## #StackBounty: #bayesian #hierarchical-bayesian #lognormal What is the full-conditional distribution for \$log(sigma) sim N(mu_sigma,…

### Bounty: 50

What is the full-conditional distribution for $$[sigma|textbf{y},mu]$$ given the following hierarchical structure?:

$$y_i sim N(mu,sigma^2)$$

$$mu sim N(mu_0, sigma^2_0)$$

$$log(sigma) sim N(mu_sigma,tau_sigma^2)$$

My work:
$$[sigma|cdot] propto (sigma^2)^{-n/2}exp(frac{1}{-2sigma^2}Sigma^n_{i=1}(y_i-mu)^2)(sigma^2)^{-1/2}exp(frac{-(ln(sigma)-mu_sigma)^2}{2tau^2_sigma})$$

$$propto (sigma^2)^{-(frac{n-1}{2}+1)}exp[frac{1}{-2sigma^2}Sigma^n_{i=1}(y_i-mu)^2-frac{1}{2tau^2_sigma}(ln^2(sigma)-2ln(sigma)mu_sigma)]$$

From here, I recognize that this somewhat resembles an inverse gamma distribution, where you let $$q=frac{n-1}{2}$$. However, I need to find the $$r$$. I know that the $$exp$$ term should follow the format $$exp[-frac{1}{rsigma^2}]$$, but I do not know how to manipulate my last line to obtain that $$r$$. Do you have any suggestions?

Edit:

I now am getting

$$[sigma|cdot] propto (sigma)^{-n-1+frac{mu_sigma}{2tau_sigma^2}}exp(frac{1}{-2sigma^2}Sigma^n_{i=1}(y_i-mu)^2-frac{ln(sigma)^2}{2tau_sigma^2})$$

This more closely resembles an inverse gamma distribution, but I still have the last pesky exponential term that is getting in the way of this being the IG kernel.

Get this bounty!!!

## #StackBounty: #bayesian #hierarchical-bayesian #lognormal What is the full-conditional distribution for \$log(sigma) sim N(mu_sigma,…

### Bounty: 50

What is the full-conditional distribution for $$[sigma|textbf{y},mu]$$ given the following hierarchical structure?:

$$y_i sim N(mu,sigma^2)$$

$$mu sim N(mu_0, sigma^2_0)$$

$$log(sigma) sim N(mu_sigma,tau_sigma^2)$$

My work:
$$[sigma|cdot] propto (sigma^2)^{-n/2}exp(frac{1}{-2sigma^2}Sigma^n_{i=1}(y_i-mu)^2)(sigma^2)^{-1/2}exp(frac{-(ln(sigma)-mu_sigma)^2}{2tau^2_sigma})$$

$$propto (sigma^2)^{-(frac{n-1}{2}+1)}exp[frac{1}{-2sigma^2}Sigma^n_{i=1}(y_i-mu)^2-frac{1}{2tau^2_sigma}(ln^2(sigma)-2ln(sigma)mu_sigma)]$$

From here, I recognize that this somewhat resembles an inverse gamma distribution, where you let $$q=frac{n-1}{2}$$. However, I need to find the $$r$$. I know that the $$exp$$ term should follow the format $$exp[-frac{1}{rsigma^2}]$$, but I do not know how to manipulate my last line to obtain that $$r$$. Do you have any suggestions?

Edit:

I now am getting

$$[sigma|cdot] propto (sigma)^{-n-1+frac{mu_sigma}{2tau_sigma^2}}exp(frac{1}{-2sigma^2}Sigma^n_{i=1}(y_i-mu)^2-frac{ln(sigma)^2}{2tau_sigma^2})$$

This more closely resembles an inverse gamma distribution, but I still have the last pesky exponential term that is getting in the way of this being the IG kernel.

Get this bounty!!!

## #StackBounty: #bayesian #hierarchical-bayesian #lognormal What is the full-conditional distribution for \$log(sigma) sim N(mu_sigma,…

### Bounty: 50

What is the full-conditional distribution for $$[sigma|textbf{y},mu]$$ given the following hierarchical structure?:

$$y_i sim N(mu,sigma^2)$$

$$mu sim N(mu_0, sigma^2_0)$$

$$log(sigma) sim N(mu_sigma,tau_sigma^2)$$

My work:
$$[sigma|cdot] propto (sigma^2)^{-n/2}exp(frac{1}{-2sigma^2}Sigma^n_{i=1}(y_i-mu)^2)(sigma^2)^{-1/2}exp(frac{-(ln(sigma)-mu_sigma)^2}{2tau^2_sigma})$$

$$propto (sigma^2)^{-(frac{n-1}{2}+1)}exp[frac{1}{-2sigma^2}Sigma^n_{i=1}(y_i-mu)^2-frac{1}{2tau^2_sigma}(ln^2(sigma)-2ln(sigma)mu_sigma)]$$

From here, I recognize that this somewhat resembles an inverse gamma distribution, where you let $$q=frac{n-1}{2}$$. However, I need to find the $$r$$. I know that the $$exp$$ term should follow the format $$exp[-frac{1}{rsigma^2}]$$, but I do not know how to manipulate my last line to obtain that $$r$$. Do you have any suggestions?

Edit:

I now am getting

$$[sigma|cdot] propto (sigma)^{-n-1+frac{mu_sigma}{2tau_sigma^2}}exp(frac{1}{-2sigma^2}Sigma^n_{i=1}(y_i-mu)^2-frac{ln(sigma)^2}{2tau_sigma^2})$$

This more closely resembles an inverse gamma distribution, but I still have the last pesky exponential term that is getting in the way of this being the IG kernel.

Get this bounty!!!

## #StackBounty: #bayesian #hierarchical-bayesian #lognormal What is the full-conditional distribution for \$log(sigma) sim N(mu_sigma,…

### Bounty: 50

What is the full-conditional distribution for $$[sigma|textbf{y},mu]$$ given the following hierarchical structure?:

$$y_i sim N(mu,sigma^2)$$

$$mu sim N(mu_0, sigma^2_0)$$

$$log(sigma) sim N(mu_sigma,tau_sigma^2)$$

My work:
$$[sigma|cdot] propto (sigma^2)^{-n/2}exp(frac{1}{-2sigma^2}Sigma^n_{i=1}(y_i-mu)^2)(sigma^2)^{-1/2}exp(frac{-(ln(sigma)-mu_sigma)^2}{2tau^2_sigma})$$

$$propto (sigma^2)^{-(frac{n-1}{2}+1)}exp[frac{1}{-2sigma^2}Sigma^n_{i=1}(y_i-mu)^2-frac{1}{2tau^2_sigma}(ln^2(sigma)-2ln(sigma)mu_sigma)]$$

From here, I recognize that this somewhat resembles an inverse gamma distribution, where you let $$q=frac{n-1}{2}$$. However, I need to find the $$r$$. I know that the $$exp$$ term should follow the format $$exp[-frac{1}{rsigma^2}]$$, but I do not know how to manipulate my last line to obtain that $$r$$. Do you have any suggestions?

Edit:

I now am getting

$$[sigma|cdot] propto (sigma)^{-n-1+frac{mu_sigma}{2tau_sigma^2}}exp(frac{1}{-2sigma^2}Sigma^n_{i=1}(y_i-mu)^2-frac{ln(sigma)^2}{2tau_sigma^2})$$

This more closely resembles an inverse gamma distribution, but I still have the last pesky exponential term that is getting in the way of this being the IG kernel.

Get this bounty!!!

## #StackBounty: #bayesian #hierarchical-bayesian #lognormal What is the full-conditional distribution for \$log(sigma) sim N(mu_sigma,…

### Bounty: 50

What is the full-conditional distribution for $$[sigma|textbf{y},mu]$$ given the following hierarchical structure?:

$$y_i sim N(mu,sigma^2)$$

$$mu sim N(mu_0, sigma^2_0)$$

$$log(sigma) sim N(mu_sigma,tau_sigma^2)$$

My work:
$$[sigma|cdot] propto (sigma^2)^{-n/2}exp(frac{1}{-2sigma^2}Sigma^n_{i=1}(y_i-mu)^2)(sigma^2)^{-1/2}exp(frac{-(ln(sigma)-mu_sigma)^2}{2tau^2_sigma})$$

$$propto (sigma^2)^{-(frac{n-1}{2}+1)}exp[frac{1}{-2sigma^2}Sigma^n_{i=1}(y_i-mu)^2-frac{1}{2tau^2_sigma}(ln^2(sigma)-2ln(sigma)mu_sigma)]$$

From here, I recognize that this somewhat resembles an inverse gamma distribution, where you let $$q=frac{n-1}{2}$$. However, I need to find the $$r$$. I know that the $$exp$$ term should follow the format $$exp[-frac{1}{rsigma^2}]$$, but I do not know how to manipulate my last line to obtain that $$r$$. Do you have any suggestions?

Edit:

I now am getting

$$[sigma|cdot] propto (sigma)^{-n-1+frac{mu_sigma}{2tau_sigma^2}}exp(frac{1}{-2sigma^2}Sigma^n_{i=1}(y_i-mu)^2-frac{ln(sigma)^2}{2tau_sigma^2})$$

This more closely resembles an inverse gamma distribution, but I still have the last pesky exponential term that is getting in the way of this being the IG kernel.

Get this bounty!!!

## #StackBounty: #bayesian #hierarchical-bayesian #lognormal What is the full-conditional distribution for \$log(sigma) sim N(mu_sigma,…

### Bounty: 50

What is the full-conditional distribution for $$[sigma|textbf{y},mu]$$ given the following hierarchical structure?:

$$y_i sim N(mu,sigma^2)$$

$$mu sim N(mu_0, sigma^2_0)$$

$$log(sigma) sim N(mu_sigma,tau_sigma^2)$$

My work:
$$[sigma|cdot] propto (sigma^2)^{-n/2}exp(frac{1}{-2sigma^2}Sigma^n_{i=1}(y_i-mu)^2)(sigma^2)^{-1/2}exp(frac{-(ln(sigma)-mu_sigma)^2}{2tau^2_sigma})$$

$$propto (sigma^2)^{-(frac{n-1}{2}+1)}exp[frac{1}{-2sigma^2}Sigma^n_{i=1}(y_i-mu)^2-frac{1}{2tau^2_sigma}(ln^2(sigma)-2ln(sigma)mu_sigma)]$$

From here, I recognize that this somewhat resembles an inverse gamma distribution, where you let $$q=frac{n-1}{2}$$. However, I need to find the $$r$$. I know that the $$exp$$ term should follow the format $$exp[-frac{1}{rsigma^2}]$$, but I do not know how to manipulate my last line to obtain that $$r$$. Do you have any suggestions?

Edit:

I now am getting

$$[sigma|cdot] propto (sigma)^{-n-1+frac{mu_sigma}{2tau_sigma^2}}exp(frac{1}{-2sigma^2}Sigma^n_{i=1}(y_i-mu)^2-frac{ln(sigma)^2}{2tau_sigma^2})$$

This more closely resembles an inverse gamma distribution, but I still have the last pesky exponential term that is getting in the way of this being the IG kernel.

Get this bounty!!!

## #StackBounty: #bayesian #hierarchical-bayesian #lognormal What is the full-conditional distribution for \$log(sigma) sim N(mu_sigma,…

### Bounty: 50

What is the full-conditional distribution for $$[sigma|textbf{y},mu]$$ given the following hierarchical structure?:

$$y_i sim N(mu,sigma^2)$$

$$mu sim N(mu_0, sigma^2_0)$$

$$log(sigma) sim N(mu_sigma,tau_sigma^2)$$

My work:
$$[sigma|cdot] propto (sigma^2)^{-n/2}exp(frac{1}{-2sigma^2}Sigma^n_{i=1}(y_i-mu)^2)(sigma^2)^{-1/2}exp(frac{-(ln(sigma)-mu_sigma)^2}{2tau^2_sigma})$$

$$propto (sigma^2)^{-(frac{n-1}{2}+1)}exp[frac{1}{-2sigma^2}Sigma^n_{i=1}(y_i-mu)^2-frac{1}{2tau^2_sigma}(ln^2(sigma)-2ln(sigma)mu_sigma)]$$

From here, I recognize that this somewhat resembles an inverse gamma distribution, where you let $$q=frac{n-1}{2}$$. However, I need to find the $$r$$. I know that the $$exp$$ term should follow the format $$exp[-frac{1}{rsigma^2}]$$, but I do not know how to manipulate my last line to obtain that $$r$$. Do you have any suggestions?

Edit:

I now am getting

$$[sigma|cdot] propto (sigma)^{-n-1+frac{mu_sigma}{2tau_sigma^2}}exp(frac{1}{-2sigma^2}Sigma^n_{i=1}(y_i-mu)^2-frac{ln(sigma)^2}{2tau_sigma^2})$$

This more closely resembles an inverse gamma distribution, but I still have the last pesky exponential term that is getting in the way of this being the IG kernel.

Get this bounty!!!

## #StackBounty: #bayesian #hierarchical-bayesian #lognormal What is the full-conditional distribution for \$log(sigma) sim N(mu_sigma,…

### Bounty: 50

What is the full-conditional distribution for $$[sigma|textbf{y},mu]$$ given the following hierarchical structure?:

$$y_i sim N(mu,sigma^2)$$

$$mu sim N(mu_0, sigma^2_0)$$

$$log(sigma) sim N(mu_sigma,tau_sigma^2)$$

My work:
$$[sigma|cdot] propto (sigma^2)^{-n/2}exp(frac{1}{-2sigma^2}Sigma^n_{i=1}(y_i-mu)^2)(sigma^2)^{-1/2}exp(frac{-(ln(sigma)-mu_sigma)^2}{2tau^2_sigma})$$

$$propto (sigma^2)^{-(frac{n-1}{2}+1)}exp[frac{1}{-2sigma^2}Sigma^n_{i=1}(y_i-mu)^2-frac{1}{2tau^2_sigma}(ln^2(sigma)-2ln(sigma)mu_sigma)]$$

From here, I recognize that this somewhat resembles an inverse gamma distribution, where you let $$q=frac{n-1}{2}$$. However, I need to find the $$r$$. I know that the $$exp$$ term should follow the format $$exp[-frac{1}{rsigma^2}]$$, but I do not know how to manipulate my last line to obtain that $$r$$. Do you have any suggestions?

Edit:

I now am getting

$$[sigma|cdot] propto (sigma)^{-n-1+frac{mu_sigma}{2tau_sigma^2}}exp(frac{1}{-2sigma^2}Sigma^n_{i=1}(y_i-mu)^2-frac{ln(sigma)^2}{2tau_sigma^2})$$

This more closely resembles an inverse gamma distribution, but I still have the last pesky exponential term that is getting in the way of this being the IG kernel.

Get this bounty!!!

## #StackBounty: #bayesian #hierarchical-bayesian #lognormal What is the full-conditional distribution for \$log(sigma) sim N(mu_sigma,…

### Bounty: 50

What is the full-conditional distribution for $$[sigma|textbf{y},mu]$$ given the following hierarchical structure?:

$$y_i sim N(mu,sigma^2)$$

$$mu sim N(mu_0, sigma^2_0)$$

$$log(sigma) sim N(mu_sigma,tau_sigma^2)$$

My work:
$$[sigma|cdot] propto (sigma^2)^{-n/2}exp(frac{1}{-2sigma^2}Sigma^n_{i=1}(y_i-mu)^2)(sigma^2)^{-1/2}exp(frac{-(ln(sigma)-mu_sigma)^2}{2tau^2_sigma})$$

$$propto (sigma^2)^{-(frac{n-1}{2}+1)}exp[frac{1}{-2sigma^2}Sigma^n_{i=1}(y_i-mu)^2-frac{1}{2tau^2_sigma}(ln^2(sigma)-2ln(sigma)mu_sigma)]$$

From here, I recognize that this somewhat resembles an inverse gamma distribution, where you let $$q=frac{n-1}{2}$$. However, I need to find the $$r$$. I know that the $$exp$$ term should follow the format $$exp[-frac{1}{rsigma^2}]$$, but I do not know how to manipulate my last line to obtain that $$r$$. Do you have any suggestions?

Edit:

I now am getting

$$[sigma|cdot] propto (sigma)^{-n-1+frac{mu_sigma}{2tau_sigma^2}}exp(frac{1}{-2sigma^2}Sigma^n_{i=1}(y_i-mu)^2-frac{ln(sigma)^2}{2tau_sigma^2})$$

This more closely resembles an inverse gamma distribution, but I still have the last pesky exponential term that is getting in the way of this being the IG kernel.

Get this bounty!!!

## #StackBounty: #bayesian #hierarchical-bayesian #lognormal What is the full-conditional distribution for \$log(sigma) sim N(mu_sigma,…

### Bounty: 50

What is the full-conditional distribution for $$[sigma|textbf{y},mu]$$ given the following hierarchical structure?:

$$y_i sim N(mu,sigma^2)$$

$$mu sim N(mu_0, sigma^2_0)$$

$$log(sigma) sim N(mu_sigma,tau_sigma^2)$$

My work:
$$[sigma|cdot] propto (sigma^2)^{-n/2}exp(frac{1}{-2sigma^2}Sigma^n_{i=1}(y_i-mu)^2)(sigma^2)^{-1/2}exp(frac{-(ln(sigma)-mu_sigma)^2}{2tau^2_sigma})$$

$$propto (sigma^2)^{-(frac{n-1}{2}+1)}exp[frac{1}{-2sigma^2}Sigma^n_{i=1}(y_i-mu)^2-frac{1}{2tau^2_sigma}(ln^2(sigma)-2ln(sigma)mu_sigma)]$$

From here, I recognize that this somewhat resembles an inverse gamma distribution, where you let $$q=frac{n-1}{2}$$. However, I need to find the $$r$$. I know that the $$exp$$ term should follow the format $$exp[-frac{1}{rsigma^2}]$$, but I do not know how to manipulate my last line to obtain that $$r$$. Do you have any suggestions?

Edit:

I now am getting

$$[sigma|cdot] propto (sigma)^{-n-1+frac{mu_sigma}{2tau_sigma^2}}exp(frac{1}{-2sigma^2}Sigma^n_{i=1}(y_i-mu)^2-frac{ln(sigma)^2}{2tau_sigma^2})$$

This more closely resembles an inverse gamma distribution, but I still have the last pesky exponential term that is getting in the way of this being the IG kernel.

Get this bounty!!!