Boise inversion answers: about 20 days per year, peaking in December. It's a thin fog, usually clear or only thin clouds overhead. Lasts several days at a stretch, until a front comes through and blows it out. The major problem is wood smoke from stoves and fireplaces. 2001 and 2002 were the worst years for this (10 and 14 days of unhealthy air). Since then, Boise has instituted a wood-burning ban when severe inversions are forecast, and that problem has gone down to an average of 2 unhealthy days per year.
References on valley air quality index:
and an overview of valley inversion/AQI history:
In each of the winter months (Dec-Feb), Boise has on average 20 cloudy days, 6 partly cloudy, and 5 of those fabulously brilliant sunny winters days you ask about.
Ref: Western Regional Climate Center
There's lots more info at the link above, if you wish.