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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Non-Dirichlet Prior for $Cat(\theta)$ parameter that can tractably be integrated out (for Latent Dirichlet Analysis)?

In LDA Topic Models, it is standard to 'integrate out' the $\theta$ parameter, which contains a document's Categorical probabilities of drawing each topic. QUESTION If one uses the standard Dirichlet ...
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Bayesian statistics: what is the variable we are integrating in?

This is a screenshot from Bayesian Data Analysis by Gelman. I am a little bit confused by Equation 1.4 (first and second lines), having read Equation 1.3. In Equation 1.3, the variable of integration ...
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Incorporating prior evidence of predictor having no effect in bayesian linear regression model

Say we start with a linear regression model of the form $$y = \beta_0 + \beta_1x_1 + \beta_2x_2 + \epsilon, \quad \epsilon \sim N(0, \sigma^2)$$ with the conjugate prior $$ \begin{align*} &\sigma^...
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Prior selection in Gaussian - an application to height measurement

Say I have just purchased ACME's Tree Height Measuring Device (THMD). ACME states that the error $\epsilon$ in tree height measurement from this device can be modelled as a normal distribution with ...
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The PDF of the Data (Marginal Likelihood) Given the Prior of a Gamma Distribution with Prior on the $ \beta $ Paraneter

Given a model where $ x_i | \beta \sim \mathcal{Gamma} ( \alpha, \beta ) $ where $ \beta \sim \mathcal{Gamma} ( \alpha0, \beta0 ) $, is there a closed form formula for the PDF of $ x_i $? Namely, what'...
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The PDF of the Data Given (Marginal Likelihood) the Likelihood and the Prior of a Normal Distribution with Prior on the Mean

Given a model where $ x_i | \mu \sim \mathcal{N} ( \mu, \sigma^2 ) $ where $ \mu \sim \mathcal{N} ( \mu_0, \sigma_0^2 ) $, is there a closed form formula for the PDF of $ x_i $? Namely, what's $ p (...
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Is there any strong argument about objective/non-informative improper prior?

Decades ago improper objective priors - e.g. $\pi(\sigma) \propto \sigma^{-1}, \sigma > 0,$ for a scale parameter - were considered problematic because some authors thought they were leading to the ...
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Beta distribution equivalence with two redondant parameters [duplicate]

context In Factor graphs on discrete variables, the parameters are contained in factors associated each with a subset of the random variables in the system. Each factor provides a different positive ...
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Kl Divergence between factorized Gaussian and standard normal

Given two distributions, one a parameterized gaussian and the other a standard normal gaussian: $q(x) \sim \mathcal{N}(\mu,\sigma)$ $p(x) \sim \mathcal{N}(0,I)$ We want to compute the KL Divergence $...
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Bayesian statistics

Assuming I have that $Y_i\mid \mu$ is an iid ~ $N(\mu,\sigma^2)$, for $i \in (1,\dotsc,n)$ with $\sigma_i$ known and improper prior $\pi(\mu)=1$ for all $\mu$. i. How can I derive a formula for the ...
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Bayes: How to use results from a single study to shape data-driven priors

One way to construct informative priors for a subsequent Bayesian analysis is to carry out a meta-analysis of previous studies. Here, substantial research has been done. However, what to do when there ...
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Derivation of posterior distribution under Dirichlet prior distribution:

suppose that $\mathbf{y}=(y_1, y_2, \cdots, y_n)$ is a vector of $n$ observed sample points drawn from a mixture of $g$ components, and $\mathbf{z}=(z_1, z_2, \cdots, z_n)$ is a vector of latent ...
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Partially specified Bayesian prior?

In bayesian linear regression for example, we may specify a model as: $$y_i \sim N(\beta_0 + \beta_1 x_i, \epsilon^2) \\\\ \beta_0 \sim N(0, \tau_0^2) \\\\ \beta_1 \sim N(0, \tau_1^2) \\\\ \epsilon \...
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Non-informative prior of a geometric distribution [duplicate]

If we are given a standard geometric distribution $(1-p)^{x-1} p$, with $0<p<1$ what would be a suitable non-informative prior for this?
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Informative priors for standard deviation (or variance)

Suppose I want to perform Bayesian estimation of the mean $\mu$ and standard deviation $\sigma$ of a Gaussian distribution. Is there a standard way to specify an informative prior over $\sigma$, ...
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How can I sample from a shifted and scaled Student-t distribution with a specified mean and sd in R?

I'm currently building some Bayesian models with the brms package and the default intercept prior is student_t(3, 0, 6.3) and so ...
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Can I transform a parameter's posterior to a different parametrization?

I have a model with several parameters. I apply Bayesian inference with a uniform prior for all of the parameters. After the process is finished, I realize that I need one of those parameters $x$ ...
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Choosing priors for the parameters of Gamma distribution

Suppose that $X_1, X_2, \cdots, X_n$ is a sample drawn from a Gamma distribution with parameter $\alpha$ and $\beta$. Then, the likelihood function can be written as follows: \begin{equation} L(\...
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Why does a GAN generate samples from a random prior?

I've been reading Goodfellow et. al.'s paper on GANs and also the conditional GAN one by Mirza et. al. While relatively straight forward, I'm not sure I understand why the prior for the generator is ...
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Probability of finding an object on the beach: Bayes theorem [closed]

Introduction Beach litter surveys are collections of observations that detail the object and quantity found of that object within a defined length of shoreline. The sampling protocol was initially ...
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Deriving log-level cofficients from level-level regression

Suppose we have estimated parameters $\hat{\beta} = [\hat{\beta}_{0}, \hat{\beta}_{1}, \hat{\beta}_{2}]$ from a level-level regression: $$\hat{y} = \hat{\beta}_{0} + \hat{\beta}_{1}X_{1} + \hat{\beta}...
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Maximum entropy prior for binomial trial, is it 1/(2n+2) and this reasonable?

I am looking into what prior probability should be assigned to an event in a binomial trial that could occur but has not yet occurred after many trials. rephrased, what probability should be assigned ...
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Can I use a Prior with Simulated data?

I have a prior about some proportion that follows a Beta distribution. Unfortunately, I do not have (yet) observed data but I was offered a thousand simulated datasets. Each dataset comes from ...
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Hierachical Bayesian modelling using brms: how to insert a prior that reflects cut-offs of Reaction Times distribution?

I am running a hierarchical Bayesian model using brms on reaction times (RTs) of a GoNogo task. The predictors are categorical and include the 3 stimuli/condition that participants observed and the 2 ...
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Choice of prior to restrict coefficient to negative values?

I'm slightly embarassed to be asking this question, but I couldn't locate a satisfying reference. I'm a complete novice in Bayesian regression, working through the textbook Statistical Rethinking. In ...
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brms: prior of categorical predictors in Multilevel Bayesian model

I want to run a Bayesian Multilevel model on reaction times using two categorical factors (conditionStimuli = 3 levels; sequenceTrials = 2 levels). Initially, I run the model with default priors: <...
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What priors are typically used for the correlation parameter in a bivariate normal?

The Bivariate normal distribution contains a correlation parameter. I want to implement an MCMC sampler to sample from the posterior distribution of the parameters of the bivariate normal distribution....
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Optimistic and Pessimistic Priors

In Bayesian Statistics, while computing posteriors, we define priors based on our existing knowledge. For instance, if we are not sure about the data, we'd probably go with a uniform prior. What if we ...
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Ridge regression, argmax of the MAP

How can i prove it? I only proved that it is equivalent to $$arg\min_w \sum_{i=1}^N(y_i-w_0-\textbf{w}^T\textbf{x}_i)^2+\lambda||\textbf{w}||^2_2$$
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With knowing the exact form of the prior density, how to improve the convergence/acceptance rate of an independent Metropolis Hasting algorithm?

I have a set of parameters $[\eta_1,\eta_2,...,\eta_k]$ (k can be a bit large, sometimes k=40) to estimate via the Bayesian MCMC method. I know that each component of $\eta_{k}$ is independently ...
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What prior distribution could one choose to model the number of survey comments?

This question has been inspired by the recent release of the results of a company survey to its employees. There were 12,000 respondents and 16,000 comments. This means that, necessarily, more than ...
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Exponential Posteriori with a Uniform Prior

I'm studyng for a final exam and found this problem from another generation, but I don't know how I should continue... I will be gratefull for any help, thanks you. Let be $X|\theta\sim U(0,\theta)$ ...
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Lasso (or Ridge) vs Bayesian MAP

This is the first time I have posted here. I am looking for some feedback or perspective on this question. To make it simple, let's just talk about linear models. We know the MLE solution for the $l_1$...
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Need help computing and sampling from a posterior distribution

I am trying to compute and draw samples from a posterior distribution. Here is what I have: My data, $\textbf{x}$, is a vectorized $N\times N$ image, i.e. it is of length $N^2$. An arbitrary shape (...
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Is a draw from the posterior always the same as a draw from the prior?

I'm reading Bayesian data analysis. On page 155, the authors state: Each of the [...] parameters were assigned independent Beta(2, 2) prior distributions. ... If the model were true, we would expect ...
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Is this Considered "Cheating" in Bayesian Modelling?

Suppose you have some data that corresponds to a single predictor variable and a single response variable. You are interested in fitting a regression model to this data : Y = B_0 + B_1 * X If you ...
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In Bayesian models, can you use Uniform(-inf, inf) as a prior?

In Bayesian models, can you use Uniform(-inf, inf) as a prior? I ask because in an class, we looked at MH MCMC sampler, and showed that to sample from a distribution, we need not explicitly solve for ...
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Why Are Empirical Bayes Methods Not Considered "Controversial"? [duplicate]

I was reading about Empirical Bayesian Methods and came across the following: My Question: As this text explains, I have often heard that the priors used in Bayesian Methods should be decided prior ...
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What kind of bayesian approach should be used for expenditure reasearch?

I must find determinants of expenditure which using baysian approach. Dependent variable: LN education expenditure Independent variables: continuous: LN income, study year of mom, commuting time to ...
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Data-informed grouping of covariates in Bayesian Hierarchical Modeling?

Is there a way to place a prior on the first stage's betas that allows the second stage groups to be determined from the data? I am working with co-exposures where I am not super confident in how they ...
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What distribution would make a good hyper-prior for a Beta distribution parameterized by mean and sample size?

I have a model which includes a Beta distribution and I am looking for guidance on how to parameterize a hyper-prior for it. For example, this post uses a Beta parameterized with a mean and ...
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What does the beta parameter tell you in the cauchy distribution?

I am doing some bayesian analysis, and I recently used the half-cauchy distribution as a prior for a variable that tracked monthly spending. My thinking is that this is a non-negative number that is ...
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How to set prior for Bayesian analysis

I am new to statistics and Bayesian analysis. Therefore, I have some problems that would like to clarify. Suppose my problem is to calculate the posterior distribution for the time of ship to spend in ...
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Choosing informative Gibbs priors for Bayesian updating

I'm trying to create some kind of iterative Bayesian algorithm, which continuously updates as more data is gathered. The aim is to iteratively update the coefficients based on the second dataset using ...
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Bayes prior in MAP estimation corresponding to $\ell^0$ penalization

I gather that in the context of penalized least squares, we can interpret a penalty term as corresponding to a prior $\pi(\beta)\propto \exp\{-\text{pen}\}.$ Is this also true for $\ell^0$ ...
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Bayesian priors associated with regularization penalties

I gather that adding a penalty term to (linear) least squares minimization typically corresponds with choosing some prior for Bayes estimation in the normal linear regression model. A couple questions ...
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The Elephant in The Room: How is Real-World Domain Knowledge Converted into Bayesian Priors?

I have been trying to look into the daunting problem within Bayesian Models: How is Real-World Domain Knowledge Converted into Bayesian Priors? Logically speaking, it seems that Bayesian Priors can be ...
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Prior predictive distribution with an improper prior for a Poisson likelihood

I have recently started exploring some bayesian statistics and I cannot seem to understand something about improper priors. In particular, the set up consists of a Poisson likelihood $ p(X|\theta) = \...
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Conjugate Hyperpriors

I heard it was possible to have a Bayesian model with likelihood, prior and hyperprior that has a posterior of closed form, by choosing a conjugate prior and conjugate hyperprior. But I struggle to ...
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Prior correlation between uncertain parameters of a distribution in the context of Bayesian updating

In the context of Bayesian analysis, I have a given (Weibull) distribution with uncertain scale $a$ and shape $k$ parameters. I assume that $a,k$ follow a bivariate lognormal distribution and I want ...
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