Thanks guys! Lots of helpful stuff in this thread. Since someone mentioned something about the use of personal knowledge appeals depends on the situation, here is the kind of situation I had in mind.
Unbeliever: So why do you believe in God?
Christian: Because [insert argument for God's existence or presuppositional defense].
Unbeliever: No, they don't work. [proceeds to "refute" such arguments--either really, because the Christian didn't word things properly, or in appearance, because the Christian can't tell that the arguments have not really been refuted].
Christian: Oh.
Unbeliever: So do you still believe in God?
Christian: Yes.
Unbeliever: Why do you then?
Christian: I personally know who God is. The Holy Spirit opened my eyes to see the truth of the matter--that I was a rebellious sinner, who among other things was repressing what should have been clear. I continue to know that God is as the Holy Spirit bears witness with the Scriptures.
Unbeliever: [Well, I guess the unbeliever could say all sorts of things at this point! But one of them would include the, "Well, anyone could say that! How do you know you're no better off than the mormon?" Another would include, "How do you know that the Spirit who bears witness is the Holy Spirit?" If I understood Turretin correctly, I think such would be answered "The marks in Scripture of the Spirit match with this Spirit showing He is indeed the Holy Spirit." Another response from the unbeliever could be, "What do you mean by personal knowledge?" or something to that effect, in which case, Philip answered that nicely.]
Another situation I had in mind could be with a Christian who is not necessarily unevenly matched with the unbeliever.
Unbeliever: So why do you believe in God?
Christian: Well, the ultimate reason is because I personally know Him: God has revealed Himself to me personally with His Holy Spirit and His Word, showing my blindness as a sinner and bringing me to repentance; indeed, that is the only way you will ever believe too. Hence, any arguments for God's existence that I can give are merely to show that such a belief is reasonable and are not the foundation of my faith, so even if you refute these arguments, I will still believe. Nevertheless, because I know that my personal knowledge of God will not persuade you, I will go ahead with such arguments anyway.
......
So thoughts about the use of personal knowledge in these cases? I tried to formulate the "faith seeking understanding" type of reasoning in the last scenario, but if I could have used better language and/or been more precise, I certainly appreciate correction (even though that isn't the main topic of this thread)! I also realise that "Why do you believe in God?" is not the most precise statement, but it is the way the question of "Why are you a Christian?/Why do you believe in the Christian God?" is usually formulated (in my experience), though it also is the way "Why do you believe God exists?" is also formulated--and "God" in that sense may or may not be the Christian and true God.
Edit:
Peairtach said:
I think that a personal testimony to our salvation that makes intelligent use of the Scriptures to interpret that testimony is not purely subjective, and may be used of God, even although the person does not have a sophisticated or unsophisticated apologetic.
Out of curiosity, what would this look like?