Md. Asif Uddin

    Chapter 5 VII.5

    Representation Learning for Biology

    The machinery of a latent-variable model is domain-neutral; what makes it biological is the likelihood and what the nuisance terms absorb.

    The likelihood is where biology enters a latent-variable model.

    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

    The ELBO is derived in Book I and does not need deriving again. What this chapter supplies is the part that is genuinely biological: a count likelihood, a library-size term, and a batch covariate that absorbs what should not be called signal.

    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. 5 — 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

    • Embeddings
    • Autoencoders
    • Variational autoencoders
    • Transformers
    • Foundation models for biology

    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.

    Kullback–Leibler divergence 0.IT.03 · The exponential family 0.PR.06 · Estimators, bias and variance 0.ST.01

    Notation

    • θAll parameters of a model, taken together
    • 𝔼Expectation
    • KLKullback–Leibler divergence
    • VarVariance
    • σThe logistic function, or a standard deviation

    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.