Could be the battery itself. My Bionic was getting 2 hrs of battery life on its original battery. I bought an aftermarket battery, lasts 18hrs -1 day on it now. Try JuiceDefender too, are you leaving wifi on? or the signal, bluetooth? Juice Defender shuts off wifi and other drainers when the phone is asleep and it detects no user. It works on my Gingerbread device really well.
Samsung are notorious for bad battery life, because of their oversized screens. Turn the screen brightness down also. Google "battery tips for S4 Mini" and "battery tips for android" they will walk you through steps to optimize battery performance. Most tips are really simple to get the most out of the battery. Don't drain it completely either, this can damage the battery. Use the charge at 20% rule.
Just found video on yt but cannot post links yet.