Questions tagged [inverse-gamma-distribution]

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

Filter by
Sorted by
Tagged with
0 votes
0 answers
18 views

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 ...
user avatar
1 vote
1 answer
52 views

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 ...
user avatar
  • 13
4 votes
3 answers
149 views

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 ...
user avatar
  • 635
0 votes
1 answer
75 views

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 ...
user avatar
  • 179
0 votes
0 answers
48 views

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 ...
user avatar
  • 101
1 vote
1 answer
36 views

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*} ...
user avatar
3 votes
1 answer
256 views

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}^{...
user avatar
0 votes
0 answers
32 views

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 ...
user avatar
  • 143
0 votes
0 answers
23 views

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?
user avatar
  • 1
0 votes
1 answer
42 views

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 ...
user avatar
0 votes
0 answers
33 views

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/...
user avatar
1 vote
1 answer
176 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 ...
user avatar
0 votes
1 answer
201 views

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)...
user avatar
  • 159
1 vote
1 answer
337 views

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 ...
user avatar
1 vote
1 answer
330 views

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 ...
user avatar
  • 461
2 votes
0 answers
129 views

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)...
user avatar
1 vote
1 answer
38 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 $\{x_{ij}\}_{j=1}^{m_i}$ ...
user avatar
  • 461
11 votes
2 answers
776 views

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 ...
user avatar
1 vote
1 answer
68 views

What exactly is a inverse $\chi^2$ distribution?

What exactly is a inverse-chi square distribution?
user avatar
1 vote
0 answers
24 views

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 ...
user avatar
0 votes
0 answers
32 views

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 ...
user avatar
0 votes
0 answers
60 views

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 ...
user avatar
  • 43
0 votes
1 answer
150 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 ...
user avatar
3 votes
0 answers
538 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}{\...
user avatar
  • 205
1 vote
0 answers
198 views

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 ...
user avatar
  • 31
1 vote
1 answer
3k 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 ...
user avatar
  • 35
0 votes
0 answers
467 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 ...
user avatar
3 votes
1 answer
469 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 $\...
user avatar
2 votes
2 answers
221 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 ...
user avatar
  • 23
1 vote
0 answers
234 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 ...
user avatar
  • 511
2 votes
0 answers
742 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 ...
user avatar
  • 1,152
3 votes
0 answers
204 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 ...
user avatar
  • 713
2 votes
1 answer
4k 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 ...
user avatar
5 votes
1 answer
7k 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 ...
user avatar
0 votes
1 answer
230 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 ...
user avatar
0 votes
0 answers
193 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 ...
user avatar
  • 171
5 votes
1 answer
366 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 $\...
user avatar
  • 103
0 votes
1 answer
3k 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^...
user avatar
  • 63
3 votes
2 answers
509 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 ...
user avatar
  • 325
0 votes
0 answers
132 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 ...
user avatar
  • 21
3 votes
2 answers
1k 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 ...
user avatar
1 vote
0 answers
135 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 ...
user avatar
1 vote
1 answer
958 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 $R$ ...
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? ...
user avatar
  • 21
0 votes
1 answer
2k 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 ...
user avatar
2 votes
1 answer
922 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 ...
user avatar
6 votes
1 answer
4k 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 ...
user avatar
  • 515
1 vote
0 answers
783 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 ...
user avatar
  • 11
3 votes
1 answer
1k 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 ...
user avatar
2 votes
2 answers
405 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) \...
user avatar
  • 525