Charles N. Pope also gave me this identity "bombshell" for this thread:
You got me going on this! If true, it would be quite a bombshell indeed. But there are some problems....
...both Louis and Maria Josepha "passed away" young, in the mid-1760's in France, when George and Martha were living in comparative obscurity at Mount Vernon.
"Comparative" obscurity perhaps, but George and Martha were not obscure figures in colonial Virginia. George was a renowned military hero in the French and Indian War as commander of the Virginia Regiment, with 1000 men under his direction. The Washington family were land speculators and slave owners, but George's fortune was basically dwarfed by Martha's empire. When she married George, Martha was a widower who owned or managed 17,500 acres and 300 slaves, which were inherited from her first husband. Their wedding in 1759 was a grand society affair.
Or in other words, George and Martha were two of the most wealthy and famous young folks in Virginia in 1765, when the Dauphin of France passed away. And furthermore, the paintings of George Washington don't look anything like the paintings of Louis the Dauphin. Martha has more of a resemblance to Maria Josepha, but Martha's hair is much darker.
From this point forward, the Bayesian analysis looks like common sense. Given H1=George Washington and Louis Dauphin were two distinct people, then all of the historical details about both Louis and George in France and Virginia respectively are reasonably plausible, P(E|H1)~=1. Whereas under H2=George and Louis were the same person, we have to account for how all of George's many acquaintances in Virginia failed to notice that George had suddenly been replaced by Louis. We need to postulate an enormous conspiracy to cover this up. Let's be generous, P(E|H2) ~ 0.1.
Now let's look at the evidence given in favor of H2.
The two couples were about the same age. This does nothing to discriminate between H1 and H2. Married couples tend to be about the same age, and lots of couples were born in the early 1730's. P(E|H1)=1, but also P(E|H2)=1.
Louis XVI wanted to help George during the war. But this would have been true for geopolitical reasons in any case. Of course the French wanted to promote rebellion in the colonial empire of arch-rival Britain. P(E|H1)=1, but also P(E|H2)=1.
Washington loved Lafayette as his son. But Washington and Lafayette fought together in the Revolutionary War, as heroic figures. Washington was old enough to have been Lafayette's father. It seems reasonable (although not inevitable) that Washington would have loved Lafayette as a son, even if this were not literally true. And a quick Google search could not turn up any evidence that Lafayette's mother, Marie Louise Jolie de La Rivière, was in any position to have an affair with the Dauphin; nor that she was anything other than a devoted wife to her husband. How common is paternity fraud in the French nobility? If "George" really was Louis, and Louis was Lafayette's father, then of course "George" loved the Marquis. But was Lafayette actually Louis's son? Probably not. I would score this as P(E|H1)=.5, but maybe P(E|H2)=0.4 at best.
Louis Dauphin and Maria Josepha were more pure blue blood than George and Martha, who were (allegedly!) from colonial upstart families. So perhaps it's unlikely that George and Martha would take such historically important roles. But one could argue just as easily that Louis Dauphin and his wife would have been slumming in the Virginia Colony, considering that they could have just as easily been King and Queen of France.
Maybe we should be suspicious that George and Martha were higher born than they seem at first glance? I score P(E|H1)=.5, but also P(E|H2)=0.5, no difference.
How to evaluate the prior probabilities? It seems to me that with hundreds of millions of people in the world in the 18th century, including many hundreds of nobility and blue bloods, the prior probability that George and Louis were two different people must be reasonably high. Whereas the probability of identity hoaxes seems pretty low to me. In fact I can't think of a single proven and historically acknowledged example of an identity hoax of this magnitude. P(H1)~=1, P(H2)<0.01 ?? Charles Pope might argue that my low estimate of P(H2) is unfair, and he might say that there are many, many examples of such frauds in ancient history. But we have to start somewhere, by finding examples where the evidence is strongly supporting the hoax.
I won't belabor the point by plugging in my estimates into Bayes' equation and multiplying out the factors.
And I feel that the analysis would be very similar for Anthony vs. Herod the Great, although the amount of surviving evidence is smaller and lower quality.