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Questions tagged [decision-theory]

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23 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
149 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
40 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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25 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
80 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
79 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
225 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
34 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
19 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
19 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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38 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
45 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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12 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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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
38 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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42 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
39 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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86 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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50 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
26 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
124 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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27 views

Finding the type II error given the type I error for a minimax decision rule with 0-1 loss

Assume a two world state ($\Omega=\left\{ \omega_{0},\omega_{1}\right\}$ ) scenario and that we are given the [continuous] ROC curve $\left\{ \left(\alpha\left(\theta\right),1-\beta\left(\theta\right)\...
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116 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 ...
2
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1answer
112 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
72 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
42 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
46 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
197 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
60 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
25 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
71 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
101 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
295 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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1answer
59 views

Minimize mis-classification - 0 - 1 output

I am studying logistic regression from the book Advanced Data Analysis from an Elementary Point of View which states the following on page 280: “We minimize the mis-classification rate by ...
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2answers
322 views

Does a density forecast add value beyond a point forecast when the loss function is given?

Density forecasts are more universal than point forecasts; they provide information on the whole predicted distribution of a random variable rather than on a concrete function thereof (such as ...
2
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1answer
38 views

Multidimensional Bayes point estimates

Consider the posterior distribution $p(\theta|x)$. We aim to find a "good" estimate of the random variable $\theta$. The Bayes risk associated with the loss function $L(\hat{\theta}, \theta)$ is ...
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3answers
301 views

MAP is a solution to $L(\theta) = \mathcal{I}[\theta \ne \theta^{*}]$

I have come across these slides (slide # 16 & #17) in one of the online courses. The instructor was trying to explain how Maximum Posterior Estimate(MAP) is actually the solution $L(\theta) = \...
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0answers
58 views

Decision tree ,information gain and overfitting

If i use the information gain in order to evaluate the best split in a decision tree, why using a binomial split reduces the risk of overfitting ? Is the information gain test misleading if we have a ...
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3answers
302 views

Does a Bayes estimator require that the true parameter is a possible variate of the prior?

This might be a bit of a philosophical question, but here we go: In decision theory, the risk of a Bayes estimator $\hat\theta(x)$ for $\theta\in\Theta$ is defined with respect to a prior distribution ...
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0answers
121 views

The math behind Spearman-Karber method

A number of methods in my field use the Spearman-Karber method to estimate the minimum level of a variable needed for 50% success on a task. In addition to the original work, I have tried to increase ...
2
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2answers
410 views

comparing distributions - bayesian decision analysis

I am attempting to use Bayesian analysis to compare distributions to help with decision analysis - when to treat a patient based on a blood measurement X. Here you can see 1000 samples from two ...
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1answer
784 views

Different definitions of Bayes risk

I'm having trouble understanding the proper definition of Bayes risk. Let the data/variate $x \sim P(X|\theta)$, $\theta\in \Theta$, $\pi$ be a distribution on $\Theta$ (prior), $\hat \theta(x)$ be ...
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0answers
251 views

Why can't the complete class theorem be easily generalized to all locally-compact spaces?

So I was reading Christian P. Robert's The Bayesian Choice, going through the constellation of results related to complete class theorems, and I don't see why all of them are necessary. In particular, ...
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0answers
86 views

Optimal strategy for a combinatoric dice game

The game can be played at https://xcvd.github.io/dice-game/ The player gets 12 throws of 3 dice and chooses a grid to place these throws in (there are 6*6*6=216 possible throws). Each throw ...
2
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1answer
199 views

Why Loss function has to be bounded from below (statistical decision theory)?

In statistical decision theory the loss function $L(\theta, a)\ge-K > -\infty$ is often chosen for technical convenience (e.g. See [1] p.3 ). Can anyone explain why the above condition is ...
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1answer
1k views

What do the thresholds on x and y axis of ROC curve represent?

There is a detailed explanation of what the AUC of an ROC curve is here. However I have searched high and low for an explanation regarding what the X and y axes of the ROC curve are. I have understood ...
2
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1answer
89 views

Machine learning methods for exploring relationships for a continuous response variable

I would like to explore a model to predict the value of a continuous response variable, from a set (around 100) of explanatory variables. I do not want to apply PCA like feature reduction, because I ...