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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Why do we reparameterize before assigning a hyperprior distribution?

I am studying hierarchical models, and trying to understand a point in the book where they try to decide on a non-informative hyperprior distribution. The hyperparameters is $\alpha$ and $\beta$ for a ...
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What are some common prior/likelihood choices for Bayesian logistic regression?

I'm not really clear on the Bayesian approach to logistic regression. From everything I've read, the prior and likelihood can be can be whatever you want them to be. Well, I've a couple things; namely,...
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Choosing reasonable priors for Poisson GLMM

I am using the package brms in R to fit a generalized linear mixed model using a Poisson distribution with log link. The model takes count data that ranges from 0 ...
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Why prior distribution is not conditioned on X?

I would like to know why in the below formula the prior distribution of theta is not conditioned on X (observations): In my understanding, the correct formula should be: P(theta | X, y) = P(y| X, ...
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How does Prior Variance Affect Discrepancy between MLE and Posterior Expectation

Suppose that $\theta\in R$ is a parameter of interest, $p(\theta)$ is our prior belief regarding $\theta$, and $\hat \theta$ is the MLE for theta derived from the data $x$. It is my understanding that ...
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Find the prior distribution for the natural parameter of an exponential family

Show that for the binomial likelihood $y$ ~$Bin(n, \theta)$, $p(\theta) \propto \theta^{-1} (1-\theta)^{-1}$ is the uniform prior distribution for the natural parameter of the exponential family. I am ...
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Estimate with known sample mean, sample size, prior mean, prior standard deviation?

I want to estimate the actual "eval" of a chess move (in this case, expected win rate - expected loss rate, ranging from -100 to +100). I have empirically calculated that on average, random ...
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posterior distribution of a Poisson mixture model

This is a Poisson-gamma model with mixture prior, thus mixture posterior. I am having some trouble finding the posterior weightings. I have the prior weightings $p_1=1/3$; $p_2=2/3$. The 2 component ...
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Regarding the use of non informative priors

I am a beginner to Bayesian analysis and I am trying to fit a logistic regression model using Bayesian approach. For the prior distribution of the $\beta$ regression coefficients , I used a non ...
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Bayesian Inference question: What was the likelihood that my observation originated from one distribution versus another?

While analyzing one of my datasets, I noticed that a subset of my data has some interesting distinguishing features. The light line represents the distributions of the blue/red/green feature for all ...
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How to describe an “incomplete” prior?

I would like to know how to describe sources of uncertainty neglected when I approximate a prior distribution $p(x)$ by a marginal distribution. Specifically, let's say that I have a marginal ...
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What is a non-informative choice of parameters for a Dirichlet distribution?

Dirichlet distribution is a conjugate prior for multinomial distribution. I want to impose a non-informative prior over sampling weights $\pi$ for a draw $x=(x_1,…,x_N)$ from a multinomial ...
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On the choice of prior in Bayesian Bootstrap

Let $d=(d_1,…,d_K)$ be a vector of all the possible values that the data $x=(x_1,…,x_N)$ could possibly take. Then, each $x_i$ is modeled as being drawn from the $K$ possible values where the ...
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How to impose restrictions on a random matrix via its prior distribution?

I am reading the paper Factor analysis and outliers: A Bayesian approach. The author starts with a factor analysis model given by $${\bf y}_i = {\bf \Lambda} {\bf z}_i + {\bf e}_i, \quad i = 1, \ldots,...
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Using a beta-binomial model to estimate the average for a uniform prior [duplicate]

Say we had a sample of 100 people who were asked how many days during the last week they drove their car. Let's say the resulting frequency table is as follows: Days, frequency 0, 1 1, 5 2, 3 3, 15 4, ...
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Information Theoretical optimality of Jeffreys Prior?

Given a "conditional distribution" $f(x|\theta)$ and its corresponding Jeffreys prior $r(\theta)$, is there some information theoretic sense (in terms of quantities like mutual information) ...
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The Bayes' Theorem Components of the Probability Output of a Classifier

Let's give a simple setup. I have $500$ photos of dogs and $500$ photos of cats, all labeled. From these, I want to build a classifier of photos. For each photo, the classifier outputs a probability ...
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Do we update a priori distribution somehow?

I'm trying to understand Bayesian statistics. Recently I asked here whether we estimate paramteres of a priori distribution in bayesian statistics. I was responded that we typically don't estimate ...
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Group level distribution for positive parameters in Bayesian multilevel models

I am doing a lot of modeling with models that require some parameters to be positive by design. However, I am struggling to figure out which approach works best when I try to use multilevel modeling ...
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What is the posterior distribution of two parameter Weibull likelihood and a uniform prior?

I've seen examples of Jeffreys' prior with a Weibull likelihood. I was wondering if $p(\alpha) = 1$, then what is the posterior distribution of alpha for two parameter Weibull in that case.
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What does it mean to have a “gaussian prior?”

When reading up on ridge regression, I saw it stated that it has a "gaussian prior." I realized that I don't know what the word prior means in this context and what it is applied to? I ...
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Setting variance of an informative prior

I am creating a Bayesian Poisson Regression model and I have access to a dataset and a previous corresponding model. I want to use the previous model to create a prior that I will combine with the ...
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Bayesian priors and probability distributions

Book "Bayesian Statistics the Fun Way: Understanding Statistics and Probability with Star Wars, Lego, and Rubber Ducks", chapter 9 "Bayesian priors and working with probability ...
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How to choose a non-informative or weakly informative hyper priors for my hierarchical bayesian model?

I am learning Bayes on "Applied Bayesian Statistics" by MK Cowles. The chapter about "Bayesian Hierarchical Models" mentioned an example that we estimate a softball player’s ...
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Bayesian estimation Prior adaptation [closed]

I have a dataset of 1 dimensional 20points as prior information, so assuming prior distribution to be Gaussian distribution we can easily find its variance and mean. Now we will use this prior finding ...
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How to multiply a likelihood by a prior?

I'm trying to wrap my brain about computations in bayesian stats. The concept of multiplying a prior by a likelihood is a bit confusing to me, especially in a continuous case. As an example, suppose I ...
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Why is this an example of a noninformative prior?

From Bayesian Data Analysis 3rd Edition [Gelman et. al], they give this as an example when introducing non-informative priors: "We return to the problem of estimating the mean θ of a normal ...
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Using prior information for a GAM smooth function to reduce standard errors

I have some data that I want to model in a GAM. However, there are few observations, generally leading to high standard errors. ...
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Problems with setting weakly-informative priors

I am very new to the topic of Bayesian inferencing and I struggle with setting priors. I do understand the underlying idea of incorporating existing knowledge/prior beliefs, however, I seem to be ...
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Sparsity-inducing priors for non-negative random variables

Which priors could be used for inducing sparsity on a random variable with non-negativity constraints?
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The prior in MAP and Bayesian interference

We can use a Normal distribution as a prior when handling a Normal distribution as likelihood in Bayesian inference However if we want to do MAP given a Bernoulli as likelihood can we use Normal ...
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In what number space do brms, lme4 etc. understand priors in a binomial (logit link) model?

I understand that predictors in a logit-linked binomial linear model are mapped onto the 0..1 probability range by the logit function. A normal distribution that would be considered flat in a ...
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Bartlett-Lindley effect

Does any have a citation to the "Bartlett-Lindley effect"? As I understand it from the context of the paper I encountered the related reference, it refers to the inability of a Bayesian posterior to ...
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Bayesian Updating Prior and Posterior, 1 Coin Toss

There was one coin flip. The chances of heads were 0.5 We have a test for whether or not it landed with the Heads side up. That test, which reports a simple positive or negative, has a 80% sensitivity ...
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How to choose a prior : family for a response with negative values?

I’m modeling percentage change in oxygen levels in the blood from a particular experiment. So my prior before seeing the data was an inverse gaussian distribution. But my data (response variable ) has ...
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How to calculate the expected value of k heads in this case?

I'm having some trouble on how to tackle the following problem $X_1$ is a random variable with probability density $f(x)$ in the range $[0,1]$. A value of $X_1$ is picked, call its value $p$. A coin ...
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Randomly positioning ordered points along line?

Assume that I have a line along which I want to randomly place (say) three points. If this were the only requirement, I could simply use independent uniform priors for all three points, and be done ...
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What are reference priors exactly? [duplicate]

They are priors that abide by a particular rule? That is, a rule that governs the relationship between the prior and the posterior? Rules include Jeffrey's Rule and KL Divergence. You can pick a ton ...
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How to plot the prior, posterior and likelihood function from given data in python [closed]

I wrote a simple bayesian program which calculates prior, posterior and likelihood in python. ...
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Bayesian Inference & Determining the Prior

I have a dataset made up of the date (YYYYMMDD) of a specific event, with the time period spanning from 1970-2015. I want to compare two time periods of 10 years each, and look at the yearly total ...
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Justification for use of non-conjugate priors?

Google searches gives no results to this question and there is the opposite question in this site, which makes me think this has an intuitive response I am missing. In most course notes and responses ...
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Parameters of beta distribution with a given HDI [duplicate]

Is there some way to calculate the parameters of a beta distribution, if the highest density interval is known. That is, given $a,b,x$ I want to have a beta distribution such that the probability of ...
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Early paper on equivalence of models, loss functions, and regularizers

I'm trying to find an early paper discussing how models, loss functions, and regularizers can all be seen as the same thing. For example, instead of changing the loss function, I can use the standard ...
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Priors and nested random effects in MCMCglmm

I am trying to construct a zero inflation poisson GLMM using MCMCglmm(). I am new to Bayesian Statistics and this function and I am struggling to understand a couple of things. For my data I am ...
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Prior of a product

Let say, we have N servers. Every server is in production for a different amount of years. For every server, we know how many times this server crashed, the total for all the years. A number of ...
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Bayes test function with a discrete prior

Let X be exponential with mean θ. Consider testing H0 : θ = 1 versus H1 : θ = 2 with a single observation. Loss function: 0-1 Loss function. So the risk of the test function φ is R(1, φ) = E1 (φ(X))...
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Bayesian prior that two parameters are identical/similar, but no information on their values?

Given two coins with respective biases $\mu_a$ and $\mu_b$, suppose that we have no information on their biases, but we believe that the two biases are identical or similar. Is there a standard/...
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For Prior definition in bayesian regression with R package MCMCglmm, how to convey different strength of believe via parameter nu?

I understand the strength of the Prior is set via parameter nu however, I can not find information what nu expresses in statistical terms, e.g. how strong would a prior that is similar to the number ...
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Joint Posterior for Binomial Likelihood and Beta Priors

Suppose we have the likelihood for known $n$ $$\mathbf{x} \vert p,k \sim \mathrm{Binomial}(n, kp)$$ with a beta prior for $p$ with known parameters $a$ and $b$ $$p \sim \mathrm{Beta}(a, b)$$ and ...
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Likelihood given g-prior

Given the regression $Y = \alpha + X\beta + \epsilon_t$ where $\epsilon_t$ ~ $N(0,\sigma)$ Also given that $\beta | σ$ ∼ $N(0, \sigma^2g(X’X)^{-1})$ is a g-prior. I want to write the likelihood $p(\...

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