# How can I get the prior of a random variable that's a function of a random variable in Bayesian data analysis?

I have a model which includes the following priors:

$\lambda_C \rightarrow \dfrac{1}{\sigma_C^2}$

and

$\sigma \sim \text{uniform}(0,500)$

Where $\sigma$ is the standard deviation and $\lambda_C$ is the precision of a normal distribution.

Being a beginner, I have problems conceptualizing the prior of $\lambda_C$. How would I even calculate the single values for the prior of $\lambda_C$?

You can think of the relationship between $\lambda$ and $\sigma$ as just a change of variables, a.k.a. a reparameterization, of the probability distribution that was initially specified in terms of $\sigma$.

We have the random variable $\sigma$ distributed as $p(\sigma)=1/500$ for $0<\sigma<500$. We want the distribution for $\lambda=\sigma^{-2}$:

\begin{equation} \begin{aligned} p(\sigma) d\sigma &= p(\lambda)d \lambda \\ \frac{\lambda^{-3/2}}{1000} d\lambda &= p(\lambda) d\lambda \\ \frac{1}{500^2} <& \lambda < \infty \end{aligned} \end{equation}

(Deriving this is involves equating the CDFs, $\Phi( \sigma) =\Phi(\lambda)$, reversing the sense of $\sigma$, i.e. remapping $\sigma \rightarrow 500-\sigma$, and then taking the derivative of the resulting CDF)

I am not sure if I really get what is troubling you, but perhaps a couple of hints will help you:

• How can I conceptualize the prior of a deterministic variable in Bayesian data analysis?

Formally, deterministic variables don't have a probability distribution (yes you guessed it, since they are not random variables!). No random variable => no prior.

• Now, as far as I understand the first is a deterministic variable and the second is stochastic.

If by first you mean $\lambda_C$, since it is a function of r.v. $\sigma$, IT IS a random variable. (just a side note: usually you reserve the term "stochastic" for random quantities involving time, such as a stochastic process)

• However, I have problems conceptualizing the prior of the deterministic variable.

I think I answered this in the first comment. But let me elaborate. In Bayesian statistics there are multiple ways how you can treat a parameter, depending on your previous knowledge. In general you assume the parameter to have an uncertain value, and consider it random. Depending on your beliefs and prior knowledge about its values you can either, use a uniform distribution, or a distribution that favors a certain value, e.g. if you know the precision is most likely centered around a certain value, you can use a Gamma($\alpha$,$\beta$) distributed prior with appropriate values $\alpha$ and $\beta$. Now here is the point that might be tripping you up. Priors are there since sometimes you have a strong belief, but usually you will not be a 100% certain about the value of a certain parameter. Technically, that case can be considered a random variable, with a point mass. But that is completely beyond the point. Priors help model uncertainty of parameters. Deterministic parameters in the model are something completely different. They are unknowns, that you decide to model as single values without any uncertainty.

• How would you, for example, plot the graph of the prior?

By plotting its probability distribution.

• How would I even calculate the single values for the prior of $\lambda_C$?

Not sure what you mean here. I hope by now you can see this question makes no sense.

HTH

• Alright, so, as far as I understand I mixed up the terminology here. Which is not surprising since I'm a beginner. I adapted the title and content of the question. Would you mind checking if it's better now or did I still miss something? – Gerome Bochmann Oct 30 '13 at 13:24