This story has been going on for years. In a 2007 article by CBS, they state, "Nobody makes the claim that all ADD and autism cases are caused by the mercury in vaccines. But many researchers believe it plays a large role in our epidemic of the 1990's." http://www.cbsnews.com/stories/2004/06/22/eveningnews/main625458.shtml
Adding fuel to the fire, Michele Bachman said on NBC's "Today" show this pat month that a woman "told me that her little daughter took [the HPV vaccine, Gardasil], that injection, and she suffered from mental retardation thereafter." http://www.politico.com/news/stories/0911/63369.html
While this claim has been disputed in many studies, the claim cannot properly be falsified. It is not ethical to randomly choose two groups and give one certain vaccinations and not give any to the other group. Additionally, it is not possible to know ahead of time which people in either of those chosen groups might have ended up with an illness anyways. The claim is also implying causation where it cannot be proven. Correlation on the other hand might be more accurate. And a third factor must be taken into consideration when looking for causation. As the population has grown exponentially, so have the number of people who receive vaccines, and so has the number of people who suffer from autism, Asperger syndrome, and other illnesses that people could be incorrectly linking to vaccines.
I believe that people are victim to Illusory Correlation, but decidedly calling it causation. People who believe in the claims are not looking at the fact that millions of people have received all of the required and recommended vaccinations and do not have any illnesses. It is easier to decide that the illness is the end result of the vaccine when an one is diagnosed after a vaccine is given, rather than the absence of an illness being the typical end result.
Nice application of the principles of scientific thinking. I think there is a lot of anecdotal evidence (used by Michelle Bachmann in the debate) that people use to provide evidence for different claims. You also point to the idea that we are only seeing the cases where it occurs, and not looking at all of the times it doesn't. It's a hard thing to think about, as a future parent. The fact is, we don't really know, so that causes a lot of people to stay on the "safe" side.
What did you think about the study with the rats? It is true that we probably cannot do an experimental study with people and not give vaccines to some groups and give it to others...I'm not sure if it is possible to do that or not. But we can do it with animals. Do you think that provides evidence?