To me, it is still unclear what exactly is the difference between Frequentist and Bayesian statistics. Most explanations involve terms such as "likelihood", "uncertainty" and "prior probabilities". Here, I'm going to show the difference between both statistical paradigms by using a coin flipping example. In the examples, the effect of showing more data to both paradigms will be visualised.

Lets start by generating some data from a fair coin flip, that is, the probability of heads is 0.5.

```
using Distributions
using Random
n = 80
Random.seed!(102)
p_true = 0.5
is_heads = rand(Bernoulli(p_true), n)
```

To give some intuition about the sample, the first six elements of `is_heads`

are

```
is_heads[1:6] = Bool[1, 0, 0, 1, 1, 1]
```

The Frequentist estimate for a one sample t-test after seeing $n$ samples can be calculated with

```
using HypothesisTests
function frequentist_estimate(n)
t_result = OneSampleTTest(is_heads[1:n])
t_lower, t_upper = confint(t_result)
(lower = t_lower, middle = t_result.xbar, upper = t_upper)
end
```

For the Bayesian estimate, we can use the closed-form solution (The Turing Language, 2020). A closed-form solution is not available for many real-world problems, but quite useful for this example.

```
closed_form_prior = Beta(1, 1)
function update_belief(k)
heads = sum(is_heads[1:k-1])
tails = k - heads
updated_belief = Beta(closed_form_prior.α + heads, closed_form_prior.β + tails)
end
beliefs = [closed_form_prior; update_belief.(1:n)]
```

```
function bayesian_estimate(n)
distribution = beliefs[n]
q(x) = quantile(distribution, x)
(lower = q(0.025), middle = mean(distribution), upper = q(0.975))
end
```

```
using Gadfly
function plot_estimates(estimate_function; title="")
draws = 2:4:80
estimates = estimate_function.(draws)
middles = [t.middle for t in estimates]
plot(y = draws,
x = [t.middle for t in estimates],
xmin = [t.lower for t in estimates],
xmax = [t.upper for t in estimates],
Geom.point, Geom.errorbar,
Coord.cartesian(xmin = 0.0, xmax = 1.0),
Guide.xlabel("Probability of heads"), Guide.ylabel("Observed number of draws"),
Guide.title(title),
layer(xintercept = [0.5], Geom.vline(color = "gray"))
)
end
```

`plot_estimates(frequentist_estimate, title = "Frequentist estimates")`

`plot_estimates(bayesian_estimate, title = "Bayesian estimates")`

If you don't have much programming experience, then you might be wondering how to come up with this pretty code which can neatly work for the Frequentist **and** Bayesian estimates. The answer is: lots of trial and error, and moving text around.

Based on these plots, we can conclude two things. Firstly, the Bayesian approach provides better estimates for small sample sizes. The Bayesian approach successfully uses the fact that a probability should be between 0 and 1, which was given to the model via the `Beta(1, 1)`

prior. For increasingly larger sample sizes, the difference between both statistical paradigms vanish in this situation. Secondly, collecting more and more samples until the result is significant is dangerous. This approach is called *optional stopping*. Around 25 samples, it would find that the data must come from a distribution with a mean higher than 0.5, whereas we know that this is false. Cumming (2011) calls this the "dance of the $p$-values".

Cumming, G. (2011). Understanding the new statistics: Effect sizes, confidence intervals, and meta-analysis. Routledge.

The Turing Language (2020). Introduction to Turing. https://turing.ml/dev/tutorials/0-introduction/.