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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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Posterior distribution for multivariate Gamma-Normal model

Let $\theta \in \mathbb{R}_{>0}^n$ be a random variable with prior distribution $p(\theta)$: \begin{equation} p(\theta) = \prod_{i=1}^n \text{Ga}(\alpha_i, \beta_i)(\theta_i), \end{equation} where $...
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Given N observations - Bayesian Posterior for Unknown Variance of a Normal Distribution with a Known Mean?

So, starting from no information besides N trials from a Gaussian with $\mu = 0$, I'd like to know the best Bayesian posterior for the unknown variance, $\sigma^2$. My approach so far as been to ...
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Conjugate prior for univariate normal with same mean and unknown sum of two variances

I have a Bayesian inference problem where the likelihood function is conditioned on two unknown variances. $$\log\mathcal{L}(d\mid \sigma_n,\sigma_s) = -\frac{1}{2} \log (\sigma_n^2 + \sigma_s^2) -\...
Riccardo Buscicchio's user avatar
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bayesian problem using inverse gamma: negative initial values

My study involves a dependent variable measuring reading times (minimum value = 0.3) and two categorical variables (y = "quick" or "slow"; t = "cute" or "ugly") ...
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Parameterization of inverse gamma prior in Bayesian methods

For a prior of $\sigma^2 \sim IG(0.01, 0.01)$, often recommended as an uninformative prior for the variance parameter in MCMC approaches and other Bayesian methods, which parameterization does this ...
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Proof of an approximation of Welch

Let $S^2 = X/\nu$ where $X \sim \chi^2(\nu)$ and define $W = \frac{1}{S^2}$. In Welch's paper, "On the comparison of several mean values: an alternative approach" (1951), he gives the ...
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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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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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Sum of Gamma distributions weighted by different multipoles

1) Introduction : I am interested in computing the variance of an observable $$ O=\frac{\sum_{\ell=1}^{N} \sum_{m=-\ell}^{\ell} a_{\ell m}^{2}}{\sum_{\ell=1}^{N} \sum_{m=-\ell}^{\ell}\left(a_{\ell m}^{...
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why Gamma inverse is the conjugate prior of normal distribution?

I am trying to understand Bayesian regression. Then in Wikipedia enter link description here, it is written that by using the following relation we get the Gamma inverse as follow for the conjugate ...
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How to calculate alpha and beta parameters from an known mean and variance in normal-inverse gamma distribution

How can I calculate the $\alpha$ and $\beta$ parameters for a normal-inverse gamma distribution if I know the mean and variance?
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How to compute quantile of a mixed distribution? [duplicate]

A mixed distribution where cumulative probability distribution function (CDF) is given by G(x)= (1-p)H(x)+pF(x) where, p=0.2 (assumed in this case as it ranges ...
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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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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 ...
Benjamin Wang's user avatar
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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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1 vote
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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)...
Maverick Meerkat's user avatar
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1 answer
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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 $\{x_{ij}\}_{j=1}^{m_i}$ ...
Nathaniel Bubis's user avatar
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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 ...
BonnieKlein's user avatar
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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 ...
MatteoSeba's user avatar
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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 ...
Yuan Yin's user avatar
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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 ...
tumidou's user avatar
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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 ...
user1971988's user avatar
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744 views

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 ...
Aishwarya Deore's user avatar
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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 $\...
fountain3's user avatar
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388 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 ...
Robert's user avatar
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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 ...
deschen's user avatar
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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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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 ...
rhonsprudel's user avatar
7 votes
2 answers
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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 ...
LeastSquaresWonderer's user avatar
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1 answer
273 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 ...
Jim Conrad's user avatar
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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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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 $\...
t-student's user avatar
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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^...
Demo's user avatar
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3 votes
2 answers
656 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 ...
peco's user avatar
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167 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 ...
Dave92's user avatar
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3 votes
2 answers
1k views

Variation on inverse gamma: $1/X^r \sim \textrm{InvGamma}({}~~,{}~~)$ if $X \sim \textrm{Gamma}(a, b). $

If $X \sim \textrm{Gamma}(a, b), $ then $1/X \sim \textrm{InvGamma}(a, b). $ Therefore if $X \sim \textrm{Gamma}(a, b), $ then $1/X^r \sim \textrm{InvGamma}({}~~,{}~~)? $ My brain is exploding I don't ...
Stefano Vespucci's user avatar
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139 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 ...
Lucivius's user avatar
2 votes
1 answer
1k 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 $R$ ...
yosef soussana's user avatar
2 votes
1 answer
1k 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? ...
cch's user avatar
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1 answer
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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 ...
NumbThumb's user avatar
2 votes
1 answer
1k 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 ...
LuckyLuke's user avatar