Improvise : music ::

a. refute : invitation

b. reiterate : garlic

c. repose : replacement

d. speculate : theory

e. vigil : watchman

I'd say the correct analogy here is D) speculate : theory.

Because if you are improvising music, it means that you don't know it by heart, you are making it up as you go. The same thing goes for speculation - it isn't certain yet, but the more you speculate, the closer you are to creating a theory.