I haven't read his work on MLK, but I've read his book on the JFK assassination, and unless one is a conspiracy nut who feels the need to demonize anyone who pokes holes in favorite conspiracy theories, it's hard to escape the conclusion that in fact Posner is one of the most painstakingly thorough, intellectually honest investigators alive. He's frank about conflicting evidence, things which can't be known, gaps in various theories (including his own), and opposing claims or explanations. He doesn't overstate his case or jump to conclusions. In fact, for the most part he simply spends years gathering massive amounts of evidence (often unearthing long-lost reports or interviews, or finding still-living witnesses to re-interview), then organizing it and presenting it in a coherent manner, letting the reader draw their own conclusions. And thanks to his incredibly thorough job in "Case Closed" (his book on the JFK assassination), his title is entirely justified. I have, quite literally, never seen such an immensely well-researched, comprehensive, convincing book on a historical dispute.
I am not saying he doesn't have some good info, but you should take into consideration the source...
I shall, after having determined for myself that this source is extremely credible, fairminded, and meticulous.