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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Random Forest explanation
I am having trouble understanding Random Forest, especially some terms.
What is a node what is node size? What are ...
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How does an estimator that minimizes a weighted sum of squared bias and variance fit into decision theory?
Okay--my original message failed to elicit a response; so, let me put the question a differently. I will start by explaining my understanding of estimation from a decision theoretic perspective. I ...
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Optimal decision process to estimate Markov chain limiting distribution
Suppose there is a irreducible, reversible Markov chain with known states $1,\ldots,N$ and unknown transition matrix $T_{ij}$ and unknown limiting distribution $\pi_i$. I am able to repeatedly ...
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What is the meaning of admissibility within a class, that every decision rule in a class is admissible in that class?
Suppose that I have that $X$ is a Poisson random variable with mean $\lambda$. Suppose a decision rule is to estimate $\lambda$ by using $\delta(Y) = aY$. Now, let $K$ be the class of all decision ...
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Non-probabilistic vs probabilistic frameworks for decision theory in metric spaces
I have a task to make a decision, say to classify an object as $X$ or $ \overline X$. However, $\overline X$ usually means everything else and you only have positive examples of $X$ and not so many ...
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Bayes decision rule and thresholding
The best possible classification is for a set of samples drawn from any probability distribution is given by the Bayes decision rule.
For any distribution, the rule is given by
$$
f(x) = 1 \quad\...
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Calculating the risk of an estimator using zero-one loss
Consider two observations where
$$P_\theta(x=\theta+1)=P_\theta(x=\theta-1)=0.5,\ \
\theta\in\mathbb{R}$$Let $\mathbb{D}=\Theta=\mathbb{R}$ the decision
space. Suppose that the associated loss ...
user72621
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Bayes factor (B) vs p-values: sensitive (H0/H1) vs insensitive data
The question of a beginner in Bayesian stats. As far as I understand, it is claimed (e.g. Dienes 2014) that B-based inference allows us to either confidently reject/accept the null, OR declare the ...
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Most likely event in a multinomial distribution setting
I'm looking at the following scenario:
$k$ categories, distributed by a multinomial ($p_1,\dots,p_k$) such that $p_1 \ge \dots \ge p_k$. Draw $n$ samples. I'm interested in estimators/lower bounds ...
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Comparison between statistical decision theory and supervised learning [closed]
What are the differences and overlaps between statistical decision theory and machine learning ?
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What is a better political voting method? [closed]
We are in a season where some major elections are happening (e.g. U.S. elections) and I find it interesting to address.
Objective
When we decide "better", we need to define an objective. To be clear,...
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Does Bayesian Statistics have no concept of statistical hypothesis testing?
I was told that the framework of Bayesian Statistics has no concept of statistical hypothesis testing or confidence intervals.
How does this make sense? Bayesian statistics only says that we ...
user46925
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Is a max Brier score really a max Brier score?
I am generating some random data to test out a function that calculates Brier and scaled Brier scores. See here for a reference (http://journals.plos.org/plosone/article?id=10.1371/journal.pone....
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Which machine learning technique is appropriate for my problem?
I'm new in machine learning topics and I've problem in modeling my environment which has multi parameters with different value ranges and a few actions to perform when value of each parameter is not ...
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Is decision tree output a prediction or class probabilities?
A Random Forest works by aggregating the results of many decision trees. Recently, I was reading about how the RandomForest aggregates the results, and it made me question whether the results from ...
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On the proof of admissibility of constant estimators under squared loss
The question concerns the discussion in Wasserman, All of Statistics, Section 13.6. He defines:
An estimator $\hat{\theta}$ is inadmissible if there exists another
rule $\hat{\theta}'$ such that ...
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Email open-rate optimization
I am trying to maximize open rates of emails by selecting between two subject headlines {h1, h2} for a marketing campaing. The hypothesis is that different customers react to different headlines.
...
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How much to pay? A practical problem
This is not a home work question but real problem faced by our company.
Very recently (2 days ago) we ordered for manufacturing of 10000 product labels to a dealer. Dealer is independent person. He ...
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Why is Wald's decision theory not universally recognized as the foundation of statistics?
This is somewhat ill-defined, but: Why is Wald's decision theory not universally recognized as the foundation of statistics? I gather (or maybe I infer) that it was formulated to put frequentist and ...
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Why is the MLE/OLS estimator so common in regression despite inadmissibility? [closed]
Why is regression so commonly used if the OLS estimator for the vector of regression coefficients is inadmissible under the squared error loss function? Is it because of its historical popularity or ...
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Quadratic loss function implying conditional expectation
I am reading Bishop's pattern recognition book. In the decision theory part he first derives that using a quadratic loss function implies that our estimate $y(x)$ should be the conditional expectation ...
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What is an appropriate machine learning model for a dice game?
I'm having trouble thinking of the correct way to pose the following problem: Say a dice game (like Yahtzee) involves throwing up to 5 6-sided die in three rounds. After three rounds, a score is ...
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Decision tree with equal points
Suppose I have a decision tree built, and in the training set there are two points, say $x_1$ and $x_2$, which are completely equal. What happens if I remove exactly one of them from the training data?...
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Struggling with payoff matrix
I've been struggling finding the loss functions, $L(\theta,d_1)$ and $L(\theta,d_2)$, for the following question:
Items I manufacture are either independently flawed with probability, $p$, or perfect....
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Random effects in Bayesian network or Decision Tree
I wonder if we can incorporate a random effect model (as it is used a function..for example linear or logistic regression) to other machine learning algorithms such as Bayes network or decision tree?
...
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Simple question on graphical representation of minmax decision rule
In the picture below, I cannot understand why the minmax decision rule is on the line $R_1=R_2$.
$R_i=R(\theta_i,d)$, where $\theta_i$ is the parameter and $d$ is the decision rule. $S$ is the risk ...
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What are easy steps of finding cutpoint in continuous variable with Time to event outcome, in Stata?
I find it painful to manually guess a dichotomized cutpoint predictor (continuous) for an time to event outcome in Simple Cox proportional hazard model.
Currently I was trying to find the cutpoint ...
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How to find the threshold that minimizes the (weighted) mis-classification rate?
To use a logistic regression model for doing prediction, let
\begin{equation}
\hat Y_i=
\begin{cases}
1 &\mbox{if $P(Y_i=1|X_i)>\alpha$}\\
0 &\mbox{if $P(Y_i=1|X_i)\leq\alpha$}
...
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How to make optimal decisions with uncertain outcomes: achieving a "Yahtzee"
The game of Yahtzee is a poker-like game played with dice. Each move consists of three rolls of five (ordinary, fair, six-sided) dice. After each of the first two rolls the player may designate any ...
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Aside from Durbin-Watson, what hypothesis tests can produce inconclusive results?
The Durbin-Watson test statistic can lie in an inconclusive region, where it is not possible either to reject or fail to reject the null hypothesis (in this case, of zero autocorrelation).
What other ...
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Admissible Estimator for Linear Regression
Is there an admissible estimator for a linear regression model with many parameters without restricting the parameter space?
Admissibility will be with respect to Mean Square Error on the regression ...
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showing that $\bar{X}$ is inadmissible by comparing with $\max(\bar{X},2)$ under squared error loss function
suppose $X_1,X_2,\ldots,X_n$ be a random sample of $N(\theta,1), \theta>2$.
how can I show $\bar{X}$ is inadmissible estimator
Compared to $\max(\bar{X},2)$ under Squared error loss function
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Does maximum likelihood minimize a kind of generalized "0-1 loss"?
A very good point was raised here about how the optimal betting strategy under 0-1 loss was to bet on the mode, while under MSE loss the optimal strategy was to bet on the mean.
Maximum likelihood is,...
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What would be an example of when L2 is a good loss function for computing a posterior loss?
L2 loss, together with L0 and L1 loss, are three a very common "default" loss functions used when summarising a posterior by the minimum posterior expected loss. One reason for this is perhaps that ...
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Why are inf and sup used in the definition of minimax estimators?
An estimator $\hat{\delta}$ is minimax iff $$\sup_\theta R(\theta,\hat{\delta})=\inf_\delta\sup_\theta R(\theta,\delta)$$
or in english iff out of all estimators it has the least maximum risk. For ...
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Choosing a line/plane to separate two classes of binned data
In high energy physics I know it is common task to find the best separation point between two classes of data, usually signal and noise. This separation point is usually determined by first binning ...
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Decision tree indicating payoffs
I need to draw a decision tree to represents these requirements :
The research and development manager in an old oil company, which is considering making some changes, lists the following courses of ...
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Minimizing the misclassification rate
I am reading the book Pattern Recognition and Machine Learning, and have a specific question from a text snippet. I'll state a few lines in the text
Suppose that our goal is simply to make as few ...
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predict sales using naive bayes and handle sparse data problem
Problem
I am trying to use naive bayes for ranking products in a search application. I would like to predict the sales of a given product given the search keyword and the category. the current formula ...
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Intuitive interpretation of Bayes risk $R(\delta, \lambda) = \int_{\Omega}R(\theta, \delta) \lambda(\theta) d\theta$
Consider the risk function R of an estimator (statistic) $\delta(X)$ trying to estimate parameter $\theta$:
$$R(\theta, \delta) = E_{X \sim P_{\theta}}[Loss(\theta,\delta(X)]$$
Which can be ...
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How are statistical decision theory and statistical learning theory related?
This paper attempts to contrast the basic elements of statistical learning theory and statistical decision theory, but I'm still confused about how the two are related.
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minimax property of sample mean
Suppose $X_1,X_2,\ldots,X_n$ are iid $\mathcal{N}(\mu,\sigma^2)$, where $\sigma$ is known, but $\mu$ is not. We wish to construct a confidence interval of length $L$ (given) for $\mu$. Is it true that ...
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Is summing posterior probabilities valid for classification problems?
A classification for two mutually exclusive problem can be formulated by having a decision hinge on whether $P_0(x) > P_1(x)$ or $P_0(x) < P_1(x)$ where $P_0(x)$ and $P_1(x)$ are posterior ...
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Incorporating Risk Aversion in Bayesian Expected Loss functions
In Berger's Statistical Decision Theory and Bayesian Analysis, he presents the following expected loss function for decision theory:
$\rho(\pi^*,a)=\int_\Theta L(\theta,a)d\pi^*(\theta)$
Where $\...
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Association rules or classifier for product modeling for queries
I have a set of products P {1...n} which are rated on a goodness scale
G ={1...100} (G10 is more good than G5). Each product has a set of features
F {1....m}, now I want to learn a model for ...
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Are these independent: the sample, randomized rule, and random variable having the prior distribution on the parameter space?
In section 1.3 of Bickel and Doksum's Mathematical Statistics 2006,
the risk function of a nonrandomized rule $d$ is the expectation of loss of the rule wrt the random sample.
$$
R(\theta, d) := E_{X\...
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Is there a distribution where the data is generated from a decision process?
Sometimes my data is the result of the decisions each individuum of the observed population made: how much invest, how much gambled etc. The outcome would be a metric variable: money spent in total by ...
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admissibility under weighted error loss implies admissibility under square error loss
Show that admissibility of a decision rule under weighted square error loss implies it's admissibility under square error loss.
Weighted Square error loss = $\frac {(d(x) - \theta)^2}{\theta*(1-\...
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admissibility of bayes rule
How to show that for a binomial(n, p) distribution, the MLE X/n is admissible under square error loss?
The Bayes rule undr square error loss with beta($\alpha, \beta$) prior is X+$\alpha$/ (n +$\...
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Bayes estimator under LINEX loss?
Suppose we are trying to estimate a real valued parameter under the linear exponential (LINEX) loss. Suppose we have some prior $g$ that has finite mean, variance and a moment generating function.
...
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