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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Getting Bayes Prior

We want to predict how many points per game LeBron James is going to get per game. Assume there is some underlying theta that does not change game to game, that predicts how many points he will get. ...
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How do I choose parameters for my beta prior?

Suppose today I'm going to flip a coin. I believe that 9 of 10 flips will come up heads. I flip the coin and 8 of 10 are heads. Is my distribution of belief beta(9+8, 1+2) beta(1+9+8, 1+1+2) beta(...
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Estimating probability or frequency with low N?

I am trying to estimate the probability of an event using a low number of observations. The naive estimator $\hat{p} =\frac{\text{number of positive observations}}{\text{total number of observations}}...
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Non-informative prior for integer restricted parameters

Does anyone have advice/references on choosing a minimally informative prior for the shape parameter of an Erlang distribution (gamma distribution with shape parameter restricted to integer space), or ...
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269 views

Bayesian estimation of tridiagonal covariance

I want to estimate covariance of a multivariate normal distribution from data using a Bayesian method. I want to force the result to be tridiagonal. I am looking for an appropriate prior or method. Is ...
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720 views

why isn't the the marginal distribution needed when using a conjugate prior?

What is a good explanation as to why you wouldn't have to integrate to find the posterior when you use a conjugate prior. Most examples (like for instance: http://www.youtube.com/watch?v=0XD6C_MQXXE) ...
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872 views

Choosing non-informative priors

I am working on a model relying on an ugly parametrized function acting as a calibration function on a part of the model. Using a Bayesian setting, I need to get non-informative priors for the ...
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854 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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priors for strictly positive index or score types of variables

Is there a prior that's commonly used for "index" or "score" type variables that are user-defined as a weighted sum of a small number of variables (sometimes with pre-defined interaction contributions)...
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What are typical values to use for alpha and beta in Latent Dirichlet Allocation?

Specifically in the case where I don't know anything about the documents I'm working with. I'm looking a specific number or number range.
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293 views

Help with Bayesian inference in OpenBugs

I have a task that involved Bayesian inference and could use some pointers and hints. I've already got some parts figured out but others remain blurred. Also, my OpenBUGS abilities are frankly limited ...
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Detect bias in subset of Bernoulli processes

I'm looking for advice on the best method to use to answer this question. General scenario: We have multiple testing machines A,B,C,D etc. each tests a identical randomly selected part and provides ...
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Help me understand Bayesian prior and posterior distributions

In a group of students, there are 2 out of 18 that are left-handed. Find the posterior distribution of left-handed students in the population assuming uninformative prior. Summarize the results. ...
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Neg Binomial and the Jeffreys' Prior

I'm trying to obtain the Jeffreys' prior for a negative binomial distribution. I can't see where I go wrong, so if someone could help point that out that would be appreciated. Okay, so the situation ...
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534 views

Selecting a Bayes shrinkage prior

I'm looking for a way to integrate prior knowledge about a parameter in a context equivalent to Bayesian hierarchical models. I come from frequentist background and I'm uninitiated in hierarchical ...
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Kalman-Bucy filter: how does a prior mean change, alter the posterior?

I have a question on Kalman-Bucy filter: The prior distribution is $g \sim N(0,σ_g^2 )$, signal is $ds=(μ+g_t )dt+σdZ_t$, posterior distribution becomes $g_t \sim N((\hat{g_t},\hatσ_t^2)$. Here,$σ_g,...
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Jeffrey's prior for variance

I'm dealing with hierarchical model where $Y_i$ are from normal distribution. About variance the formulation is the following: Similarly, the data contain substantial information about the measurement ...
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315 views

How to derive Bayes prior given large amounts of sample data from different population members

In Poker, it is important to judge the likelihood of an opponents action, but for unknown opponents, we have few samples available. One method proposed in http://www.husng.com/content/interpreting-...
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Jeffreys prior for inverse gamma distribution

Does anybody have the experience of dealing with Jeffreys prior? I am working with hierarchical model at the moment where the parameter σ^2 from normal distribution is said to be chosen according to ...
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Prior for Bayesian Inference on Failure Rate in Poisson Distribution

I'm trying to derive the posterior distribution for the failure rate (lambda) of a process with poisson distribution. I have tried the use of an improper uniform distribution on lambda by letting the ...
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Priors for parameters of normal distribution leading to same results as frequentist formula

Given a sample vector $x$ of size $N$ from a normally distributed population. With frequentist methods the population mean is estimated as $\hat{\mu}=\frac{\Sigma{}x_i}{N}$, population sigma is ...
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Difficulties with a Bayesian formulation of a model for human timing data

The Wing-Kristofferson model is a simple model of the behavior of a human trying to drum out a steady beat (that is, trying to mimic a metronome). Let $y_i$ be the $i$th interval between two drum ...
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Is a diffuse prior always weak?

There are countless examples of diffuse priors being used to 'allow the data to speak.' However, what if one's past experience leads you to be skeptical of new data, without necessarily having a ...
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Bayesian batting average prior

I wanted to ask a question inspired by an excellent answer to the query about the intuition for the beta distribution. I wanted to get a better understanding of the derivation for the prior ...
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How concerned should I be about the appropriateness of my prior?

As I understand it, selecting a prior provides something of a starting point for your analysis. From there, the distribution is shaped by the observed data. Obviously, the more data you observe, the ...
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"forgetfulness" of the prior in the Bayesian setting?

It is well-known that as you have more evidence (say in the form of larger $n$ for $n$ i.i.d. examples), the Bayesian prior gets "forgotten", and most of the inference is impacted by the evidence (or ...
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How detailed should a data-driven Bayesian prior be?

My exact problem is this. I have a number of sources of traffic with different conversion rates. I have good evidence that conversion rates vary based on the source. For each traffic source I have ...
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Using priors in classification for regression

I am ultimately trying to perform a regression task - for this example, let's say I'm trying to determine the height (in pixels) of a person in an image. However, rather than doing regression, I am ...
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Taking expectations for Fisher information/Jeffreys' prior

I've been working through a few examples of the Jeffreys' prior; however, I'm a little confused by one section. I was hoping that somebody could provide some clarification. If the Jeffreys' prior is ...
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Nonparametric Bayesian priors on mean $0$ distributions?

Is there any standard way of putting a prior on the mean $0$ distributions? I'm interested in this from the perspective of robustly modelling the error distribution in a regression. So for instance I ...
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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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7k views

What prior distributions could/should be used for the variance in a hierarchical bayesisan model when the mean variance is of interest?

In his widely cited paper Prior distributions for variance parameters in hierarchical models (916 citation so far on Google Scholar) Gelman proposes that good non-informative prior distributions for ...
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504 views

Bias of variance/precision estimator using Gamma prior

Assume I have $N$ samples $x_1, \cdots, x_N$ from a Gaussian random variable $X\sim N(\mu, \sigma^2)$ where both $\mu$ and $\lambda = 1/\sigma^2$ are unknown. If I apply MLE, I have $\mu_{MLE} = \...
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A re-formalization of a conjugate prior?

It is quite easy to prove that if $p(\theta)$ is a conjugate prior to some likelihood then the following: $$q(\theta') \propto p(\theta)I(\theta \in A)$$ where $A$ is a subset of the parameter space ...
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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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Why is the Jeffreys prior useful?

I understand that the Jeffreys prior is invariant under re-parameterization. However, what I don't understand is why this property is desired. Why wouldn't you want the prior to change under a change ...
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Specifying the Form of Prior, Likelihood and Posterior Distributions for Bayesian Analysis

I have recently begun to look into Bayesian Analysis, and, although I'm beginning to get to grips with the general framework (i.e. $\text{posterior} \propto \text{likelihood} \times \text{prior}$), I'...
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560 views

Creating a correlated prior

I would like to create a "weakly informative" prior distribution for a couple parameters. They both could theoretically take any value between 0 and 1, but I have reason to think that they should be ...
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Parameters without defined priors in Stan

I've just started to learn to use Stan and rstan. Unless I've always been confused about how JAGS/BUGS worked, I thought you always had to define a prior ...
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Natural interpretation for LDA hyperparameters

Can somebody explain what is the natural interpretation for LDA hyperparameters? ALPHA and BETA are parameters of Dirichlet ...
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223 views

Weighted Distributions in Predictive Model

I'm not a statistician but do work with large datasets and have a problem I'd like to use a predictive model for. I have two datasets that I'd like to use together to build predictions. The first ...
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Bayes factors with improper priors

I have a question regarding model comparison using Bayes factors. In many cases, statisticians are interested on using a Bayesian approach with improper priors (for example some Jeffreys priors and ...
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Hyperprior distributions for the parameters (scale matrix and degrees of freedom) of a wishart prior to an inverse covariance matrix

I'm estimating several inverse covariance matrices of a set of measurements across different subpopulations using an wishart prior in jags/rjags/R. Instead of specifying a scale matrix and degrees ...
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How to specify a non MLE prior for a naiveBayes model in R?

I am just learning R, and am aware that the package e1071 has a naiveBayes method that takes in predictor and class membership, and estimates the class prior using ...
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How is the inverse gamma distribution related to $n$ and $\sigma$?

Given that the posterior estimate of $\sigma'^{2}$of a normal likelihood and an inverse gamma prior on $\sigma^2$ is: $$\sigma'^{2}\sim\textrm{IG}\left(\alpha + \frac{n}{2}, \beta +\frac{\sum_{i=1}^n{...
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Dealing with a partially pooled intercept in a Bayesian model

I have a multilevel logistic regression (Rasch), fit in JAGS. Specifically, $\text{logit} (P(\text{Win})) = \alpha_p + \gamma_w + \delta_{p,w}$ The prior on $\alpha_p$ is partially pooled, $\...
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How do I complete the square with normal likelihood and normal prior?

How do I complete the square from the point I have left off at, and is this correct so far? I have a normal prior for $\beta$ of the form $p(\beta|\sigma^2)\sim \mathcal{N}(0,\sigma^2V)$, to get: $...
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704 views

Gamma-normal distribution as prior

When I study the Bayesian econometrics, the book firstly introduces Gamma-Normal distribution as (conjugate) prior, then the posterior will have the same distribution as the prior. But my question is, ...
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How to sample from the prior predictive distribution in jags? [closed]

Is there an example available of how I can sample from the prior predictive distribution (without data) in jags? I would like to get a better sense for the contribution of the prior in a multilevel ...
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False discovery rate of a Bayesian classifier: scaling based on prior odds?

I am trying to assess the performance of my Bayesian classifier. One measure that I calculate is the false discovery rate (FDR): FP / (FP + TP), where FP = False Positive and TP = True Positive. ...