USCG-published Colregs state navigation
lights are to be displayed between sunset and sunrise, as well as during other periods of restricted visibility. Also, proper sounds are to be made during periods of restricted visibility.
Restricted visibility isn't defined, however. So, it is a matter of prudent seamanship.
For most of us, the lights are to have a minimum range of 1 and/or 2 miles while the sound device have a range of at least half a mile. See pages 42-43 and 154-155 of http://www.navcen.uscg.gov/pdf/navRules/navrules.pdf
Showing lights and making sounds during restricted visibility have marginal usefulness outside the range of these devices. Thus, my rule
of thumb is to turn on the navigation
lights when restricted visibility is less than two miles and the "fog" horn at less than half a mile. It's not a matter of whether the restricted visibility is caused by fog
, mist, smoke, or whatever.