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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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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24 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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11 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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13 views

sparsity prior infer.net

I have implemented a network inference model using infer.net. In my model I put all possible weights : Y(i) = Beta(i) * WeightedSumOf(Y(j)) : for all j I need to infer these weights - involved in ...
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31 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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21 views

Sparsity priors

I have implemented a network inference model using infer.net. In my model I put all possible weights : Y(i) = Beta(i) * WeightedSum(Y(j)) : for all j I need to infer these weights. I have created my ...
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20 views

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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54 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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20 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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1answer
27 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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63 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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2answers
87 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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204 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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1answer
183 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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1answer
69 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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46 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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1answer
83 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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4answers
151 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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1answer
52 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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1answer
102 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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16 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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37 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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35 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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1answer
32 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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37 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
31 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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46 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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2answers
163 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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1answer
67 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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40 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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2answers
50 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 ...
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44 views

Gibbs sampling with Log-Normal observations

I am writing a Gibbs sampler for data that is Log-Normal (LN) distributed, with unknown mean and variance. There is a wealth of information on inference for LN models when either the mean or variance ...
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21 views

Prior over transition matrices of markov chain

I want to be able to marginalize over the transition matrix of a markov model. The goal is to get the marginal likelihood on the number of states necessary to explain the data. Something that would ...
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492 views

How can an improper prior lead to a proper posterior distribution?

We know that in the case of a proper prior distribution, $P(\theta \mid X) = \dfrac{P(X \mid \theta)P(\theta)}{P(X)}$ $ \propto P(X \mid \theta)P(\theta)$. The usual justification for this step is ...
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50 views

Are discrete single value prior distributions always lost in MAP estimation?

Iā€™d like to illustrate my problem with a little (heavily abbreviated) excercise. I think it will help a lot to stress my point. Meet Mary, Tom and Jane. They all are programmers. Mary is a decent ...
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1answer
52 views

Non-binomial posteriors for a binomial prior?

Let's assume we have a discrete binary random variable K (K=0 or K=1) for which the prior distribution is binomial. My understanding of Bayesian statistics tells me that regardless of the likelihood, ...
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1answer
37 views

How should I construct a prior distribution with a particular kind of count data

For context I will first explain the overall problem that I am working on. I am given a catalog of product names and I am also given a large text dataset that may contain mentions of these catalog ...
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363 views

You observe k heads out of n tosses. Is the coin fair?

I was asked this question with $(n, k) = (400, 220)$ in an interview. Is there a "correct" answer? Assume the tosses are i.i.d. and the probability of heads is $p=0.5$. The distribution of the ...
4
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2answers
378 views

Ridge regression ā€“ Bayesian interpretation

I have heard that ridge regression can be derived as the mean of a posterior distribution, if the prior is adequately chosen. Is the intuition that the constraints as set on the regression ...
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1answer
46 views

Distribution of a Mean and Variance

Say we have observations $x_1 \dots x_n$ and we have some sort of Bayesian framework where we would like to estimate a distribution for the mean $\mu$ of our observations and the variance $\sigma^2$ ...
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215 views

Hyper-prior for negative binomial in hierarchical model using JAGS/BUGS

Below I'm using a negative binomial because it is more flexible than a simple poisson model. The data are counts $y$ of events for 16 individuals $x$. There are 14 counts (i.e. counting periods) for ...
3
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1answer
52 views

Univariate priors for the parameters of a Beta distribution

I need a rather a prior on the parameters of a Beta distribution (i.e. $\alpha$ and $\beta$). I have an external constraint that requires me to use univariate priors, one for $\alpha$ and one ...
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12 views

Applying Beta distribution for calculation latent variables

I would like to find the probability distribution function for the below scenario,its similar to Computer Adaptive technique (IRT) I need to estimate the ability of a student from answered questions ...
2
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99 views

Non-informative prior for Poisson/gamma density

In the Albert book on Bayesian computation with R, exercise 4.8.5 (p.83), it is suggested to use $$ p(a, b) \sim (a \times b)^{-2} $$ as the non-informative prior for the Poisson/Gamma model: $$ ...
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26 views

Calculating posterior and prior odds

Question: Now, I'm confused about assigning probabilities here. I find $P(A^c|E) = (.001)(.99) = .00099$ and $P(E|A) = .99$, but what about the first two sentences? Does that mean that $P(E) = .001$ ...
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87 views

Can likelihood be changed when the prior changes?

I have a data which follows gamma distribution and want to know the uncertainty of the parameters of this data. $\text{Data} \sim \text{Gamma} (\alpha, \beta)$ Parameters $\alpha \sim \text{Gamma} ...
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27 views

Marginal Likelihood Prior

I have a model with probability matrix for a distribution of $x$, $y=\{0,1\}$, $p(x,y|w)$ where $w=[w_1,w_2,w_3,w_4]$ $p(x=0,y=0)=w_1$, $p(x=0,y=1)=w_2$ $p(x=1,y=0)=w_3$, ...
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1answer
58 views

How can a uniform prior make the posterior mean different from the MLE?

I read the following in Machine Learning: A Probabilistic Perspective: How can a uniform prior move the posterior mean? Isn't a uniform distribution supposed to not bias the result? Are there any ...
5
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1answer
92 views

Log odds as prior

I have this problem: In betting situations one is often interested in odds, referring in the thumbtack tossing $\theta / (1 - \theta)$. Alternatively one may consider the log-odds: ...
2
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2answers
125 views

Prior Gamma distribution: Select appropriate alpha given beta and median

I am trying to programatically select a prior distribution from the Gamma family of distributions. The primary criteria that I need to satisfy is that the median of the distribution should be a given ...