Exploring parameter estimation through coin flips: learning how MLE and MAP differ, and why Bayesian priors make estimates more robust.

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Introduction

In this lab, I studied two foundational techniques for estimating model parameters: Maximum Likelihood Estimation (MLE) and Maximum A Posteriori Estimation (MAP).

Both aim to estimate unknown parameters (like the probability of getting heads in a coin toss), but they take different philosophical approaches:

• MLE assumes a single “true” parameter value and chooses the one that maximizes the likelihood of the observed data.
• MAP combines observed data with prior beliefs (expressed as a distribution), producing more stable estimates — especially with small datasets.

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Key Steps Covered

• MLE on Unbiased Coin
  • Estimated probability of heads with θ̂ = α_H / (α_H + α_T).
  • Example: 7 heads, 3 tails → θ̂ = 0.7.
• MAP on Unbiased Coin
  • Incorporated prior knowledge using a Beta distribution.
  • Example: Prior mean 0.5 with observed 7 heads, 3 tails → θ̂ = 0.6 (closer to true fairness).
• Biased Coin Simulation
  • Ran experiments with biased coins (true θ = 0.6).
  • Showed how MLE converges to the true value with many samples, but fluctuates with fewer.
  • MAP produced more stable estimates by incorporating prior knowledge.
• Multiple Experiments
  • Compared MLE estimates across multiple trials.
  • Visualized convergence trends with line plots.

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Takeaway

This lab highlighted the trade-off between MLE and MAP:

• MLE is straightforward and works well with abundant data.
• MAP is more robust with limited data, as prior beliefs smooth out fluctuations.

Together, they form the backbone of statistical parameter estimation — one rooted in frequentist thinking, the other in Bayesian reasoning.

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🔗 View the full Lab Notebook on GitHub
▶️ Run in Google Colab