
48 views

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 $\...
2
votes
1answer
67 views

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 $\...
1
vote
0answers
43 views

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
0answers
56 views

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 ...
0
votes
1answer
20 views

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 ...
0
votes
1answer
22 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 ...
3
votes
1answer
112 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 ...
0
votes
1answer
29 views

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 ...
2
votes
1answer
69 views

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 ...
0
votes
1answer
70 views

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\...
2
votes
1answer
41 views

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 ...
1
vote
1answer
124 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!
0
votes
1answer
88 views

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 $\...
0
votes
0answers
18 views

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 ...
2
votes
0answers
37 views

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 ...
0
votes
0answers
46 views

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 ...
1
vote
1answer
132 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 ...
4
votes
1answer
76 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 ...
0
votes
0answers
13 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
votes
2answers
83 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 ...
7
votes
3answers
367 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 ...
1
vote
0answers
22 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,
0
votes
0answers
71 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 , ...
0
votes
0answers
17 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
votes
1answer
38 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
votes
1answer
71 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 ...
0
votes
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
votes
1answer
36 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
votes
1answer
26 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
votes
1answer
77 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
votes
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
votes
1answer
115 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 ...
1
vote
0answers
130 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
votes
0answers
124 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
votes
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 ...
1
vote
0answers
16 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
vote
0answers
43 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
votes
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
vote
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 ...
1
vote
0answers
53 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 ...
0
votes
2answers
56 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', '...
0
votes
0answers
71 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
votes
1answer
147 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
votes
0answers
201 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
votes
1answer
100 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
vote
0answers
29 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
votes
1answer
203 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 ...
2
votes
1answer
263 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 ...

1
2 3 4 5 6