I just found this again, a nice summary of the method by which we can establish that Shakespeare really did write Shakespeare, and which illustrates some of the ideas I developed in my last post.
The method is to construct a number of ‘description clusters’, and then demonstrate that these clusters can only be satisfied by a single historical individual. A description cluster is a number of sub-descriptions of which it is beyond question that they are satisfied by a single individual. Thus we have the ‘William Shakespeare the author’ cluster, consisting of the subdescriptions “Shakespeare the author of Titus Andronicus”, “Shakespeare the author of Henry VI Part 2”, “Shakespeare the author of Romeo and Juliet”, etc. We know (or are at least very certain) these sub-descriptions apply to the same person, because those plays were published with the name ‘William Shakespeare’ in his own lifetime.
Then we have the description cluster William Shakespeare the actor, consisting of the subdescriptions ‘Shakespeare who played in the King’s Men, 19 May 1603’, ‘Shakespeare who appeared on Sir George Home’s list of "Players"’ etc. We are very certain they are uniquely satisfied because it is unlikely the same company would have had two actors of exactly the same name.
Similarly we have description clusters for ‘William Shakespeare the Globe-sharer’, consisting of descriptions found in legal titles to shares in the globe theatre. Finally we have a cluster for ‘William Shakespeare of Stratford’. Each of these clusters is satisfied by a single person. To prove that they are all satisfied by a single person, rather than four separate people, we look for little overlaps, where it seems highly like that two or more sub-descriptions in different clusters are satisfied by a single person. For example, a sub-description in the ‘William Shakespeare of Stratford’ cluster contains the information ‘is legally entitled to be called “gentleman”’. So does a subdescription in the ‘William Shakespeare the Globe-sharer’ cluster. Similarly, a subdescription in ‘William Shakespeare of Stratford’ matches a subdescription in the ‘William Shakespeare the author’ cluster.
The available evidence, at least if the website linked to is correct, is that because of a significant number of overlapping subdescriptions, the four big clusters are satisfied by a single person. There is actually a fifth cluster (or perhaps it's a subcluster of the 'Will Shakespeare of Stratford' cluster) of descriptions attached to Shakespeare's monument in Stratford church. These refer to Shakespeare (i.e. the subject of the monument) as a great writer within a few years of his death.