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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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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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 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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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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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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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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277 views

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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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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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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Why are beta distributions commonly chosen for priors? [duplicate]

Is there any specific reason why a Beta distribution would be chosen as a prior, other than that it is conjugate for the Binomial?
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location/scale invariant priors

I'm trying to understand what's the motivation behind these priors, and why they are used. I understand that for location parameters of some distribution, you want it to be invariant of movement. e.g....
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Setting priors for bivariate regression

I would like to perform a bivariate MCMC regression with boldness scores as the continuous response variable, aggression ranks as the ordinal response variable, trial numbers as fixed effect and ...
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Point estimate and 95% credible interval

The text of the problem as follows: The data follows a normal distribution with $\mu$ and $\sigma^2$ unknown. We wish to perform inference on the mean selling price $\mu$. And our sample data are (...
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Suppose we have a Bayesian Model on the data $Y_i = \alpha \beta^2 X_i^3$. What priors cause the model to not be identifiable?

Suppose $$ Y_i = \alpha \beta^2 X_i^3 \ \ \text{for $i$ in $1, \ldots, N$} $$ Both $Y_i$ and $X_i$ are positive real numbers. If we were to put a uniform prior on $\alpha$ and $\beta$, why would ...
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What does it mean for the uniform prior? [closed]

I wonder about the meaning of uniform prior of an unknown parameter. Any argumentation with detail explanation would be much appreciated.
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What is the distribution of a r.v. if the reciprocal is distributed gamma?

I want to find the posterior distribution of σ^2 when (X1, X2, ..., Xn) ~ N(μ,σ^2), μ is known, and 1/(σ^2) ~ Gamma(α,β), but I'm not sure how to find the prior of σ^2 given the prior of 1/(σ^2). ...
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How to calculate the confidence interval with weighted data?

I've done some search for similar questions, but they're not the same as what I'm trying to get. Assume that there's a server that handles requests $r$ and returns a set of items $I_{r}$ of random ...
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1answer
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Using prior knowledge about correlated variable in ridge regression

I am wondering what methods are available for incorporating prior knowledge of some variable that is correlated with the unknown regression coefficients in a ridge regression. I have a sparse matrix ...
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1answer
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How to sample a total variation prior?

I am interested in drawing samples from a total variation prior: $$\pi_{\mathrm{pr}}(\boldsymbol{x})=\left(\frac{\alpha}{2}\right)^n\exp\left(-\alpha\sum_{j=0}^{n-1}\vert x_{j+1}-x_{j}\vert\right)=\...
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1answer
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Bayesian prior update feedback

Consider a standard Bayesian MCMC inference problem with $\theta_n$ free parameters. I know very little about their distributions, so I solve using uniform priors. Then I take, for example, the mean ...
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1answer
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Computing Mode of Prior

How do you compute the mode of a prior with beta distribution $(\alpha, \beta)$?
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Prior Predictive Distribution for hierarchical model

What I need to do is figure out the prior predictive distribution for the following mixed model: $$\begin{align*} y_{ij} & \sim \mathsf{Normal}(\alpha_j + \beta x_i, \sigma^2)\\ \alpha_j & \...
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1answer
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why Beta(1,1) is an improper prior

While I am looking for the sun-rise problem, in part of prior selection they said that ... in a section of Beta distribution https://en.wikipedia.org/wiki/...
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Conjugate priors calculation [duplicate]

I am following the Bayesian Methods for Machine Learning course on Coursera. Unfortunately, it glosses over many details, and I am struggling to understand how to check if a distribution is a ...
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Does quadratic loss find the median of the prior distribution?

Does quadratic loss find the median of the prior distribution? Someone told me linear loss finds the mean, all-nothing loss function finds the mode of the prior.
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Pick a prior for my bayesian generalised linear model with binary outcomes

I need help in my choice of a prior for a bayesian model. I have data from a set of participants responding to a set of yes/no questions. Answers are correct or incorrect. I suspect some questions ...
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No operational difference between a prior density $f(\theta)$ and $f(x \vert \theta)$?

I am currently studying the textbook In All Likelihood -- Statistical Modelling and Inference Using Likelihood by Yudi Pawitan. Section Bayesians versus frequentists of chapter 1 says the following: ...
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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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