My questions is about how jump proposals are made when using the Metropolis algorithm.
I understand that the motivation for using the metropolis algorithm is to deduce the most probable value of some parameters by sampling from the un-normalized target distribution. This is done by generating a random walk about the parameter space and evaluating the un-normalized target distribution at those random parameter values.
One of the requirements of the random walk is that parameters with larger density values should be sampled more, generally a normal distribution is used to achieve this.
At each time step the current parameters are evaluated and then a jump is proposed to another paramter value.
The equation i am given for how the next parameter value is selected is:
θ(proposed) = θ(current) + Δθ
How is Δθ derived? From what i have read it is randomly generated from the normal distribution, but if the point of the random walk is to meander around, the proposed jump can be to the right or left (if our parameter space is 1 dimensional) but since the normal distribution only outputs positive values, its co-domain is [0,infinity] how will our θ(proposed) ever move to the left if Δθ is always positive.
TLDR; Is Δθ = normal(randomly generated x), if not, what exact function is giving Δθ, the inverse normal?