Statistical Analysis Stack Exchange
Statistical Analysis Stack Exchange

Questions tagged [inverse-gamma-distribution]

The inverse gamma distribution is a right-skew, continuous distribution for a random variables taking positive values.

The inverse gamma distribution is a right-skew, continuous distribution on the positive half of the real line.

It is the distribution of the inverse (reciprocal) of a random variable which is gamma distributed. Explicitly, the density function takes the form:

$$p(x;\alpha,\beta) = \frac{\beta^\alpha}{\Gamma(\alpha)} x^{\alpha-1} \exp \left( -\frac{\beta}{x}\right)$$

It is often used as a prior distribution used in Bayesian statistics. For example, it is a conjugate prior for the variance parameter of a normal distribution.

Reference: Wikipedia on Inverse Gamma

76 questions
11
votes
2 answers

What kind of distribution does this have?

Currently I'm trying to figure out the distribution of the following: $X \sim \frac{\sqrt{n}}{\sqrt{Gamma(n,\beta)}}$ where the denominator follows a $Gamma(n,\beta)$ distribution. I've checked out these links: Square root of an inverse gamma…
BonnieKlein
  • 391
  • 1
  • 7
7
votes
1 answer

Weibull distribution with the negative shape parameter

Just wondering why in the literature Weibull distribution is always defined for positive shapes, whereas the extension in the negative direction is possible and has many useful properties. Suppose $X \propto \mathrm{Weibull}(\theta, \lambda)$, i.e.…
alyst
  • 71
  • 2
7
votes
2 answers

Relationship between inverse gamma and gamma distribution

I have the following posterior distribution for $v$ $$f(v)\propto v^{-p/2}\exp\left(-\frac{1}{v}\frac{s}{2}\right)$$ and so clearly $$v\sim\text{Inverse-Gamma}\left(\frac{p}{2}-1,\frac{s}{2}\right)$$ Now can I say that…
user30490
7
votes
1 answer

What does inverse-chi-square in Fisher method (classifying) exactly do?

In the book “Programming Collective Intelligence” Segaran explains the Fisher method for categorizing text as an alternative to Naive Bayes classifier. The Fisher method uses inverse-chi-square-distribution, which I do not really understand. I…
aufziehvogel
  • 203
  • 1
  • 5
6
votes
2 answers

How to determine the marginal pdf, the posterior?

How to get the marginal pdf of $p(y)$? Do you just integrate out $p({\sigma}^{2})$? Say, the following joint distribution for $y \in {{R}^{d}}$ and ${{\sigma }^{2}}\in {{R}^{d}}$ IG: means inverse Gamma $${{\sigma }^{2}}\sim IG(\alpha ,\beta…
user1061210
  • 1,005
  • 3
  • 13
  • 19
6
votes
1 answer

Sampling from an Inverse Gamma distribution

I am using Gibbs sampling in the MCMC estimation of a stochastic volatility model. One of the posterior distributions is an Inverse Gamma distribution.I was struggling with the sampling procedure or to be precise with the link to the Gamma…
mscnvrsy
  • 515
  • 6
  • 15
5
votes
1 answer

Why do we use inverse Gamma as prior on variance, when empirical variance is Gamma (chi square)

Let $$X_i\sim \mathcal{N}(0,\sigma^2)$$ than we know that $$\sum_{i=1}^N\frac{X_i^2}{N}\sim\Gamma(\frac{N}{2},\frac{2\sigma^2}{N})$$ that the empirical variance follows a Gamma distribution. How do we reconcile this, with the fact the the inverse…
LeastSquaresWonderer
  • 223
  • 2
  • 6
5
votes
1 answer

Expectation of Sum of Gamma over Product of Inverse-Gamma

Let $X_1, X_2, \cdots, X_n \sim Gamma(\alpha, \beta)$. How do we compute $E\left(\cfrac{\sum_1^n X_i}{(\prod_1^n X_i)^{1/n}}\right)$ ? I am stuck on how to compute this expectation. I know that $\cfrac{1}{X_i}$ follows inverse-gamma but how to we…
t-student
  • 103
  • 3
5
votes
1 answer

MCMC for infinite variance posteriors

My question originated from Xi'an's suggestion to check integrability against the posterior in my nonlinear hierarchical model. I did not check it, but had possible infinity in mind and found out that one of my conditionals (which is inverse gamma…
Tomas
  • 1,133
  • 8
  • 16
4
votes
1 answer

Posterior distribution of linear regression with normal and inverse gamma prior

If I have the following model: $$y\sim N_n(X\beta, \sigma^2 I_n)$$ with prior distributions: $$\beta\sim N_n(\beta_0, B_0)$$ and $$\sigma^2 \sim IG(\alpha_0/ 2, \delta_0/2)$$ What would be the posterior of $\theta=(\beta,\sigma^2)$?
user2246905
  • 203
  • 2
  • 8
4
votes
2 answers

What is the distribution of norm induced by an inverse Wishart?

Suppose $S$ is distributed as a Wishart matrix with $n$ degrees of freedom and scale matrix $\Sigma$, and let $\vec{a}$ be a fixed vector. It is well known that $\vec{a}^{\top}S\vec{a}$ is equal to $\vec{a}^{\top}\Sigma\vec{a}$ times a Chi-square…
shabbychef
  • 10,388
  • 7
  • 50
  • 93
4
votes
1 answer

Is there an analytic distribution for the sum of random variables distributed IID inverse gamma?

How about their ratio? I have looked at the related distributions section on Wikipedia and tried playing with the pdf's by hand. I could have been a little more specific about the case that is currently of interest to me: Both inv-gamma…
rcorty
  • 438
  • 4
  • 13
3
votes
2 answers

Confidence Interval for Inverse Gamma Distribution

I would like to understand if there exists any method to find confidence interval for the parameters of inverse gamma distribution.
user44241
  • 31
  • 2
3
votes
1 answer

Jeffreys prior for inverse gamma distribution

Does anybody have the experience of dealing with Jeffreys prior? I am working with hierarchical model at the moment where the parameter σ^2 from normal distribution is said to be chosen according to inverse gamma distribution from Jeffreys…
Oleg
  • 611
  • 4
  • 14
3
votes
1 answer

Is a power of an inverse gamma random variable itself inverse gamma?

If $X$ is an inverse gamma distributed random variable, then would $X^p$ also be distributed as inverse gamma? I found someone asked about the square root of inverse gamma, but I didn't find a direct answer to that question.
Jingjings
  • 1,173
  • 1
  • 9
  • 13
1
2 3 4 5 6