AcK I think you made a good point about it not being clear whether the effect of personal prayer is placebo as believed by skeptics or divine as believed by the religious. However I think to solve this we can look at prayer studies where the recipient of prayer is not aware they are being prayed for. Surely this shouldn't matter to God, yet there is no noticeable difference in the recovery of the prayer recipient if he does not know he is being prayed for.
I don't know if this is an actual study that exists, but I can see logically how, if it does, it could cause one to favor the placebo idea. But what I also think, is that God is not likely a willing lab rat who doesn't realize people might be doing studies on his ways. Imagine if you were the supreme being. What would you think of this? If you were the supreme being that the Atheists think you should be, then obviously you would cooperate with the study in hopes that more people would believe in you. But in my experience, and that of most people, God (assuming he exists) is not the God that the Atheists think he should be, agreed? He seems more than willing to allow people to continue in disbelief, if that is what they want.
In support of this there is Deuteronomy 6:16: "You shall not put the Lord your God to the test, as you tested Him at Massah." This was also quoted by Jesus when tempted by the Devil.
Also Matthew 14:1-4 The Pharisees and Sadducees came to Jesus and tested him by asking him to show them a sign from heaven. He replied, "When evening comes, you say, 'It will be fair weather, for the sky is red,' and in the morning, 'Today it will be stormy, for the sky is red and overcast.' You know how to interpret the appearance of the sky, but you cannot interpret the signs of the times.
A wicked and adulterous generation looks for a miraculous sign, but none will be given it except the sign of Jonah." Jesus then left them and went away.
Also try a thought experiment. Shows like the 700 Club regularly claim to have prayer-cured people of cancer, but they never cure people of obvious afflictions, like a loss of a limb. Surely this is within God's power yet you know that if every single good Christian prayed for someone's leg to grow back it would never happen. Cancer is a much deadlier problem that God supposedly cures all the time, why cannot he not grow back a limb?
Well, I cannot say that I believe what I see on the 700 Club. I think there are a lot of people out there making money by manipulating people and their desire to know God. God does things for reasons, and I don't know what they are. I believe that God could grow a limb back, and can cure cancer, but I do not know what his reason or timing are all about. You hear all around people who claim to have been cured, but you apparently don't have a bunch of public documentation of these things. There are instances in the NT when Jesus heals someone and tells them not to go telling everyone. Why is that? I don't exactly know, but you can see the consistency. It seems to me God is primarily interested in individuals, and is not interested in giving skeptics a sign. The Pharisees were right there and saw so much of what he did, and continued to choose to disbelieve. I do not think a sign is even close to enough to convince someone of something they don't want to believe, and I don't think God is interested in doing that anyway.
In terms of prayer I tend to agree with Feuerbach who saw prayer as humanity giving ear to its own desires.
What about a man who prays for humility? A man who prays for his enemies? A man who is sick praying for healing, but adding in that he pray for God's will above all, regardless of what it is? I think you and Feuerbach are confused as to what Christianity teaches.