All Questions
Tagged with prior improper-prior
18
questions
9
votes
2
answers
426
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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 ...
2
votes
0
answers
28
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Distribution families whose likelihoods integrate to $+\infty$ for some sample values
I've recently started learning about Bayesian statistics, and I came across this very nice answer by Xi'an https://stats.stackexchange.com/a/129908/268693, which [in my slight paraphrasing] says the ...
2
votes
1
answer
209
views
For multivariate normal posterior with improper prior, why posterior is proper only if $n\geq d$
This is related to Gelman's BDA chapter 3 section 5's noninformative prior density for $\mu$.
Let $\Sigma$ be fixed positive definite symmetric matrix of size $d$ by $d$. Let $y_1,\dots, y_n$ be iid ...
1
vote
1
answer
107
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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 ...
0
votes
1
answer
42
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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 ...
1
vote
1
answer
317
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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) = \...
0
votes
1
answer
102
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Showing that a posterior is Normal given improper prior
I am having difficulty showing the following problem and I suspect it has something to do with my lack of understanding of the question. The question is this:
Suppose we have an improper prior ...
3
votes
2
answers
2k
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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 ...
2
votes
1
answer
2k
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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 ...
4
votes
1
answer
212
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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 ...
4
votes
2
answers
1k
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 ...
5
votes
2
answers
5k
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Haldane's prior Beta(0,0) - Part 1
This article$^1$ on p.16 specifies Haldane's prior as:
$$p(\theta) = \frac{1}{θ(1−θ)}$$.
However, other$^2$ source on p.6 specifies Haldane's prior as proportional to $\frac{1}{θ(1−θ)}$, i.e.
$$p(\...
1
vote
1
answer
1k
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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
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Difference between non-informative and improper Priors
I wonder what is the difference between these two kind of priors:
Non-informative
Improper