1D Lab: one coin, two views
These two charts use the exact same binomial formula. Only the axis changes. Drag the sliders and watch how probability is a bell over outcomes, while likelihood is a curve over the bias θ.
Scroll into view to watch a short demo: the observed heads k will move and both charts will update.
k. Likelihood over θ is not a probability distribution — notice the area under the likelihood curve is not 1.
2D Lab: two independent coins
With two coins the likelihood becomes a surface over a 2D parameter space (θ₁, θ₂). Drag the data sliders and watch the contour peak move to (k₁/n₁, k₂/n₂).
Scroll into view to watch a short demo: the contour will sweep as the observed counts change.
Coin 1
Coin 2
3D Lab: three coins, one high-likelihood cloud
Now the likelihood lives in 3D parameter space. The isosurface below encloses the region where the likelihood is above a chosen fraction of its maximum. The MLE point is at (k₁/n₁, k₂/n₂, k₃/n₃).
Scroll into view to watch a short demo: the camera slowly orbits around the cloud while the threshold gently breathes.
Coin 1
Coin 2
Coin 3
4D Lab: four coins, a 4D parameter space
We cannot see four spatial dimensions, so we use four different projections at once: a parallel-coordinates plot, a scatter-plot matrix (SPLOM) of all 2D projections, a 3D isosurface slice where θ₄ is the slicing variable, and a rotating wireframe tesseract representing the 4D hypercube [0,1]⁴. Scroll into view to watch θ₄ sweep: the 3D slice, parallel-coordinates and SPLOM highlight the current slice, and the tesseract rotates.
Coin 1
Coin 2
Coin 3
Coin 4
Scroll into view to watch a short demo: the 3D slice will move as θ₄ changes.
(k₁/n₁, k₂/n₂, k₃/n₃, k₄/n₄) is the single line that should be brightest. Everything else is just a lower-likelihood point in a 4D hypercube.
Forensic likelihood ratio: glass from the car or the dishes?
This is the prosecutor's-fallacy test. The evidence E is: the glass fragments on the suspect's clothes match the broken car window. The prosecution hypothesis Hp says the glass came from the car. The defense hypothesis Hd says it came from the broken dishes at home.
Scroll into view to watch a short demo: the random-match probability and the prior move, and the posterior probability curve updates.
The forensic findings provide weak support for Hp relative to Hd.
The LR measures how much the evidence discriminates between the two hypotheses. It is not the probability of guilt. The final probability of guilt still depends on the prior. This is why a tiny random-match probability P(E | Hd) can be misleading by itself. Values < 1 support the defence; report the inverse (1/LR).
| Likelihood ratio | Verbal equivalent |
|---|---|
| 1 | No support for either proposition |
| 2 – 10 | Weak support |
| 10 – 100 | Moderate support |
| 100 – 1,000 | Moderately strong support |
| 1,000 – 10,000 | Strong support |
| 10,000 – 1,000,000 | Very strong support |
| ≥ 1,000,000 | Extremely strong support |
Verbal scale adapted from ENFSI (2016), Guideline for Evaluative Reporting, Methodology p. 1. A statement of “weak support for Hp relative to Hd” means the findings are up to 10 times more probable under Hp; it does not mean strong support for Hd.
Verdict
| Feature | Probability P(X | θ) |
Likelihood L(θ | X) |
|---|---|---|
| Variable space | Data X | Parameters θ |
| Fixed | Parameters θ | Observed data X |
| Integrates to 1? | Yes, over data space | No, over parameter space |
| Use case | Predict future outcomes | Estimate parameters (MLE) |
They share the same numerical evaluation, but because they vary over different spaces they obey different rules. Probability is a distribution; likelihood is a relative plausibility score.