An ordered binary record contains two successes and one failure. Compare the candidate success rates p=1/2 and p=2/3. Their probabilities for that particular vector are 1/8 and 4/27. The second is larger, but neither is a probability that the parameter is correct. We have evaluated two different sampling laws at the same observed outcome. This operation produces a likelihood function. Holding data fixed reverses our usual computational direction without reversing the conditional probability statement.
Predict
Predict whether retaining only the total changes the ratio of these two likelihoods. The binomial likelihood includes a factor of three for the possible arrangements, so both likelihoods are multiplied by the same constant. Their ratio remains(4/27)/(1/8)=32/27. Parameter-free factors can be dropped for maximization and likelihood ratios. Parameter-dependent factors cannot. In particular, a support indicator depending on the parameter may be decisive even when it looks like an inconvenient technical detail.
Experiment
Build a small likelihood table for the fixed record. Compare each entry with the middle candidate, then inspect how the likelihood tends to zero near either endpoint because this record contains both outcomes.
Candidate p
Likelihood p²(1−p)
Ratio to p=1/2
1/3
2/27
16/27
1/2
1/8
1
2/3
4/27
32/27
Differentiate log L=2 log p+log(1−p). Its derivative 2/p−1/(1−p) vanishes at 2/3. Its second derivative is negative throughout the open interval. The calculus agrees with, but is stronger than, the three-row comparison: a finite grid cannot establish a global optimum by itself.
Understand
For observed x, likelihood is L(θ;x)=fθ(x), regarded as a function of θ. A log likelihood is useful because logarithms turn products into sums and preserve order wherever likelihood is positive. Zero likelihood must still be handled explicitly. With continuous observations, fθ(x) is a density; it can exceed one and has units inverse to those of the observation. Ratios using a common reference measure avoid treating a density height as an event probability.
Identifiability asks a different question: do distinct parameter values determine distinct probability laws? In the family N(θ²,1), θ and−θ are observationally indistinguishable. No amount of data can distinguish their signs within that model. Restricting θ to nonnegative values resolves this particular ambiguity. Numerical optimizers do not resolve structural nonidentifiability; they may simply select one of several equivalent answers.
Apply
Consider a mixture consisting of weight w on a normal component with mean a and weight 1−w on a component with mean b, both of known variance one. Interchanging(a,b,w) with (b,a,1−w) leaves the density unchanged. Propose the convention a<b and explain what happens when a=b or a weight is zero. The convention addresses label ambiguity in the distinct, positive-weight case, but not every degeneracy of the model.
Finally, diagnose “normalize the likelihood and call it probability.” A posterior needs a prior and a parameter measure. A prior flat in θ usually is not flat after a nonlinear transformation. Normalizing an unspecified likelihood is therefore adding an unstated modeling choice, not extracting a parameter distribution from the sampling model alone.
Continue
Record three separate conclusions: which observed-data candidate is favored, whether the model is identifiable, and whether the model itself is credible. None implies the other two. The next lesson compares estimators across repeated samples rather than comparing parameter values for one sample. Continue to Bias, variance and risk. Bayesian decisions return later with priors and loss functions made explicit, so their probability statements can be interpreted without ambiguity.
Apply · explain · revise
Practice and reasoning
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A binary vector has three successes and one failure. What is L(3/4)/L(1/2)?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
For Poisson observations 1,2,3, what is the sample mean and interior likelihood maximizer?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Show the family X∼N(θ²,1), θ∈R, is not identifiable and give an identifiable restriction.
Structure your argument and preserve revisions
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Explain why likelihood values for continuous data need not lie between zero and one.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Why does normalizing L(θ) over θ not automatically produce a parameter probability justified by the sampling model?
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
A two-component normal mixture permits swapping component labels. Explain the resulting nonidentifiability and a possible convention.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
