Questions tagged [improper-prior]

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2
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1answer
71 views

Integrating out parameter with improper prior

I got this problem while I was reading the book "Machine Learning: A Probabilistic Perspective" by Kevin Murphy. It is in section 7.6.1 of the book. Assume the likelihood is given by $$ \begin{...
7
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3answers
349 views

verifying a posterior is proper

There's a homework problem in a textbook that asks to verify propriety of a certain posterior distribution, and I'm having a little trouble with it. The setup is you have a logistic regression model ...
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0answers
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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 ...
3
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2answers
358 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 ...
0
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0answers
75 views

Nonparametric Bayesian estimation of several black-box functions of different variables from their noisy sums

In order to introduce my problem, let’s start with the nonparametric estimation of a single unknown/black-box function $f:{\Omega _f} \to \mathbb{R}$ of a discrete variable $x$ in a finite domain ${\...
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0answers
24 views

Is there any reference about the almost sure convergence of a posterior distribution to the posterior with non-informative prior?

I'm trying to show $$ \pi(\theta_n|X) \overset{a.s.}{\to} \pi(\theta|X) $$ where $\theta_n \sim N(0, n)$, and $\pi(\theta) = 1$(improper), and $X$ is normal. Is there any reference, or hint to ...
1
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2answers
818 views

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(\...
2
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1answer
67 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 ...
2
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2answers
137 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 ...
0
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1answer
105 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? ...
9
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2answers
2k views

Difference between non-informative and improper Priors

I wonder what is the difference between these two kind of priors: Non-informative Improper
2
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0answers
115 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{...
15
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2answers
1k 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 ...
3
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2answers
970 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
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1answer
141 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 ...
12
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1answer
308 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)...