I need a good reason why improper uniform prior could be use as a prior in Serial Numbered Population (SNP) problem. Or maybe someone can tell me about improper uniform prior itself. An pdf link might be very helpful for me.
I attached the journal.
Hi, most Bayesian textbooks will discuss this. Check for example Bayesian Data Analysis by Gelman and coauthors. This book is one of the most cited and any uni library will have a copy. If you want to look for docs online, look for "Jeffreys prior improper", depending on your background you may read what you find accessible. I just did a brief search query and found for example a link to this doc: http://isites.harvard.edu/fs/docs/icb.topic353981.files/unit2.pdf
Good luck
An improper prior can be used when the resulting posterior is proper. The uniform distribution is an "uninformative" prior for a location parameter of a continuous variable (it is obtained by the Laplace postulate as well as the Jeffrey's prior as well as the reference prior determined by the method of Berger-Bernardo).
"Improper" simply means that the distribution does not integrate to 1, so it is actually not a probability distribution. The uniform prior can be seen as the limiting case of the normal distribution for the variance going to infinity -> the shape of the posterior simply equals the shape of the likelihood, the normalization makes the posterior have a unit integral. Thus, based on the uniform prior, the confidence interval for a location parameter (determined from the likelihood) equals the central credibility interval (excluding similar abounts of creadability at either side from the posterior).
See the link for further details.
http://www.ime.unicamp.br/~veronica/ME705/paper2.pdf
This is a simple answer!
In Bayesian statistics, the prior is the probability distribution that expresses uncertainty before taking evidence (data) into account. A uniform distribution means that any value is equally likely – that is you have little strong prior belief about the size of the population – you are letting the data and the model (and its assumptions) largely determine the result.
This distribution is improper because it has an infinite integral and is really not a probability density at all. However, I presume that when combined with the data via the likelihood it provides a proper posterior distribution of the parameter – the estimated size of the population (except it appears when n = 1). The innovation here is that they are able to derive a closed formula for its calculation that gives the result without fancy techniques like MCMC simulation.
A good answer by Kelvin (good to meet you here online - I took once a class from you).
I would like to add that even though diffuse priors like a uniform prior on a standard deviation parameter in a hierarchical model have attractive features, there may be some drawbacks as well. In the example i just gave, there is a known issue of miscalibration.
One could also specify a uniform distribution over the log od standard deviation parameter but then we obtain an improper posterior.
All these issues are discussed in detail in this seminal article by Gelman: http://projecteuclid.org/download/pdf_1/euclid.ba/1340371048
I'm not sure if Jochen's answer is entirely correct; "An improper prior can be used when the resulting posterior is proper." I'd modify it to say that an improper prior can be used when the resulting posterior can be achieved with a *proper* prior and the limits of the distribution, after the posterior is calculated, is taken to infinity and the result is shown to be the same. For example, you can use a proper, bounded uniform prior and then, after you calculate the posterior, take the limits as the bounds grow to infinity to show that if you used the improper prior in the first place it would give the same answer.
Seen this way, the improper prior is a short-cut to make the analysis cleaner. For example, I personally find E T Jaynes' analyses very clear because he uses improper priors (when he can!), and warns that one has to be careful to do it properly if there is even a hint of trouble. I find a paper like Bretthorst's "Difference of Means" paper to be a much harder read because he uses proper priors throughout, even in cases where I think the improper prior would work.
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