Entrusting your money to a bank once seemed strange and risky. Similarly, entrusting all of your data to a company and letting its algorithms build a detailed model of you from it might seem to be an odd or even dangerous idea, but we’ll all soon take it for granted.

A decade from now, your personal model will be more indispensable than your smartphone, and the company that provides it may well be the world’s first trillion-dollar business. So it is time to start getting acquainted with our digital alter egos—and what they’ll...