BIOS6902 Statistical Inference
The Complete Exam Bible · Sem A 2026/27BIOS6902 Overview
- City University of Hong Kong
- Semester A 2026/27
- Postgraduate
- Statistics
- Eleven lectures
What this course is
This is a systematic and modern introduction to the key ideas and methods of inference, with weight given to what is useful in biostatistics and data science.
- Two floors, not one average Continuous assessment must reach 40% and the final examination must reach 35%, separately.
- Sixty per cent before the final Three assignments and a midterm decide most of the grade before the examination period.
- Closed book, one sheet The final runs two hours with one A4 sheet of handwritten notes permitted.
- Assumptions earn the marks The course states that it assesses interpretation and appropriate use rather than long algebraic derivation.
How BIOS6902 is assessed
| Component | Weight | Format |
|---|---|---|
| Assignments (3) | 30% | Continuous assessment. Three written assignments, each submitted as a single document with an appendix of code. No per-assignment weight is published. |
| Midterm examination | 30% | Continuous assessment. No date, duration or format is published in the course materials. |
| Final examination (2 hours) | 40% | Two hours. Examinations are closed book, with one A4 sheet of handwritten notes permitted. |
Continuous assessment, meaning the three assignments plus the midterm examination, totals 60% and the final examination is 40%. The pass condition is not an overall average: to pass, students must obtain at least 40% of the total possible marks in continuous assessment and at least 35% of the total possible marks in the final examination, and both conditions apply. No assignment due dates are published in the course materials, and no examination date is published, so take every date from the course site.
What BIOS6902 covers
Eleven source-led chapters move from estimators and the bootstrap through likelihood, sufficiency and Bayesian updating to testing, causal inference and trustworthy experimentation.
Models, Estimators and the Bias-Variance Trade-off
Name the target, build an estimator for it, and measure how badly it can missWhat the first lecture is actually setting up
Inference, as this course defines it, is what you do when observations are used to work out something about whatever produced them.
- Parametric against nonparametric models, with the four biostatistical families
- Statistical functionals: mean, variance, median, regression function
- Bias, variance and mean squared error as one budget
- Consistency, and why a shrinking bias is compatible with it
- Hoeffding and normal-based confidence intervals for a prevalence
- The correct and incorrect ways to read a ninety-five per cent interval
Empirical Distributions, Functionals and the Bootstrap
Estimate the whole distribution first, then read any quantity you want off the estimateThe staircase that estimates an entire function
The first lecture estimated one number at a time. This one estimates the whole distribution function and then reads whatever number is wanted off the estimate.
- The empirical distribution function and its pointwise mean and variance
- Glivenko-Cantelli uniform consistency
- The simultaneous confidence band and what it promises
- The plug-in principle, and linear functionals as sample averages
- The resampling algorithm and the bootstrap variance
- Normal, percentile and pivotal intervals, and when each fails
Method of Moments and Maximum Likelihood
Two ways to fit a parametric model, and the information that prices the answerNarrowing the question, and two ways to answer it
A parametric model replaces the question “what is the whole distribution” with the much smaller question “what is the parameter”. Two answers dominate this course.
- Moment equations and the estimators they produce
- The likelihood, the log-likelihood, and three reasons to prefer the logarithm
- Invariance, consistency and asymptotic normality of the maximiser
- Score, Fisher information, and information as curvature
- Standard errors and asymptotic intervals from the information
- The delta method as slope times spread, worked on the logit
Choosing Between Methods and Diagnosing Failure
Match the assumptions to the problem, and recognise the four ways these tools breakA workshop, not a method
By the fourth lecture the course owns a small workshop: a staircase estimate of the whole distribution with a simultaneous band, a plug-in rule for any functional, moment and likelihood estimators for parametric models, and three ways to attach uncertainty to each.
- The assumption table across distribution-free, normal-based and resampling intervals
- The rule of thumb linking sample size and tail weight to a choice
- Coverage failure of a normal interval for a proportion near a boundary
- Under-coverage of a symmetric interval around a skewed statistic
- Why resampling reproduces rather than removes a boundary bias
- Distribution-free coverage under normal and heavy-tailed data
Sufficient Statistics and the Factorization Theorem
Reduce the data to a summary that keeps every scrap of information about the parameterThrowing data away without losing anything
Anything computed from the observations alone, with no unknown quantity inside it, counts as a statistic: the ordered values, their average, their spread, the smallest, the largest.
- Statistics, and what distinguishes a sufficient one
- The conditional-distribution definition of sufficiency
- The Factorization Theorem as the practical test
- Sufficient statistics for the Bernoulli, Poisson, Normal and uniform models
- Why the likelihood can be written from the summary alone
- Rao-Blackwell improvement by conditioning
Bayesian Updating, the Cramer-Rao Bound and EM
Combine a prior with a likelihood, learn the floor on precision, and handle missing labelsPutting a distribution on the parameter
The Bayesian move is small to state and large in consequence: let the parameter itself carry a distribution before the data arrive, then condition on what was observed.
- Posterior proportional to likelihood times prior
- Conjugate updating for a proportion, read as imaginary counts
- Prior sensitivity at small sample sizes
- The precision-weighted posterior mean for a normal mean
- Fisher information examples and the Cramer-Rao lower bound
- Efficiency, and why the bound only restricts unbiased estimators
Hypothesis Testing: Errors, Power and Neyman-Pearson
Turn an estimate into a decision, and account honestly for both ways of being wrongEight cups, and the whole apparatus
The story the course opens with concerns a colleague of Fisher who insisted she could taste which liquid had gone into the cup first.
- The tea-tasting experiment as the original test
- Null and alternative, simple and composite
- Type I and Type II error, and the power function
- Size against level, and why discrete data separate them
- The Neyman-Pearson lemma and its narrow scope
- Wald tests, and which standard error belongs in the denominator
P-values, Model Fit and Permutation Inference
Report evidence rather than a verdict, and build a reference distribution when no formula fitsWhy a p-value replaces a pre-committed threshold
Fixing a significance level in advance forces a binary decision, and different choices can send identical data in opposite directions.
- The p-value as the smallest level at which the null would be rejected
- One-sided and two-sided tail probabilities
- The power of a Wald test and the shift term that controls it
- Four standard misreadings of a p-value, with their corrections
- Pearson's chi-square statistic and its degrees of freedom
- Two-sample Wald and likelihood-ratio tests
Potential Outcomes, Randomisation and A/B Tests
Ask what would happen under an intervention, and design a study that can answer itWhy eight lectures of estimation were not enough
Frozen-dessert sales and shark bites climb together because both climb when it is hot. Children with larger feet read better because both improve with age.
- Three classic associations that are not causal
- Potential outcomes and the outcome that is never observed
- The average treatment effect against the observed contrast
- How randomisation makes assignment independent of potential outcomes
- The difference-in-means estimator and its standard error
- Sharp-null re-randomisation as an exact test
Confounding, Propensity Scores and Adjustment
What survives when nothing was randomised, and what you must assume to get itThe comparison that invents an effect
Whether tobacco use causes cancer of the lung. Whether a change in schooling policy raises attainment. Whether antibiotics given early in life shape the microbiome years later.
- Why some causal questions cannot be randomised
- A confounding story in which the treatment does nothing
- Three symptoms that precede any causal sentence
- Ignorability, and the positivity condition that travels with it
- Stratification and standardisation for low-dimensional covariates
- The propensity score and its balancing property
Trustworthy A/B Testing: Peeking and Multiple Metrics
Decide in advance how many chances to be wrong you are buyingWhere trustworthy experiments actually break
Controlled online experiments run in software products, in digital health and behaviour-change programmes, in pricing and promotion, and increasingly in public-health work.
- Where online experiments are used, and what the environment adds
- Five named pitfalls and what each does to the error rate
- The arithmetic of repeated looks in the independent case
- Multiple metrics as the same accounting problem
- Dividing the level by the number of planned comparisons
- Sequential designs that pay for interim looks properly
Over eleven lectures it covers point and interval estimation, the evaluation of estimators, likelihood methods, the bootstrap, sufficiency, elements of Bayesian inference, hypothesis testing including power, likelihood-ratio and permutation procedures, and an introduction to causal inference and A/B testing.
The stated emphasis throughout is on conceptual understanding, practical interpretation and the appropriate use of methods, supported by concrete examples and illustrations in R.
That emphasis should change how you revise. The course says in its own materials that it places weight on understanding assumptions, interpreting results and recognising when a method is appropriate, rather than on lengthy algebraic derivation.
A memorised derivation earns little here. A correctly named assumption and an accurate reading of a number earn a great deal.
How the eleven lectures fit together
The material falls into three arcs.
The first four lectures build estimation: models and estimators, the empirical distribution and the bootstrap, moment and likelihood methods, and a lecture spent entirely on choosing between them and diagnosing failure. The next two supply structure: sufficiency and the Factorization Theorem, then Bayesian updating, the Cramer-Rao bound and the expectation-maximisation algorithm.
Lectures seven and eight build hypothesis testing and the p-value, including goodness of fit and permutation procedures.
The last three lectures are the destination. They introduce potential outcomes and the average treatment effect, randomised experiments and their analysis, confounding and adjustment in observational studies, and the process failures that break real experiments.
The course materials describe these as modern causal and experimental ideas that are increasingly important in biostatistics and industry, complementing the classical core developed earlier.
What this guide contains
Eleven chapters follow the eleven lectures in teaching order, each with worked examples, practice items with full solutions, figures computed from their own stated data, and a glossary of the terms that chapter introduces.
Two closing sections follow: a mixed practice paper drawn across all three arcs, and a preparation section on what belongs on the single permitted sheet of notes and how to spend two hours in the examination room.
Everything about the course itself, including the assessment weights, the pass condition and the topic sequence, was read from the University's own syllabus and lecture materials for this offering.
The explanations, worked solutions, practice questions, figures and any mark allocations shown on practice items are ours. The marks beside our practice items are a study aid and are not the University's marking scheme, which publishes no per-question breakdown.
Where the course actually gets difficult
Three places, reliably.
The first is the interpretation of a confidence interval: once the data are in, the interval and the parameter are both fixed numbers, so the stated confidence describes the procedure rather than the particular interval. The second is the p-value, which is computed assuming the null hypothesis and therefore cannot be read as the probability that the null is true, nor a large value as proof that it is.
The third is the step from association to causation, where the same arithmetic means one thing after randomisation and something quite different without it.
All three are named explicitly in the lectures, which is a reliable signal: a misreading the teaching materials go out of their way to warn against is a misreading the assessment is likely to probe.
Rehearse the correct sentence for each until it is automatic, because under time pressure the wrong version is the one that arrives first.
Worked example · free
A prevalence estimate, its interval, and the sentence that goes in the report
- 1Estimate the prevalence from the counts.
- 2Compute the estimated standard error for a proportion.
- 1Assemble the approximate 95% interval.
- 2Write the interpretation, and name the misreading you are avoiding.
Key terms
- Statistical Inference
- Working out something about whatever produced a set of observations, from those observations alone, by estimating a quantity, bounding it or testing a claim.
- Estimator
- A function of the observed data used to stand in for an unknown feature of the distribution, judged by its bias, variance and mean squared error.
- Confidence Interval
- A random interval built so that, across repeated studies, a stated proportion of such intervals contain the fixed parameter.
- Likelihood
- The probability or density of the observed data read as a function of the parameter, maximised in logarithmic form to fit parametric models.
- Bootstrap
- A resampling method that estimates the sampling behaviour of a statistic by drawing repeatedly with replacement from the observed data.
- Sufficient Statistic
- A summary of the data that carries all the information about the parameter, so the raw observations may be discarded without loss.
- P-value
- The probability, computed under the null hypothesis, of a statistic at least as extreme as the one observed.
- Statistical Power
- The probability that a test rejects the null when a specified alternative is true, rising with sample size, effect size and information per observation.
- Average Treatment Effect
- The expected difference between the outcome a unit would show under treatment and the outcome it would show under control, averaged across units.
- Confounding
- Distortion of a treatment-outcome comparison by a common cause of both, which randomisation removes by design and adjustment attempts to remove by assumption.
- Propensity Score
- The probability of receiving treatment given the measured covariates, used to reduce a high-dimensional adjustment to a single number.
BIOS6902 FAQ
How is Statistical Inference at City University of Hong Kong assessed?
There are three components. Three assignments together carry 30%, a midterm examination carries 30%, and a two-hour final examination carries 40%. The assignments and the midterm are treated as continuous assessment and total 60%.
The pass condition is not an overall average: students must obtain at least 40% of the total possible marks in continuous assessment and at least 35% of the total possible marks in the final examination, and both conditions apply independently.
Is the final examination open book?
No. The course materials state that examinations are closed book, with one A4 sheet of handwritten notes permitted. That single sheet is the most valuable preparation artefact in the course, and it rewards conditions of validity, the shapes of standard errors and the critical values rather than full derivations.
How much mathematics does this course expect?
Real mathematics, but not lengthy derivation. You will differentiate log-likelihoods, compute Fisher information, apply the delta method and work with chi-square and normal critical values.
The course materials say explicitly that emphasis falls on understanding assumptions and interpreting results rather than on long algebraic manipulation, and a background document covering calculus, probability and common distributions is provided before the first lecture.
What topics does the course actually cover?
Eleven lectures: models and estimators with the bias-variance trade-off; the empirical distribution, functionals and the bootstrap; method of moments and maximum likelihood; comparing methods, Fisher information and the delta method; sufficient statistics and the Factorization Theorem; Bayesian ideas, conjugate priors and the Cramer-Rao bound; hypothesis testing; p-values, likelihood-ratio tests, goodness of fit and permutation tests; causal inference and A/B testing; observational studies, confounding and propensity scores; and modern A/B testing practice.
What are the three assignments like?
Each is out of one hundred points and splits into a theory part and a practical part, submitted as a single document with an appendix of R code.
The first covers estimation, standard errors and bootstrap intervals; the second covers likelihood-ratio and Wald tests, power, and Bayesian updating under different priors; the third is a causal and experimental mini-project with a choice between a simulated experiment and an illustration of confounding. No due dates are published in the materials.
Why do the last three lectures look different from the rest?
Because they answer a different kind of question. The first eight lectures estimate features of a distribution; the last three ask what would happen under an intervention, which no amount of estimation answers on its own. The course materials describe them as modern causal and experimental ideas increasingly important in biostatistics and industry, complementing the classical core developed earlier.
What is the single most common mistake students make in this subject?
Attaching a probability to a fixed number. After the data arrive, a confidence interval is two fixed numbers and the parameter is one fixed number, so the stated confidence describes the procedure across repeated studies rather than this particular interval. The same error reappears with p-values, which are computed assuming the null and therefore cannot be read as the probability that the null is true.
Questions students ask in each BIOS6902 chapter
- Should an estimator centred on its target always be preferred?
- No, and the question is raised in the opening lecture precisely because the intuitive answer is wrong. What matters is mean squared error, which adds the squared bias to the variance. An estimator that sits slightly off-centre but barely moves from sample to sample can easily beat an unbiased estimator that swings widely. The largest observation from a uniform distribution is the standard illustration: it is biased at every sample size and still converges to the truth. Chapter 01 · Models, Estimators and the Bias-Variance Trade-off
- When would I bootstrap instead of using a formula?
- Whenever the target has no convenient analytic standard error. Medians, quantiles, correlations, ratios of expectations and robust scale measures all fall into that category, and the course uses each of them. The method also earns its place when a formula exists but rests on an approximation you do not trust, because the resampled distribution can be inspected directly rather than assumed. Chapter 02 · Empirical Distributions, Functionals and the Bootstrap
- Do the moment and likelihood estimators always agree?
- No. They agree on several familiar models, including the Bernoulli and the Normal, which is why the coincidence feels like a rule. On skewed families such as the Gamma the two procedures solve genuinely different equations and return different numbers, and the course asks you to compare them on a small sample for exactly that reason. Chapter 03 · Method of Moments and Maximum Likelihood
- How do I decide between a narrow interval and a guaranteed one?
- Ask what a false claim would cost and how small the sample is. A distribution-free interval holds at every sample size and for every true value, and that unconditional validity is worth the extra width when the consequence of being wrong is serious or when the data are visibly not normal. When the sample is large and roughly symmetric the guarantee buys almost nothing and the narrower interval is the better report. Chapter 04 · Choosing Between Methods and Diagnosing Failure
- Is the sample mean always sufficient?
- No, and the counter-example is the uniform model on an interval starting at zero. There the likelihood is flat wherever every observation falls below the upper endpoint and zero otherwise, so the data touch the parameter only through the largest observation. The maximum is sufficient and the sample mean is not, which is exactly why the course keeps returning to this family. Chapter 05 · Sufficient Statistics and the Factorization Theorem
- Does the Cramer-Rao bound apply to every estimator?
- No, and treating it as a universal limit is a standard error in written answers. It restricts unbiased estimators only. A biased estimator, including every Bayes estimator with a non-degenerate prior, can have a smaller mean squared error than the bound permits for unbiased competitors, which is one concrete reason unbiasedness is not treated as a goal in this course. Chapter 06 · Bayesian Updating, the Cramer-Rao Bound and EM
- Why is the Type I error controlled rather than the Type II error?
- Because the two mistakes fall on different parties and carry different costs. Rejecting a true null means announcing an effect that does not exist, which propagates into practice, into later research and into decisions; the convention fixes that rate in advance and then buys as much detection ability as the budget allows. It is a convention about who bears the risk, not a mathematical necessity. Chapter 07 · Hypothesis Testing: Errors, Power and Neyman-Pearson
- Does a large p-value mean the null hypothesis is true?
- No. It means the data are compatible with the null, which is also what a small study of a genuine effect produces. The correct sentence is that the evidence against the null is insufficient at this level. Report the estimate and its interval alongside the p-value so a reader can see which values were ruled out and which remain plausible. Chapter 08 · P-values, Model Fit and Permutation Inference
- Why can both potential outcomes never be observed for one unit?
- Because each unit receives exactly one assignment and the observed outcome is the one attached to it. The other is a counterfactual, a description of a world that did not occur, and no measurement can recover it. That is why the individual effect is not an estimable quantity and why the target is an average, where the two unobservable halves can be estimated from different groups of people. Chapter 09 · Potential Outcomes, Randomisation and A/B Tests
- Why can ignorability never be tested from the data?
- Because it is an assertion about the unobserved half of each unit's outcomes. Any test would need the counterfactual outcomes that the design never supplies, so no statistic can confirm or refute it. It is defended by subject-matter argument about what actually drove treatment decisions, supported by a sensitivity analysis, and never by a goodness-of-fit measure. Chapter 10 · Confounding, Propensity Scores and Adjustment
- Why does an early inspection break the error rate?
- Because every inspection is a fresh opportunity to cross the threshold and the rule is built to halt at the first crossing. The guarantee attached to a single test assumes one comparison, and taking several without adjustment spends the error budget many times over. Fixing the decision rule in advance, or using a design that allocates the budget across planned looks, restores the guarantee. Chapter 11 · Trustworthy A/B Testing: Peeking and Multiple Metrics
How to study for the exam
Work in the order the lectures run, because the course is cumulative in a specific way: the standard error built in the likelihood chapter is what the testing chapters divide by, and the permutation argument built for two-sample testing is what the causal chapters reuse on a randomised experiment.
For each chapter, do the worked example before reading its solution, then write one line onto your single permitted sheet: the formula or the condition you had to look up. That sheet, built chapter by chapter, is worth more than any amount of rereading. Finish with the mixed practice paper under timed conditions and mark yourself only on whether a reader could act on your answers.
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