I understand that it is crucial to convey what you think during a coding interview. Because even if you failed to solve the problem, discussing your thinking process and logic are also big parts for the interviewer to judge your ability. Most of the "coding interview guide" emphasized that talking to your interviewer is very important.

My question is, am I allowed to think silently for a minute or two before I come up with a viable solution/direction? Is it harmful?

I found it hard to keep talking while you're thinking, especially at the early phase of solving a problem. During that phase your brain is rapidly searching for a possible solution and is also rapidly eliminate bad ideas. If I force myself to talk about what I am thinking, it breaks my thinking process and usually causes a mess. I tend to keep my mouth shut until I figure out a viable way to solve the problem. Then I start to talk about the solution, fix details, discuss, and code.

On the other side, during my few interview experiences, I always felt awkward to be silent in a coding interview. A part of my brain was telling me that I need to concentrate on finding a solution; the other part was worrying about not making a sound for too long. I could feel the stares from the interviewer and maybe he/she was already having bad impression on me because I took one or two minute to think on my own without talking/discussing.

What is your advice on this? Let's say if you can solve a coding problem in 15 minutes, what is the optimal time management?