Md. Asif Uddin

    Chapter 2 VII.2

    Biological Data

    Biological counts are overdispersed and sparse, and a method that assumes otherwise fails in a direction it will not report.

    Counts are overdispersed and sparse, and a method that assumes otherwise fails silently.

    How this chapter is built

    M2Substantive

    The derivations are the chapter. A reader who skips the algebra has not learned it.

    basics2/9what the words mean
    concept2/2what to picture
    theory0/2why it works, and when it does not
    mathematics0/9derive it, then compute it
    practice0/6build it, break it, read the papers

    Five strands, not one. Mathematics is the spine; the other four are the body. A chapter cannot pay its way out of teaching with problems, nor out of problems with teaching.

    Before you start

    The problem

    Sequencing produces counts, not measurements. Their noise is not Gaussian, their zeros are not absences, and using a model that assumes either produces confident wrong answers rather than obvious errors.

    Where scVI departs from a textbook autoencoderThree rows comparing an ordinary variational autoencoder with scVI: the likelihood is a negative binomial rather than a Gaussian, batch identity conditions both encoder and decoder rather than being corrected afterwards, and library size gets its own latent instead of contaminating the cell state.textbook VAEscVIcount likelihoodGaussian, squared errornegative binomialbatchcorrected afterwardsconditioned on, in both halvessequencing depthleft in the latentits own scaling factorAny method that ignores these produces a beautiful embedding of your experimental logistics.Three independent evaluations, different teams, different data, put this eight-year-old VAE ahead.
    Fig. 2 — Three departures from a textbook autoencoder, each one a fact about the assay: a count likelihood, batch as a conditioning variable, and library size given its own latent.

    What this chapter covers

    • Sequencing
    • Expression
    • Single-cell data
    • Imaging
    • Spatial transcriptomics
    • Perturbation data

    Apparatus

    The mathematics this chapter leans on, held in Book 0 so it can be assumed here without being taught here. Not a gate — follow a link when a step stops making sense.

    Distributions, discrete and continuous 0.PR.01 · Variance and covariance 0.PR.03 · The exponential family 0.PR.06

    Notation

    • 𝔼Expectation
    • VarVariance
    • X (random)A random variable
    • ℝThe reals

    Propositions

    Not yet written. The topics above are the plan for this chapter; each will become a proposition with its own figure.

    Worked problems

    0/3 problems0/3 variants0/6 exercisesowes 9 more

    Not yet written. At M2 this chapter owes 3 worked problems across 3 distinct variants, and 6 exercises, every one with a published solution. The build enforces that from the day the chapter is marked published.