[RPG] Does Bard’s Jack of all Trades increase passive checks

Yes, the Bard adds 1/2 proficiency to initiative.

In 5e all checks are ability checks. This is why every check in published materials is listed as Ability (Name), such as Strength (Athletics).

Initiative is a Dexterity check. Under Initiative in the PHB:

every participant makes a Dexterity check to determine their place in the initiative order. (p. 189)

If you can add an ability modifier to a roll, it's an ability check (A saving throw or an attack is a separate kind of d20 roll). If you have a Skill related to that Ability, and you are proficient in that skill, you add your proficiency bonus.

Initiative is a Dexterity check with no other Skill attached to it. Things that modify Dexterity checks work. Other things modify your ability checks as well - such as adding all or half of your proficiency bonus. The Bard has one example of this.

The Champion Fighter's L7 feature has another:

Starting at 7th level, you can add half your proficiency bonus (round up) to any Strength, Dexterity, or Constitution check you make that doesn’t already use your proficiency bonus. (BD&D 26)

Since Initiative is an ability check that uses your Dexterity, the Champion Fighter also gets half their proficiency to added to their initiative roll.

As of 2019, the new Sage Advice Compendium clearly states that they don't interact.

These aren't allowed to work together on a single check.

Can the rogue’s Reliable Talent feature be used in conjunction with Remarkable Athlete or Jack of All Trades?

No. Each of these features has a precondition for its use; Reliable Talent activates when you make an ability check that uses your proficiency bonus, whereas the other two features activate when you make an ability check that doesn’t use your proficiency bonus. In other words, a check that qualifies for Reliable Talent doesn’t qualify for Remarkable Athlete or Jack of All Trades. And Remarkable Athlete and Jack of All Trades don’t work with each other, since you can add your proficiency bonus, or any portion thereof, only once to a roll.

An earlier Jeremy Crawford tweet restates this:

Making sure this is clear: these features work together on a character. They don't work together on the same ability check.

This is definitive, and contradicts earlier apparent rulings, i.e. from the 2017 SA compendium.