Occam's razor is
a scientific and philosophic rule that entities should not be multiplied unnecessarily which is interpreted as requiring that the simplest of competing theories be preferred to the more complex or that explanations of unknown phenomena be sought first in terms of known quantities.
"Occam's Razor" Merriam-Webster.com. Merriam-Webster, n.d. Web. 26 Sept. 2015. (emphasis ours)
In the world of scholarly inquiry, regardless of the field, the degree to which one can say that something has been proven is when the currently available empirical evidence has been examined, tested or replicated by others, and everyone's results agree.
Wouldn't the "only true church," the only church recognized by the one true god, have the most simple, basic answers for proving its truth? Would an all-loving father want trickery and obfuscation when dealing with humans and proving his truths to them? Would it be his way to muddy things and then have humans, to whom he gives logical-thinking brains, use convoluted arguments, with low probabilities of being the real answer, to make things clear? This is a problem the devout and apologists of the Church of Jesus Christ of Latter-day Saints must deal with when trying to prove the truth claims of their church.
The devout often claim that god does not require humans to have proof, but rather faith. Faith is based on belief in the supernatural. When they attempt to argue in behalf of some idea, and things are not going well by appealing to empirical evidence, they may resort to pulling their trump card: "With God all things are possible." (Matthew 19:26) To them, it apparently does not matter how improbable something is because God is omnipotent and can do all things. But, in general, many critics of the LDS Church do not believe in the supernatural and all such appeals are immediately regarded by them as having a low to zero probability of being the answer.
In addition to using "any possibility" as a potential answer, the devout may regard such issues as falling into the realm of "truths that are not very useful" (as explained by Elder Boyd K. Packer, "The Mantle is Far, Far Greater Than the Intellect," Address to the Fifth Annual CES Religious Educators' Symposium, 1981). But if that is the case, why does the LDS Church and its apologists spend so much time, effort and money trying to track down empirical evidence to prop up their positions? For example, if the Book of Mormon is "the most correct book," why does the LDS church and its defenders spend so much time searching for archaeological evidence to support it?
Typically, critics follow the idea of Occam's Razor and contend that the explanation for most LDS policies, positions, doctrine, historicity, etc. that does not appeal to the supernatural and is the simplest explanation, is most likely the correct interpretation. In other words, the truth.
For a single issue that has been identified as problematic for the truth claims of the LDS Church, there is a possibility that x is the answer that conclusively resolves said issue. But, what is the probability, especially when removing supernatural explanations? Especially when the answer becomes more and more complicated?
This leads to the critic using Occam's Razor to claim that the Church is most likely based on fraudulent claims made by Joseph Smith and some of his contemporaries. This creates the fewest problems needing resolved using only natural explanations.
Mormon apologists and the Church accept the following as facts:
1) Joseph Smith practiced polyandry (marrying women who were already married) with at least seven married women. (As acknowledged by the LDS Church in the essay, "Plural Marriage in Kirtland and Nauvoo," footnote 30.)
2) The Book of Abraham is not the literal translation of an ancient papyri as it purports to be in the introduction to the Book of Abraham. (As acknowledged by the LDS Church in the essay, "Translation and Historicity of the Book of Abraham.".)
Applying Occam's Razor, what is the most likely explanations to each of these two issues?
1) Joseph Smith married these women for the purpose of having sexual relations with them.
2) Joseph Smith pretended to "translate" the papyri by simply making up the content.
However, the apologist accepts a priori that Joseph Smith was a man of god and that all of his motives were from god to help him further god's purposes. So the apologist must make the two cases above turn out favorable for Joseph Smith, which could take countless pages to try and explain away this behavior. (To see how apologists attempt to answer these questions, see information about polyandry and also Book of Abraham at the FAIRMormon site.)
There is ample archaeological proof that great civilizations have existed throughout history. It is reasonable to expect that if the Book of Mormon is a historical record of real people living in the Americas, then there should be archaeological proof that they existed and that they used the things the book claims they used. No such definitive proof exists.
Apologists create elaborate narratives for why such evidence cannot be found. Or they take any sliver of possibility and try to make it a probability (for example, see their discourse on Nahom and the NHM inscription; see also Philip Jenkins explanation of why such a discussion does not cut it in the scholarly world).
Although there is no empirical evidence directly supporting the historicity of the Book of Mormon, there is still the possibility that such evidence may one day be found.
If there was one piece of empirical evidence backing the historicity of the Book of Mormon, then there is a probability that that the Book of Mormon is a historical record. The more empirical evidence that can be found, the greater the probability that it is a historical record.
If there is enough incontrovertible empirical evidence, its historical claims will be declared as proven.
But currently there is zero incontrovertible empirical evidence supporting it. The devout cannot accept that, so they continue to grasp at any inkling of a straw that might provide a shred of empirical evidence. And they make convoluted arguments and excuses for why they cannot find the empirical evidence. Or they rely on the supernatural.
Whereas the critic simply applies Occam's Razor and believes that the Book of Mormon was most likely a work of fiction made by Joseph Smith, possibly with the help of others. And the devout would most likely say the same thing about any other work that had an identical track record that was not linked to Mormonism, such as James Strang's Voree Plates or Chris Nemelka's The Sealed Portion: The final testament of Jesus Christ.
Both the probability and possibility of the LDS Church's truth claims continue to diminish as more issues that have improbable natural (as opposed to supernatural) explanations accrue.