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At what point does a body of evidence become massive enough to count as proof? When has a question been answered enough times that it can be put to rest?

When it comes to the safety and effectiveness of vaccines, it sometimes seems as though public health advocates must constantly roll the burden of proof toward a mountaintop that never comes into view.

The latest salvo against vaccinations came courtesy of Robert Kennedy Jr. and Robert De Niro. At a joint appearance this week, Kennedy offered $100,000 to anyone who could turn up a study showing that it is safe to administer vaccines to children and pregnant women, with a specific call out to concerns about mercury. De Niro was there to lend his endorsement and a patina of Oscar-winning gravitas.

Both men have an unreliable history when it comes to their views about vaccinations. Kennedy's reference to mercury alludes to thimerosal, a preservative once used in vaccines, which he has long maintained can lead to autism. (It doesn't.) A meeting earlier this year between then-President-elect Donald Trump (who has hair-raising views of his own about vaccines) and Kennedy caused grave concern within the medical community, including me. Kennedy claimed Trump asked him to helm a commission on vaccine safety (even though the U.S. already has a vaccine safety commission), but it has yet to materialize.

De Niro came under fire for endorsing a film that purports a link between vaccinations and autism, though instead of mercury, it blames the measles/mumps/rubella (MMR) vaccine. The Tribeca Film Festival, which De Niro co-founded, included the anti-vaccine documentary in its 2016 lineup. After complaints, it was pulled from the schedule, but the actor subsequently touted its message and urged people to see it. Unfortunately, it's clear he hasn't changed his mind since then.

Like most people, I can think of many handy uses for a spare hundred grand, and would gladly sit down and share my experience as a pediatrician with De Niro and Kennedy at great length. It's nearing two decades since I graduated from medical school, and in that span of time I've immunized thousands of patients. Not once have I encountered a case where those immunizations could be plausibly linked with autism.

In the off chance that my word alone isn't sufficient to collect the $100,000, I'm happy to proffer lots of studies that support the safety of vaccines. Studies never seem to settle the question for anti-vaccine activists, but they are the best evidence we could ever have, based on millions of people and using many different types of comparisons, that vaccination is safe for kids.

The explanation for the bogus vaccine-autism link is a constantly shifting target. As noted, both the MMR vaccine and thimerosal have been blamed, and the anti-vaccine movement happily gloms onto both explanations despite the fact that they are completely unrelated. That the various theories never really cohere doesn't seem to give the movement pause. Blurring dark but vague threats, anti-vaccine activists blend them into a miasma through which no given study can hope to penetrate. Uncertainty is good for stoking fear.

When studies show that the MMR vaccine doesn't cause autism, and when the original study suggesting a link is exposed as a fraud? It must be thimerosal! Other studies show no association between thimerosal and autism, and thimerosal isn't even used anymore? The combination of all the vaccines at once is the problem! Produce evidence to support the safety of the current vaccination schedule, and the boogeyman simply adopts another form.

Because much of the evidence in support of vaccines comes from the Centers for Disease Control and Prevention, detractors seize on corruption as an explanation for studies with findings contrary to their beliefs. The anti-vaccine movement affords the CDC roughly as much respect as you'd typically give cardsharps and second-rate grifters, and anything the CDC produces is dismissed out of hand. But even if the CDC were a hotbed of malign pharmaceutical industry influence, that doesn't explain why large studies demonstrating the safety of vaccines come from places like Denmark or the United Kingdom, where the CDC doesn't have a lot of pull.

All of this information is readily available to anyone who chooses to look for it — 350 health organizations recently reaffirmed the safety of vaccines and highlighted more than 40 of the most respected studies in an open letter to President Trump — and yet still Kennedy and De Niro are happy to pretend none of it exists.

Conversely, a growing body of evidence suggests brain differences associated with autism may be found early in infancy — well before children receive most vaccines. Changes in the volume of certain brain areas found by MRI may help predict autism in infants with an older sibling who has the diagnosis, according to a recent study in the journal Nature. Other studies have found that alterations in brain cell development related to autism may occur before birth. These findings are clearly inconsistent with vaccines as a cause of autism.

But none of this emerging research seems to have dampened the fires burning within the anti-vaccine movement. I could resurrect Edward Jenner and Jonas Salk for joint TED talks about the benefits of vaccination, and somehow I doubt it would make any difference at this point. Despite Kennedy's disingenuous plea for evidence of safety, it's not evidence he really cares about. If it were, he could find more than enough for free.

However, if either De Niro or Kennedy read this article and change their mind, I'm happy to take the $100,000 anyhow.

Dr. Daniel Summers is a pediatrician in New England. He wrote this article for the Washington Post.