Questions tagged [prior]

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 distribution represents a greater degree of information.

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Implementing Predictive Posterior Distribution Using Stan

Background I had an example that sought to demonstrate the posterior predictive distribution in the context of a normal measurement model. The data that was used is as follows: ...
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Horseshoe priors and random slope/intercept regressions

I'm interested in using the horseshoe prior (or the related hierarchical-shrinkage family of priors) for regression coefficients of a traditional multilevel regression (e.g., random slopes/intercepts)....
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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 ...
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Including feature-dependent priors on output class, in bayesian logistic regression

When doing logistic regression with data $D_N = \{(x_i, y_i)\}_i^N$ with $x_i \in \mathbf{X}^N$ (each data point has N features) and $y_i \in \mathbf{Y}$ being assigned output classes, in a Bayesian ...
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67 views

Bayesian prior over long probability vectors

Suppose you have i.i.d. variables $x_i$ in ${1,\ldots,K}$ modeled as $$P(x_i = k) = \theta_k$$ and and you want to infer the probability vector $\theta$. A Bayesian approach puts a prior over $\...
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401 views

Prior for $\lambda$ is LASSO prior?

I have a regression model with regression coefficients $\beta_j$, $j=1,...,n$, and I would like to use a LASSO prior for $\beta_j$, this is: $$\beta_j \sim Laplace(0,1/\lambda),$$ where the Laplace ...
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118 views

Examples of usage of community of priors or why aren't they used more commonly?

Kass and Greenhouse (1989) proposed using "community of priors" (see also Fayers et al, 1997; 2000). As described by Spiegelhalter (2004), they can be seen as a range of viewpoints that should be ...
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581 views

Noninformative prior for variance, understanding and coding

I have three questions regarding the understanding behind and implementation of a noninformative prior for variance. I'm attempting to build a Metropolis sampler and I'm trying to sample from a ...
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408 views

AR(1) model - which prior to use?

I want to use the following univariate model: $y_t = \mu_t + \epsilon_t, \ \epsilon_t \sim N(0,1)$ $\mu_t = \phi \mu_{t-1} + \omega_t, \ \omega_t \sim N(0,\sigma_\omega^2)$ That is, $\mu_t$ follows ...
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240 views

Prior elicitation with Normal-Gamma or Normal-Inverse-Gamma

I am looking for a way to have experts elicit a prior for a Normal-Inverse-Gamma Bayesian linear regression model. Is there any material suggesting intuitive ways to go about this?
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852 views

Variance of marginal posterior distribution

Suppose $Y_1,\dots,Y_n\mid\mu,\sigma^2 \sim \text{ iid } N(\mu,\sigma^2)$ and suppose the priors $\mu \mid \sigma^2 \sim N(\mu_0, \sigma^2 / \kappa_0)$ and $1/\sigma^2 \sim \text{gamma}(\nu_0/2, \nu_0 ...
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315 views

Deriving priors for MCMC implementation

I have been working on an assignment lately wherein the object is to implement an MCMC approach to simulate from a generated posterior distribution. The posterior distribution is generated from a ...
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Decision Theory: Why is it called a “least favorable prior”?

I'm currently reading the chapter on Statistical Decision Theory in Larry Wasserman's "All of Statistics". Reading the section 13.4 about Minimax Rules he introduces the so called Least ...
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Why are $\mathbb{E}( \ln(x))$ and $\mathbb{E} ( \ln(1 - x))$ reasonable descriptions of knowledge about a beta distribution?

The max entropy philosophy states that given some constraints on the prior, we should choose the prior that is maximum entropy subject to those constraints. I know that the Beta($\alpha, \beta$) is ...
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Reproducing a didactic example of Lindley (1993)

Lindley (1993) discusses the following mixed discrete and continuous prior for the tea tasting lady experiment, where $\pi$ is probability of a correct classification: $p(\pi=0.5) = 0.8$ (discrete ...
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988 views

Bayes-Poincaré solution to k-sample tests for comparison and the Behrens-Fisher problem?

I’d like to share and submit for (dis)approval and discussion yet another, simple but original (to the best of my knowledge) Bayesian solution to the classical problem of comparing k samples or groups,...
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Can someone provide a non-technical explanation of how Cholesky Covariance priors work?

I am looking for an explanation of how Cholesky Covariance priors work in the context of mixed effects regression. In particular, when they are applied to the correlations among random effects. What ...
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186 views

Using Bayesian Lasso with an informed prior

I'm looking for advice on how best to go about setting an informative prior for the Bayesian Lasso and BART (I'm applying these in R using the rjags and bartMachine packages) I have 3 proteomics ...
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244 views

Intuition regarding Bayesian pseudo-priors

One approach to model comparison in a Bayesian framework uses a Bernoulli indicator variable to indicate which of two models is likely to be the "true model". When applying MCMC-based tools for ...
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141 views

Combining non-independent priors

I've been working on a stats package that includes Bayesian survival models. The user is allowed to write priors directly for all the parameters involved. However, I think it's pretty difficult for ...
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533 views

Benchmark priors for Bayesian ridge regression

Consider a Bayesian linear regression model $$\mathbf{Y=X\beta} + \boldsymbol{\varepsilon}$$ where $\mathbf{Y} \in \mathbb{R}^n$ and $\mathbf{X} \in \mathbb{R}^{n,p}$ are given, $\boldsymbol{\...
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Gibbs sampling for spike and slab priors

In Spike and slab variable selection (equation 4) there is a model setup of the form $\beta_k | \lambda_k, \tau_k \sim \text{Normal} (0, \lambda_k \tau_k^2)$ $\lambda_k | \nu_0, w \sim (1-w)\delta_{\...
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502 views

How to write unnormalized posterior when prior is a mixture of continuous and discrete

Suppose I want to do bayesian inference on the regression problem $\beta$ for Y = X$\beta$ + $\epsilon$ for $\epsilon_i \sim N(0,\sigma^2)$. The complication is that the prior for each component $\...
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Specifying the priors for multivariate MCMCglmm mixed model in R (Poisson distribution)

I am trying to build a model using MCMCglmm. Ideally, I would use a negative binomial distribution for my response; however, this is not an option in MCMCglmm. I don't know of any open-source ...
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160 views

Linear regression with prior on $\arctan \beta_1$

Suppose we have $\hat{y} = \beta_1 x + \beta_0$ (I ask only for the univariate case.) A typical Bayesian approach might involve Normal priors on both parameters. I was thinking today about a ...
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127 views

How to use information about likelihood of classes in a classifier?

General question: How can information about the likelihood of classes be used to improve a classifier? Suppose that the probability of each class is known quite precisely (from a very large sample), ...
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167 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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183 views

Maximum entropy priors in infinite dimensional spaces

Has the idea of a maximum entropy probability distribution been explored for function spaces, and if so what are some key papers, books, or terms to look for? For $\mathbb{R}^n$ (and discrete spaces),...
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296 views

What prior distributions are used in mcmcsamp() from lme4?

The mcmcsamp() function generates simulations from the posterior distributions of a Bayesian mixed model fitted with the lmer() ...
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Metrics for assessing the quality of prior distributions

Clarification: My purpose is to compare different methods for selecting/creating priors (or perhaps I should refer to them as predictive distributions for a quantity of interest/parameter). I do not ...
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163 views

How do I choose a prior for this hierarchical model? (Kruschke book)

I am working through Kruschke's "Doing Bayesian Data Analysis", currently working on the Hierarchical models chapter. The book uses JAGS for MCMC. One of the exercises asks the reader to compare two ...
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30 views

Informative priors from frequentist regression

Say I run a standard frequentist regression on a subset of my data (for example using lm in R) and obtain some values for the coefficients of the model, I have fairly large data set ~100k samples. Now ...
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42 views

Help with prior selection

Say I have two samples of univariate data, both of equal size: $X_1, \ldots, X_n \overset{\text{iid}}{\sim} \text{Normal}\left(\mu_1, \sigma^2_1\right)$ and $Y_1, \ldots, Y_n \overset{\text{iid}}{\sim}...
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Using Bayesian formula to apply uncertainty to topic probability given length of document

After computing topics ($z$) over a word network, I'm assigning topic probability to documents, following: $$p(z|d) = \sum_{w_i}p(z|w_i)p(w_i|d)$$ with $d$ being a document composed by $w_{i...n}$ ...
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How to prove that the prior for which Bayes rule is also the minimax rule, is the least favorable prior?

I have read in the book Mathematical Statistics: A Decision Theoretic Approach by Thomas Ferguson that The prior for which the Bayes rule is also minimax rule, then that prior is Least favorable prior....
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What's the role of the scale matrix for the Inverse-Wishart and Wishart distributions?

What's the role of the scale matrix for the Inverse-Wishart and Wishart distributions? The purpose of finding this information is to enlighten me on how should one decide on a prior for a positive-...
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Moments of the horseshoe prior?

Are the first two moments well defined for the horseshoe prior? I would say that the expectation is zero but the variance does not exist. Using the following argument. Let $$\beta_i \mid \lambda_i, \...
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What is the limit of this expression?

If $\det(\Lambda_0) \to 0$, what does $$ \exp\left(-\frac{1}{2}\text{trace}\left(\Lambda_0 \Sigma^{-1}\right)\right)\det\left(\Lambda_0\right)^{-1/2} $$ approach? I was trying to answer the ...
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101 views

Supervised document classification with prior distribution on some features

There is a somewhat off-the-beaten-path supervised document classification problem I am trying to tackle. For the sake of simplicity, let's say we are using the bag-of-words approach. Usually, we ...
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330 views

Weakly-informative priors, complete separation and identifiability in Bayesian logistic regression

Where complete separation may result in non-identifiability of parameter estimates in Bayesian logistic regression, Gelman et al (2008) recommend using weakly-informative priors using a Cauchy ...
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How to set up a proper prior sensitivity analysis?

Working with Bayesian econometrics I am regularly faced to the (valid) question: How does your prior choice affect the results? Given my parameter space is of rather low-dimension I may come up ...
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Understanding the influence of the prior distribution on the original parametrization

Be $y_1,y_2,..y_2$ a simple random sampling from $p(y|\theta)$. Be $\theta$ a parameter with a given prior distribution $p(\theta)$. A way to understand how much informative is $p(\theta)$ is to plot ...
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80 views

Priors on variable ordering and/or percentile ranking

Consider a set of variables $\mathbf{X}$ = $X_1 \ldots X_n$ where each variable is $\in [0,1]$. I am modeling an inference problem on these variables. Among other things, I have the following prior ...
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174 views

Does a proper prior lead always a proper posterior?

Does a proper prior lead always a proper posterior? I cannot check whether the posterior is proper, so I was wondering if this assumption is always satisfied .
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Incorporating population priors into MLE fits with few/limited samples

I am fitting Beta distributions to data resulting from each of many experiments using maximum likelihood. My goal is for each experiment, given iid data $y_{1:k}$, fit a Beta distribution, and then ...
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55 views

prior for integer-valued random variable taking values 1 or greater

In my model I have an integer-valued random variable which should only take values one or greater. I would like to specify an appropriate prior for this which has most of the mass say around 1 to 5 ...
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164 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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274 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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70 views

How to include prior knowledge that a model might be able to figure out itself

I have a problem where I want to predict the outcome of a sequence given another sequence online. Let $(x_1, x_2, ... x_T)$ be denoted by $x_{1:T}$, then I am estimating: $$ p(y_T|x_{1:T}) $$ where $...
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Question about foundations of the uniform shrinkage prior

I am collecting papers about the uniform shrinkage prior for hierarchical Bayesian model. In "A prior for the variance in hierarchical models" of Michael J. Daniels it is stated at the end of page two ...

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