Questions tagged [decision-theory]

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40 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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9 views

Is there a risk function only dependent on bias of an estimator? [closed]

Is there a risk function implementable in software that is only a function of bias? So that when minimized, the coefficients in my regression are all unbiased? If a risk function that only considers ...
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10 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
37 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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40 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 ...
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1answer
115 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 ...
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0answers
40 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 ...
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1answer
54 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 ...
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0answers
26 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: ...
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1answer
172 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
103 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
28 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 ...
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0answers
23 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{...
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1answer
43 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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15 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 ...
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0answers
35 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 $...
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4answers
177 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 ...
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1answer
44 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, ...
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0answers
26 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 (...
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0answers
81 views

Two class bayesian decision theory

I'm new to decision theory, but in the many "intro to bayesian decision theory" tutorials, the two-category classification example is usually given. It boils down to deciding action 1 if it's risk is ...
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1answer
89 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 ...
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2answers
376 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 $...
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1answer
44 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 ...
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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
20 views

Is it rational to select a parameter posterior value because it maximizes utility, even if probability is low?

I did Bayesian parameter estimation and I have now an estimate of the posterior distribution for my model parameters (say I have 2000 samples). Now I would like to make the optimal decision under my ...
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0answers
23 views

Admissible and Inadmissible actions

Consider the following loss matrix. $\begin{array}{|c|c|c|c|} \hline & \alpha_1 & \alpha_2 & \alpha_3 \\ \hline \theta_1 & 1000& -300& 4000\\ \hline \theta_2 & -1000&...
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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 ...
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1answer
55 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 ...
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0answers
14 views

What are some approaches in online decision making under uncertain input data?

For example, I observe a set of measurements $n=\{n_1,n_2,n_3,n_4,....,n_k\}$. Here, a subset of measurements $\{n_1,n_2,n_3\}$ are assumed to be uncertain and are not trustworthy. As a decision maker,...
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0answers
9 views

Finding most effective sequence of treatments

I am looking for (any) pointers on how to approach the following abstract problem. Not: my statistics background is very limited, so I might very well be missing something obvious. We have subjects ...
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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 $...
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55 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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0answers
45 views

How to prove that the prior for which Bayes rule is also the minimax rule, is the least favorable prior?

I have read in the book Mathematical Statistics: A Decision Theoretic Approach by Thomas Ferguson that The prior for which the Bayes rule is also minimax rule, then that prior is Least favorable prior....
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0answers
110 views

Questioning the axiom of continuity in Von Neumann–Morgenstern utility theorem

In my previous question, I aksed about they we care so much about expected utility, rather than e.g., the variance in utility (Why care so much about expected utility?) From the helpful answers, I ...
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0answers
54 views

bayesian decision making - comparing expected loss

The problem is like this: Suppose that I am considering which country should I invest on, country A and country B, based on their GDP growth rate $\alpha$. There are two possible choices for each ...
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1answer
27 views

Choose one of two normal distribution that will give the probability of biggest value when sampling it

Suppose you have two (or more) normal distributions with different mean and variance. You can draw only one sample of only one of the available distributions. Your goal is to get the biggest value ...
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1answer
172 views

What is minCases in C5.0Control using R

from Package (C5.0 Decision tree Using R ) definition "minCases : an integer for the smallest number of samples that must be put in at least two of the splits." I very confuse about it . Please ...
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144 views

Decision Tree from Agglomerative Clustering

I have agglomerative clustering done. I want to convert it to a decision tree so I can figure out the cluster very quickly. How to do so? A tedious approach (bad, I know): Take the top ...
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1answer
177 views

Admissibility does not imply minimax

The answer to minimax estimator explains why minimax does not imply admissibility. The relevant statement is from https://www.stat.berkeley.edu/~yuekai/201b/lec6.pdf which says, minimaxity does not ...
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1answer
104 views

DECISION TREE : How to calculated for repeat decision noded such as this picture (C5.0 Algorithm -Decision tree)

I confused about decision tree such as this picture why repeat decision node.Could you please explain that decision tree. thank you
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0answers
45 views

How many sunrises are worth observing?

The one-sun version of Laplace's sunrise problem provides a Bayesian argument that, if on all $n$ mornings in recorded history the Sun has risen, its probability of doing so tomorrow is $\frac{n+1}{n+...
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1answer
25 views

How to categorize data as others if training set is not available?

I run into a problem. I am using the decision tree to classify the incident category based on the short description the user has used while logging the ticket. I have the training data only for 5 ...
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0answers
51 views

Model fitting vs minimizing expected risk

I'm confused about the mechanics of model fitting vs minimizing risk in decision theory. There's numerous resources online, but I can't seem to find a straight answer regarding what I'm confused about....
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2answers
262 views

Why is the risk function defined to be the expectation of loss function?

In decision theory, we define the risk associated with a particular predictor function as the expected value of the loss function. Since the input and output are considered random variables therefore ...
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1answer
61 views

Decision tree without the “tree”

I would like to construct something like a decision tree. However, instead of using "recursive partitioning" to build a tree, I would like to find an optimal set of "global" splits. For example, in a ...
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0answers
28 views

Can additional iterations of backward induction as described affect optimal policy?

Consider a game with the following properties: Single player Finite number of game states (after the player arrives at a terminal state, he or she can begin again from the start state; the player can ...
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2answers
77 views

Hypothesis testing using spectra

How does hypothesis testing work when a measurement is not a single number, but an entire spectrum? For instance, suppose we want to distinguish a species of plant based on its absorption spectrum. ...
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1answer
121 views

What problem or game are variance and standard deviation optimal solutions for?

For a given random variable (or a population, or a stochastic process), mathematical expectation is the answer to a question What point forecast minimizes the expected square loss?. Also, it is the ...
2
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1answer
408 views

Bayes estimate with weighted square error loss

First, let $T(x)$ be an estimator of $g(\theta)$ and assume we have a square error loss function defined as $$L[g(\theta),T(x)]=[g(\theta)-T(x)]^2$$ Then the posterior expected risk of $T$ is $$\...

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