Christian philosopher Robert Larmer, who specializes in miracles, was kind enough to provide feedback on my Lourdes draft, which I'm posting with permission.
The question is whether in any sample group of 200 million people who pray for miraculous healing, there's a comparable percentage of unexplained cures.
I have no basis upon which to answer that question. I do have a few comments.
1. "Spontaneous remission" is not an explanation of why someone gets better. It is the admission that no explanation is known. It is probable that some events labelled as 'spontaneous remission' are answers to prayer, but that the attending doctors will not countenance a supernatural explanation. I am not claiming there are no spontaneous remissions that have a natural cause.
2. I agree that some events cannot be plausibly thought to be explicable in terms of natural causes.
3. The criteria for an event being called a miracle at Lourdes are extremely strict. Stanley Jaki in his "Miracles and Physics" references a case where a compound fracture, i.e. bones sticking through the skin, was instantaneously healed, but it did not meet Lourdes' criteria for calling something a miracle because a medical doctor was not in attendance. Jaki quotes a commentator to the effect that one does not need to be a tailor to tell if a coat is full of holes.
4. I do not think that healing miracles have to happen at certain special sites, but it does not bother me if God's providence includes people coming to certain locations to experience healing. If I need to be healed then God may require me to exhibit enough faith to go to a healing meeting being held in a certain location.
5. I think God may well perform miracles at Lourdes. That does not to my mind provide strong evidence for Marian doctrine, given that He also performs miracles for people who do not accept Marian doctrine. Both George Whitefield's and John Wesley's ministries were distinguished by events I view as miraculous, but Whitefield was Calvinist and Wesley was Arminian. Miracles are evidence of God's mercy and power, but in His mercy God does not require that we get all our doctrines totally right before He grants a miracle. When Jesus fed the five thousand he did not first ask who accepted him as the Messiah and who did not.