Module Review: Estimation Theory
Congratulations on completing the Estimation Theory module! You now possess the tools to infer population parameters from sample data.
Key Takeaways
- Estimators are Random Variables: Because they depend on sample data, estimators have their own probability distributions (Sampling Distributions).
- MLE is the Gold Standard: Maximum Likelihood Estimation finds the parameters that make the observed data most probable. It is asymptotically efficient (lowest variance).
- MAP incorporates Priors: Bayesian MAP estimation combines the Likelihood with a Prior distribution, acting as a regularizer.
- Bias-Variance Tradeoff: Total error (MSE) = Bias2 + Variance + Irreducible Error. Simple models have high bias; complex models have high variance.
- Method of Moments: A simple alternative to MLE that equates sample moments to population moments.
Interactive Flashcards
Test your knowledge of the key concepts.
Maximum Likelihood Estimation (MLE)
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A method that estimates parameters by maximizing the Likelihood Function L(θ) = P(Data | θ).
Bias
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The difference between the expected value of an estimator and the true parameter value. Bias = E[θ̂] - θ.
Consistent Estimator
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An estimator that converges in probability to the true parameter value as the sample size n → ∞.
Maximum A Posteriori (MAP)
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A Bayesian estimation method that maximizes the Posterior distribution: Likelihood × Prior.
Method of Moments
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A technique that equates sample moments (e.g., sample mean) to population moments (e.g., expected value) to solve for parameters.
Mean Squared Error (MSE)
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A measure of estimator quality that combines Bias and Variance: MSE = Bias2 + Variance.
Estimation Cheat Sheet
| Concept | Formula / Definition | Key Property |
|---|---|---|
| Likelihood | L(θ) = Π P(x<sub>i</sub> \| θ) |
Not a probability (doesn’t sum to 1). |
| Log-Likelihood | l(θ) = Σ ln P(x<sub>i</sub> \| θ) |
Maximized at same θ as L. Prevents underflow. |
| MLE | argmax l(θ) |
Asymptotically Unbiased & Efficient. |
| MAP | argmax (l(θ) + ln P(θ)) |
Incorporates Prior. Acts as Regularizer. |
| Bias | E[θ̂] - θ |
Systematic error. |
| Variance | E[(θ̂ - E[θ̂])<sup>2</sup>] |
Sensitivity to data fluctuations. |
| MSE | Bias<sup>2</sup> + Variance |
Total error metric. |
| Sample Mean | x̄ = (1/n) Σ x<sub>i</sub> |
Unbiased estimator for Population Mean μ. |
| Sample Variance | s<sup>2</sup> = (1/(n-1)) Σ (x<sub>i</sub> - x̄)<sup>2</sup> |
Unbiased estimator for Population Variance σ<sup>2</sup>. |
Further Reading
- Statistics Glossary: Detailed definitions of all terms.
- Next Module: Regression Analysis (Coming Soon)
- Related: Probability Distributions
Ready for the next step?
Now that you can estimate parameters, the next logical step is to test hypotheses about them. This leads us to Hypothesis Testing and Confidence Intervals.