Introduction E9F5FC

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Readings

• Statistics Reading Group: Reading List
• Evan Patterson. The algebra and machine representation of statistical models.
• Peter McCullagh. What is a statistical model?
• Workshop Categorical Probability and Statistics, 5–8 June 2020 (online) and June 2021 (in person). See tutorial videos.
• Nathaniel Virgo on Markov kernels
• Juan Pablo Vigneaux. Topology of statistical systems. A cohomological approach to information theory.

Videos

• Evan Patterson's presentation
• Peter McCullagh: Categorical notions in statistics
• Peter McCullagh: Methods & Theory: Statistical sparsity
• Evan Patterson: The algebra of statistical theories and models

Wikipedia

• Markov kernel
• Dempster-Shafer theory

Algebra of Statistical Models

• Monoidal category (tensor product)
• Symmetric monoidal category (maximally symmetric braided tensor product)
• Cartesian category
• Regular category (regular logic: exists, and, true)
• Coherent category (coherent logic: exists, and, true, or, false)
• Elementary topos (first-order logic)
• Cartesian category (tensor product is categorical product)(finite products)
• Cartesian closed category (lambda calculus, product types)(terminal object, products, exponentials)(internal Hom functor is adjoint functor to the product)
• Bicartesian closed category (lambda calculus, product types, sum types)
• Elementary topos

Concepts

• Whether is an object: its identity morphism.
• What it is as an object: the functor F. (Builds on validation of the rest.)
• How it is as an object: the Homset of morphisms. (Builds on validation of the rest.)
• Why it is an object: the entire category of morphisms along with their composition rules and identity rules.

Thoughts

• You have data - you have evidence of a probability distribution.
  • If the model is known - if it is a well understood process - then you want to apply the given model.
  • If the model is not known - if you want to find the most illuminating model - then you need to have a procedure for choosing the most meaningful model, that model which best balances insightfulness and faithfulness. You need to identify candidate models, distinguish competing models and have criteria for deciding amongst them.
• Patterson's goals: Understanding the internal structure of statistical models, comparing competing models, arguing that a certain model supports or fails to support some scientific theory.
• Extraction problem: Given a set of measurements of a phenomenon - a recurring activity - how can you extract the model of the process that is generating it. Note that there is likewise a guided coding by the process into what may be quite a simple model (binomial distribution) though it is generated by extremely complicated implementation (such as the factors determining the height of women, or the factors underlying the shapes of coins).
• Hierarchy of reproductive advantages. Reproduction is facilitated by a variety of advantages which order themselves in a hierarchy as to which most directly impact the probability of reproduction. Thus the properties that attract or repulse a mate are perhaps highest; those that facilitate competition with rivals, etc.

Ideas

• A type indicates the range of values that a variable may have. A probability distribution assigns a probability to every subrange, as well as its complement, thus to every subtype. See Subtyping.
• Questions-covariates-conscious. Answers-responses-unconscious. Model should relate the language of questions and the language of answers.
• Learn about relational frame theory.
• Consider game theory equilibria.
• Leibniz's Identity of indiscernibles.
  • "Those things are which show themselves to be."
  • Linguistics: things are the same if they always occur together. Things are the same if they never occur together.

Statistical concepts

Measure

• Lebesgue integration is important for measure theory and probability. The range is important, not the domain. It is the backwards of a function. What is the basis for this asymmetry? What does that say? How does it relate to the asymmetry of the category Set?
• Measure is additive (summing to 1), it is internal structure.

Probability distribution

• The relation between {$F_q$} and {$F_1$} is that between a list and a set, between a probability distribution (where there is choice) and a function (where this is a single choice of 1, a choice that is no choice).

Nondeterminism

• Relationships between nondeterminism and forgetting: elimination of variable (in automata theory), forgetful functor.

Randomness

• Randomness as an inverse functor.

Monad

• Choice as a wrapper.
• Coordinate system (observer) as a wrapper.
• Endofunctor. Introduction of order.

Bayesian statistics - only one methods of fitting a model (Bayesian inference)

Frequentist statistics - a single model may be fit by many different models

Bayesian statistics and Frequentist statistics are initial and terminal objects.

Frequentist estimators are usually defined as solutions to optimization problems. Efficient algorithms are often algorithms that are not guaranteed to converge to the intended estimator.

Fitting a statistical model to data

• 1) Specification of a statistical model.
• 2) Method of estimating the model's parameters.
• 3) Algorithm for computing the estimator.

Notes

• Statistics relates the ideal and the empirical, thus bridges the ideal and empirical Houses of knowledge.
• https://en.wikipedia.org/wiki/Bootstrapping_(statistics)
• Checking model fit: Could you simulate fake data for your model and see if it looks like your real data?
• Andrew Gelman: Bayes, statistics and reproducibility
• Gelman about Cantor: "All the action is on the diagonal." The diagonal is the fixed point, the identity morphism.

Gelman: Three fundamental problems in statistics

• Generalizing from sample to population
• Generalizing from treatment to control group
• Generalizing from your measurement to your underlying constructive interest

Thermodynamics and statistical physics

• Adiabatic process
• Brownian motion in time
• (small deviations) Ito-Stratanovich (big deviations) fluctuations, stochastic processes
• Levy process, Levy signal

Random matrices

• Tai-Danae Bradley. At the Interface of Algebra and Statistics
• https://www.math3ma.com/blog/matrices-probability-graphs
• Random matrices https://arxiv.org/abs/1712.07903

Sevensome

• I suspect that the seven-fold equation (fraction of differences, as with Mobius transformations) in Shu-Hong's thesis organizes seven natural bases that express different ways of looking at probability and confidence in a self-standing system.
• Richard Feynman. Negative probability.

https://en.wikipedia.org/wiki/PCP_theorem Every decision problem in the NP complexity class has probabilistically checkable proofs

Study choice, probability, statistics.

Spaces of states

• nLab: State
• Classical bit: line segment [0,1]
• Qubit: shaped like an American football

{$\begin{pmatrix} a & b+ic \\ b-ic & d \end{pmatrix}$}

Think of probabilities {$a, 1-a$} and mediator {$b \pm ic$}. We have {$a^2+b^2+c^2\leq a$} and {$a^2+b^2+c^2 = a$} for pure states. Rotate {$a-a^2$} from 0 to 1 around the a-axis.

How are games in game theory (with incomplete information, partial information) characterized by probability distributions.

A qubit specifies the relation between affirmation and negation of probabilities. In matrix form, it provides a complex number which is the coefficient that gets multiplied to the negation (in calculating the new affirmation) and whose conjugate gets multipled to the affirmation (in calculating the new negation). In classical bits, this coefficient is simply zero.

Schroedinger's cat

• Do the probabilities of a superposition evolve as per the phasor?
• Do they run through all possibilities between wavelengths?
• And does that mean that the decision of whether Schroedinger's cat is awake or asleep depends on the exact moment that we make the measurement?
• And what does that say about time? and superpositions?
• Is this a valid interpretation of superposition?
• Foundation of statistics is models. Distinguish the signal and the noise. Ignore the noise.
• http://stat.vadoveliai.lt/

https://www.johnmyleswhite.com/notebook/2013/03/22/modes-medians-and-means-an-unifying-perspective/