How much ‘gut-driven’ decision making do you encourage vs. purely based on qual / quant?

I think it depends on the context. Typically I have a hypothesis from my gut/experience and then I like to use data to validate (or not) that subjective idea. I do find that over the years I will lean more on my gut vs data if I'm dealing with psychographic issues because often data can tell you the what, but not the why.

I do not think that the question is how much. I would rather go with when to complement your decision based on guts. Let me give you an example. We have been trying to close a deal with a customer who really knows what he wants and knows very well how to extract the last drop of juice of a lemon.... Yesterday they sent us an email stating that they will not consider an offer we had been working jointly for several months.  Our sales rep told me that in the way the email was written, something in her guts told her that the deal was not over. My reaction was, go with your guts. It paid off.
In short, in my belief there are times where qual/quant is a must to know where you are, but instinct, knowledge of people, environment external an internal, competition, you name it, whatever of these appeals to your gut, just follow it. You might be wrong but if you are right this will for sure improve your skills and ability to read situations in a non-standard way.
In my case I often make gut decision, after qual/quant due diligence has been met. 
Beware that in some cases qual/quant vs guts is not a simple decision.
Hope this helps. 

I'm not quite sure of the context (I work in product marketing), but this is one of those "depends" kind of issues. In a start-up where data is non-existent or limited, you have no choice. When in a larger enterprise, I always try to encourage the use of data. If the data is not available or hasn't been tracked, you're out of luck, but it does give you an understanding of how to set up your metrics so that you have it the next time (quarter, project, etc.). I'm not a fan of "gut" decisions simply because they are often based on older data (market, competitors, tactics applied that worked for other, diffferent orgs, etc.). If you have checked those kinds of factors and feel that you know the situation well enough, despite not having hard data, then that's about as good as you can get. Is it "gut" at that point? Probably not. Hope this helps.