The EPROM is the erasable programmable read only memory - the computer chip that is at the heart of the onboard ECM - Electronic Control module. It is the devise that all of your sensors feed into, and adjusts engine performance accordingly. The EPROM stores fault codes. You can pull those fault codes using a special service connector (Part number 070PZ-ZY30100) plugged into the red, 4-prong service connector plug under the electronics cover on the front of your engine. Alternatively, you can use a paper clip to shunt the lime-green/white wire to the black wire on that plug. The fault code, or codes, are given by the number of blinks on the MIL (malfunction indicator light) on your console. If this is all very new to you, I would recommend taking the engine to a dealer and have them run diagnostics with their computer.
Not cheaper to replace the T-stats, but may be easier to do.
Forget what I said about the exhaust pipes - that problem was for the BF200 and BF225, only.