B747-400 flight crew reported a malfunction in the IRS system contributing to an airborne conflict while in descent.

Narrative

We were cleared to descend via the arrival into ZZZ. We were descending in VNAV. The fix has a crossing restriction of 9;000 ft. We crossed at 9;000 ft and were descending to meet our next crossing of at 8;000. ATC called traffic; a 737 that had just departed ZZZ1. As I was looking for the traffic between cloud layers; ATC advised us to maintain our current altitude. We stopped our descent immediately and the traffic passed 1;000 ft below us. ATC then advised us to call them [with] reference [to] a possible pilot deviation. I copied the number. We were then given a vector for final and a descent to 5;000 ft. We then landed uneventfully and completed our flight with no further issues. Upon completion of our flight deck duties I called ATC and they advised me that the airplane descended below 9;000 ft about a mile prior to the fix. We were all surprised because the airplane was descending in SPD/LNAV/VNAV PATH and this is a highly unusual and unlikely occurrence. Company procedures and SOP were followed at all times. All three pilots involved had no idea how this could have possibly occurred. However as part of our procedures; we record the IRS positions and send them to the company. It was noted that the #1 IRS was 17 miles off course. This is highly unusual; particularly on a relatively short flight from ZZZ2 to ZZZ. It was entered into the logbook page as a discrepancy and Maintenance was notified. Upon completion of the LOGBOOK page I notified the Chief Pilot to warn the company of the system failure; and notified the safety department advising them that a [report] would be forthcoming. The #1 IRS system was off course by 17 miles. When that (rarely but occasionally occurs;) the system takes out the off course IRS and we get a 'split IRS OPERATION FMC message to warn of a faulty or failing IRS system. Because we didn't err in operating the aircraft; it is my suspicion that the IRS fault allowed us to be out of position to what our indications had shown us to be. We reviewed the event as a crew and none of us could find any error on how we operated the aircraft. I am unaware of the preventative maintenance program or what the follow up solution that was performed by our maintenance department.I was shocked because we reviewed as a crew our arrival and how the aircraft was configured and our actions and found that we followed procedures as we had done before on this arrival. When I went to the IRS monitors as part of our post flight duties; [I] was amazed to see the IRS so far out of position. Normally on a short 4 hour flight like this it may be a mile or two out of position; or on a 16 hour flight 3 or 4 miles out; but I have never seen one 17 miles out. I don't know any procedures that need to be changed; however I have a new 'technique ' to observe any cross track errors prior to joining an arrival. Additionally; perhaps more avionics maintenance would have prevented this."