They have different priors and according to CDC a prior exposure to Covid yields better protection than vaccine alone, and vaccine in addition to prior exposure does not yield statistical significant reduction in hazard ratio.

> They have different priors and according to CDC a prior exposure to Covid yields better protection than vaccine alone, and vaccine in addition to prior exposure does not yield statistical significant reduction in hazard ratio.

I wouldn't read too much into that:

https://apnews.com/article/coronavirus-pandemic-science-heal...

> The new study’s findings do make sense, said Christine Petersen, a University of Iowa epidemiologist. She said a vaccine developed against an earlier form of the coronavirus is likely to become less and less effective against newer, mutated versions....

> Another thing to consider: The “staunchly unvaccinated” aren’t likely to get tested and the study only included lab-confirmed cases, Wherry said....

> CDC officials noted other limitations. The study was done before the omicron variant took over and before many Americans received booster doses, which have been shown to dramatically amplify protection by raising levels of virus-fighting antibodies. The analysis also did not include information on the severity of past infections, or address the risk of severe illness or death from COVID-19.

> — 6-fold lower in California and 4.5-fold lower in New York in those who were vaccinated but not previously infected.

> — 29-fold lower in California and 15-fold lower in New York in those who had been infected but never vaccinated.

> — 32.5-fold lower in California and 20-fold lower in New York in those who had been infected and vaccinated.

So 29 fold lower rates in California for unvaccinated with prior infection, how much of that do you want to attribute to behavior vs actual immunity? Why is it so unbelievable that getting infected actually gives immunity?

Except when they are not similar, that is. This is more common than you seem to think, e.g. TBE - tick-borne encephalitis - is a nasty piece of work of a disease which can leave the victim with permanent disabilities (around 25%), dead (around 2.5% for the strain common here in Sweden, far higher for the strain endemic in Russia), cognitively impaired and more. I got it in 2005 (which made me the "index patient" for this part of the country, it had not been seen before in this area) and it was... unpleasant. Now that I had the disease I am immune for life. My wife and children got vaccinated to avoid my fate. The vaccination consists of three initial doses followed by a new dose every 3d to 5th year, depending on age (the older, the more frequent).

The current crop of SARS2 vaccines comes of far worse than the TBE vaccine with immunity waning in a period counted in weeks, not years [1,2,3]. Natural immunity is "from 6 to 27 times stronger" depending on which researcher you listen to. Here's part of the abstract of the largest study (~700.000 test subjects) on these matters:

SARS-CoV-2-naïve vaccinees had a 13.06-fold (95% CI, 8.08 to 21.11) increased risk for breakthrough infection with the Delta variant compared to those previously infected, when the first event (infection or vaccination) occurred during January and February of 2021. The increased risk was significant (P<0.001) for symptomatic disease as well. When allowing the infection to occur at any time before vaccination (from March 2020 to February 2021), evidence of waning natural immunity was demonstrated, though SARS-CoV-2 naïve vaccinees had a 5.96-fold (95% CI, 4.85 to 7.33) increased risk for breakthrough infection and a 7.13-fold (95% CI, 5.51 to 9.21) increased risk for symptomatic disease. SARS-CoV-2-naïve vaccinees were also at a greater risk for COVID-19-related-hospitalizations compared to those that were previously infected.

[1] https://www.science.org/content/article/having-sars-cov-2-on...

[2] https://www.westernjournal.com/johns-hopkins-doc-says-natura...

[3] https://www.medrxiv.org/content/10.1101/2021.08.24.21262415v...