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# Questions tagged [improper-prior]

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2 answers
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### How is data generated when using an improper prior

Let $X$ be an $\mathcal{X}$ valued random variable. We are doing Bayesian statistics. Suppose that $\theta$ is a $\Theta$ valued random variable with known prior distribution $\Pi$ and that the ...
• 456
1 vote
0 answers
15 views

• 699
3 votes
2 answers
2k views

### 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 ...
• 655
2 votes
1 answer
2k views

### 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 ...
3 votes
2 answers
223 views

### Can an improper prior distribution be informative?

I have just worked through an example where, with an improper prior, the bayesian estimator equals the maximum likelihood estimator, leading me to believe that improper priors are uninformative. But ...
• 1,276
1 vote
1 answer
1k views

### Finding the posterior distribution given an improper prior

Let $X \sim N(\theta, \sigma^2)$ where $\sigma^2$ is known. Let the prior density $\pi(\theta) =1, \theta \in \mathbb{R}$ to be the improper uniform density over the real line. Find the posterior ...
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3 votes
1 answer
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• 571
2 votes
1 answer
304 views

### Compute $\pi(H_0|x)$ with Jeffreys prior for a family $N(\theta,1)$

Given a random sample $x = (x_1,\ldots,x_n)$ taken from a family $\{N(x|\theta,1):\theta \in \mathbb{R}\}$. And consider the hypothesis test: $H_0: \theta = 0$ vs $H_1: \theta \in \mathbb{R}$ (this ...
• 133
2 votes
2 answers
258 views

### How are does software compute posterior distributions from improper (flat) priors?

In Bayesian statistics, how do software packages compute the posterior distribution when the prior is improper (flat)? If I understand correctly, this can't be done analytically so how is it done ...
• 2,176
1 vote
1 answer
1k views

### Bayesian Biased Prior Formula

I know that for a Bayesian uniform/flat prior, the formula is 1/n (and n=1), as each value has an equal chance of being chosen. However, is there an equation for when the prior is biased/informative? ...
10 votes
2 answers
4k views

### Difference between non-informative and improper Priors

I wonder what is the difference between these two kind of priors: Non-informative Improper
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2 votes
0 answers
260 views

### Understanding my posterior with an uninformative prior with a poisson likelihood. Am I thinking about this correctly?

I have a problem to which I am trying to apply a Bayesian model. My data is generated as follows \begin{align} N_i \mid \mu &\sim \text{Poisson}(\mu) \\ Y_i \mid N_i, \theta_i &\sim \text{...
• 1,029
5 votes
1 answer
82 views

### Bayesian hierarchical linear model with uninformative priors

I have the model $y_{i,t}=x_{i,t}'\beta_{i} + \epsilon_{i,t}$ where $x_{i,t}$ is a k-dimensional vector of explanatory variables and $\beta_{i}$ is a k-dimensional parameter vector, where \$\beta_{i}=\...
• 521
17 votes
2 answers
3k views

### Sampling from an Improper Distribution (using MCMC and otherwise)

My basic question is: how would you sample from an improper distribution? Does it even make sense to sample from an improper distribution? Xi'an's comment here kind of addresses the question, but I ...
• 15.8k
5 votes
2 answers
2k views

### Does "improper" posterior or prior refer to a density function that does not integrate to 1 or to one that does not integrate to a finite value?

I am a bit confused about improper priors and posteriors. I have seen references that classify a prior or posterior probability density function as "improper" if the integral over infinite support ...
• 2,415
2 votes
2 answers
368 views

### How to choose a importance density for Jeffreys prior?

I want to draw Bayesian inference via importance sampling and I do not come up with a good idea of an importance density for $$p(\sigma)\sim\frac{1}{\sigma}.$$ Is there a way to sample from this ...
• 1,330
12 votes
1 answer
638 views

### When should I be worried about the Jeffreys-Lindley paradox in Bayesian model choice?

I am considering a large (but finite) space of models of varying complexity which I explore using RJMCMC. The prior on the parameter vector for each model is fairly informative. In what cases (if any)...
• 1,057
13 votes
1 answer
5k views

### Bayes factors with improper priors

I have a question regarding model comparison using Bayes factors. In many cases, statisticians are interested on using a Bayesian approach with improper priors (for example some Jeffreys priors and ...
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