So does Bayes’ theorem reduce to ordinary prejudice?
In some circumstances, you could say that, you could say that priors are arguably just rooted in personal biases, but not all applications of bayes theorem have that aspect of generating priors from personal bias.
For example, this problem:
The probability of breast cancer is 1% for a woman at age forty who participates in routine screening. If a woman has breast cancer, the probability is 80% that she will get a positive mammography. If a woman does not have breast cancer, the probability is 9.69% that she will also get a positive mammography. A woman in this age group had a positive mammography in a routine screening. What is the probability that she actually has breast cancer?
Bayesian reasoning can help us answer this question easily, but no subjective personal biases come in - the priors that go into the calculation can all come from reliably gathered statistics.