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1answer
13 views

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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0answers
14 views

Optimizing two decision trees with dependent labels

I might need help phrasing the question. I'll start with a simplified example. I have 20 cars with color, height, year, and width attributes. My attributes are in two groups: A (color, height) and B ...
2
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0answers
20 views

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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0answers
51 views

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 $P(Y_i=1|X_i)\leq\alpha$} ...
3
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1answer
79 views

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 ...
8
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2answers
206 views

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 ...
3
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2answers
314 views

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 ...
4
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1answer
65 views

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
2
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0answers
35 views

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 ...
7
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1answer
116 views

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 ...
2
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1answer
80 views

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 ...
2
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1answer
51 views

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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0answers
22 views

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 ...
0
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1answer
57 views

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 ...
3
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1answer
60 views

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 ...
2
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1answer
75 views

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.
2
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1answer
116 views

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 ...
1
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1answer
49 views

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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0answers
29 views

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 ...
0
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1answer
37 views

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 ...
0
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0answers
18 views

Is a Bayes randomized rule always a Bayes nonrandomized rule?

Given a prior distribution on the parameter space, is a Bayes rule among all randomized rules (nonrandomized rules are special randomized rules) always a nonrandomized rule? I.e. Is a Bayes randomized ...
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0answers
32 views

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) := ...
1
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1answer
41 views

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 ...
1
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0answers
24 views

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) - ...
-1
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1answer
77 views

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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0answers
10 views

Optimal intertemporal allocation of bets based on estimated returns

Each day, an investor predicts the day's stock-market return in the morning, generating a point estimate and prediction interval, and chooses to bet an amount $B_t$ on the market that day based on her ...
1
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1answer
166 views

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. ...
0
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0answers
23 views

Existence of Minimax rule

If a minimax point does not exist, a minimax rule will not exist? But can a non-randomized rule be minimax in this situation? My set of risk points are: S = {(10,5),(8.4,5.4),(9.6,5.6), (8,6)).
1
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1answer
25 views

Bayesian calculation of parameter multiplying normal

I am interested in a model like: $y_{i} = \sum_{k\in K}{\beta_{k} z_{k}}$, with $z_{k} \tilde{} N(\mu_{k}, \sigma_{k})$. where $\beta \equiv(\beta_{k})_{k\in K}$ is not known, but all else is. I ...
0
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1answer
82 views

Detecting a consistent pattern in a dataset via Decision Trees and cross-validation

Assume a classification problem where there are two classes and the aim is to detect a consistent pattern which successfully separates the input dataset regardless of how we divide it into training / ...
2
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2answers
127 views

Is 'fair statistics' a thing?

Given that statistics can often be abused to deliberately present 'facts' to support a pre-existing viewpoint. (Lies, damned lies and statistics). And given confirmation bias. Is there an ...
1
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0answers
14 views

Optimal Number of Entries in Contest of Skill

Objective: I'm looking for the optimal number of unique entries in a contest of skill with monetary prizes. Description: The contests vary from as little as 20 entries up to 10,000+ entries. You ...
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0answers
60 views

Intuition behind the Stein's paradox [duplicate]

I had read wiki and some sources. jmanton's blog, Wasserman's blog the background is that: You have Xi ∼ N(θi, 1), and we want to estimate the each θi. Where Xi are ...
5
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2answers
251 views

Drawing numbered balls from an urn

PROBLEM There is an urn with a set of balls where each ball is labeled with a different integer. The numbers on the balls are known and are not a range of integers. For example the set of balls could ...
3
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2answers
233 views

Bayes decision theory: Classification error probability

In Bayesian decision theory: Given $\omega_1$ and $\omega_2$ as two classes for classification, $P\left( \omega_1 \right)$ and $P\left( \omega_2\right)$ their prior probabilities, $x$ the feature ...
4
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1answer
85 views

Loss function that relates ROPE with HDI?

In Doing Bayesian Data Analysis (link to the book) and Bayesian Estimation Supersedes the t-Test, J. Kruschke proposes using the following criterion to reject or accept the null hypothesis in a ...
2
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1answer
69 views

Decision function problem based on the logistic function

Suppose we have a bunch of a sampled pairs $(x_1,y_1)...(x_n,y_n)$ with the $y_i =\pm1$. Then consider the decision function $h(x) = -1$ if $p(x)=\frac{1}{1+e^{-x}}\leq0.5$, and $h(x) = 1$ if $p(x) ...
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0answers
122 views

Given 3 discriminant functions I was able to classify 2D patterns, but how do I plot decision boundaries via matplotlib?

I have implemented the discriminant function and was able to classify the 2D patterns (via Python), but I have troubles thinking about an approach to plot the decision boundaries. Hope anyone has an ...
7
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1answer
101 views

Formal justification of Bayesian inference as a model for belief

I remember a proof that Bayesian probability theory is the only valid method for representing beliefs, it went something like we represent belief by some non-negative function over some domain of ...
2
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0answers
31 views

Admissibility and domination for estimators

Watching a video by the "mathematicalmonk" on the web, I was wondering how to answer this kind of questions: Given $X_1,\ldots,X_n\sim \mathcal{N}\left(\mu,\sigma^2\right)$. Assume that $\mu$ is ...
2
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0answers
43 views

What is a robust way to find the max of $n$ independent, non-identical random variates?

Suppose I observe $n$ random variates along with their variance (but not mean) and I'd like to select the one with the largest mean as frequently as possible. The procedure must be memoryless--you ...
3
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1answer
50 views

Given $n$ different univariate non-normal sample sets calculate for a new sample, $x$, which it most likely belongs to [duplicate]

Say you have $n$ different, non-normal, potentially overlapping data sets of samples. Maybe their densities look something like: and you are given a new sample $x$, how would you decide to which of ...
5
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0answers
110 views

Maximizing returns - A Bayesian approach

I want to design a Bayesian model for a simple asset allocation problem. Say I can buy $a_i$ amounts of $N$ assets. The return values of these assets are given by random variables $r_i$ with known ...
2
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1answer
210 views

How to incorporate constraints in random forest output

Suppose I am doing random forest classification of labels $A$,$B$,$C$,$D$. There is some theoretical ordering to this output such that when $A$ is more likely than $B$, $B$ is also more likely than ...
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0answers
173 views

Computing Confidence in A/B testing

First of all, I'm fairly new to Decision Theory and my question might be rather too simple for the experts. I am having some issues in computing confidence for A/B Testing. I will first explain the ...
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0answers
27 views

determining the number of variables in the kernal of a U-stat

I'm working through some problems on deriving U-statistics, and I am unsure about determining the number of samples (random variables?) that are needed for the kernel of a given U-stat. For example: ...
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0answers
137 views

Showing that a statistic is ancillary for a parameter

Working through a HW problem, and a hint is that for a decision rule $$T(X) = \frac{X_{(1)} + X_{(n)}}{2}$$ Then $$T - \bar{X} $$ is ancillary. Intuitively this makes complete sense, but I am ...
2
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1answer
282 views

How do I combine multiple prior components and a likelihood?

Lets imagine I am comparing two groups of animals (treatment/control). There is previous data from cell cultures indicating the treatment should have a positive effect. This gives me "prior component ...
2
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1answer
114 views

Markov Decision Process and its generality

My major is CS and I have a question about Markov decision process. I have been reading a book, planning with markov decision process an AI perspective. While reading it, I have a question regarding ...
2
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1answer
202 views

Prove/counter example: A minimax decision rule is always Bayes wrt some proper prior

Not sure whether the claim is true or false. If claim is true, intuitively, it might have something to do with "least favorable priors", but am not able to figure out the connection. If claim is ...