2D Optimization: Finding the Best Line

Now with two parameters: intercept (β₀) and slope (β₁)

1D (Mean Only) 2D (Line Fit) 3D (Multiple Regression) 4D+ (Beyond Visualisation) Multi-Optima Bayesian MCMC Terrain = Likelihood Uncertainty
Linear Regression: y = β₀ + β₁x

Data + Current Fit

Log-Likelihood Contour Surface

Find the Peak!

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

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

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Intercept (β₀)
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Slope (β₁)
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Log-Likelihood
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What's Different in 2D?

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

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

And notice how quickly the surface falls away from the peak — steeply in some directions, gently in others. Hold that thought: that steepness turns out to be the standard error of each estimate.