How do insurers justify raising premiums after no-fault claims?

The probability of a claim is based on any and every factor they can use -- if statistics say that people who own dogs are more likely to make a claim, they'll consider adding a question about dog ownership to the insurance application.

By far the strongest indication that someone will make a claim in the future is that they've made a claim in the past. This covers all sorts of "hidden factors" that the insurance company can't account for with the other statistics they look at. For example, maybe there's something about how you park that makes it harder for others to judge where your car is, or maybe you've surrounded your house with eucalyptus trees.

At least here in the US there's also another factor at play: Many insurance companies offer claim-free discounts. Making a claim means you're no longer claim free. (Or there can be a middle ground--with my auto insurance if they recover every dollar they pay me then it's not counted as a claim. When an idiot didn't look and hit me I chose to fight it myself because they could easily end up in a position where something small wasn't worth the time and thus not recover every dollar. Clear-cut fault is irrelevant.)

From your question and comments, I think you may be incorrectly reasoning about probability and specifically over-generalizing from the rule "occurrence of random events don't predict anything about future events".

Let's take a simple demonstration with dice, which hopefully shows how the insurer thinks about random events. Let's say we have 100 6-sided dice, and we've done everything we can to make sure they're all identical, short of actually rolling them. E.g., they all answered the same to various questions like age, income, etc.

We did a pretty good job, so let's say 99 of these dice are completely normal, fair dice, and one of them is bad and always rolls a 6. And we'll play a game where rolling a 6 is the same as having an insurable event and making a claim to the insurance company.

It's useful to think about what that "bad die" represents -- this is a person that looks the same as a normal person, but for whatever reason just makes a lot more claims. Maybe they're a jerk that lies about no-fault claims, maybe they keep parking their car under a rock-slide-prone area and can't be convinced otherwise.

Anyways, let's roll all the dice. As expected, of the 99 normal dies, around 16.5 should roll a 6, let's say exactly 16 do, and of course the bad die also rolls a 6. These dice all "make a claim" to the insurance company.

Now, what rates should we be charging to the 2 different groups: the 17 dice that rolled a 6 (we still don't know which one is the bad die), or the 73 dice that rolled something else? For the 73 dice that haven't filed a claim, the probability of filing a claim next time is still 1/6 = 16.7% as usual.

For the 17 that did file a claim, the expected number of claims on the next round is (1/6 * 16 + 1) / 17 = 21.5%. That difference in probability definitely matters to the insurance company who has millions of customers.

Overall, note that the probability of an individual die didn't change, and for a non-rigged die, nothing about the past rolls says anything about the future. That is, being struck by lightning didn't make anyone more likely to get struck by lightning in the second round. However, merely by filing a claim, you necessarily display externally-visible signals which are the same as people who are very expensive to insure (e.g., those who file a lot of claims), so to the insurer, it makes sense to lump you in with everyone who filed a claim.