Monte Carlo simulation is a simple but powerful idea: use randomness to approximate values.
A classic example is the value pi. You generate random points inside a square, check whether they fall within a circle, and use the proportion of hits to estimate pi. As the number of samples increases, the estimate becomes more stable and gets closer to the true value.
But there is a limitation. This approach is often presented as a single number, and that number changes every time you run the simulation. So the key question becomes: How uncertain is this estimate at any point in time?
In this setup, each sampled point is either a success or a failure. This allows us to use a Bayesian approach to model uncertainty. Instead of reporting one estimate, we describe a distribution of plausible values for pi given the data and update it as more data is collected.
The visualization below shows this process. On the left, sampled points are added step by step. On the right, the distribution of pi is updated continuously, together with a 95% credible interval. At the beginning, the range of plausible values is wide. As more samples are collected, the distribution becomes narrower and centers around the true value.
This example is based on an article by Pedro Pessoa, who explores this idea in more detail. The graph shown here is also taken from his article:
github.com/PessoaP/blog/blob…
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