I wrote another comment here clarifying my point, if you're interested: https://news.ycombinator.com/item?id=47566033

In Bayesian inference, the population ratio is seen as a quantity that can take different values each with a associated probability (i.e. a random variable), and the result of Bayesian inference is an estimate of the probability distribution of the population parameter, in this case the population ratio. Now, in reality the population ratio is a concrete number, say 9-to-10, meaning that there are 9 men for every 10 women in the population. But Bayesians don't care. They'll tell you that the population ratio is a random variable which can take many values, and that the probability that it is equal to 9-to-10 is whatever number between 0 and 100%.

This is nonsense because the population ratio is NOT a random variable. People don't come in and out of existence randomly, right? In a way, they're saying there are infinitely many possible universes, each with a different population ratio, and then they come up with an estimate of the probability that the universe in which the ratio is 9-to-10 has whatever probability of occurring. This is absolutely BIZARRE. (I hope you agree). And it's wrong because it's impossible to know how likely one universe is compared to all other possible universes, since we live in our universe and this is all we can hope to observe.

The problem with this approach is that we can only observe ONE level of treatment effectiveness, i.e., the level of treatment effectiveness that the treatment actually possesses. All other possible levels of effectiveness are entirely hypothetical. There's no data about all these other possible levels of effectiveness because they don't occur in reality. So the data cannot possibly tell you anything about how likely is the observed outcome, because the observed outcome is the only outcome that you observe. I

This criticism was made over 100 years ago, and Bayesians still don't have an answer. They just keep going as if nothing happened, but the reality is their methodology is utterly and fatally flawed.

To be more precise, in Bayesian statistics a parameter is random variable. But what does that mean? A parameter is a characteristic of a population (as opposed to a characteristic of a sample, which is called a statistic). A quantity, such as the average cars per household right now. That's a parameter. To think of a parameter as a random variable is like regarding reality as just one realisation of an infinite number of alternate realities that could have been. The problem is we only observe our reality. All the data samples that we can ever study come from this reality. As a result, it's impossible to infer anything about the probability distribution of the parameter. The whole Bayesian approach to statistical inference is nonsensical.

By the way, doing a better job than the average human is NOT a sign of intelligence. Through history we have invented plenty of machines that are better at certain tasks than us. None of them are intelligent.

    population_data <- data.frame(
        uniform = runif(10000, min = -20, max = 20),
        normal = rnorm(10000, mean = 0, sd = 4),
        binomial = rbinom(10000, size = 1, prob = .5),
        beta = rbeta(10000, shape1 = .9, shape2 = .5),
        exponential = rexp(10000, .4),
        chisquare = rchisq(10000, df = 2)
    )
    
    histogram(~ values|ind, stack(population_data),
              layout = c(6, 1),
              scales = list(x = list(relation="free")),
              breaks = NULL)
    
    take_random_sample_mean <- function(data, sample_size) {
        x <- sample(data, sample_size)
        c(mean = mean(x), sd = sqrt(var(x)))
    }
    
    sample_statistics <- replicate(20000, sapply(population_data, take_random_sample_mean, 60))
    
    sample_mean <- as.data.frame(t(sample_statistics["mean", , ]))
    sample_sd <- as.data.frame(t(sample_statistics["sd", , ]))
    
    histogram(sample_mean[["uniform"]])
    histogram(sample_mean[["binomial"]])
    
    histogram(~values|ind, stack(sample_mean), layout = c(6, 1),
              scales = list(x = list(relation="free")),
              breaks = NULL)