I'm a newbie when it comes to MCMC. I'm working with a program that is used to fit a number of parameters given some data. The documentation is leaving me a bit confused, I'm hoping someone here could help put me on the right track.
For example, say I am fitting 3 parameters, $\alpha$, $\beta$, and $\gamma$. I have some knowledge of the underlying distributions of $\alpha$ and $\beta$, but not $\gamma$.
This program allows me to input my prior knowledge of these distributions by indicating the underlying distribution of each parameter. For example, I can tell the program:
$\alpha$~$N(0,1)$
$\beta$~$exponential(3)$
$\gamma$~$uniform(0,\infty)$
After writing up a control file indicating these parameters and distributions, I need to set up the 'MCMC' run. There are a number of different moves I can choose from, including (copied from the manual):
log uniform proposal: A Metropolis-Hastings move for continuous scalar variables that are constrained to be positive only (or negative only). If x is the current state then the proposed state is $xe^U$, where $U$ is simulated from a continuous uniform distribution with range 0 ± half-width. This move is equivalent to the uniform proposal on the logarithmic scale.
logit uniform proposal: A Metropolis-Hastings move for continuous scalar variables that are constrained between 0 and 1. If x is the current state then the proposed state is $\frac{1}{1+e^{-log{\left(\frac{x}{1-x}\right)-U}}}$ where $U$ is simulated from a continuous uniform distribution with range 0 ± half-width. This move is equivalent to the uniform proposal on the logit scale.
uniform proposal: A Metropolis-Hastings move for continuous scalar variables. If $x$ is the current state then the proposed state is $x$ + $U$, where $U$ is simulated from a continuous uniform distribution with range 0 ± half-width.
Now I'm confused by this, because it appears the moves are determined by the MCMC proposals, not the underlying distributions that I previously defined.
I'm hoping someone can help me understand how (if) the underlying distributions inform the proposed jumps in the MCMC.
Thanks!