I have never heard of this term. But evidently some DMs use this as a check without telling their players.  I guess.  

How have you used Passive Investigation in the past?

Yes. It’s sometimes used that way, sometimes used to determine if a PC is clever enough to recognize a clue for what it is, even if the player isn’t. I wouldn’t use it to solve a puzzle for the party, or even to reveal what the clue means, but I sometimes have as a means of indicating that there’s something about a particular item/marking/whatever that is tickling the PC’s mind, but they can’t quite put their finger on it. Like, to see if the PC’s smart enough to earn the player a gentle hint or something.

My home group’s main GM is traditionally a big fan of utilizing passive scores for just about any skill as a rough guide to whether or not the PC is capable of X. If the passive score is above a certain threshold he determines there’s no chance of failure and therefore resolves X without requiring a roll.

Passive scores can be as useful (or useless) as any given DM wants, but technically RAW there’s one for every Skill in the game.

Investigation has always seemed like an "active" skill to me. And I only see "passive investigation" referenced once in the PHB. 

Do you have examples from DMing that you recall rolling passive Investigation for your players?  If a player lacks the Investigation skill or a high INT score, is it even worth thinking about?

That’s the thing, you don’t need to roll for passive scores. One example was when the players all got stuck on what to do because they had discounted a clue and gotten suckered by a red herring. I looked at their passive Investigation scores and one was pretty high, so I told that player something about the clue was scratching at the back of their PC’s mind, and that the clue they were fixating on was starting to “smell fishy.” That prompted the group to actively reconsider their accumulated clues and start thinking of alternative lines of investigation. Ultimately it worked out because once they were off the rut they had gotten themselves into they started to shake more clues loose that revealed unequivocally that the one other clue was indeed a red herring.

What is it that you consider when doing passive checks?

THat is, what is the inciting event that causes such a thing to happen, and if there is a roll involved, what is the mitigating factor for such (the thing that makes it possible and the thing that makes it harder)?

The Observant feat refers to passive Intelligence (Investigation) scores and i don't recall if any adventures specifically do.

As DM i use passive checks mainly to secretely determine the outcome of an ability check without rolling dice or to represent the average result for a task done repeatedly..

I specifically use passive Intelligence (Investigation) scores mainly to secretly determine if a character recall and deduce information or figure something hidden or concealed without openly calling for an Intelligence check to alert everyone of something. 

I generally ask for an active check when a character actively search for a hidden object, search lore in books or try to figure a way to open a secret passage discovered for exemple. Repeatedly looking for it again would rely on passive check instead for simplicity.

Like, how “common knowledge/sense” something is when it comes to mental skills. If it’s common enough, and their passive score is high enough, then they just succeed. Or, as in the case of my example about the clues, how “obvious” I think it should be as compared to how “clever” the character was. I didn’t want to just give them the answer, just a nudge is all. I used that as my excuse.

Kinda like, using the rules for jumping/falling, if a particular PC is making a careful, deliberate leap across a chasm with a slightly higher landing point of loose rocks and scree, and their passive Dex (Acrobatics) score is 18+, are they really all that likely to fail that check to stick the landing? Pro’ly not, no. So I wouldn’t ask for a roll in that instance. Makes sense, right? Or if their passive Str (Athletics) score is 18+, are they real all that likely to fail the check to see if the mild elevation is going to be a problem for that PC? Pro’ly not, no, so I don’t ask for the check. I don’t necessarily use passive scores as a guaranteed floor for their checks by any means, just more of a guideline. If that same PC was in a hurry mid combat with limited time and space then there’s still a chance in the math of failing those simple DC 10 checks, so a roll is called for. But with all the time in the world and nothing extraneous making the task more challenging then I just assume they are their Wheaties that day and press onward. I basically apply the same principles to all passive scores.

That makes sense. Making fewer rolls does save time, doesn't it?  

I'm going to start using passive checks more. 

I honestly think WotC should use passive scores more too. Like, jump distances for example. Instead of having their own calculations they could just use half a PC’s passive Athletics and be done with it, stuff like that. If WotC made passive scores more relevant then so would everyone else.

Another situation in which using passive scores is handy is when adjudicating those terrible, horrible, baby eating “PvP” situations. More than once have I had a player say something like “my character quietly gets up and goes for a short walk to check the perimeter” [or whatever], to which another player asks if their PC noticed that or not. In situations like that I’ll ask the first player if their PC was actively trying to be stealthy or just not specifically drawing attention to themselves, and I’d ask the second player if their PC was actively paying attention to what the other PCs were doing or just wanted to know if they just happened to notice it. If neither was intentionally trying to Stealth or Perceive, then I just compare passive scores to determine the outcome of the situation. Very handy, and nobody has any real reason to get butthurt about the situation. (Not that that’s much of an issue at my home table anyway, we’ve all been friends for 20+ years at this point.)  

WotC doesn't seem sure what passive checks are for, and the playtest looks suspiciously like they've just done away with them, as passive perception is no longer the target for stealth, and that's the main canonical use of passive checks. However, my main problem with passive investigation is that I have a problem with investigation skill in general, it's just not at all clear what it actually does.

Quote from Pantagruel666 >>

WotC doesn't seem sure what passive checks are for, and the playtest looks suspiciously like they've just done away with them, as passive perception is no longer the target for stealth, and that's the main canonical use of passive checks. However, my main problem with passive investigation is that I have a problem with investigation skill in general, it's just not at all clear what it actually does.

 
That’s part of why I liked the Study action from one of the 1DD UAs so much.

The way I see it is that Perception is for noticing stuff with the four of the five main senses: sight, hearing, sound, smell, touch.

Investigation I use for anything that requires logic, analysis, or intuition (when not a social situation).  I use Insight for social intuition. 

Quote from Song_of_Blues >>

The way I see it is that Perception is for noticing stuff with the four of the five main senses: sight, hearing, sound, smell, touch.

Investigation I use for anything that requires logic, analysis, or intuition (when not a social situation).  I use Insight for social intuition. 

 
Yeah, that’s basically how I treat it too.