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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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Defining custom Bayesian priors in R (BayesFactor package)

I'm performing some Bayesian analyses in R using the BayesFactor package, and was wondering whether it is possible to specify priors for the alternative not centered on zero (the current defaults ...
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19 views

Prior predictive distribution usage

I understand the mechanics and math behind prior predictive distributions, but I don't understand its practical uses. Theoretically and application wise, what is its purpose?
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12 views

What's the role of the scale matrix for the Inverse-Wishart and Wishart distributions?

What's the role of the scale matrix for the Inverse-Wishart and Wishart distributions? The purpose of finding this information is to enlighten me on how should one decide on a prior for a positive-...
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1answer
141 views

Joint posterior distribution of $(\mu,\sigma^2)$ in the Normal model

Find the joint posterior of $(\mu, \sigma^2)$ given Normal data. I've found the joint prior of $\mu$ and $\sigma^2$ (where $\displaystyle\sigma^2\sim\chi^{-2}(v_o,v_os_o^2)$ and $\mu|\sigma^2\sim N(\...
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21 views

Gaussian Processes, Prior function of constant, Linear and Polynomial Kernel

In my university course slide, regarding the exponential and Gaussian kernel there are these two relevant pictures. They show the prior function of each kernel: You can see that the resulting prior ...
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1answer
30 views

Gaussian Processes, basic question about how the prior is computed

I'm approaching the topic of GP, and I have a question regarding how functions are sampled. On my textbook is stated that to represent a distribution over a function (the prior): we only need to ...
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1answer
17 views

Prior/degree of belief/degree of lack-of-information/algorithms/complexity

For a long time I had a bit of difficulty understanding what "degree of belief" means. Recently I had some thoughts about it and I wonder if they make any sense, or is there some literature about ...
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1answer
72 views

Likelihood raised to a power; how to set the power?

Suppose ${\bf{\theta}} = (\theta_1 , \ldots, \theta_d)$ and you have a posterior as below: $$\pi(\theta | D ) \propto L(\theta |D ) \pi(\theta)$$ Suppose we are in active learning setting and need ...
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2answers
42 views

Why is the normal distribution a default choice for a prior over a set of real numbers?

I found this question without any context while preparing for a job interview. The only reason I can think of is that the Normal distribution's support are the real numbers. Why wouldn't we use a ...
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13 views

Bayesian approach for exponential and Cox proportional hazard model in R

I have difficulty understanding and applying the Bayesian approach to survival analysis. I assume that data follow exponential life time likelihood and that failure rate lambda can be expressed in ...
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1answer
37 views

How to justify using Beta distribution as a prior distribution in the following problem

Let $\theta$ be the proportion of people who are ready to quit smoking within 6 months. Let's say we perform a survey in $2017$ with a $n$ volunteers who ask people this question until they obtain yes ...
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1answer
20 views

What prior distribution of parameters in the Bayesian estimation of a GARCH model?

In the case of the Bayesian estimation of GARCH(1, 1) model with Student–t or a Skewed distributions for innovations, is it more correct to assume a uniform distribution for the parameters or to ...
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1answer
15 views

What is the interpretation for the priors in the derivation of Laplace smoothing?

Laplace smoothing has a generalisation that can be justified with the use of Bayes formula. Let $f(x;\alpha,\beta)$ be the (non-normalised) beta distribution, i.e. $$f(x;\alpha,\beta) = x^{\alpha-1}(...
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1answer
27 views

How to show that $\frac{1}{\theta}$ is a flat prior for $\log\theta$? [duplicate]

I do not really understand what the statement even means. To my understanding, a prior $p(\theta)$ is said to be flat if $p(\theta) =$ constant $\forall \theta,$ where $p(\theta)$ is the prior ...
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36 views

Biassing priors to improve confusion matrix

I have a text classification problem to solve. I need to classify a given sample of text into one of two classes A or B. My training set has about 30% A and 70% B. This is my prior. Now, when I ...
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15 views

Incorporating prior knowledge into feature selection in the setting of multicollinearity?

Background: I'm trying to find the optimal combination of two parameters for finding the first peak meeting some criteria in a signal. The filtering is a bit simplistic, there's a threshold (...
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1answer
42 views

Ockham's Razor in Bayesian Modelling

This question might be a little philosophical / generate discussion. I hope, there may still be some useful answers. I am currently thinking about how Ockham's Razor relates to Bayesian statistical ...
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1answer
80 views

Binomial distribution as likelihood in Bayesian modeling. When (not) to use it?

I am currently trying to figure out some strangeness about using the Binomial distribution in Bayesian modeling to define the likelihood. To make an example assume I have two conditions, and in each ...
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2answers
82 views

Bayesian approach to report simulation studies?

I am running a simulation study where we want to estimate a proportion $p$. We are reporting the coverage of credible intervals with a uniform prior, and we are doing $500$ Monte Carlo simulations. We ...
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1answer
34 views

How to derive the noninformative prior for location parameters and scale parameter?

I am reading this paper, in it: I have a lot of confusion reading it, I will list it one by one: Let $X$ be distributed as $f(x-\theta)$, which is a location invariant density. Q1: The sentence <...
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30 views

Dirichlet distribution as conjugate prior to Multinomial distribution [closed]

I stumbled upon the following exercise: Show that Dirichlet distribution with parameter $\alpha$ is a conjugate prior to the Multinomial distribution as likelihood. Derive the parameters to the ...
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1answer
43 views

Understand VAE with VamPrior

I am currently reading this paper. The authors propose to use as an prior this expression: $$ p_\lambda(z) = \frac{1}{K} \sum^K_{k=1} q_\phi (z\mid u_k) $$ where $q_\phi$ is the encoder, and $u_k$ is ...
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Doing cumulative bayesian analysis in R: shouldn't results converge?

I'm in the process of designing a Bayesian simulation of a cumulative bayesian analysis. In short, I am attempting to use the posterior information from a t-test to provide the priors for a subsequent ...
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26 views

Objective Bayesianism: Jeffreys priors vs reference priors vs principle of transformation groups

According to this answer, José Bernardo has produced an original theory of reference priors where he chooses the prior in order to maximise the information brought by the data by maximising the ...
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1answer
40 views

Choice of Gaussian process in non-parametric regression

I have been trying to understand non-parametric regression using Gaussian processes (GP), which are used to represent prior distributions over the space of functions. The linear model considered is $$ ...
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17 views

What is the significance of conjugate joint prior for normal distribution?

For example, for a normal distribution N(μ,σ2), it is possible to place, say, a separate normal prior on μ and another separate IG prior on σ2, but what is the prrof for conjugate joint prior ...
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1answer
40 views

what do you mean by writing out the posterior distribution of µ up to a normalizing constant? [duplicate]

I think i am stuck at the basic what do you mean by writing out the posterior distribution of µ up to a normalizing constant ? How to compute the value of C here?
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1answer
106 views

Show posterior mean can be written as a weighted average of the prior mean and MLE

Suppose $Y_1, \dots Y_n$ are exponentially distributed: $Y_i | \lambda \sim Exp(\lambda)$. Find the conjugate prior for $\lambda$, and the corresponding posterior distribution. Show that the posterior ...
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1answer
179 views

Some Questions about reference measures and maximum entropy priors (from The Bayesian Choice)

I am relatively new to statistics and Bayesian theory but I am trying to understand it by working through a few books. There are some things I am confused about. ( As I believe my analysis and such is ...
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1answer
154 views

Conditional distribution for Gibbs sampling for Gaussian mixture

If we draw $n$ i.i.d. points $x_1,x_2,\dots,x_n$ from the following Gaussian mixture: $$ \frac 12 \mathcal N(x \mid \mu_1,1) + \frac 12 \mathcal N(x\mid \mu_2,1) $$ and the prior $p(\mu_1 , \mu_2 )$ ...
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2answers
86 views

Why are weakly informative priors a good idea?

There are many solutions to the problem that typically not enough information is available to fully specify a prior. For all approaches (I know) but weakly informative priors I kind of understand ...
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2answers
215 views

Gaussian mixture model - does an improper uniform prior give a proper posterior?

We draw $n$ i.i.d. points $x_1 , x_2 , ..., x_n$ from the following Gaussian mixture: $$p(x|\mu_1,\mu_2) = \frac{1}{2} \text{N} (x|\mu_1,1) + \frac{1}{2} \text{N} (x|\mu_2,1).$$ The prior is the ...
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1answer
18 views

Predictive classification including varying information about classes

I'd appreciate help conceptualizing a problem. I constructed a supervised training set where user inputs carry the correct classification. In building a classification model, I'll strip out the ...
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1answer
47 views

Applying Bayesian Gaussian movement question

I have a question from my stats class that I am confused about how to proceed with. I have a general idea of what I am to do but I am not sure how to start. The question is about a car that is moving ...
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1answer
44 views

Why Does the $\propto$ Symbol Replace the $=$ Symbol When Using Bayes' Rule to Convert Posterior Density to Unnormalised Posterior Density?

My textbook says the following: In order to make probability statements about $\theta$ given $y$, we must begin with a model providing a joint probability distribution for $\theta$ and $y$. 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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1answer
41 views

Metropolis algorithm to Bernoulli likelihood and beta prior (Kruschke 7.3.1)

This question pertains to a specific line written in the book Doing Bayesian Data Analysis by John K. Kruschke. In section 7.3.1, he applies Metropolis algorithm to a case with: $prior = beta(\...
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55 views

Setting up the priors for Bayesian Multilevel Multinomial Mixed Model using BRMS

I am new (understatement!) to Bayesian statistics in general and to brms(). More specifically, I am confused as to how to specify the priors for nested random effects in a multinomial mixed model. ...
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1answer
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When is the posterior distribution equal to the prior?

So I have heard that if the prior distribution is in the subexponential class, applying Bayes rule does not change the belief. I have been trying to find an example of this but I am unable to do so. I ...
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478 views

Can a proper prior and exponentiated likelihood lead to an improper posterior?

(This question is inspired by this comment from Xi'an.) It is well known that if the prior distribution $\pi(\theta)$ is proper and the likelihood $L(\theta | x)$ is well-defined, then the posterior ...
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1answer
46 views

Diffuse priors Bayes Factor

In textbooks I always read that it is necessary to have a proper prior on the parameter that we want to test with Bayes factor, otherwise we would always posteriori favor the model with less ...
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1answer
65 views

What is the conjugate prior distribution? [duplicate]

I am new to the bayesian statistics and I most frequently see the conjugate prior distribution. Can you explain it with clear example? I would be very thankful.
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1answer
236 views

MAP estimation as regularisation of MLE

Going through the Wikipedia article on Maximum a posteriori estimation, it got confusing after reading this: It is closely related to the method of maximum likelihood (ML) estimation, but employs ...
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0answers
23 views

Updating a Normal Prior with Realized Data

So I am trying to remember how Bayesian updating works and reading the Wikipedia page on conjugate priors. I'm reading a lot about how either variance or the mean must be known. In what cases of ...
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1answer
123 views

Deriving Posterior Binomial Density from Uniform Prior

I'm trying to derive the posterior density of the probability parameter of a binomial random variable, given one realization of the random variable and a uniform prior density on the probability ...
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16 views

Uniqueness on bayesian factor model's loading matrix

I'm doing uniqueness on factor loading matrix in a factor model. $ y = \Lambda f + \epsilon$ where $ f \sim N(0,\Sigma)$ , $\epsilon \sim N(0,\Omega) $ and $\epsilon \perp f$. It's well known that ...
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47 views

Restriction in bayesian likelihood expression

I'm doing MCMC simulation but I'm confused in some part of my model. I dont know which of my likelihood expression is right. My model is as following. $\gamma$ = $(\gamma_{1},\cdots,\gamma_{K})$ and ...
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1answer
151 views

Bayesian Statistics. Please help me to find an example where posterior variance is greater than prior variance

Suppose we observe y successes in n trials where the probability of success in each trial is θ. If we choose a Beta(1, 1) (Uniform) prior then the posterior variance will be smaller than the prior ...
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Moments of the horseshoe prior?

Are the first two moments well defined for the horseshoe prior? I would say that the expectation is zero but the variance does not exist. Using the following argument. Let $$\beta_i \mid \lambda_i, \...
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1answer
26 views

constrained least squares by fixed ratio in the coefficients

For multiple least squares linear regression, can we actually specify a ratio between the coefficients as the prior? For example, for following linear model: $y = b_1*x_1 + b_2*x_2 + b_3*x_3$ ,can we ...