2D Optimization: Intercept + Slope

Linear Regression: y = β₀ + β₁x

Adjust both sliders to find the maximum likelihood. Or click directly on the contour plot.

Intercept (β₀): 0.0

Slope (β₁): 0.00

Tip: Watch both the regression line and the likelihood surface. The red dot should climb toward the peak!

Adjustments made: 0

Intercept (β₀)

Slope (β₁)

Log-Likelihood

Iterations

With two parameters, the likelihood surface becomes a 2D landscape shown as contour lines:

Contour lines connect points with equal log-likelihood (like elevation lines on a map)
The peak (MLE) is at the center of the elliptical contours
Ellipse shape shows how intercept and slope estimates are correlated
Gradient descent follows the steepest uphill path to the peak

Notice how the regression line changes as you move across the likelihood surface. Finding the best fit means climbing to the highest point!

2D Optimization: Finding the Best Line