The article is entitled Religion in the Life, Thought, and Presidency of James Madison. There's much good in it and the entire thing is worth a read (please do). However I am going to focus on the part that discusses Madison's personal creed.
First Munoz summarizes the enigmatic dynamic to Madison's religious beliefs. The following seems to be the most telling passage:
The disagreement over Madison’s personal faith results, in part, from the fact that after 1776 Madison wrote almost nothing about his religious convictions—in the words of William Lee Miller, “he kept his mouth shut” about his religious beliefs.9
Keeping one's mouth shut (as did GW) is a telltale sign that someone has something to hide (i.e., unnacceptably heretical-heterodox beliefs; the very same beliefs that J. Adams, Jefferson and Franklin explicated in detail in their private letters).
Of course Munoz notes Madison's letter William Bradford, 25 September 1773 that Christian America proponents oft-cite where Madison supports his friend's desire to enter the ministry. As Madison wrote:
I have sometimes thought there could be no stronger testimony in favor of Religion or against temporal Enjoyments even the most rational and manly than for men who occupy the most honorable and gainful departments and are rising in reputation and wealth, publicly to declare their unsatisfactoriness by becoming fervent Advocates in the cause of Christ, & I wish you may give in your Evidence in this way.
But as Munoz aptly observes: "Such statements disappear from Madison’s writings after 1776. Whether he maintained his belief in an afterlife beyond his youth he does not say...."
Although Munoz doesn't cite him, Bishop Meade who knew Madison personally thought at around this time (1776) Madison ceased strict adherence to orthodox Christianity based on his political associations other "infidel" Founding Fathers (he doesn't mention them but was certainly referring to at least Jefferson):
His political associations with those of infidel principles, of whom there were many in his day, if they did not actually change his creed, yet subjected him to the general suspicion of it.
Meade also notes his personal impression of Madison was that he didn't strictly believe in the Bible:
I was never at Mr. Madison’s but once, and then our conversation took such a turn—though not designed on my part—as to call forth some expressions and arguments which left the impression on my mind that his creed was not strictly regulated by the Bible.
See the original here.
Munoz's most interesting analysis is of James Madison's letter to to Frederick Beasley, dated 20 November 1825 where Madison appeals to the Arian Anglican Divine Samuel Clarke (NOT John Witherspoon) for authority and otherwise speaks as a dyed in the wool philosophical rationalist. As Madison wrote:
DEAR SIR I have duly recd the copy of your little tract on the proofs of the Being & Attributes of God. To do full justice to it, would require not only a more critical attention than I have been able to bestow on it, but a resort to the celebrated work of Dr. Clarke, which I read fifty years ago only, and to that of Dr. Waterland also which I never read. . . .
The finiteness of the human understanding betrays itself on all subjects, but more especially when it contemplates such as involves infinity. What may safely be said seems to be, that the infinity of time & space forces itself on our conception, a limitation of either being inconceivable; that the mind prefers at once the idea of a self-existing cause to that of an infinite series of cause & effect, which augments, instead of avoiding the difficulty; and that it finds more facility in assenting to the self-existence of an invisible cause possessing infinite power, wisdom & goodness, than to the self-existence of the universe, visibly destitute of those attributes, and which may be the effect of them. In this comparative facility of conception & belief, all philosophical Reasoning on the subject may perhaps terminate.18
And Munoz's analysis of said passage:
Madison posits that philosophical reasoning can deduce two possible alternatives to explain the cause of existence: an invisible self-caused cause that itself is the cause of all that exists or, alternatively, the infinite self-existence of the universe. The mind, he says, “prefers at once” the former. It “finds more facility” in assenting to belief in an invisible cause possessing “infinite power, wisdom, and goodness” than it does to the self-existence of the universe without such attributes.
But why? Why, we might ask, does the mind prefer the self-existing cause possessing infinite power, wisdom, and goodness?...
Madison says the possibility of an infinite series of cause and effect “augments, instead of avoiding the difficulty.” Perhaps Madison means to suggest that belief in the eternal existence of the universe with an infinite series of cause and effect fails to offer a satisfactory resolution to the question of how existence itself came into being since our finite minds struggle to contemplate infinity. If this is correct, then it is the finiteness of our minds that leads it to prefer belief in an invisible self-caused cause over the eternal existence of the world—that is, Madison does not claim that reason itself sides with belief in an invisible cause possessing infinite power, wisdom, and goodness over belief in the eternal existence of the world. This conclusion would seem to be confirmed by Madison’s statement that “in this comparative facility of conception & belief, all philosophical Reasoning must terminate.” Madison suggests that philosophical reasoning alone cannot arbitrate between the possibility of the eternity of the world and the existence of a self-caused cause. In short, Madison’s position seems to be that reason suggests the possibility of but does not confirm the existence of a creator God possessing infinite power, wisdom, and goodness.
Strikingly, what we do not find in Madison’s writings in an explicit appeal to Scripture to decide the question. We have copies of the notes Madison took from his study of the Bible as a young man,19 but as far as I can tell, Madison never cites Scripture to resolve questions pertaining to the existence or nature of God.
Munoz then notes one time in the Federalist Papers (Federalist 37) where Madison alluded to revelation, and, according to Munoz, "Madison seems to question the certainty with which man can apprehend the meaning of divine revelation." As Madison wrote:
When the Almighty himself condescends to address mankind in their own language, his meaning, luminous as it may be, is rendered dim and doubtful by the cloudy medium through which it is communicated.20
Examining the available evidence, Munoz concludes:
On theological matters, Madison was first and foremost a rationalist. The starting point (and perhaps the end point) of his reflections seems to have been unaided philosophical reasoning—not so much reason aided by faith but human reason simply. [Bold mine.]
Munoz notes there is a strong mystery in how unaided philosophical reason lead Madison to belief in God's existence, but simply, that it did: "In the aforementioned response to Beasley, Madison also states,
"But whatever effect may be produced on some minds by the more abstract train of ideas which you so strongly support, it will probably always be found that the course of reasoning from the effect to the cause, 'from Nature to Nature’s God,' Will be the more universal & more persuasive application.21
"Madison seems to reveal the type of reasoning that he himself found most persuasive—'from Nature to Nature’s God.'"