Questions tagged [empirical-bayes]

Procedures for statistical inference in which the prior distribution is estimated from the data.

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Meaning of “log-likelihood” in terms of models in nlme [closed]

When R says that the log-likelihood of a model using nlme is some number, what does that mean? The log-likelihood is an abbreiation for the log of the likelihood function dependent on parameters $x_1,...
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Beta-Binomial regression or Poisson-Gamma model to account for uncertainty in (empricial Bayesian) prior? Explained in simple terms?

I have a dataset of $m$ individuals. For each individual $m$ I have $n_m$ (binomial ) observations with $s_m$ corresponding to the number of 'successes'. I use this data to fit a beta-binomial ...
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Admissible Empirical Bayes Examples

I would like to hear about a few simple empirical bayes estimators that are admissible for high (i.e. at least 3) dimensional parameter space. What are some textbook lollipop examples to study for ...
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Accounting for uncertain information (few observations) in a prior (empirial Bayes)

I did not really know how to choose an adequate title for this question, so please feel free to change it. I have a weird case wherein frequentist and Bayesian philosophies come together. I am ...
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Likelihood of linear mixed effects model

Consider the following model $$\left \{ \begin{array}{l} y_i = x_i\beta + z_ib + \varepsilon_i,\\\\ b_i \sim \mathcal N(0, \Sigma), \quad \varepsilon_i \sim \mathcal N(0, \sigma^2), \end{array} \right....
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Replicating a Tweedie corrected experiment from Computer Age Statistical Inference

I have been attempting to replicate an experiment from Computer Age Statistical Inference by Bradley Efron and Trevor Hastie on page 411. In this experiment 100 datasets are populated normal random ...
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How to calculate Empirical Bayes for continuous variable in Hierarchical Linear Models?

I am currently working with the radon dataset of Gelman for Hierarchical Linear Models. I have heard of Empirical Bayes and how to compute it for the Poisson-Gamma case from the data but didn't find ...
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Is appropriate to use empirical Bayes (EB) in this way?

Background. I have data from a study where participants make a series of judgments (a series of decisions with a binomial outcome, either $y=1$ or $y=0$). I have a model of the underlying decision-...
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Do Bayesian credible intervals treat the estimated parameter as a random variable?

I read the following paragraph on Wikipedia recently: Bayesian intervals treat their bounds as fixed and the estimated parameter as a random variable, whereas frequentist confidence intervals treat ...
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Modeling the Probability of Falling Asleep

Problem The goal of this exercise is to estimate $p = \text{Pr}(\text{fall asleep})$, the probability of falling asleep imminently, as a mental exercise for those nights when sleep doesn't seem to be ...
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Poisson-gamma posterior mean expectation

Let's have a gamma prior $\lambda\sim \operatorname{Gamma}(a,b)$ (mean: $\frac{a}{b}$) With Poisson data $Y\mid \lambda\sim \operatorname{Pois}(N\lambda)$ (mean: $N\lambda$) The posterior is $\...
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Get posterior distribution of categorical variable given empirical continuous-categorical priors?

Suppose I have categorical variable $Z \in D$ defined for some finite domain $D$. I also have a continuous variable $X \in \mathbb{R}$ which is observed. From historical data samples I have the ...
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Empirial Bayes for means

I have data for US ~3100 counties where the variable is a mean score based on a sample. However, for many small counties, the sample size is quite small (like 5), and so these mean values fluctuate a ...
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Is expectation maximization an example of empirical Bayes?

I don't think I truly understand what methods are classified as "empirical Bayes". Is expectation maximization considered an example of this?
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How do Bayesian hierarchical models adaptively learn the prior?

It seems the main difference between a hierarchical and a non hierarchical model is that the hierarchical model learns the prior. That is it adaptively comes up with a regularizing prior to be applied ...
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Learning prior distribution from data

Suppose I have a dataset. How can I learn the prior distributions of the parameters of a model from this data? I want to learn the prior from this data in order to use them in a Bayesian model. Sorry ...
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question about empirical Bayes(EB)

In David Robinson's blog:Understanding empirical Bayes estimation (using baseball statistics) he used the hit ratio to fit the beta distribution as a prior distribution,Where hit rate = hits/total. ...
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How to pick starting parameters for MASS:fitdist() with the beta distribution?

I have data set of ~700k yes/no events that I want to first aggregate on various features (e.g. group by average), always resulting in a 34 length vector. From there, I want to fit a beta ...
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Empirical Bayes (In)Admissibility

Sticking to a pure Bayesian approach to statistics with proper priors most of the time leads to admissible estimators. Nevertheless there is good reason to use Empirical Bayes in many cases, and the ...
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Can “cross-validation” be used to choose a prior?

To be clear, I doubt I am using the term "cross-validation" correctly here; what I am suggesting also seems similar to "boot-strapping" and "hyperparameter tuning". Terminology is not my strength. ...
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Why the local Bayes fdr is greater than the Bayes FDR?

My question is related to empirical Bayes and large-scale inference. It is explained that the local Bayes false discovery rate (fdr) is greater than the Bayes false discovery rate (FDR). It is argued ...
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How to calculate the inverse cdf for a sample with unknown distribution?

I'm stuck with an exercise that I'll just write down first: Based on a sample $x_1,...,x_{47}$, that can be considered coming from an unknown distribution, we study a qq-plot where the empirical ...
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Empirical Bayes: method of moments

The model for the data is $X_{i}$ ~ $Bin(n_{i},\theta_{i})$ (iid, $i=1,...,k$). The prior distribution is $\theta_{i}$ ~ $Beta(\alpha,\beta)$. How do we choose (and deduct) moment estimators for $\...
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Hyper-parameter estimation for Beta-Binomial Empirical Bayes

I am reading a paper Illustrating empirical Bayes methods and in the paper the author uses method of moments to derive the value of an estimate. In equation 17 the author gives the following marginal ...
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Robbins estimate Empirical Bayes

From the compound sampling model where: $Y_i | \theta_i \sim Poi(\theta_i)$ The marginal distribution of $\theta_i$ is $G$, non-parametric. We get that the Bayes estimate of $\theta_i$ under ...
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Empirical Bayes Estimation

Suppose $X_i$ conditioned on $\mu$ is iid $N(\mu, \sigma^2)$ and $\mu$ is distributed as $N(\mu_0, \tau^2)$. Is there a way to estimate $\tau^2$?
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James-Stein Estimator with unequal numbers in groups

In the book Computer Age Statistical Inference the James-Stein estimator is introduced. Brad Efron runs through an example where batting averages are estimated from each players 90 at-bats. $$p_i\sim ...
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How is empirical Bayes valid?

So I just finished reading a great book Introduction to Empirical Bayes. I thought that the book was great, but building priors from the data felt wrong. I was trained that you come up with an ...
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Using empirical Bayesian estimation (Gamma-Poisson) to analyze high arrival counts (n ~= 5000)

Here's a problem I'm currently working on, as well as the empirical Bayesian approach I'm using. I'd like to make sure my approach is grounded in solid statistical theory. I have a set of entities $e=...
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Expectation of inverse chi-square random variable (Efron, 2010)

Let $z_i|\mu\sim N(\mu,1)$ and $\mu\sim N(B,A)$ for $i=1,\dots,N$, the implication is that $z_i|B\sim N(B,A+1)$. Define $S=\|\textbf{z}\|^2$, and let $B=0$, then $S\sim(A+1)\chi^2_N$ since $\|\...
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Why does Empirical Bayes work in my simple case?

I have a problem where I am trying to classify data into two groups using a single parameter. The distribution of this parameter is Gaussian for two groups, so what I'm dealing with is two overlapping ...
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Bootstrapping the data to set up a prior

I am using a Gaussian model with a conjugate Normal-Inverse-Wishart (NIW) prior, as described here. The advantage of this approach is that the marginal likelihood $p(y)$, which is what I am interested ...
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Empirical Bayes vs “non-informative” priors

I am familiar with the mechanics with both methods, but don't know what factors I should consider when choosing between these two approaches for adjusting a prior. I would imagine that, on a case by ...
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What is the empirical Bayes estimator for a gamma-Poisson with more than 1 observation for each Poisson parameter?

I am looking at the Wikipedia entry for empirical Bayes, but it's a bit confusing - it seems to me the solution must apply only to the case in which there's only $n=1$ sample $y$ for each $\theta$ and ...
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Empirical Bayes/MCMC references

I'm interested in references for running empirical Bayes (EB) in conjunction with MCMC. The closest thing I've found to what I'm looking at is a surprisingly recent and somewhat obscure paper ...
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hierarchical Bayesian models vs. empirical Bayes

Would you consider the HBM vs EB to be two alternatives in which the hyperparameters are "in the game" of being sampled/estimated/etc.? There is clearly a connection between these two. Would you ...
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Is there a connection between empirical Bayes and random effects?

I recently happened to read about empirical Bayes (Casella, 1985, An introduction to empirical Bayes data analysis) and it looked a lot like random effects model; in that both have estimates shrunken ...
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Cross-validation vs empirical Bayes for estimating hyperparameters

Given a hierarchical model $p(x|\phi,\theta)$, I want a two stage process to fit the model. First, fix a handful of hyperparameters $\theta$, and then do Bayesian inference on the rest of the ...