I’m going to reorder your points a little bit as I address them:
I claim that prayer has changed me. Can I measure that change? I’m not sure how to measure it, but I believe I am a different person than what I would be if I did not pray. I believe I am a different person from the person I was before I prayed. How can I prove it? Why must I prove it? The point is I believe it, and if I believe that prayer has changed me, then of course it is distinguishable from what I think I would be if I did not pray
Yes, of course prayer has changed you, just as meditation changes a Zen Buddhist, just as initiation rituals change the tribal individuals who underwent them, and just as finding a rare stamp changes the person who collects stamps.
Everything is changing all the time, and of course every act that we do has the effect of “changing” us in some regard (usually in tiny, unnoticeable ways). Ritualized acts that the mind attaches siginificance to will, naturally, produce changes that the mind is more impressed by.
No one is denying that ritualized behavior of all kinds has an effect on the mind and perhaps, correspondingly, the behavior.
The issue at stake here, however, is not just whether prayer produces psychological/behavioral effects, which is exactly what we would expect if prayer were nothing more than talking to yourself or meditating. The issue at stake is whether prayer can cause change to occur in the world through some supernatural causal means (such as, for example, through the intervention of a supernatural god). This is explicitly the topic of the thread, since the OP is asking how we can distinguish “granted” prayers (“I prayed to find my car keys and then I found them!”) and coincidence (you would have found those keys whether you prayed or not).
Why do you maintain that if you can’t demonstrate that prayer has a measureable effect, then it’s indistinguishable from prayer not working at all?
You’ll see the point pretty easily if we replace “prayer” with pretty much anything else.
For example, let’s say that I claim that doing a rain dance causes it to rain. So I do a rain dance one day, and within fifteen minutes after I finish, it starts to rain. Question: am I really justified in believing that it was
my raindance that caused the rain? Or could it be that it was just a coincidence?
This one event cannot itself answer this question; if we don’t want to avoid mistaking a coincidence for a causal event, we need to test it.
What I’m saying is that a claim about affecting the external world, like “This rain dance can make it rain” needs to be able to demonstrate its claim – and in this case, since we can measure the results very accurately, it needs to demonstrate them to a statistically significant degree – before we can accept it. Imagine if I did a rain dance every day for a specified period, and we recorded the results: some work and some don’t (and, of course, I’ll say that “The conditions weren’t right” or “My power was weak” or some other excuse when the ritual doesn’t work): in the end, we find that the amount of rainfall that came over that period is within the amounts of rainfall that we would expect from weather data.
So I claim that this rain dance has an “effect” – and indeed, sometimes I do the dance and it rains shortly thereafter – but when we actually take the time to examine the claim, we find that the amount and pattern of rainfall that “results” from my dance is more or less exactly what we would expect if I had never done the dance at all.
That’s what I mean by “indistinguishable from not working at all.” Now just replace the rain dance with prayer, and you’ve got the crux of my argument.