Questions tagged [inverse-gamma]

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

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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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27 views

Why does the inverse gamma distribution look essentially the same when increasing the variance?

I have a function in R which for a given mean $\mu$ and variance $\sigma^2$, spits out the parameters for the shape, $\alpha$, and rate, $\beta$ of the inverse gamma distribution with that mean and ...
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121 views

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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158 views

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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146 views

how to get joint pdf of mixed random variables

I would like to know how the joint probability density function $p(b,r,\sigma^2)$ can be calculated for the following graph. Random variable $b$ is a latent binary variable, and random variable $\...
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2answers
113 views

Should updating one data point at a time or all change the posterior of a normal-inverse-gamma?

I have implemented the normal inverse gamma distribution per section 3 of https://people.eecs.berkeley.edu/~jordan/courses/260-spring10/lectures/lecture5.pdf in some code. However, I've noticed ...
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92 views

Bayesian learning - how to update an inverse gamma distribution

I'm trying to implement a Bayesian Learning/Updating Model (multi-armed bandit) in the following way: I'm conducting a survey where respondents can rate items on a 5-point scale. I have a total set ...
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235 views

MLE estimation: Inverse gamma duration model with exogenous variables

I observe a cross section of durations, $t_1, \ldots, t_n$ which are all strictly positive and a corresponding vector of exogenous variables $x_i$ which are assumed not to change from time $T=0$ until ...
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101 views

Problems Estimating Parameters for the Inverse Gamma Distribution

I am trying to estimate the parameters of an inverse gamma distribution such that a given amount of probability mass lies above and below some specified threshold. If $x$ is an inverse gamma ...
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1answer
784 views

Simulation of t copula in Python [closed]

I am trying to simulate a t-copula using Python, but my code yields strange results (is not well-behaving): I followed the approach suggested by Demarta & McNeil (2004) in "The t Copula and ...
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1answer
2k views

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 ...
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134 views

Inverse Gamma Posterior variance derivation in Tobit model

I have a doubt about the posterior distribution of the variance parameter for the Tobit model as provided by Koop, Poierier, Tobias (2007) in "Bayesian Econometrics Methods" page 221. Posteriors for ...
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75 views

Why do I get 'Inf' value while doing quantile mapping in MATLAB?

I am doing bias correction of rainfall data simulated by Global Climate Model. Since, rainfall data generally follows gamma distribution, I am estimating parameter ...
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1answer
217 views

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 $\...
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1k views

Conjugate prior for inverse Gamma with known scale parameter

Suppose $Y \sim \text{Inverse Gamma}(\alpha, \beta)$ with scale parameter $\beta$ known, and $\alpha$ unknown, and the pdf is given by $$f(y) = \frac{\exp(-1/\beta y)}{\Gamma(\alpha) \beta^\alpha y^...
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163 views

Merits of reparameterizing the Gamma and inverse Gamma

Wikipedia states that the PDFs for the Gamma distribution is: $$ f(x|\alpha,\beta) = \frac{\beta^\alpha}{\Gamma(\alpha)}x^{\alpha-1}\exp(-\beta x) $$ However, in Rasmussen 2000, the pdf for the ...
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90 views

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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301 views

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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115 views

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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388 views

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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1answer
693 views

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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566 views

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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1answer
410 views

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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2k views

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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514 views

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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1answer
503 views

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 ...
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156 views

Inverse gamma distribution definition

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) \...
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1answer
2k views

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 ...
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1answer
76 views

Is this an Inverse Gamma? [duplicate]

My professor wrote in an assignment that a random variable with an Inverse-Gamma 1 distribution has density function $$f_{ig}(\sigma|d,s) = C_g^{-1}(d,s)\cdot \sigma^{-(d+1)}\cdot \exp\left(-\frac{1}{...
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93 views

The purpose of scaling the normal variance in NIG-distribution

The Normal-Inverse-Gamma distribution is often written as $N(\phi | \mu, \sigma^2 \Sigma) IG(\sigma^2 | \alpha, \beta),$ and used as a conjugate prior for a linear model given observations $y_t \...
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638 views

Farlie-Gumbel-Morgenstern copula

I have the Farlie-Gumbel-Morgenstern copula and I want to generate two gamma marginals and find an expression for the linear correlation. I understand that to get the random variates $(u,v)$ I need to ...
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1answer
331 views

what is “Minimum Length Least Square”

I am in the process of implementing Bayesian Lasso with Normal-Gamma prior; In section 3.3 mention The prior for the scale parameter $\gamma$ conditional on $\lambda$ is given by $v_\beta = 2 \lambda ...
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1answer
269 views

Hierarchical Bayesian Regression, Can an Inverse-Gamma distributed Variance look Normal or t?

Using Peter Hoff's book, A First Course in Bayesian Statistical Methods, I used some of my own data to fit a Hierarchical Bayesian Regression following his example. In his book, he utilized a Gibbs ...
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1answer
848 views

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 ...
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1answer
154 views

Bayesian Linear Model Posterior as Sum of Squares?

As part of a homework, I am asked to do the math from the Normal-Inverse Gamma linear regression model. Starting from priors $N(\beta_0, \sigma^2 A)$ and $IG(\alpha_0, \delta_0)$ and with the help ...
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404 views

Excel: Still have to use the normal approximation with GAMMA.INV for high values of alpha?

The GAMMAINV function from Excel 2003 had a propensity for generating NA error messages for high alpha values as the iterative process failed to reach convergence. It was therefore common to use a ...
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1answer
86 views

Which distribution do I get?

Be $X\sim N(\mu,1)$ and $Y\sim Inverse-Gamma(\alpha,\beta)$. For the Inverse-Gamma, I usually use the parameterization which leads to the following probability distribution function for Y: $f(y;\...
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91 views

Interpretation of priors in example

Suppose you have 3 variances $W_{1},W_{2},W_{3}$ that can be expressed as $W_{j}=q_{j}V$ with $j = 1,2,3$. According to one model, $W_{3}$ should be pronounced and $W_{1}$, $W_{2}$ should be small to ...
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131 views

Confusion about nonlinear transformation of gamma and inverse-gamma distributions

I have a question about the variance of a transformed random variable, illustrated by a particular example. Let $X_1, ..., X_n$ and $Y_1, ..., Y_n$ be independent random variables drawn from an ...
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879 views

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 ...
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1answer
231 views

Pearson 5/Inverse Gamma/ Double Pareto

My current research deals with landslide size frequency distribution which has been fit to a three-parameter inverse gamma function and a double Pareto function by previous researchers. I am trained ...
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2answers
627 views

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.
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1answer
2k views

How to generate random variables from a defined density via R?

Given probability distribution: $f_x(x)=(x+1.5)^{-1.75}e^{-x/400}$, Let $t=x-1.5$. (I want to generate a random set $x$ from this distribution) Generate ...
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1k views

Conjugate inverseGamma prior and Multivariate normal?

I do know that inverse gamma is a conjugate prior for univariate normal distribution. I guess it's also a conjugate prior for multivariate normal distribution. I'm trying to get the closed-form ...
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2answers
8k views

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 $$v\sim\text{Inverse-Gamma}\left(\frac{p}{2}-1,\frac{s}{2}\right)$$ ...
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1k views

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 ...
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1answer
250 views

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 ...
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543 views

Thompson sampling with multivariate posterior distribution

I'm implementing Thompson sampling for a multi armed bandit problem (see http://en.wikipedia.org/wiki/Thompson_sampling). The underlying Bayesian model is a Bayesian Linear Regression, which has a ...
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915 views

Why doesn't the method of moments work when calculating the variance of the inverse gamma distribution?

I'm trying to calculate the variance of the inverse gamma distribution using the method of movements. According to wikipedia the variance should be: $$\sigma^2 =\frac{\beta^2}{(\alpha-1)^2(\alpha-2)}$...
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1answer
329 views

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 ...