I haven't read The Design Inference, but I have printed out several pages of his seminal thesis (upon which the book and his subsequent writing about Complex, Specified Information is based) for reference. When I first started reading the arguments from the Intelligent Design community I was immediately interested in their claim (based on Dembskis claim) that it is being used in Forensic Science and Archaeology. It was a short order for me to cook up an argument probing my interlocutors as to exactly how the Design Inference would differentiate between four scenarios. Needless to say, they merely told me I should read the Design Inference (a seeming tacit admission that they haven't read it, or at least haven't understood it). Taking their bait, that's how I came to accessing Dembskis thesis (my library didn't have a copy of The Design Inference, but did have a copy of The Design Revolution which I was unimpressed by, to say the least). I've tried my best to comprehend it, but as I mentioned even Dembskis academic peers have criticized his writing for being misleading or hard to comprehend (and not because it's a difficult topic). Suffice it to say, those actually involved in Forensics and Archaeology aren't impressed. See: Chapter 8, written by Gary S. Hurd in 'Why Intelligent Design Fails'.
From my surface understanding of the Design Inference, it is nothing more than a Designer of the Gaps argument backed by bad math. His main thesis seems to be "it is too improbable, therefore magic". Of course, internally his whole thesis hinges on getting the probabilities right, and from what I have seen he uses the Sharpshooter fallacy where probabilities are calculated after an event has occurred (which is the wrong way to do it).
In a nutshell, if we look at evolutionary pathways down to the fine grained level of individual mutations then every evolutionary pathway is extremely improbable. The problem for Dembski's thesis is that the mere fact of the arrow of time guarantees that a highly improbable evolutionary pathway will occur due to the simple fact that mutations occur in large populations of organisms. It is equivalent to a sharpshooter painting the bullseye over his bullet hole.
If you look at Mt Rushmore (a favourite ID example) you don’t go I can’t think of a natural explanation, it must be designed - but you might well go humans carved that to honour famous people
If an archaeologists is digging in the ground and finds a potshard and an earthworm, which one does he take back to the museum as evidence of an intelligence? The potshard. If Dembski wants to claim that archaeologists are using the Design Inference, then he needs to explain why this is.
Yes, that is a potential flaw of the method but I haven’t seen Dembski do a good enough job of applying his method for that to be a real issue in practice. If Dembski ever gets to calculating the probabilities of evolutionary paths (and he may have, I haven’t looked at his work in the last few years) that might be an issue. But I haven’t seen him even try it, even when he should.
Then it ends up being a moot point. It would seem that the Explanator Filter is incapable of explaining biology, the very thing that ID is focused on.