Hassan Ijaz
Ai, Web & Design
Parameter estimation (MLE, MAP)
Interactive likelihood function visualizer where users guess parameters and see likelihood surface with gradient ascent animation to find MLE
Concept Overview
Parameter estimation involves using sample data to estimate unknown population parameters. Two main approaches are Maximum Likelihood Estimation (MLE) and Maximum A Posteriori (MAP) estimation.
Maximum Likelihood Estimation (MLE)
Find parameter values that maximize the probability of observing the data
L(θ) = ∏ P(x_i | θ)
Likelihood function
θ̂_MLE = argmax L(θ)
MLE estimate
- Asymptotically unbiased and efficient
- Invariant under transformations
- Often solved using calculus (set derivative to 0)
- May not exist or be unique in some cases
Maximum A Posteriori (MAP) Estimation
Incorporates prior knowledge about parameter values
P(θ|x) ∝ P(x|θ) × P(θ)
Posterior ∝ Likelihood × Prior
θ̂_MAP = argmax P(θ|x)
MAP estimate
- Reduces to MLE when prior is uniform
- Can incorporate domain expertise
- Provides regularization against overfitting
- Sensitive to prior specification
Common Examples
Normal Distribution
MLE for μ: sample mean, MLE for σ²: sample variance (with n denominator)
Bernoulli Distribution
MLE for p: proportion of successes in sample
Exponential Distribution
MLE for λ: 1/sample mean
Properties of Estimators
Unbiased
E[θ̂] = θ (expectation equals true value)
Consistent
θ̂ → θ as n → ∞ (converges to truth)
Efficient
Achieves Cramér-Rao lower bound
Sufficient
Uses all information in the data
Practical Tip: MLE often requires numerical optimization for complex models. Modern statistical software uses algorithms like Newton-Raphson or EM algorithm to find MLEs.
The interactive likelihood visualizer below lets you guess parameters and see the likelihood surface. Watch the gradient ascent animation find the MLE automatically.
Interactive Visualization
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Interactive likelihood function visualizer where users guess parameters and see likelihood surface with gradient ascent animation to find MLE