1D Optimisation: Estimating a Single Parameter

The simplest case: estimating the mean of some data

1D (Mean Only) 2D (Line Fit) 3D (Multiple Regression) 4D+ (Beyond Visualisation) Multi-Optima Bayesian MCMC Terrain = Likelihood
Finding the Maximum Likelihood Estimate

Data + Current Fit

Log-Likelihood Surface

Find the Peak!

Drag the slider or click on the curve to set your estimate. Can you find the maximum?

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Tip: Watch the log-likelihood value — higher is better!

Adjustments made: 0
Current Estimate
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Log-Likelihood
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Iteration
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True MLE
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Analytic Solution

For the Gaussian mean, we can solve directly: the MLE is simply the sample mean $\bar{y}$. No iteration needed!

What You're Seeing

The curve shows the log-likelihood function $\ell(\theta)$ for different values of the parameter $\theta$. Higher values mean the parameter explains the data better.

$$\ell(\theta) = \sum_{i=1}^{n} \log f(y_i \mid \theta)$$

The red dot shows the current estimate. Watch how different algorithms navigate to the peak:

  • Analytic: Jumps directly to the solution (when a formula exists)
  • Newton-Raphson: Uses curvature information for fast convergence
  • Gradient Ascent: Follows the slope uphill, step by step

Why This Matters

In 1D, finding the maximum is trivial — you can see it! But real models have many parameters, creating surfaces we can't visualise. Understanding how algorithms work in 1D builds intuition for what happens in higher dimensions.

For the theory behind these surfaces — likelihood, optim() and Fisher information — see Likelihood and Simulation Theory on JonStats.