If you have varying account sizes, you know right away the price isn’t an evenly weighted average of beliefs, but I don’t see a huge problem with what Nate Silver wrote given his purposes. He just said they were proportional. Is someone claiming that market prices are *inversely* proportional to traders’ perceptions? Otherwise, this specific point borders on pedantry even if the broader point that market structure frustrates any interpretations of price is correct. Also I don’t really agree with the statement that the current price “tells us nothing about the beliefs of traders who are not party to this transaction.” Presumably if the other traders thought the last price was horribly off, they would move it. I agree that you can’t infer much about beliefs from the orders that are away from the market because often that is just posturing.
Some of this was addressed in what was probably the first skeptical PM paper per se, and subsequent discussion:
http://www.aeaweb.org/annual_mtg_papers/2006/0106_1015_0703.pdf
