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 integrate to obtain prior distributions?

In the test for difference in means with unknown variance, it's stated that in order to obtain prior distributions, we have to take the double integral with respect to mean and to the variance? ...
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Do you specify priors according to the link function's transformed space?

Suppose I'm developing a model where the response variable is weight measured in pounds and is Gamma distributed. I would like to specify a prior on my intercept coefficient using other information ...
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Comparing posteriors when the same data are examined with different priors

I have two Bayesian regression models: both use the same data $D$ and all same specification $M$ but the different priors $p_1(\theta)$ and $p_2(\theta)$. Can I interpret the posteriors from the two ...
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Bayesian random slope model with divergent transitions. Help in setting stronger priors

I have a model with which I have convergence problems. This are the model specifics (brms package): ...
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Maximum entropy prior for r.v. supported on real line with no other constraints?

What would be a suitable maximum entropy prior for a random variable supported on the real line with no other constraints (i.e. unknown mean, unknown variance, unknown bounds)? All kinds of answers (...
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Are “improper uniform priors” in Bayesian analysis equivalent to maximum likelihood estimations?

The improper uniform distribution for parameter $\theta$ is : $p(\theta)=1,\ for -\infty<\theta<\infty$. It is called "improper" since it does not integrate to 1. Because Bayesian theorem is ...
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Why bayesian needs prior and neural net does not? [closed]

Bayesian requires prior distribution that should be magically taken from somewhere. Neural nets does not require that magical foreknowledge. Why Bayesian requires it, why can't it work without it ...
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What is the effect of “bounding the range” for a prior distribution?

Below is a material about the prior disribution for the proportions. The appropriate prior distribution for the parameter $\theta$ of a Bernoulli or binomial distribution is one of the oldest ...
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Using the data to obtain a prior for an analysis of that same data?

I am refereeing a physics paper in which the authors first analyze their data using a low-accuracy method, and then use the result of that method as a prior to re-analyze the data using a higher-...
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Bayesian ZIF negative binomial regression priors

I'm currently searching for some literature on the efficacy of various non-informative priors on zero-inflated negative binomial regression. However, most of the research I can find provides ...
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Selecting informative priors

I am questioning myself on how to chose the priors for a bayesian analysis in Rstudio. I'm trying to investigate the chances of relapse in a set of patients. These patients are all affected by a ...
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Weibull Distribution with priors on shape and scale diverges

I have a variable that is Weibull Distributed Duration ~ dweib(Shape,Scale) The Shape and Scale parameters are distributed to log-normal and Weibull ...
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What is the difference between prior vs bias? (term usage)

I'm confused about the usage of the term "prior" vs "bias". They often seem to be synonymous? Example 1: I'm estimating the probability for heads of a coin, and use a Beta(20, 20) prior. Then you ...
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Custom topic priors in LDA

I've been working with LDA (Latent Dirichlet Allocation topic model) for a while now and I believe I have an intermediate understanding of it. The unsupervised nature of LDA is one of its big ...
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In “A Topology Layer for Machine Learning,” are the topological priors learned by the network or imposed by humans?

In this paper by Gabrielsson, Nelson, et al. the authors "present a differentiable topology layer that can, among other things, construct a loss on the output of a deep generative network to ...
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The posterior distribution of Bt is Bernoulli

I'm trying to follow the math of Estimating Heston's and Bates’ models parameters using Markov chain Monte Carlo simulation in Journal of Statistical Computation and Simulation, but I'm having trouble ...
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How to choose the parameters of a prior distribution based on a range for the variance?

How can I use R to calculate the parameters of a prior distribution if I want the variance to fall within a specific range? For instance, I have a variable that follows the inverse gamma distribution ...
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Is it a proper prior specification for estimating MCMCglmm?

I am working with my doctoral thesis and trying to fit a generalized linear mixed effects model by using ‘MCMCglmm’ package in R. I have repeatedly read Jarrod's helpful tutorial materials. However, ...
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How to assign a prior distribution to a loading matrix that has restrictions?

I came across the paper Fast Variational Bayesian Linear State-Space Model. They work with the following model: $$\begin{align} {\bf{x}}_n &= {\bf A} {\bf{x}}_{n-1} + {\text{noise}} \\ {\bf{y}}_n &...
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Non-informative prior for the covariance matrix

I'm currently working on a project around the Bayesian approach to portfolio selection, and I can't manage to wrap my mind around the specification of the non-informative (diffuse) prior. Assuming ...
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Generating data from the posterior distribution

Let $$p(D \mid \mu,\sigma^2) \sim \mathcal{N}(\mu,\sigma^2)$$ where $D=(x_1\ldots x_n)$ is my data. I imposed a normal prior on the mean as $$\pi(\mu) \sim \mathcal{N}(\mu_0,\sigma_0^2)$$ Using Bayes, ...
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dirichlet distribution and excessively large numerator

what I am trying to do is calculating posterior probability using dirichlet distribution as my prior. the situation is like this. a web log have three variables A, B, C, and each variable's value is ...
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Is assigning an inverse-Wishart distribution to a diagonal matrix problematic?

I'm reading the paper Bayesian Vector Autoregressions by Thomas Wozniak. He considers the model $$y_t = \mu + A_1 y_{t-1} + \cdots A_k y_{t-k} + u_t$$ where each $y_i$ is a $N$-vector, each $A_j$ is a ...
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Calibrating LASSO prior (how to select the scale hyperparameter)?

I want to use a LASSO prior (Laplace prior) for a location parameter $\mu$ $$\pi(\mu \mid s) = \dfrac{1}{2s}\exp\left(-\frac{\vert \mu \vert}{s}\right).$$ However, I do not know to calibrate this ...
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Finding the posterior distribution of a Bayesian analysis prior

I have a prior distribution $f(x)=\pi cos(\pi x) $ where $x$ is the probability of getting tails in a coin toss. Should a coin toss result in tails, how would this be reflected in the posterior ...
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Finding the posterior distribution for Beta likelihood with unknown alpha [duplicate]

If $(y_i|\theta)$ is distributed as $Beta(y_i,\theta,\beta)$ then what prior distribution do I use? My initial thought was to use a Beta prior. Is this right? I found the likelihood but I'm not sure ...
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Posterior as prior for correlated parameters [closed]

I want to use the posterior distribution of the model parameters $\theta$ given data in the time frame $[0,t]$ days, $P(\theta|y_{0:t})$; as a prior for the parameters in the time frame $[t+1, t+n]$ ...
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Significance of parameterisation invariance of Jeffreys prior

I often hear it said that the Jeffreys prior is well-motivated because it is invariant under reparametrization. The proof of this is quite straight-forward (I know the proof on e.g., wiki). I'm a bit ...
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Explanation of the Posterior Derivation of the Gaussian Distribution

I'm reading through my notes and I don't quite understand this bit: I understand how the likelihood was calculated but no more than that.Can anyone explain the steps and exactly how they go from one ...
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Formal Bayesian justification of conditional modelling

I'm having some trouble following the logic of this passage from Chapter 14 in Bayesian Data Analysis, A. Gelman: The numerical 'data' in a regression problem includes both $X$ and $y$. Thus, a ...
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Independence test with priors

Say you are selling a product, and you know from experience that the green version of the product sells better than the blue version. But you have two types of customer A and B, and you want to know ...
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Constraints on choice of marginal distribution and likelihood

For some time I have been reading into Bishop's Pattern Recognition and Machine Learning. Coming back to some earlier chapters the following got me confused and I am interested where, formally I go ...
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1D Bayesian Inference clarification

I'd like some help making sure I understand a 1D Bayesian inference problem. Say I have a data vector which is an array of the number of flu cases reported weekly in California for the past 10 years. ...
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Crew selection: ranking rowers by letting them race against each other

Seat selection is a common practice in competitive rowing and I would be curious about more solid statistical underpinnings: there are more rowers in a team than the 8 seats in the crew boat. So the ...
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How do Bayesian hierarchical models adaptively learn the prior?

It seems the main difference between a hierarchical and a non hierarchical model is that the hierarchical model learns the prior. That is it adaptively comes up with a regularizing prior to be applied ...
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Automatic selection of plot bounds for Normal pdfs in a combined chart

Suppose an app's user specifies several Normal priors by setting the mean and variance for each. I'd like to display a combined plot with the prior pdfs, like that on the figure. . Q: What is a good ...
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How to set a Bayesian prior on a set with a large but unknown number of elements?

Let us suppose that we are trying to analyze a given starfish. We would like to know which species does the starfish belong to. We have a list of 1000 starfish species, but we know that there is an ...
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Compute conjugate prior from the sample distribution

I feel like this question might be marked as duplicate because I see many similar incurring in that fate but I'll try anyway. I would say I did not find anything similar. I have been thought a ...
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What is the posterior distribution of a Bernoulli prior that gets updated with a continuous uniform signal?

I'm trying to figure out what the distribution of the posterior is after I update a Bernoulli prior with a continuous uniform signal, say: P(D=G|u)=x where D{G,I} and u is uniformly distributed on ...
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How does L2 penalize large weights

The L2 regularization term is useful because it penalizes large weights over smaller weights which is good to prevent overfitting. I'm having a hard time understanding how exactly it does this. This ...
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In a Hierarchical Bayesian Model, how can we sample and see how a prior distribution looks like if it contains hyperparameters with hyperpriors?

I have a Bayesian Hierarchical Model that looks like: \begin{equation} Y_i \sim N(\mu, \sigma^2) \\ \mu \sim N(\mu_0, \sigma_0^2) \\ \sigma^2 \sim Gamma(1,1) \\ \mu_0 \sim N(0,1) \\ \sigma_0^2 \sim ...
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Valid highly informative prior for proportion

I am trying to find a prior distribution for a proportion $\theta$ that is highly informative i.e. it is almost point mass at $\theta$ but I am not able to find such distribution that is valid for a ...
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In a Multi-level Bayesian Hierarchical Model, would higher level parameters be affected by how they are jointly modeled in lower levels?

Suppose we have a Multi-level Hierarchical Model where: $$ \begin{equation} Y_{0i} \sim Bin(\theta_{0i}, n_{0i}) \\ Y_{1i} \sim Bin(\theta_{1i}, n_{1i}) \\ \theta_{0i} \sim Unif(0,1) \\ log\left(\...
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Plot Scaling Problem

I am trying to implement the following code in R $\theta \sim Beta(a=1,b=1)$ $x|\theta \sim Bin(N=5,\theta)$ $\theta|x \sim Beta(a+\sum x_{i},b+\sum N-\sum x_{i})$ ...
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How many missings are too many to impute data with the AMELIA II package?

I have a very large longitudinal dataset. I only want to impute 3 variables individually using the AMELIA package in R. The problem is that some individuals have a lot of missing values, so I want to ...
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Understanding the Bayesian question [closed]

I have a Bayesian question here: In a drug experiment, patients with a chronic condition are asked to choose between two drugs, C (control), and T (new treatment). (You may assume for the purpose ...
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125 views

Learning prior distribution from data

Suppose I have a dataset. How can I learn the prior distributions of the parameters of a model from this data? I want to learn the prior from this data in order to use them in a Bayesian model. Sorry ...
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Is this Bayesian model averaging?

A classical example of Bayesian model averaging (BMA) is the regression setup where the choice of different sets of covariates corresponds to different models $\mathcal{M}_k$, $k = 1, \ldots, K$, ...
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Finding mode of posterior using Newton method in R

I am attempting to approximate the posterior $\tilde{\pi_{G}}(z|\theta,Y)$ which is the Gaussian approximation to the full conditional of $z$, and in order to do this I need to find the mode $z^{*} \...
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Best estimate of underdetermined system using prior

I have measured two variables which depend on the same set of four parameters. I want to know the parameters which best explain my measurements. Of course, I cannot solve for four unknowns from just ...