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.

```
begin
import CairoMakie
using AlgebraOfGraphics: Lines, Scatter, data, draw, visual, mapping
using HypothesisTests: OneSampleTTest, confint
using Turing
using Random: seed!
end
```

`n = 80;`

`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]`

`[true, false, false, true, true, true]`

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

```
function frequentist_estimate(n)
t_result = OneSampleTTest(is_heads[1:n])
middle = t_result.xbar
lower, upper = confint(t_result)
return (; lower, middle, upper)
end;
```

For the Bayesian estimate, we can use the closed-form solution (https://turing.ml/dev/tutorials/00-introduction/). 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)
return updated_belief
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)
return (; lower, middle, upper)
end;
```

Now we can write a function to plot the estimates:

```
function plot_estimates(estimate_function; title="")
draws = 2:4:80
estimates = estimate_function.(draws)
middles = [t.middle for t in estimates]
lowers = [t.lower for t in estimates]
uppers = [t.upper for t in estimates]
df = (; draws, estimates, P=middles)
layers = data(df) * visual(Scatter)
df_middle = (; P=fill(0.5, length(draws) + 2), draws=[-1; draws; 83])
layers += data(df_middle) * visual(Lines) * visual(linestyle=:dash)
for (n, lower, upper) in zip(draws, lowers, uppers)
df_bounds = (; P=[lower, upper], draws=[n, n])
layers += data(df_bounds) * visual(Lines)
end
axis = (; yticks=0:20:80, limits=((-0.2, 1.2), nothing), title)
map = mapping(:P => "Probability of heads", :draws => "Observed number of draws")
draw(layers * map; axis)
end;
```

And plot the Frequentist and Bayesian estimates:

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

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

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".

AlgebraOfGraphics 0.6.0 CairoMakie 0.6.6 HypothesisTests 0.10.4 Turing 0.18.0