Statistical note

Likelihood and log-likelihood

View observed data as evidence for competing parameter values and understand maximum-likelihood estimation.

11 August 20266 min read

What quantity is being estimated, from which population, and what uncertainty follows from the sampling process? This note turns that question into a practical analysis workflow.

Conceptual guide to Likelihood and log-likelihood

Core idea

View observed data as evidence for competing parameter values and understand maximum-likelihood estimation. The method is useful only when its target, data-generating assumptions, and unit of analysis match the scientific question.

$\ell(\theta)=\sum_i\log p(y_i\mid\theta)$

The equation is a compact statement of the target; it is not a substitute for checking design and data quality. State what each observation represents, how it entered the sample, and which sources of dependence remain.

Practical workflow

Write the scientific question and estimand in one sentence.
Identify biological units, technical replicates, nesting, repeated measurements, and exclusions.
Visualise raw observations and group structure before fitting the method.
Check assumptions using design knowledge, diagnostic plots, and sensitivity analyses.
Report the estimate, uncertainty, effect magnitude, sample sizes at every level, and limitations.

import numpy as np
from scipy import stats

x = np.array([8.2, 8.7, 9.1, 9.4, 9.8])
estimate = np.mean(x)
se = stats.sem(x)
interval = stats.t.interval(0.95, len(x)-1, loc=estimate, scale=se)
print(estimate, interval)

Interpretation and failure modes

Do not read a software output as an automatic scientific conclusion. Ask whether dependence was modeled, whether preprocessing used information from validation data, whether missingness or selection is informative, and whether the reported uncertainty covers every level of sampling. Prefer estimates and intervals to threshold-only language.

Microscopy case study

A Poisson likelihood can estimate an event rate from spot counts while respecting the count-valued outcome.

Connections

Use the tags above to continue to notes on design, assumptions, uncertainty, diagnostics, and complementary methods. A robust analysis normally combines several concepts rather than selecting one test in isolation.