Explain

Intuition

We want to update our belief about the probability of a coin coming up heads ($\pi$) after observing some coin flips ($\mathcal{D}$). Bayes' rule is the mathematical formulation of this "belief updating".

Prior: Starting out with a "uniform prior" means we have no idea whether the coin is biased or not. We think any probability between 0 and 1 is equally likely.

Likelihood: This is what the data tells us. If we flip the coin $n$ times and get $s$ heads, the likelihood is $\pi^s(1-\pi)^{n-s}$. The data "pulls" our belief towards the observed proportion of heads.

Normalization: To make sure our new belief (posterior) represents valid probabilities and sums to 1, we divide by the integral of all possibilities. The identity provided is just a mathematical shortcut to calculate this area. We end up with a Beta distribution.

When $n=1$, we just flipped the coin once.

• If it landed Heads ($s=1$), our updated belief increases linearly towards $\pi=1$. It's more likely now that the coin favors heads.
• If it landed Tails ($s=0$), our updated belief leans towards $\pi=0$.

graph TD
    A[Uniform Prior: No assumption] -->|Observe Data D| B(Likelihood matches data shape);
    B -->|Normalize area to 1| C[Posterior: Updated belief about pi];