The Center for Disease Control's page on Autism and Vaccines received significant online attention after an update last week. It struck a nerve with me too based on my research history of struggling to prove the absence of something. (Perhaps these struggles are worth writing about some day.)
The claim "vaccines do not cause autism" is not an evidence-based claim because studies have not ruled out the possibility that infant vaccines cause autism.
Good scientists would not make a claim like this because it would require proving a negative: something doesn't exist. There's a huge barrier to proving the non-existence of something. Unlimited evidence would be required.
Statistics has a term for infinite evidence. These are called populations. The reality though is that infinite evidence is almost never available. Scientists collect data from samples, small subsets drawn from the population. The sample evidence is necessarily limited because only a portion of the population was tested.
Framing this issue in terms of likelihood or probabilities is a healthier way of thinking. Is a link possible? Sure. That's the realm of pure abstraction though. Is a link likely? Given our current evidence, the vaccine-autism connection seems unlikely.
Following the same logic, we can't provide 100% certainty about the non-existence of other phenomena. Can we prove the non-existence of bigfoot, aliens from outer space, and the Loch Ness monster? No. There's always that possibility that bigfoot is hiding in a cave of the remote Canadian wilderness. We just haven't found him yet. The better question: Are these creatures likely to exist? Given past hoaxes and lack of solid evidence, the answer is no.
A better scientific statement is positive: Is there evidence that vaccines cause autism? So far, the evidence has been little to none. It's not for lack of effort. This topic has been the subject of significant scientific research. A Google Scholar search for vaccine and autism yields 80,000+ hits. If there was a significant link, positive evidence should have been found by now.
The lack of compelling evidence for a vaccine-autism link is not a strong justification for continuing more research on this topic. There is a very real danger that valuable time and energy get spent on fruitless investigations.
Productive thinkers need to be looking for positive evidence. When the positive evidence is missing, we cannot conclude that the phenomenon is absent, yet we must also be mindful that it could be truly absent. In other words, don't waste your time looking for bigfoot!