The highest recognition that the Catholic Church gives to an alleged miracle is that it is "worthy of belief." Investigations of reported miraculous events (which, in the case of weeping statues, typically includes DNA testing of the tears, among other things) may result in a rejection if the event is determined to be fraudulent or lacking in supernatural character.
Alternatively, the Church may declare that there is nothing contrary to the faith in a supposed miraculous phenomenon – but without making a determination on whether a supernatural character is present.
"The Catholic Church is very cautious with these matters and employs science where possible to ferret out hoaxes and other non-supernatural explanations," said "Miracle Hunter" Michael O'Neill, who extensively researches Catholic miracles.
"Tears are collected and tested to see if they are human (pig's fat has been found in some false cases) and statues may be x-rayed to rule out any internal mechanism used to fraudulently mimic the flow of tears," O'Neill told CNA in e-mail comments.
"Some weeping icons have been shown to have natural causes - condensation or leaking ductwork in the wall behind them. On a few very rare occasions these lachrymations (tears) have been found to have no explanation and are worthy of belief as being miraculous."
There have been many claims of weeping statues or icons of Mary and other saints throughout history, with a few of them being deemed worthy of belief by the Church.
One of the best known and most recent examples occurred in Syracuse, Italy in 1953. An Italian woman, Antonina Janusso, was cured from pains while witnessing a weeping plaque of the Madonna in the home of the parents of her husband, Angelo.
The tears purportedly were the source of many miracles throughout Italy. In a 1954 radio message Venerable Pius XII approved of the miraculous weeping after the tears were found by four different doctors to be human.
Another more controversial case occurred in the 1970s and early 1980s in Akita, Japan, where Sister Agnes Sasagawa and her order, the Handmaids of the Eucharist, reported 101 lacrimations (flows of tears) emanating from a bleeding, weeping wooden statue of Mary seen by hundreds of witnesses in various instances.
Tests from Christian and non-Christian doctors found the blood on the statue to be type B and the sweat and tears type AB. The nun's claims, including the messages, were originally rejected by her archbishop, but then accepted by the local bishop, John Shojiro Ito of Niigata, who on April 22, 1984, after years of extensive investigation, declared the tears to be of supernatural origin and authorized veneration of the Holy Mother of Akita. The Vatican has not issued a formal statement on the matter.
Unlike Marian apparitions, where the Blessed Virgin appears to a member of the faithful with a message, weeping statues require the faithful to seek their own interpretations of the miracle, O'Neill stated.
"Weeping statues often cause quite a stir and inspire people to reflect on the meaning of such a phenomenon," he said.
While waiting for the result of official investigations from the Vatican, O'Neill said that the faithful can pray and reflect on the tears as a symbol of suffering.
"As is the case of all miracles, the purpose is certainly to draw people closer to Christ and considering the image of the Sorrowful Mother's tears, it is sensible to reflect on the suffering and death of Jesus Christ and as well as our own sins. Such prodigies can foster introspection in all of us and can bring us toward a conversion of heart."
Update 7/31/17: A previous version of this story said that Sister Agnes Sasagawa received 101 messages, rather than reported 101 lacrimations, from the statue of Mary. It has since been corrected.