What is the point of using a Bayesian prior? I do struggle with the most basic starting point of Bayesian statistics: why is using a prior useful?
It seems to me that if anything they hurt much more than help.
Moreover, Bayesians always say things like "the more evidence you get the less important a prior becomes". So why use them in the first place?
Especially if you have start with a prior that is really far-off, then you will be hurting your estimates.
To me a "Frequentist" approach seems much simpler and more straightforward.
I would like to discuss here an example that is very typical in introductory Bayesian course/explanations (e.g. this, this, or this)
Example 1 -- Are French people Rude?

Imagine that I am interested in estimating if French people are rude (in a binary way: Rude vs Non-Rude).
Imagine that the true parameter $\theta$ is 0.3, 30% of French are rude.
I have a random sample of 100 people with data on "rudeness".
A straightforward "Frequentist" approach would be: compute your confidence intervals on the sample and ... you done. We know what is the probability that the true $\theta$ lies within our CI, and we know that our sample average, on average, thanks to the Central Limit Theorem, will be close to the true parameter.
    true_theta = 0.3
    set.seed(111)

    # population
    X = rbinom(n = 10000, size = 1, prob = true_theta)

    # sample of 100
    x = sample(X, size = 100)
    p_hat = prop.table(table(x))[2]
    margin_error = 1.96 * sqrt( (p_hat*(1-p_hat)) / 100  
    ci_low = p_hat - margin_error
    ci_high = p_hat + margin_error
    data.frame(p_hat, ci_low, ci_high)

Now Bayesian will try to incorporate priors into this. Why?
We can imagine that most people will have beliefs that French are rude. (Using a Beta distribution) based on personal experience, someone would typically have a prior distribution like this: they know 10 French people and 7 are rude. So that is their prior.
The results are shown in the Figure above. On the bottom panel, I draw the Confidence Intervals around the sample mean.
It seems to me that the Frequentist approach will gives us a more precise and immediate answer to this question.
What am I missing here?
Example 2 -- Forecast Knock Out
Imagine I am interested in forecasting Knocked out (KO) in the Ultimate Fighting Championship (UFC).
My basic Frequentist approach would go like this.
Study the sport and see what variables play a role in predicting a fighter knocking another fighter (e.g. age of the fighters, winning streak, home advantage, …).
Then I will sample 30 UFC events and start build my model. I would use a simple logistic regression with maximum likelihood.
I don’t have prior per see, but I have a model of reality that I built on theoretical knowledge, like any scientists do, Bayesian or not.
My initial model would be that age and style of fighting (Muay Thai fighter vs BJJ) would be the most important factors predicting a KO.
I retrieve estimates from my logistic regression. Then I would cross-validate, take another sample and apply my initial model to the new data and see how it perform.
Let’s imagine that it doesn’t perform too well. I will then try to improve my model in studying more closely the sport. I find that accounting for an interaction between injury and age of the fighter is one of the most important predictors a KO. I re-run my regression with this and again cross validate, and now the model perform well.
Now I have a forecasting model telling me the probability of a KO and I can use it to bet money.
Furthermore, I know that certain special events, like a New Year's Eve event, will have more KOs, so I can adjust my model according to that fact.
Where would a prior, in the Bayesian sense, be useful here? Frequentists like all scientists use model to test reality and improve their models over time, but the difference is that they don’t put a formal initial probability on things. I still do not see why, you would want to do that.
    # code for the Figure
    theta_range <- seq(0, 1, by = 0.01)
    prior <- dbeta(x = theta_range, shape1 = 7, shape2 = 3)

    # observed success
    success_obs = table(x)[2]
    failure_obs = table(x)[1]

    # sampling distribution #
    likelihood <- dbinom(x = success_obs, size = 100, 
                           prob = theta_range) * 100

    # posterior 
    posterior_density <- likelihood * prior

   df <- data.frame(theta = theta_range,
                     likelihood = likelihood,
                     prior_dist = prior,
                     posterior_distribution = posterior_density)

    df_melt = melt(df, id.vars = 'theta')

    df_melt %>%
      ggplot(aes(x = theta, y = value, color = variable)) + 
      geom_line(size = 1.2) + 
      scale_x_continuous(breaks = seq(0, 1, by = 0.1)) + 
      geom_vline(xintercept = p_hat, size = 1.5) + 
      geom_vline(xintercept = ci_low, colour = 'gray', 
                                      size = 1.5) +
      geom_vline(xintercept = ci_high, colour = 'gray', 
                             size = 1.5) + theme_minimal() +
      ggtitle("Bayesian + Frequentist Confidence Intervals")


There are many excellent thoughts here.  There is a short answer to the question.  If you want to gauge relative evidence you can sometimes get away without using any outside information.  Relative evidence can be summarized by a likelihood ratio in the likelihoodist school of statistics.  For example one may use study data to compute the likelihood ratio assuming that true mean blood pressure is 120mmHg vs. the mean being 140mmHg.  Or you can quantify evidence more indirectly using p-values (evidence against something, only).  If you want to quantify absolute evidence there is no mathematical way to compute "absolute" probabilities without having a prior distribution.  So if you wanted to compute the probability that the true mean blood pressure is between 135mmHg and 145mmHg you would need a prior.  Likewise if you wanted to compute the probability that a medical treatment lowers mortality instead of just using a frequentist hypothesis test to compute the probability of getting data stranger than ours if the treatment does nothing about disease risk you would need a prior.
The most compelling stories I've seen about the Bayesian approach are Nate Silver's The Signal and the Noise and Bernoulli's Fallacy by Aubry Clayton.  More thoughts are here and here.
An analogy in medical diagnostic testing is often useful.  Sensitivity and specificity are sometimes used as test characteristics.  These condition on the actual disease status so only provide relative information.  To turn them into absolute information (probability of disease) one must use Bayes' rule to factor in disease prevalence (the prior).
So the prior distribution is the mathematical cost of being able to make direct statements that are not just about relative evidence.
