Likelihood, estimation and model checking
What this concept is
A likelihood treats observed data as fixed and the parameter as variable. For independent observations with common density f_θ, L(θ)=product f_θ(x_i). A log-likelihood turns the product into a sum. It is not automatically a probability density over θ.
Why it matters here
Calculus can find the best-fitting parameter within a specified family, but cannot establish that the family describes the observations.
Concrete example
For n independent uncensored exponential durations with rate λ>0, log L(λ)=n ln λ−λS, where S is the sum of durations. If S>0, its derivative n/λ−S vanishes at λ̂=n/S; second derivative −n/λ² is negative. For observations [1,2,3], λ̂=1/2 in inverse-time units. Converting seconds to milliseconds divides the rate by 1000.
Common confusion / misconception
The rate is not the mean; the mean is 1/λ. All-zero durations give an unbounded likelihood as λ increases, not a finite MLE. Recorded zeros may indicate coarse timing. Removing timeouts or censored observations invalidates the uncensored likelihood.
How to use it
Write sampling, independence, support and censoring assumptions before differentiating. Compare predicted survival exp(−λ̂t) with empirical exceedance fractions over several thresholds. Agreement in the fitted sample is a diagnostic, not a goodness-of-fit certificate. Keep a held-out check if prediction is the objective. This module does not claim unbiasedness or provide an uncertainty interval for λ̂.
Check yourself
Why does multiplication of densities require independence in this model? Is a very large fitted likelihood evidence of causality?
Mini drill or application
Derive the Bernoulli MLE k/n using k ln p+(n−k)ln(1−p), handling k=0 and k=n as boundary solutions. For durations [2,4,6], the exponential rate is 1/4. State two ways real latency data might violate this model.
Read this only if stuck
Primary source support. See the extension guide for practice, scope and evidence requirements. Local examples and lab code are original.