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Questions tagged [decision-theory]

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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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17 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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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
38 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
36 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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31 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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28 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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63 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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39 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
24 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
54 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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0answers
72 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
90 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
43 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
21 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
41 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
142 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
53 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
23 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
67 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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0answers
68 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 ...
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1answer
213 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
57 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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0answers
13 views

Predicting Value versus predicting likelihood of value in auction

I am in a situation where I am trying to estimate what price will win an auction. I am trying to decide whether I should build a model to a) predict the price directly, or b) given a price as data, ...
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2answers
284 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 ...
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1answer
33 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
255 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
47 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
291 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
97 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
349 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
599 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
214 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
83 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 ...
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0answers
18 views

Controlling the accuracy of a classifier by giving no answer

I have a multi-class classifier based on a feature vector $x$ (for example using logistic regression). Say I have accuracy $a\in[0;1]$. Now my classifier is allowed to sometimes refuse to answer. I ...
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1answer
122 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
784 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
88 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 ...
2
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1answer
55 views

Choose parameters ,such that MSE of an estimator is constant

I have an estimator : $X = (X_1,X_2,...,X_n)$ are iid and have distribution $B(1,\theta)$ $T(X) = X_1 + X_2 + ... + X_n$ I need to find such value of constants $\alpha$ and $\beta$ s.t MSE of ...
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1answer
68 views

Decision making under uncertaintly

In decision making under uncertainly we have these criterion 1- maximin criterion 2- minimax criterion 3- maximax criterion Now I want real life example to illustrate all of these criterion (I ...
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5answers
338 views

Why care so much about expected utility?

I have a naive question about decision theory. We calculate the probabilities of various outcomes assuming particular decisions and assign utilities or costs to each outcome. We find the optimal ...
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1answer
112 views

Comparing estimators of equal risk

I'm attending a course in mathematical statistics and it seems the lecturer tacitly assumes that given estimators $T_1,T_2 : \Omega \to \Lambda$ of a parameter $g : \Theta \to \Lambda$, a loss ...
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0answers
54 views

Bayesian Decision Making (for particular problem)

I've read several papers why p-values should be replaced by Bayes factors and trying to use them. What I have: say, I have matrix of 2000 rows and 1000 columns. In each column I need to make a ...
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0answers
66 views

Where can I find a measure-theoretic statement of the Von Neumann-Morgenstern axioms?

Most of the discussion of the Von Neumann-Morgenstern axioms I have seen isn't fully formal - in fact, a lot of it only applies to finite probability spaces. Where can I find a more general discussion,...
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75 views

How to bet on a binary event based on the markov transition matrix, state probabilities and the odds

There is a coupon full of football matches for a given day from a bookkeeper. I have scrapped another website and i have aquired continuous history of a particular match between ...
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3answers
834 views

How are estimators like the Horvitz-Thompson Estimator derived?

The Horvitz-Thompson Estimator is usually given by: $$ \hat{Y}_{HT} = \sum_{i=1}^n \pi_i ^{-1} Y_i $$ The proof that it is unbiased is trivial to do. In additional, there exists other estimators out ...