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Explain

Intuition​

We're trying to find the "best" guess for the coin's bias π\pi. Different methods define "best" differently:

  1. MLE (Maximum Likelihood): Asks "What value of π\pi makes the data we actually saw most probable?" The answer is exactly the fraction of heads we saw: s/ns/n.
  2. MAP (Maximum A Posteriori): Asks "Given our data AND our starting assumption (prior), what is the single most probable value of π\pi?"

The Uniform Prior Connection A "uniform prior" is mathematically saying "I have absolutely no preference for any value of π\pi; they are all equally likely before I flip the coin."

Because we brought zero initial bias/knowledge to the table, our final peak belief (MAP) is dictated 100% by the data we saw. Therefore, MAP gives the exact same answer as MLE.

Is one better? MLE and MAP (with uniform prior) give the same number, but the Bayesian expected value (from part c, s+1n+2\frac{s+1}{n+2}) is often preferred in practice. Why? Because the peak (mode) of a skewed distribution isn't always the best representative value. The expected value (mean) takes the whole shape of the uncertainty into account, preventing extreme overly-confident predictions when we haven't seen much data yet.