Seven perspectives
Methods divide for more than one reason. The question differs, certainly. But so do the assumptions placed on the data, the criteria for what counts as good, and the constraints on computation. Change any one of those and a different method suits the same prediction task.
The seven are not mutually exclusive; they look at different dimensions. The same equation often reads in several of them at once. Ridge regression, for instance, is a penalty in the geometric view, a MAP estimate under a Gaussian prior in the Bayesian view, and a control on complexity in learning theory. In the tables, a session that fits more than one perspective has the related perspectives named alongside it.
Geometry (least squares)
Minimize the sum of squared deviations. Concerned with how the structure of the data and the model is represented.What it looks at: how structure is represented
Frequentist (sampling variation, estimation, testing)
Judge the quality of an estimate or a decision by how it varies when the sample is drawn again.What it looks at: how uncertainty is handled
Bayesian
Combine what was known before observing with the data, and update belief into a posterior distribution.What it looks at: how uncertainty is handled
Learning theory
Keep the loss on data not yet seen in view, and balance fit against complexity.What it looks at: how predictive performance is judged
Information theory
Treat a short encoding as a good explanation.What it looks at: evaluation in terms of information
Games and adversaries
Read learning as the interaction of several agents in competition.What it looks at: interaction between agents
Causality
Ask what would happen under an intervention — a counterfactual question.What it looks at: the kind of question asked
→ Read the seven perspectives in equations A child page setting out the equation at the centre of each one, and how to read the symbols. Beyond it sit twenty-one pages on the methods used in actual research, grouped by perspective.
How to read the tables
The number of rows does not stand for how thoroughly something is covered. Some rows gather several sessions together, and the breadth of a perspective differs from item to item. For comparison, read the stage column.
Geometry (least squares)
Minimize the sum of squared deviations. Concerned with how the structure of the data and the model is represented.What it looks at: how structure is represented
| Course | Session | Method or topic (related perspectives) | Stage | Mathematics used | Connects to | Note |
|---|---|---|---|---|---|---|
| Statistics I | S2 · S3 | Averages and spread | Computation | — | Lin I | Summarize a distribution with sums and sums of squares. Reading them as norms comes later |
| Statistics I | S5 | Correlation and causation Causality | Computation | — | Lin I | Read correlation as covariance. The causal argument is handed on to Statistics II |
| Statistics I | S10 | Simple and multiple regression (OLS) Frequentist | Computation | — | Lin ILin II | The normal equations and projection. Assume normal errors and it agrees with maximum likelihood |
| Statistics I | S13 | Principal component analysis | Computation | — | Lin II | What the eigendecomposition means is covered; computing it numerically is not |
| Social Research | S5 | Using official statistics | Concepts | — | Lin I | Read aggregate tables. Treating them as matrices comes later |
| Applied Info. Literacy | S4 · S6 | Distributions, averages, descriptive statistics | Computation | — | Lin I | Summarize, in numbers, data the students gathered themselves |
| Applied Info. Literacy | S9 | Graphs and scatter plots | Computation | — | — | Read a relationship between two variables by eye. No equations |
| Behaviormetrics | S8 | Correlation and scatter plots | Computation | — | Lin I | Compute the correlation coefficient in R |
| Intro. to Information | S6 · S7 | Descriptive statistics and reading data | Computation | — | — | Aggregation in a spreadsheet. No equations |
| Intro. to Information | S10 | Reading graphs | Concepts | — | — | Designing axes and comparisons. Figures that mislead |
| Introduction to Data Usein preparation | S10 | Basics of descriptive statistics | Computation | — | Lin I | Mean, variance and standard deviation in Excel |
| Introduction to Data Usein preparation | S12 · S13 | Reading graphs, fake statistics Causality | Concepts | — | — | Designing figures, and misreading them. The trap of stratification is touched on |
| Big Data Analysisin preparation | S6 · S7 | Multiple regression Frequentist | Computation | — | Lin ILin II | The design matrix and the normal equations. Numerical decomposition comes later |
| Big Data Analysisin preparation | S8 · S9 | Principal component analysis | Computation | — | Lin II | Eigendecomposition and SVD |
| Big Data Analysisin preparation | S10 · S11 | k-means | Computation | — | Lin I | Minimizing the within-cluster sum of squares |
| Big Data Analysisin preparation | S12 · S13 | PLS regression Learning theory | Computation | — | Lin II | Regression in a latent space. Chosen by cross-validation |
| Big Data Apps Iin preparation | S5 · S6 | PCA and biplots | Computation | — | Lin II | Visualizing dimension reduction |
| Big Data Apps IIin preparation | S2 | Coordinate and projection transforms | Computation | — | Lin I | Transforming spatial data is matrix arithmetic |
| Big Data Apps IIin preparation | S9 · S11 · S12 | PCA, k-means, PLS | Computation | — | Lin ILin II | Compressing features, and finding the groupings in them |
| Speech Processingin preparation | S5 | MFCC (mel filterbank, DCT) | Computation | — | Lin I | Compressing features by an orthogonal transform |
| Statistics II | S11 | Confounding and spurious correlation (primarily Causality) | Computation | — | Lin ILin II | Adjusting for covariates. Checked by computing a partial correlation |
| Applied Info. Literacy | S7 | Cross-tabulation (primarily Frequentist) | Computation | — | Lin I | The joint distribution of categories. The doorway to the chi-square test |
| Big Data Apps Iin preparation | S7 · S8 | k-means and stability assessment (primarily Learning theory) | Computation | — | Lin I | Whether a clustering reproduces. Generalization made tangible |
| Big Data Apps Iin preparation | S9 · S10 | PLS-DA and cross-validation (primarily Learning theory) | Computation | — | Lin II | Separate training from validation. See overfitting as a number |
Frequentist (sampling variation, estimation, testing)
Judge the quality of an estimate or a decision by how it varies when the sample is drawn again.What it looks at: how uncertainty is handled
| Course | Session | Method or topic (related perspectives) | Stage | Mathematics used | Connects to | Note |
|---|---|---|---|---|---|---|
| Statistics I | S6 | Samples and the central limit theorem | Concepts | — | Calc ICalc II | Felt through simulation. Not proved |
| Statistics I | S7 | Point and interval estimation | Computation | Calc I | Calc II | Check the idea of coverage by building the intervals |
| Statistics I | S8 · S9 | t-tests and ANOVA | Computation | — | Lin ILin II | Decomposing a sum of squares. Reading it as a projection comes later |
| Statistics II | S4 · S5 | Binomial, Poisson, normal distributions | Concepts | Calc I | Calc I | Set up a probability model. Integrating the density is not covered |
| Statistics II | S6 | Sampling distributions and the CLT revisited | Concepts | — | Calc II | Break the independence the standard error formula assumes, and see what follows |
| Statistics II | S7 | The theory of interval estimation | Computation | Calc I | Calc II | The t distribution and degrees of freedom. Coverage seen by simulation |
| Statistics II | S8 · S9 | Multiple comparisons, effect size, criticism of p-values Learning theory | Computation | — | Calc I | What repeating a test does. Reporting the size of an effect |
| Social Research | S10 | Census and sample survey | Concepts | — | Calc I | Sampling error and the sample size needed. Selection bias sits outside the equations |
| Applied Info. Literacy | S7 | Cross-tabulation Geometry | Computation | — | Lin I | The joint distribution of categories. The doorway to the chi-square test |
| Behaviormetrics | S9 | The chi-square test | Computation | — | Calc I | The gap from the expected frequencies |
| Behaviormetrics | S10 · S11 | t-tests and paired t-tests | Computation | — | Lin I | Comparing two groups. The structure of the data changes the test |
| Behaviormetrics | S13 · S14 | ANOVA and multiple comparisons | Computation | — | Lin ILin II | Decomposing a sum of squares, and adjusting for the number of comparisons |
| Introduction to Data Usein preparation | S4 | The history of social research and the birth of statistics | Concepts | — | Calc I | How the frequentist view came about, followed through its history |
| Introduction to Data Usein preparation | S11 | Sample and population, the law of large numbers, the CLT | Concepts | — | Calc I | The doorway to inferential statistics |
| Big Data Apps IIin preparation | S5 | Kernel density estimation | Computation | — | Calc II | Choosing the bandwidth is choosing how much to smooth |
| Speech Processingin preparation | S7 | HMM, GMM, self-supervised learning Bayesian | Computation | — | Calc IILin I | Maximizing likelihood by EM. Not derived |
| Statistics I | S10 | Simple and multiple regression (OLS) (primarily Geometry) | Computation | — | Lin ILin II | The normal equations and projection. Assume normal errors and it agrees with maximum likelihood |
| Big Data Analysisin preparation | S6 · S7 | Multiple regression (primarily Geometry) | Computation | — | Lin ILin II | The design matrix and the normal equations. Numerical decomposition comes later |
Bayesian
Combine what was known before observing with the data, and update belief into a posterior distribution.What it looks at: how uncertainty is handled
| Course | Session | Method or topic (related perspectives) | Stage | Mathematics used | Connects to | Note |
|---|---|---|---|---|---|---|
| Statistics I | S12 | The doorway to Bayesian statistics | Computation | — | Calc ICalc II | With discrete hypotheses, the posterior is arithmetic |
| Statistics II | S2 · S3 | Conditional probability and Bayes’ theorem | Computation | — | Calc I | How much the prior bites, checked by counting in the finite case |
| Statistics II | S12 · S13 | Information cascades and Condorcet’s jury theorem Games | Computation | — | Calc I | Sequential updating, and group judgement once independence has broken, by simulation |
| Speech Processingin preparation | S7 | HMM, GMM, self-supervised learning (primarily Frequentist) | Computation | — | Calc IILin I | Maximizing likelihood by EM. Not derived |
Learning theory
Keep the loss on data not yet seen in view, and balance fit against complexity.What it looks at: how predictive performance is judged
| Course | Session | Method or topic (related perspectives) | Stage | Mathematics used | Connects to | Note |
|---|---|---|---|---|---|---|
| Big Data Apps Iin preparation | S7 · S8 | k-means and stability assessment Geometry | Computation | — | Lin I | Whether a clustering reproduces. Generalization made tangible |
| Big Data Apps Iin preparation | S9 · S10 | PLS-DA and cross-validation Geometry | Computation | — | Lin II | Separate training from validation. See overfitting as a number |
| Speech Processingin preparation | S8 | Conformer and Whisper | Concepts | — | Calc II | Large-scale training and generalization. Backpropagation is not derived |
| Speech Processingin preparation | S9 | Evaluation metrics (WER, CER) | Computation | — | — | Designing what counts as an error. Compared against the human error rate |
| Statistics I | S11 | Model selection and AIC (primarily Information theory) | Concepts | Calc I | Calc I | Balancing fit against complexity. The code-length argument is not covered |
| Statistics II | S8 · S9 | Multiple comparisons, effect size, criticism of p-values (primarily Frequentist) | Computation | — | Calc I | What repeating a test does. Reporting the size of an effect |
| Big Data Analysisin preparation | S5 | Approximation algorithms (primarily Information theory) | Computation | Calc I | Calc I | Sketches and probabilistic data structures. Trading error against capacity |
| Big Data Analysisin preparation | S12 · S13 | PLS regression (primarily Geometry) | Computation | — | Lin II | Regression in a latent space. Chosen by cross-validation |
| Speech Processingin preparation | S13 | Deepfake detection (primarily Games and adversaries) | Concepts | — | Calc II | The chase between generation and detection, read as a min–max |
Information theory
Treat a short encoding as a good explanation.What it looks at: evaluation in terms of information
| Course | Session | Method or topic (related perspectives) | Stage | Mathematics used | Connects to | Note |
|---|---|---|---|---|---|---|
| Statistics I | S11 | Model selection and AIC Learning theory | Concepts | Calc I | Calc I | Balancing fit against complexity. The code-length argument is not covered |
| Big Data Analysisin preparation | S5 | Approximation algorithms Learning theory | Computation | Calc I | Calc I | Sketches and probabilistic data structures. Trading error against capacity |
Games and adversaries
Read learning as the interaction of several agents in competition.What it looks at: interaction between agents
| Course | Session | Method or topic (related perspectives) | Stage | Mathematics used | Connects to | Note |
|---|---|---|---|---|---|---|
| Speech Processingin preparation | S13 | Deepfake detection Learning theory | Concepts | — | Calc II | The chase between generation and detection, read as a min–max |
| Statistics II | S12 · S13 | Information cascades and Condorcet’s jury theorem (primarily Bayesian) | Computation | — | Calc I | Sequential updating, and group judgement once independence has broken, by simulation |
Causality
Ask what would happen under an intervention — a counterfactual question.What it looks at: the kind of question asked
| Course | Session | Method or topic (related perspectives) | Stage | Mathematics used | Connects to | Note |
|---|---|---|---|---|---|---|
| Statistics II | S10 | Introduction to causal inference (RCTs and observational studies) | Concepts | — | Lin I | The difference between intervening and observing. Identification assumptions handled in words |
| Statistics II | S11 | Confounding and spurious correlation Geometry | Computation | — | Lin ILin II | Adjusting for covariates. Checked by computing a partial correlation |
| Intro. to Information | S12 | Simpson’s paradox | Concepts | — | Lin I | Stratify and it reverses. A first encounter with confounding |
| Statistics I | S5 | Correlation and causation (primarily Geometry) | Computation | — | Lin I | Read correlation as covariance. The causal argument is handed on to Statistics II |
| Introduction to Data Usein preparation | S12 · S13 | Reading graphs, fake statistics (primarily Geometry) | Concepts | — | — | Designing figures, and misreading them. The trap of stratification is touched on |
Outside the statistical mathematics this map covers
Sessions concerned with design, ethics and communication, or with a different kind of mathematics, rather than with the mathematics of a method.
| Course | Session | Content | How it is handled |
|---|---|---|---|
| Social Research | S9, S11–S14 | Qualitative research (interviews, observation, ethnography) | Handled as questions of design and ethics, of recording and interpretation |
| Applied Info. Literacy | S1–S3, S10–S13 | Planning a survey, publishing and presenting the results | Designing a survey, and communicating it |
| Big Data Analysisin preparation | S1–S4 | The five Vs, columnar storage, the CAP theorem, DAGs | Designing the data platform. A different kind of mathematics is involved here: distributed systems, and computational complexity |
What this table can and cannot show
Can show: nothing reaches the derivation stage
Almost every session sits at concepts or at computation. Deriving the equations and proving the properties is not part of any course currently taught here. Whether some other part of the curriculum carries that role is worth checking.
Can show: the games perspective has few points of contact
Only session 13 of Speech Processing. The information cascade in Statistics II sits close to it, in that it is about group judgement once independence has broken.
Cannot show: that fewer rows means thinner coverage
Geometry has many rows partly because descriptive statistics, visualization and matrix computation are all gathered under that one perspective. The scope of the classification and the unit of aggregation are mixed together here, so the number of rows cannot tell you how thoroughly a topic is covered.
Cannot show: that something is missing because the mathematical prerequisites are missing
That cannot be separated from what happens to be assigned to this teacher, from the hours available, from the stated goals, or from how the tags were applied. This table alone does not settle the cause.
There are stages within reach even without the prerequisite mathematics
With finitely many hypotheses, a Bayesian update is arithmetic. Take the probability of heads to be either 1/4 or 3/4, with a prior of 1/2 on each, and suppose three independent tosses all come up heads: the posterior probability of 3/4 is (3/4)³ ÷ ((1/4)³ + (3/4)³) = 27/28. Normalizing takes addition; no multivariable integration is needed.
Information theory can be introduced the same way, through the entropy of finitely many outcomes, with logarithms and a weighted average. Generalization, too, asks for different prerequisites depending on whether a bound is being proved or the gap between training and unseen performance is being watched in an experiment. The question is not whether that mathematics has been studied, but which stage one chooses to work at.
The order the source document sets out, and what these courses touch
The order in which the source document places the mathematics, year by year. What the courses touch is noted alongside it. Whether that order is itself the right one is not examined here.
Year 1: sums of squares and inner products → normal equations and projection → extrema in one variable
Introduction to Data Use and Applied Information Literacy handle descriptive statistics and the idea of a sample, at the computation stage. Reading them as projections is not touched.
Year 2: QR, Cholesky, SVD → multivariable gradient and Hessian, IRLS
Statistics I goes as far as regression, AIC and PCA; Statistics II as far as probability, Bayes, and the doorway to causal inference. Computing the decompositions numerically is not covered.
Year 3: generalized eigenvalues, duality and KKT → kernels → alternating optimization in EM
Speech Processing handles EM, and the big-data courses handle PCA, k-means and PLS from the side of implementation. SVMs and kernel methods appear in no course here.
Year 4: Laplace approximation, variational inference, HMC → large-scale computation → backpropagation and SDEs
Session 8 of Speech Processing touches the training of large models. Bayesian computation, Gaussian processes, and the mathematics of generative models appear in no course here.
Source document: “Mathematical techniques and their relation to statistical methods and machine learning” (Onohara, September 2025). It is a set of notes Onohara wrote privately in order to think through the curriculum of the Faculty of Information Science, and it speaks for no one but its author. The seven perspectives and the mathematics tags come from it. The distinction between stages, and the assignment of each session, were worked out for this page.
The ideas themselves are explained in A Guide to Statistical Thinking, and which session of which course covers what is set out in the course map.