Likelihood and the Maximum Likelihood Estimate
Given data and a family of models, the likelihood is how probable the data would be under each candidate. Choosing the candidate that maximises it is where most fitting procedures come from, including the least squares line.
What a learner can do afterwards
- Write the likelihood of a run of coin flips as a function of the bias
- Maximise a simple likelihood and check the answer against intuition
- Explain why the likelihood is a function of the parameter, not of the data
1 · Read
Fix the data, then score each candidate bias by how likely it makes those flips. That score is the likelihood. It varies the bias while the flips stay put, so it is a function of the parameter, not of the data.
Seven heads in ten flips gives likelihood p to the 7 times (1 minus p) to the 3. Try p = 0.5 and the score is low. Try p = 0.7 and the score peaks. The maximizing bias is seven tenths, which matches your gut: seven out of ten.
Most fitting works this way. You write how likely the data looks under each setting, then pick the setting that maximizes it. The least squares line is one famous child of this rule.
Hold the data still and move the parameter. Any sentence that varies the flips while fixing the bias has the roles backwards.
Score each bias by the fixed data, then pick the bias with the top score.
2 · Watch
Take it off screen
Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.