This is for use on questions about empirical Bayes methods in statistics.
This is for use on questions about empirical Bayes methods in statistics.
Questions tagged [empirical-bayes]
22 questions
6
votes
1 answer
Inverse propagation of information from the PDF of $Y=f(X)$ to the PDF of $X$
Assume a non-linear relation between the random variables $\mathbf{Y} = f(\mathbf{X})$, where $\mathbf{Y}\sim p_Y$ takes values $\mathbf{y} \in \mathbb{R}^M$ and $\mathbf{X}\sim p_X$ takes values $\mathbf{x} \in \mathbb{R}^N$, with $M\leq N$. My…
Quillo
- 2,260
4
votes
0 answers
Generalizations of the Robbins lemma and Gaussian integration by parts
The Robbins lemma, named after Herbert Robbins, says that if $X\sim\operatorname{Poisson}(\lambda)$ and $g$ is a function for which $\operatorname{E}(|X g(X)|) < \infty,$ then $$\operatorname{E}(Xg(X)) = \lambda…
2
votes
2 answers
Binary classification, Bayes classifier, Bayes decision boundary
I have recently come across this problem from a friend I help with stats occasionally. This however stumped me completely. I have looked online on basically every single website you can find but what I did find either I did not fully understand or I…
zzoxx
- 23
1
vote
0 answers
Why does the uniform law of large numbers hold with non-i.i.d. random variables in Bayesian experimental design?
This paper, Asymptotic theory of information-theoretic experimental design, studies Bayesian experimental design where in each round $n$, the experimenter selects a stimuli $X_n$ that maximizes mutual information between the parameter $\theta$ and…
Qcer
- 49
1
vote
0 answers
True Loss for Bayes Classifier with Two Classes
For a Bayes classifier of two classes (say 0 and 1), I'm not understanding how the largest possible true risk would be 0.5? I'm assuming that we assign a 0 loss for a correct classification and a loss of 1 for a misclassification. So does that mean…
M. Fire
- 41
1
vote
1 answer
Bayesian Estimation of CDF
i'm getting pretty confused by the following problem, hope anyone can clarify my mind:
Using a bayesian approach obtain a posteriori and interval estimations for $\mathbf{F}_{X}(x)$ using a Uniform(0,1) prior distribution for the parameter…
Alonso Rangel
- 325
1
vote
0 answers
literature on total expectation Poisson-Gamma
Having the poisson gamma model:
$$\lambda\sim Gamma(\alpha,\beta)$$
$$N|\lambda \sim Poiss(\lambda)$$
the expectation of N is $$E(N)=E(E(N|\lambda))=E(\lambda)=\frac{\alpha}{\beta}$$
Is there some literature, where I can learn, how to work with…
Rafael
- 96
1
vote
2 answers
Why Bayesian Approach don't use test data for model validation?
Up to I know the usual way of thinking in machine learning approach is to split the data in a train and test subsets. The first one is for fitting the model (with the support of a validation subset) and the second one is for compute model…
1
vote
0 answers
Inverse gamma distribution general question
I am reading a paper in the genomics field (Adjusting batch effects in microarray expression data using empirical Bayes methods. from W. Evan Johnson, Cheng Li), where they try to correct for some noise related to the experimental procedure. What…
Babas
- 131
1
vote
1 answer
Use Bayes Model to Forecast
if I have the following dataset and I'm interested to find the probability of it raining today given the last 2 days were raining. It seems like I could use Bayesian model to get the probability to forecast that.
I'm not sure how do I start.
1) the…
tkj80
- 387
1
vote
1 answer
Bayesian Rating
Suppose we're in a situation where we have a website and people rate products on that site. The total number of people on the site is $n$. The total number of people who rated a product $i$ is $c_i$, and the arithmetic mean of the rating of $i$ is…
Tobi Alafin
- 1,267
0
votes
0 answers
Dual problem of a convex problem from discrete mixing distribution
I'm studying the paper [1] and trying to understand the primal-dual problems introduced in the middle of page 4 of [1]. Below is the problem in question:
Let $A$ be a fixed $n \times m$ matrix, where each element is positive.
Consider the following…
Kim Minwoo
- 13
0
votes
0 answers
HPD interval in piecewise posterior distributions
In Bayesian analysis, suppose the posterior distribution is given by
$$
P(\theta|x)=\begin{cases}
p_1(\theta|x) & ; & 0<\theta<1 \\
p_2(\theta|x) & ; & \theta>1 \\
\end{cases}.
$$
There are several possible configurations of $p_1$ and $p_2$. For…
Ludwig
- 317
0
votes
0 answers
Proving observed value is minimax for normal mean
Suppose $X_i \sim N(\theta_i, \tau_i^2)$. The goal is to estimate $\theta_i$ under squared loss.
What is the easiest way to prove that the realization of $X_i$, call it $x_i$, is minimax?
One approach I've thought of:
We can think of $x_i$ as the…
ryankessler
- 39
0
votes
0 answers
Bayes Method How to draw coefficients summing to 1 and each of them follow exponential distribution
There is a question that how to draw the estimators (ratio estimator) if we know the prior for the numerator or the ratio estimator.
Assume that we have three coefficients: $a_1,a_2$ and $a_3$ . And we beleive that all the three coefficients follow…
Mike
- 1