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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Distribution of $1/\sqrt{X}$ when $X$ is a gamma variate [duplicate]

It is a question about finding the posterior parameters. $X$ follows $\text{Gamma}(a,b)$ ----(Prior) $y \vert x$ follows $\text{normal}(\mu,1/x)$ ----- (likelihood), that is variance $(\sigma^2) = 1/...
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How to write a gamma equation from these coefficients R?

A similar question is out there enter link description here, but I find that the answer was not comprehensive enough to cover other scenarios. ...
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46 views

Expectation of inverse square under multivariate standard normal

In one of the steps in my lecture notes, the following result was used without proof: Given $X$ is a $p$-dimensional multivariate normal distribution, where $p\ge 3$, centred on zero, with covariance ...
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29 views

Finding the Method of Moments in two Parameters

Let $X_1, X_2, \ldots, X_n$ be a random sample of size $n$ from the following distribution $$f(x;\mu, \lambda) = \sqrt{\frac{\lambda}{2\pi x^3}}e^{\frac{-\lambda(x - \mu)^2}{2\mu^2 x}}$$ where $x, \...
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Deriving posterior distribution for variance of normal distribution

I have a task to derive posterior distribution for parameter $\sigma^2$, given that the data vector $y^t = (y_1,...,y_t)$ is from $N(0,\sigma^2)$. The uninformative prior for $\sigma^2$ is $h(\sigma^2)...
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Derive cauchy distribution as a scale mixture of normal distributions

I doing Bayesian modelling these days. I found that cauchy distribution can be written as a scale mixture of normal based on following source. Link So I started to derive this. Somehow, I am not ...
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Bayesian estimation of the variance

The mean of the Gamma distribution is $\alpha/\beta$, while the mean of the Inverse Gamma is $\beta/(\alpha-1)$. Similarly, the mode of the Gamma is $(\alpha-1)/\beta$, but the mode of the Inverse ...
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Posterior distribution of $\sigma^2$

In chapter 9 of Jim Albert's Bayesian computation with R it's mentioned that, in the context of Normal Linear Regression, the posterior joint density is: $$g(\beta, \sigma^2 | y) =g(\beta|y, \sigma^2)...
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31 views

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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Approximate or exact distribution of the sum of inverse gamma variables

The random variable ${\left| {H\left( {n,m} \right)} \right|^{ - 2}} \sim Inv - Gamma\left( {{\omega },\frac{\Omega }{{\omega }}} \right)$and independent of each other. What distribution does its sum ...
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Show that Inverse-gamma prior in Weibull distribution is conjugate

Weibull distribution: $fx(x) = \frac{k}{\lambda}\left(\frac{x}{\lambda}\right)^{k-1}\text{exp}\left\{-\left(\frac{x}{\lambda}\right)^k\right\}$ Inverse gamma: $fx(x)=\frac{\beta^\alpha}{\Gamma(\alpha)...
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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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106 views

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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2k 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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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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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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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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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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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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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
3k 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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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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210 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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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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289 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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2k 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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351 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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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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2answers
654 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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130 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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732 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
972 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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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
749 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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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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1answer
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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 ...
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268 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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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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108 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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135 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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707 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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495 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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368 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 ...