I only had time to double-check one of the scary links at the top, and I wasn't too impressed with what I found:

In 2010, a careful review showed that published industry-sponsored trials are four times more likely to show positive results than published independent studies, even though the industry-sponsored trials tend to use better experimental designs.

But the careful review you link to claims that studies funded by the industry report 85% positive results, compared to 72% positive by independent organizations and 50% positive by government - which is not what I think of when I hear four times! They also give a lot of reasons to think the difference may be benign: industry tends to do different kinds of studies than independent orgs. The industry studies are mainly Phase III/IV - a part of the approval process where drugs that have already been shown to work in smaller studies are tested on a larger population; the nonprofit and government studies are more often Phase I/II - the first check to see whether a promising new chemical works at all. It makes sense that studies on a drug which has already been found to probably work are more positive than the first studies on a totally new chemical. And the degree to which pharma studies are more likely to be late-phase is greater than the degree to which pharma companies are more likely to show positive results, and the article doesn't give stats comparing like to like! The same review finds with p < .001 that pharma studies are bigger, which again would make them more likely to find a result where one exists.

The only mention of the "4x more likely" number is buried in the Discussion section and cites a completely different study, Lexchin et al.

Lexchin reports an odds ratio of 4, which I think is what your first study meant when they say "industry studies are four times more likely to be positive". Odds ratios have always been one of my least favorite statistical concepts, and I always feel like I'm misunderstanding them somehow, but I don't think "odds ratio of 4" and "four times more likely" are connotatively similar (someone smarter, please back me up on this?!). For example, the largest study in Lexchin's meta-analysis, Yaphe et al, finds that 87% of industry studies are positive versus 65% of independent studies, for an odds ratio of 3.45x. But when I hear something like "X is four times more likely than Y", I think of Y being 20% likely and X being 80% likely; not 65% vs. 87%.

This means Lexchin's results are very very similar to those of the original study you cite, which provides some confirmation that those are probably the true numbers. Lexchin also provides another hypothesis for what's going on. He says that "the research methods of trials sponsored by drug companies is at least as good as that of non-industry funded research and in many cases better", but that along with publication bias, industry fudges the results by comparing their drug to another drug, and then giving the other drug wrong. For example, if your company makes Drug X, you sponsor a study to prove that it's better than Drug Y, but give patients Drug Y at a dose that's too low to do any good (or so high that it produces side effects). Then they conduct that study absolutely perfectly and get the correct result that their drug is better than another drug at the wrong dosage. This doesn't seem like the sort of thing Bayesian statistics could fix; in fact, it sounds like it means study interpretation would require domain-specific medical knowledge; someone who could say "Wait a second, that's not how we usually give penicillin!" I don't know whether this means industry studies that compare their drug against a placebo are more trustworthy.

So, summary. Industry studies seem to hover around 85% positive, non-industry studies around 65%. Part of this is probably because industry studies are more likely to be on drugs that there's already some evidence that they work, and not due to scientific misconduct at all. More of it is due to publication bias and to getting the right answer to a wrong question like "Does this work better than another drug when the other is given improperly?".

Phrases like "Industry studies are four times more likely to show positive results" are connotatively inaccurate and don't support any of these proposals at all, except maybe the one to reduce publication bias.

This reinforces my prejudice that a lot of the literature on how misleading the literature is, is itself among the best examples of how misleading the literature is.