OK, quick stastistics lesson. Suppose you want to know whether a vaccine or covid itself causes long covid, and you're in a country such as Belgium, with about

OK, quick stastistics lesson. Suppose you want to know whether a vaccine or covid itself causes long covid, and you're in a country such as Belgium, with about 11 million inhabitants, of which about 10 million were vaccinated. What you do then is look for some "before" groups. In this case maybe you'd find 10k people who spent a night in hospital for covid and weren't vaccinated, 10k people who spent a night in hospital and had gotten the AZ vaccine, 10 ditto Biontech vaccine, etc. You make sure that the groups are similar (e.g. the same percentage of teenagers, same percentage of overweight people, same percentage of people who play sports, same percentage of people who've been admitted to hospital in the past years, etc). Your assumption that "everyone got the jab" is wrong, even for a country like Belgium: small and with a high vaccination rate. Finding ten thousand unvaccinated people isn't a problem. Then you query the data for the "after", in this case whether people got long covid. If you find out that, say, 30% of the people who had covid and a particular vaccine got long covid, while 30% of the people who had covid and no vaccine got long covid, you then conclude that this vaccine neither protects against nor causes long covid, ie. long covid is entirely caused by covid itself. There are statistical ways to be estimate how confident you can be in a conclusion and lots of other bells and whistles, but the part above is really the core.