# 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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### 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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### What's a good prior distribution for degrees of freedom in a t distribution?

I want to use a t distribution to model short interval asset returns in a bayesian model. I'd like to estimate both the degrees of freedom (along with other parameters in my model) for the ...
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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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### Multiple priors in Bayesian estimation

Typical Bayesian estimation equation is: Estimate = ( SampleSize * SampleEstimate + PrioriEstimateWeight * PrioriEstimate) / ( SampleSize + PrioriEstimateWeight ) Typically, the PrioriEstimate is ...
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### Would a posterior distribution with a flat prior look identical to the likelihood?

Graphically, let us assume that we have a flat prior for a normal distribution (a horizontal line at y=1 over all real numbers). Then, we have a likelihood function that resembles a normal ...
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### Putting prior on a function of parameters

Suppose that we have a likelihood for a conditional distribution $p(y|X,\theta)$. For clarity purposes we can consider linear regression with homescadastic errors. It is clear to me how one will put a ...
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### How to formally express low confidence in data in Bayesian framework?

I am trying to build a Bayesian regression model, and I am looking for a formal way to express low confidence in noisy data. I know that should be reflected in the prior specification.. I am using ...
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### Posterior distribution dependent on two variables make inferences about one

If I have some model for X that depends on $\theta_1, \theta_2$ and has a posterior $P(\theta_1, \theta_2 | x_1, ... x_n)$, how would I make inferences just about $\theta_1$? What I am thinking so ...
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### Prior, Posterior and Bayes rule for discrete random variables. Calculating Posteriors?

For the discrete case in the image below below, could someone explain why a density, $f(x)$, is used rather than a pmf, $p(x)$. My notes say that, for most cases the value of the parameter takes ...
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### Covariance of two random variables with random parametres with a prior distribution

Let $N_1, N_2$ be conditionally independent random variables with probability distributions $Pois(t_1\theta)$ and $Pois(t_2\theta)$ respectively. Constants $t_1, t_2$ are known. The parameter $\theta$ ...
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### Bishop: Understanding the prior and posterior for a curve fitting example (1.2)

In Bishop's Pattern Recognition and Machine Learning Book, he uses an example of fitting a polynomial to data collected from a sinusoidal curve with Gaussian noise. The goal is to find the most ...
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### half-cauchy prior for scale parameter

I am looking for a prior for a scale parameter for which I have prior knowledge such that: "$\sigma$ typically does not exceed 100." ("typically" meaning that occasionnally this can happen). In such ...
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### Deriving Marginal Distribution of Poisson [duplicate]

How do you find the marginal distribution of a Poisson distribution given a gamma(a,b) prior?
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### Informative priors from frequentist regression

Say I run a standard frequentist regression on a subset of my data (for example using lm in R) and obtain some values for the coefficients of the model, I have fairly large data set ~100k samples. Now ...
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### Quantify the output variance of a neural network classifier

Lately at work we are dealing with a theoretical problem concerning the output variance of a neural network classifier. To set the scene, suppose you have an image classifier, which takes an image as ...
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### Accounting for uncertain information (few observations) in a prior (empirial Bayes)

I did not really know how to choose an adequate title for this question, so please feel free to change it. I have a weird case wherein frequentist and Bayesian philosophies come together. I am ...
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### How to Change prior probabilities for predicted variable in neural networks and other methods in SPSS Statistics

i am trying to find right model for predicting categorical variable with two values. Problem is that ratio of cases in group 1 and group 2 is not equal but rather in ratio of 2:1. When i try to find a ...
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### What are the best ways to generate Bayesian prior estimates using beliefs of non-statisticians?

I work with a lot of qualitative researchers and designers. Many of whom interact with users and develop strong, often accurate intuitions about how the data should look. I frequently try to quantify ...
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### Sampling a proposed value with a limited range target when running MCMC [duplicate]

I want to do an MCMC algorithm and need to sample a proposed value from a proposed distribution. In the Metropolis algorithm, people usually use a normal distribution as proposal. But if the prior ...
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### Linear regression - Bayesian Predictive distribution

I am trying to answer a question about linear regression but i am stuck: $y=w \cdot x + \epsilon, \epsilon \sim N(0,\alpha)$ i am also given a prior: $w\sim N(0,\beta)$ from which i was able to ...
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### Random-walk prior with ridge-like regularizarion?

I am working with a model that contains a large number of coefficients, arranged in an ordered vector $\beta_1, \dots, \, \beta_N$. I have some prior knowledge that could be used to improve the ...
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### Clarifying a proof of a particular paper on Steins Estimator

I am trying proving result (5.4) of the following paper. Its a paper on Steins estimator on spherically symmetric cases. The doubt is a s follows: Given $$X|\theta\sim \mathcal{N}(\theta,I)$$ ...
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### What prior would lead to $\ell_\infty$ regularization of model weights?

Gaussian prior on weights of a GLM lead to Ridge / $\ell_2$ squared regularization. Laplace prior leads to $\ell_1$ regularization Question What prior would lead to $\ell_\infty$ regularization ?
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### Bayes-Poincaré solution to the Behrens-Fisher problem 2: calculations for Jeffreys’ priors [closed]

In a previous post Bayes-Poincaré solution to k-sample tests for comparison and the Behrens-Fisher problem?, the classical Bayesian and likelihoodist solutions to 2-sample tests for comparison and the ...
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### Jeffreys prior for continuous uniform distribution

A nonnegative random variable $x$ has a continuous uniform distribution in the interval $(0,\theta)$. Therefore, the likelihood is given by: $f(x|\theta) = \frac{1}{\theta}I(x\leq\theta)$, where $I$ ...
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### How to select variables when using shrinkage priors?

I am fitting a linear regression model using shrinkage priors (Horseshoe and Laplace/LASSO). This shrinks many of the variables close to zero, but I would like to select the variables. Can I use the ...
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### Why is Half-Cauchy, Half-Student-t as prior for variance parameters better than a normal distribution?

Gelman often refers to using half-cauchy or half-student-t distributions for variance parameters. Why is it better than using a vague normal distribution such as N(0,10)? Can somebody explain me the ...
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### Why in Hamiltonian MCMC do we multiply the posterior distribution by the likelihood?

So maybe I am misunderstanding what the author is staying, but I am reading Chapter 14 of Kruschke's Doing Bayesian Analysis. I am reading about the software Stan and how it uses the Hamiltonian MCMC ...
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### What kind of a priori distribution for the Markov Switching models?

Why in the Markov-Switching models is chosen as prior distribution for the probability of the transaction as follows: f(P) \propto \prod_{i=1}^K \left(\prod_{j=1}^K p_{i,j}\right) I \left\{0 < ...
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### Uniform vs Beta(1,1) prior

Is there any difference in applying a uniform prior or a Beta(1,1) prior for your Bayesian analysis ?In which conditions is one preferred over the other ?
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### Is the inductive bias a prior?

Wikipedia defines it like this: The inductive bias (also known as learning bias) of a learning algorithm is the set of assumptions that the learner uses to predict outputs given inputs that it has ...
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### What do these equations on Bayesian regression (MAP) from Chapter 3.3 in PRML by Bishop mean?

This was taken from Ch 3.3 on Bayesian Linear Regression from Pattern Recognition in Machine Learning by Bishop. Apparently the posterior can be described by eq 3.49. Eq 3.48 represents the prior ...
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### Explanation of Equation 5.3 from Gaussian Processes for Machine Learning

I am currently reading through C. E. Rasmussen & C. K. I. Williams' Gaussian Processes for Machine Learning and was going through chapter 5. I could not exactly understand the derivation of ...
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### Shrinkage priors

I am building a Bayesian model where I to put shrinkage priors such as spike and slab, horseshoe prior, etc on some parameters for feature selection, but I am not able to decide which one is the best. ...
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### Bayesian prior via cross validation

I have a particular problem where I am using Bayesian techniques to estimate parameters of a distribution of a random variable. I would like to use an external source of data to determine an ...
I am trying to solve the following problem. If $y | \beta \sim N(X \beta, I_n)$ and $\beta \sim N(0, g^{-1}(X^t X)^{-1})$ for $g>0$. Find $\pi(\beta|y)$ and show that $E(\beta|y)$ is a function ...