Why does the P-3 Orion use an engine driven compressor instead of bleed air for pressurization? In another question, an answer mentions that the P-3 Orion uses a gearbox driven compressor for cabin pressurization, even though the aircraft uses T-56 turbine engines that supply bleed air in other designs.
(How are turboprops and other propeller planes pressurized?)

Brief research suggests that the "EDC" exists, but I can not find any rationale for it. 
(http://blueaero.com/wp-content/uploads/2016/11/Heico_P3_EDC_Repair.pdf)

Does using gearbox driven compressors help in some way with the mission, possibly in dealing with the environment? Is it an anachronism/carryover from the L-188 Elektra? Is it not actually gearbox driven, but somehow an exchanger for bleed air?
The T-56 engine, like all turboprops, is much more efficient at converting fuel into rotational energy than compressing air.  It takes approximately 50-100 shaft horsepower to run an EDC, depending on altitude and how cold of a temperature you select. Bleeding a small amount of air off the engine to run engine anti-ice costs you a 9% loss of SHP, which in flight, would be anywhere from 150-250 SHP.  I don't know the exact volume of bleed air you would need to run a pressurization system, but it is definitely more than the engine-anti ice system.  I'm making that assumption base on the size of the ducting.  The anti-ice tubing is much smaller than the EDC tubing, or the main bleed air manifold for that matter.
All in all, I think it's due to efficiency.  The EDC is more efficient for that motor, but is more complicated and introduces more possible malfunctions.  A P-3 requires a LOT of cold air to keep the computers from overheating, more than a C-130 (same motor) for sure.  I'm not sure if the C-130's back in the day also had EDC's, but they use bleed air nowadays.  So the designers went with whatever can move the most air.  I don't have my Orion Service Digests handy, but I bet there's a more thorough answer in there.  I'll check whenever I go back to work.