# Questions tagged [inverse-gamma-distribution]

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

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### Shifted Inverse Gamma still conjugate

Suppose '''X~Inverse Gamma(a,b)''' over '''[0,\infty]''' and we have '''X+c''' over '''[c,\infty]''' for '''c>0''' known constant. The likelihood is normal with unknown variance, and the variance ...
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### How to write the derivative of the inverse gamma function?

I have recently been writing an R program on the inverse of the gamma function and the derivative of the inverse function. Now there is some confusion I would like to ask for advice. I have written ...
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### inverse gamma (0.001,0.001) prior on the variance in the Bayesian hierarchical model

This 8 schools data is from Gelman 2006 paper: http://www.stat.columbia.edu/~gelman/research/published/taumain.pdf. In Figure 1 (c), the prior density of inverse gamma (0.001,0.001) was overlain on ...
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### Moments of the natural statistics of the normal gamma

I am trying to find the Moments of the natural statistics of the normal-gamma distribution. $$(X,T) - NormalGamma(\mu, \lambda,\alpha,\beta)$$ I found on its Wikipedia page that the moments of the ...
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### What's the difference between using negative inverse gamma vs. inverse gamma as the conjugate prior distribution in bayesian analysis?

My current understanding is that inverse gamma is used as the conjugate prior distribution when the likelihood function is a normal distribution with known mean and unknown variance. What's the effect ...
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### Rejection sampling with inverse-gamma-like density [closed]

I would like use rejection sampling to sample from a density, $f_y$ on $(0, \infty)$ satisfying $$f_y(y) \propto \frac{y^{-1}}{1 + y^{-1}}e^{-by^{-1}}$$ I made a first observation that \begin{align*} ...
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### Posterior varince for multiple normal variables with identical variance

Suppose one is given $n$ random normal variables, all with the same variance but with different means: $$X_i\sim N(\mu_i, \sigma^2)$$ Now suppose we observe $m_i$ observations $\{x_{ij}\}_{j=1}^{m_i}$ ...
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### 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 ...
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### What exactly is a inverse $\chi^2$ distribution?

What exactly is a inverse-chi square distribution?
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### Interpretation of interactions in inverse gamma glmm

i want to calculate the effects that the interaction between zone (4 levels), species (2 levels) and distance from contact zone (continuous) has on the pulse repetition period - song rate of the two ...
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### How to evaluate the ratio of (large) gamma functions in the normal inverse gamma marginal likelihood?

I am doing some Bayesian modeling where I am using Normal-Inverse-Gamma as prior for the unknown mean and variance, and Normal for likelihood. In my application, I need the marginal likelihood, where ...
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### how to determine a priori probability distribution of sigama2 in montecarlo simulation?

1、the monte carlo simulation code in SAS： Example1: https://support.sas.com/rnd/app/stat/examples/BayesStd/new_example/index.html ...
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### mismatch in sampling between t distribution and normal-inverse-gamma distribution

I am looking at equivalence of sampling between t distribution and normal-inverse-gamma (NIG) distribution in python. The results don't match, and I want to see if there's a mistake in how I am ...
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### Posterior distribution for Weibull scale parameter with censored data

I am modeling a survival problem using a Weibull distribution with known shape parameter $k$ and unknown scale parameter $\theta$. The PDF and CDF are given by, \begin{align} f(t|k,\theta)=\frac{k}{\...
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### Closed form posterior for product of inverse Gamma and Normal distribution

I am currently reading a book about mixture analysis, and in the textbook a posterior for the parameters $\mu1,\mu2,\sigma_1^2,\sigma_2^2$ of a two-component gaussian mixture is derived as follows (S ...
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### Interpretation of the rate parameter of a Gamma distribution

I am toying with mixture models, especially in a bayesian context and the Gamma (or the inverse Gamma) distribution appears quite often. For example, inverse Gamma is used as a conjugate prior for the ...
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### hypothesis testing - gamma distribution

Let W = Y/B0 be a Random variable that has a gamma(2n,1) distribution. [Y has a gamma(2n,B) distribution and W = Y/B]. i) Suppose you want to test H0 : B ≤ B0 against H1 : B > B0 for some B0 > 0. How ...
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### Effect of sample reduction in normal-inverse gamma

I am trying to interpret/explain a result that I obtained while generating a posterior distribution, and maybe add some informations to what I had so far. The environment that I am using is the ...
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### Variation on inverse gamma: 1/X^r ~ inv_Gamma(?, ?)

(it's the first time that I write here, sorry if miss some convention) If X ~ Gamma(a, b) then 1/X ~ inv_Gamma(a, b) therefore if If X ~ Gamma(a, b) then 1/X^r ~ inv_Gamma(?, ?) My brain is ...
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### Gamma likelihood with InverseGamma prior

I've got a gamma likelihood $\Gamma(\tau_c | \alpha_k, \frac{\alpha_k} {\tau_k})$ (parameterized with shape and rate) with an InverseGamma prior $IG(\tau_k|a_0, b_0)$. I know that the resulting ...
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### inv-gamma distribution as prior for multivariate normal distribution

it's known the conjugate priors for multivariate normal distribution are the normal & inverse-whishart distributions. but i'm interest in very specific case where the correlation matrix is $R$ ...
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### What is the correct to model inverse gamma distribution [closed]

I tried to use below R code to model inverse gamma distribution (alpha=1,beta=1). However, the resulting histogram is not alike the one plotted in the wiki. Could anyone provide any hint about this? ...
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### Calculating the parameters of a Normal distribution using alpha and beta from Inverse-gamma (conjugate prior)

How is it possible to calculate the variance $\sigma^2$ for the Normal distribution if only $\alpha$ and $\beta$ (based on data) from the Inverse-gamma distribution are available? I followed the ...
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### Why is Inverted Gamma (3,0.0005) a diffuse prior when variance of this random variable is small?

A few times I came across the statement that an Inverted Gamma (3, 0.0005) prior for the variance is quite diffuse but proper. Hence, my understanding is that the prior variance of this random ...
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### 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 ...
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### Python Bayesian invgamma.rvs - joint posterior of normal distribution sampling

(My question is inspired by this blog post: The Bayesian analysis of normal distributions with Python. If you read it, you will get a good background on what I am asking.) I am trying to model the ...
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### Show that $\frac{X}{X+Y}\sim Beta(\alpha,\beta)$ [duplicate]
Let IG denote Inverse-Gamma distribution Inverse-Gamma. If $X\sim IG(\alpha,1)$ and $Y\sim IG(\beta,1)$. Show that $\frac{X}{X+Y}\sim Beta(\alpha,\beta)$ I tried with jacobian transformation ...
Wikipedia says the pdf for the gamma function is: $X \sim \operatorname{Gamma}(\alpha,\beta) \implies \Pr(X=x) \propto x^{\alpha-1}e^{-\beta x}$ If $Y = 1/X$, then \[ \Pr(Y=y) = \Pr(X=1/y) \...