Someone claiming that they have evidence for a thing is already evidence for a thing, if you trust them at all, so you can update on that, and then revise that update on how good the evidence turns out to be once you actually get it.

For example, say gwern posts to Discussion that he has a new article on his website about some drug, and he says "tl;dr: It's pretty awesome" but doesn't give any details, and when you follow the link to the site you get an error and can't see the page. gwern's put together a few articles now about drugs, and they're usually well-researched and impressive, so it's pretty safe to assume that if he says a drug is awesome, it is, even if that's the only evidence you have. This is a belief about both the drug (it is particularly effective at what it's supposed to do) and what you'll see when you're able to access the page about it (there will be many citations of research indicating that the drug is particularly effective).

Now, say a couple days later you get the page to load, and what it actually says is "ha ha, April Fools!". This is new information, and as such it changes your beliefs - in particular, your belief that the drug is any good goes down substantially, and any future cases of gwern posting about an 'awesome' drug don't make you believe as strongly that the drug is good - the chance that it's good if there is an actual page about it stays about the same, but now you also have to factor in the chance that it's another prank - or in other words that the evidence you'll be given will be much worse than is being claimed.

It's harder to work out an example of evidence turning out to be much stronger than is claimed, but it works on the same principle - knowing that there's evidence at all means you can update about as much as you would for an average piece of evidence from that source, and then when you learn that the evidence is much better, you update again based on how much better it is.