In Bayesian statistics a prior distribution formalizes information or knowledge (often subjective), available before a sample is seen, in the form of a probability distribution. A distribution with large spread is used when little is known about the parameter(s), while a more narrow prior ...

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32 views

How is data generated in the Bayesian framework and what is the nature on the parameter that generates the data?

I was trying to re-learn Bayesian statistics (every time I thought I finally got it, something else pops out that I didn't consider earlier....) but it wasn't clear (to me) what the data generation ...
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17 views

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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8 views

Creating a model for a webshop

I'm going to create a Multi-armed bandit algorithm to handle recommendations for a large scale webshop. I'm going to use Thompson sampling (http://en.wikipedia.org/wiki/Thompson_sampling) and would ...
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26 views

Prior for the coefficients of a linear regression model

I have a linear regression model $\bf Y=\bf{X}\bf{\beta}+\epsilon$. I want to assign a prior on $\bf\beta$ in order to derive the posterior predictive model $p(y_{predictive}|\bf{y},\bf{X},\beta)$. ...
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22 views

Iteratively solving for prior probabilites.

I'm using Bayes theorem to classify data into two groups, where the conditional probability is known but the prior is not. So I assume that the ratio of prior probabilities is 1 and calculate the ...
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12 views

How can i get prior information using my few data set from the whole data? [duplicate]

I have a data set (x1...x500, y1....y500 ) I want to know about bayesian regression I want to know the prior information , few data set(400) from the whole data (500) using MCMCregress( packages in ...
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14 views

Is this notation for the improper uniform prior correct?

Can I write: $\mu \sim U(0,\infty)$ ? Or do I have to use the notation $p(\mu) \propto 1$? Thank you.
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19 views

Can improper priors be implemented in some way?

I'm new to bayesian inference. I've just discovered that improper priors can't be specified in WinBUGS/OpenBUGS. I was wondering if this is common or not in bayesian inference. Are there same cases in ...
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17 views

Definition of weakly informative prior [duplicate]

According to Gelman, a weakly informative prior is defined in the following way: We characterize a prior distribution as weakly informative if it is proper but is set up so that the information ...
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26 views

Doubt about conditional conjugate priors

I've just read the definition of conditional conjugate prior in this discussion but I have still some doubts. According to the definition given, it seems that the prior distribution of $\theta$, ...
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58 views

Bayes Linear regression- logarithmic transformation of prior distribution of the variance

I have a Bayesian version of a linear regression with 3 covariates. The model is given by \begin{align*} Y\sim N(\mu,\tau)\end{align*} \begin{align*} \mu=\alpha + \sum\beta_{i}x_{i}\end{align*} where ...
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23 views

Explanation that the prior predictive (marginal) distribution follows from prior and sampling distributions

While I have a vague intuition that this makes sense, I am interested in the formal demonstration that the prior predictive distribution in Bayesian inference is equal to the integral over $\theta$ of ...
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51 views

How to elicit prior distribution parameters?

A random sample of 300 women aged 60–69 years whose immediate families have had histories of cancer are to be screened for breast cancer. Let $y_i$ be 1 if woman i has a positive test, and 0 if not, ...
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15 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 ...
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39 views

define prior probabilities in naive bayes with unbalanced classes and asymetric cost

I'm trying to apply Naive bayes to the following supervised problem: It's a binary classification problem The classes are unbalanced. The target class represents the 0.004266432 of the total and the ...
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30 views

What are examples of “flat priors”?

For example, for p as the parameter to a binomial or bernoulli, or a Poisson, what would a flat prior p be? What does it mean to be "flat" - does this refer to diffuse?
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72 views

Truncated Von Mises-Fisher distribution

I am putting a von Mises-Fisher prior on my data. The data does lie on a unit sphere, but the only problem is that my data is always positive. So I feel like I am wasting my prior on unnecessary ...
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27 views

p-values, prior probabilities

I've got a set of N normal independent normal distributions, each representing a signal. I also got a new data sample, a vector $v$ of size Nx1. Now let's say I compute the p-value using the ...
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25 views

Constructing gamma prior from Poisson

So if we have a Poisson distribution with a rate lambda we know that the prior is a gamma with alpha,beta. But suppose we didn't know that the prior was a gamma. How would we do the derivation please? ...
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12 views

The validity of using truncated PDFs as prior distributions?

I am trying to implement an ABC (Approximate Bayesian Computation) rejection-sampling algorithm in R. I am currently working with a six-parameter model and for each of the parameters I have specified ...
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70 views

Interpretation of priors in example

Suppose you have 3 variances $W_{1},W_{2},W_{3}$ that can be expressed as $W_{j}=q_{j}V$ with $j = 1,2,3$. According to one model, $W_{3}$ should be pronounced and $W_{1}$, $W_{2}$ should be small to ...
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96 views

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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79 views

Undefined real result error at WinBUGS

I am currently working on my thesis and interested in estimating a multilevel differential item functioning model and I using at WinBUGS. Until I had done model check-up, there are no errors. However, ...
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16 views

Combining two estimates of p in a binomial estimation

I have an estimation problem for a binomial data. I got a sample and from that I can get an estimation. But I also have a kind of prior information about the p. But mind it, this prior is just a ...
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32 views

How can I use ratios to set priors on multinomial probabilities?

I have a vector, $k$, that determines allocation to five pools. I'd like to set priors on these probabilities, and I can provide informative priors on a few of the ratios, e.g.: $$ \frac{k1}{k2} ...
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Picking noninformative priors using pivotal quantities

In 'Bayesian Data Analysis' (Gelman, Carlin, Stern and Rubin) on page 64 it reads: "If the density of $y$ is such that $p(y-\theta|\theta)$ is a function that is free of $\theta$ and $y$, say $f(u)$ ...
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63 views

Selecting priors based on measurement error

How do you calculate the appropriate prior if you have the measurement error of an instrument? This paragraph is from Cressie's book "Statistics for Spatio-Temporal Data": It is often the case ...
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28 views

Covariance for a multivariate Bayesian Additive Regression Tree

Chipman, George, and McCullogh (2010) state that: One can also extend the sum-of-trees model to a multivariate framework such as: $$ (29) \qquad\qquad Y_i = h_i\left( x_i \right) + ...
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32 views

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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67 views

How does one interpret the distribution over parameters in bayesian estimation?

I am new to Bayesian estimation. The assumption that the parameters are random variables seems a little unsettling to me. For example when considering a model for data, what physical interpretation ...
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103 views

What is the mathematical difference between using a un-informative prior and a frequentist approach?

Un-informative priors are preferred in instances where bias is not acceptable (ie. courtrooms, etc.) However, it seems to me that it would just make sense to use a frequentist approach instead. Why ...
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267 views

MCMC packages in R

Is there an R package for MCMC that can accept my self-defined (log)likelihood function (can be done in MCMCpack) and lets the user define contraints to the ...
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277 views

Deriving the posterior density for a lognormal likelihood and Jeffreys's prior

The likelihood function of a lognormal distribution is: $f(x; \mu, \sigma) \propto \prod_{i_1}^n \frac{1}{\sigma x_i} \exp \left ( - \frac{(\ln{x_i} - \mu)^2}{2 \sigma^2} \right ) $ and Jeffreys's ...
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151 views

Specifying a Laplace prior using a Gaussian random variable with Gamma variance

I need to place a Laplace prior on a random variable, however, I want to use a Gaussian distribution whose variance is Gamma(1,1) distributed, i.e., \begin{align} x &\sim N(\mu,\sigma^2)\\ ...
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49 views

Finding the most “uniform” or “least concentrated” density function, subject to moment constraints

Background I want to find a probability measure for a continuous random variable, subject to moment constraints, that is maximally "uniform", as defined below: Definition: Maximally Uniform ...
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101 views

In Bayesian hypothesis testing, do the prior model probabilities have to be equal?

The posterior odds is the product of the Bayes factor and prior odds: $\frac{p(M_1|data)}{p(M_2|data)}=\frac{p(data|M_1)}{p(data|M_2)}\times\frac{p(M_1)}{p(M_2)}$. I was under the impression that ...
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171 views

Is there a desription in the literature of a Normal hierarchical model with hyperparameters for both the mean and the standard deviation?

I'm looking for a comprehensive description of and justification for a Normal hierarchical model where both the means of the groups and the standard deviation are modelled. It is common to find ...
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65 views

Normal prior for Binomial likelihood [closed]

Pardon my ignorance, i am new to Bayesian Analysis. I am trying to use Normal prior for a binomial likelihood, which of these are most likely candidates ( $\bar{x} $, $ \mu $, $ \sigma $ ) ...
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119 views

Prior on a non identifiable parameter-MCMC integration

To introduce the problem I will explain the Projected normal distribution. Let $\mathbf{z}_i=(z_{i1},z_{i2})$ be a bivariate vector distributed as a bivariate normal with vector mean ...
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33 views

Setting up this generative model for inference: uniform priors

I am trying to set up a generative model where I have two images $x$ and $y$ and it is assumed that $y$ can be generated by applying some unknown transformation to $x$ i.e. $$ y = t(x, w) + e $$ ...
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42 views

Priors for discriminative methods?

Say we want to build a classifier for a binary classification problem using a discriminative method (e.g. SVM) and be able to impose a prior on the classes. For example, let's assume that we want to ...
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49 views

Conjugate Prior for Probit likelihood function

I am trying to do a Bayesian analysis in which my likelihood function is a probit function on two parameters. From various sources, I found out that Normal distribution is a conjugate prior to probit ...
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34 views

How to model the prior distribution of several Gaussians with known parameters

I might be wrong, I just feel that the following case is different from the problem of modelling observations with a conjugate prior: Suppose I have $n$ different Gaussians each with a different (but ...
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54 views

Haar prior for von Mises distribution

Ok, Let me tell you that this is the very first time that I have no idea with the question below. I can not find a solution or anything that will lead me to it. I say this to prevent comments "what ...
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1answer
32 views

Weakly informative priors for “r.v. $\mathbf{x}$ is not close to $\mathbf{x}_0$”?

Given some fixed vector $\mathbf{x}_0\in\mathbb{R}^d$, I want to put a prior on a random variable $\mathbf{x}\in\mathbb{R}^d$ so that "it's not very close to $\mathbf{x}_0$". For the moment I put a ...
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67 views

Defining constraint on prior with potential class

I have written an MCMC code in order to estimate parameters Xpos, Ypos, MASS and concentration with a set of input data gal_pos, ...
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189 views

What does it mean to integrate over the posterior?

I have been reading a book that cites an example where a uniform distribution is the initial prior, and then a person scores 9/10 on a test. Then the resulting posterior becomes the prior ...
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74 views

How to construct a reasonable prior and likelihood for Bayes modelling?

To apply Bayes inference for data analysis or machine learning, we have to construct prior and likelihood, right? But if I fail to come up with a reasonable prior and likelihood, then the Bayes model ...
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43 views

learning hyper parameters: are we allowed to touch the prior parameters after observing the data?

There are many algorithms/applications that aim to learn the hyper parameters i.e. the parameters of a prior distribution from the observed data. A typical algorithm works in an iterative function ...
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61 views

Use of the Jeffreys prior in multidimensional models

Suppose a model, $$x_{i} \sim N(\theta_{i}, \phi), \text{ for } i=1,\ldots,n$$ Furthermore, suppose the variance parameter, $\phi$, is some known constant. The multidimensional Jeffreys prior is ...