# Philosophy of statistics

Statistical analysis is very important in addressing the problem of induction. Can inductive inference be formalized? What are the caveats? Can inductive inference be automated? How does machine learning work?

## Introduction to the foundations of statistics

### Problem of induction

A key issue for the scientific method, as discussed in the previous outline, is the problem of induction. Inductive inferences are used in the scientific method to make generalizations from finite data. This introduces unique avenues of error not found in purely deductive inferences, like in logic and mathematics. Compared to deductive inferences, which are sound and necessarily follow if an argument is valid and all of its premises obtain, inductive inferences can be valid and probably (not certainly) sound, and therefore can still result in error in some cases because the support of the argument is ultimately probabilistic.

A skeptic may further probe if we are even justified in using the probabilities we use in inductive arguments. What is the probability the Sun will rise tomorrow? What kind of probabilities are reasonable?

In this outline, we sketch and explore how the mathematical theory of statistics has arisen to wrestle with the problem of induction, and how it equips us with careful ways of framing inductive arguments and notions of confidence in them.

### Early investigators

• “Ibn al-Haytham was an early proponent of the concept that a hypothesis must be supported by experiments based on confirmable procedures or mathematical evidence—an early pioneer in the scientific method five centuries before Renaissance scientists.” - Wikipedia
• Gerolamo Cardano (1501-1576)
• Book on Games of Chance (1564)
• John Graunt (1620-1674)
• Jacob Bernoulli (1655-1705)
• Ars Conjectandi (1713, posthumous)
• First modern phrasing of the problem of parameter estimation1
• See Hacking2
• Early vision of decision theory:

The art of measuring, as precisely as possible, probabilities of things, with the goal that we would be able always to choose or follow in our judgments and actions that course, which will have been determined to be better, more satisfactory, safer or more advantageous.3

### Foundations of modern statistics

• Central limit theorem
• Charles Sanders Peirce (1839-1914)
• Formulated modern statistics in “Illustrations of the Logic of Science,” a series published in Popular Science Monthly (1877-1878), and also “A Theory of Probable Inference” in Studies in Logic (1883).5
• With a repeated measures design, introduced blinded, controlled randomized experiments (before Fisher).
• Karl Pearson (1857-1936)
• The Grammar of Science (1892)
• “On the criterion that a given system of deviations…” (1900)6
• Proposed testing the validity of hypothesized values by evaluating the chi distance between the hypothesized and the empirically observed values via the $$p$$-value.
• With Frank Raphael Weldon, he established the journal Biometrika in 1902.
• Founded the world’s first university statistics department at University College, London in 1911.
• Ronald Fisher (1890-1972)
• Fisher significance of the null hypothesis ($$p$$-values)
• “On an absolute criterion for fitting frequency curves”7
• “Frequency distribution of the values of the correlation coefficient in samples of indefinitely large population”8
• “On the ‘probable error’ of a coefficient of correlation deduced from a small sample”9
• Definition of likelihood
• ANOVA
• Statistical Methods for Research Workers (1925)
• The Design of Experiments (1935)
• “Statistical methods and scientific induction”10
• Jerzy Neyman (1894-1981)
• Egon Pearson (1895-1980)
• Neyman-Pearson confidence intervals with fixed error probabilities (also $$p$$-values but considering two hypotheses involves two types of errors)
• Harold Jeffreys (1891-1989)
• objective (non-informative) Jeffreys priors
• Andrey Kolmogorov (1903-1987)
• C.R. Rao (b. 1920)

### Probability

Probability is of epistemic interest, being in some sense a measure of inductive confidence.

TODO:

• Kolmogorov axioms
• Probability vs odds: $$p/(p+q)$$ vs $$p/q$$
• Carnap: “Probability as a guide in life”23

### Expectation and variance

Expectation:

$\mathbb{E}(y) \equiv \int dx \: p(x) \: y(x) \label{eq:expectation}$

Expectation values can be approximated with a partial sum over some data or Monte Carlo sample:

$\mathbb{E}(y) \approx \frac{1}{n} \sum_s^n y(x_s) \label{eq:expectation_sum}$

The variance of a random variable, $$y$$, is defined as

\begin{align} \mathrm{Var}(y) &\equiv \mathbb{E}((y - \mathbb{E}(y))^2) \nonumber \\ &= \mathbb{E}(y^2 - 2 \: y \: \mathbb{E}(y) + \mathbb{E}(y)^2) \nonumber \\ &= \mathbb{E}(y^2) - 2 \: \mathbb{E}(y) \: \mathbb{E}(y) + \mathbb{E}(y)^2 \nonumber \\ &= \mathbb{E}(y^2) - \mathbb{E}(y)^2 \label{eq:variance} \end{align}

The covariance matrix, $$\boldsymbol{V}$$, of random variables $$x_i$$ is

\begin{align} V_{ij} &= \mathrm{Cov}(x_i, x_j) \equiv \mathbb{E}[(x_i - \mathbb{E}(x_i)) \: (x_j - \mathbb{E}(x_j))] \nonumber \\ &= \mathbb{E}(x_i \: x_{j} - \mu_i \: x_j - x_i \: \mu_j + \mu_i \: \mu_j ) \nonumber \\ &= \mathbb{E}(x_i \: x_{j}) - \mu_i \: \mu_j \label{eq:covariance_matrix_indexed} \end{align}

$$$\boldsymbol{V} = \begin{pmatrix} \mathrm{Var}(x_1) & \mathrm{Cov}(x_1, x_2) & \cdots & \mathrm{Cov}(x_1, x_n) \\ \mathrm{Cov}(x_2, x_1) & \mathrm{Var}(x_2) & \cdots & \mathrm{Cov}(x_2, x_n) \\ \vdots & \vdots & \ddots & \vdots \\ \mathrm{Cov}(x_n, x_1) & \mathrm{Cov}(x_n, x_2) & \cdots & \mathrm{Var}(x_n) \end{pmatrix} \label{eq:covariance_matrix_array}$$$

Diagonal elements of the covariance matrix are the variances of each variable.

$\mathrm{Cov}(x_i, x_i) = \mathrm{Var}(x_i)$

Off-diagonal elements of a covariance matrix measure how related two variables are, linearly. Covariance can be normalized to give the correlation coefficient between variables:

$\mathrm{Cor}(x_i, x_j) \equiv \frac{ \mathrm{Cov}(x_i, x_j) }{ \sqrt{ \mathrm{Var}(x_i) \: \mathrm{Var}(x_j) } } \label{eq:correlation_matrix}$

which is bounded: $$-1 \leq \mathrm{Cor}(x_i, x_j) \leq 1$$.

The covariance of two random vectors is given by

$\boldsymbol{V} = \mathrm{Cov}(\vec{x}, \vec{y}) = \mathbb{E}(\vec{x} \: \vec{y}^{\mathsf{T}}) - \vec{\mu}_x \: \vec{\mu}_{y}^{\mathsf{T}}\label{eq:covariance_matrix_vectors}$

### Cross entropy

TODO: discuss the Shannon entropy and Kullback-Leibler (KL) divergence.24

Shannon entropy:

$H(p) = - \underset{x\sim{}p}{\mathbb{E}}\big[ \log p(x) \big] \label{eq:shannon_entropy}$

Cross entropy:

$H(p, q) = - \underset{x\sim{}p}{\mathbb{E}}\big[ \log q(x) \big] \label{eq:cross_entropy}$

Kullback-Leibler (KL) divergence:

\begin{align} D_\mathrm{KL}(p, q) &= \underset{x\sim{}p}{\mathbb{E}}\left[ \log \left(\frac{p(x)}{q(x)}\right) \right] = \underset{x\sim{}p}{\mathbb{E}}\big[ \log p(x) - \log q(x) \big] \label{eq:kl_divergence} \\ &= - H(p) + H(p, q) \\ \end{align}

### Uncertainty

#### Quantiles and standard error

TODO:

• Quantiles
• Practice of standard error for uncertainty quantification.

#### Propagation of error

Given some vector of random variables, $$\vec{x}$$, with estimated means, $$\vec{\mu}$$, and estimated covariance matrix, $$\boldsymbol{V}$$, suppose we are concerned with estimating the variance of some variable, $$y$$, that is a function of $$\vec{x}$$. The variance of $$y$$ is given by

$\sigma^2_y = \mathbb{E}(y^2) - \mathbb{E}(y)^2 \,.$

Taylor expanding $$y(\vec{x})$$ about $$x=\mu$$ gives

$y(\vec{x}) \approx y(\vec{\mu}) + \left.\frac{\partial y}{\partial x_i}\right|_{\vec{x}=\vec{\mu}} (x_i - \mu_i) \,.$

Therefore, to first order

$\mathbb{E}(y) \approx y(\vec{\mu})$

and

\begin{align} \mathbb{E}(y^2) &\approx y^2(\vec{\mu}) + 2 \, y(\vec{\mu}) \, \left.\frac{\partial y}{\partial x_i}\right|_{\vec{x}=\vec{\mu}} \mathbb{E}(x_i - \mu_i) \nonumber \\ &+ \mathbb{E}\left[ \left(\left.\frac{\partial y}{\partial x_i}\right|_{\vec{x}=\vec{\mu}}(x_i - \mu_i)\right) \left(\left.\frac{\partial y}{\partial x_j}\right|_{\vec{x}=\vec{\mu}}(x_j - \mu_j)\right) \right] \\ &= y^2(\vec{\mu}) + \, \left.\frac{\partial y}{\partial x_i}\frac{\partial y}{\partial x_j}\right|_{\vec{x}=\vec{\mu}} V_{ij} \\ \end{align}

TODO: clarify above, then specific examples.

See Cowan.25

### Bayes’ theorem

$P(A|B) = P(B|A) \: P(A) \: / \: P(B) \label{eq:bayes_theorem}$

• Extended version of Bayes theorem
• Example of conditioning with medical diagnostics

### Likelihood and frequentist vs bayesian probability

$P(H|D) = P(D|H) \: P(H) \: / \: P(D) \label{eq:bayes_theorem_hd}$

• Likelihood

$L(\theta) = P(D|\theta) \label{eq:likelihood_def_x}$

To appeal to such a result is absurd. Bayes’ theorem ought only to be used where we have in past experience, as for example in the case of probabilities and other statistical ratios, met with every admissible value with roughly equal frequency. There is no such experience in this case.28

### Curse of dimensionality

• Curse of dimensionality
• The volume of the space increases so fast that the available data become sparse.
• The ordinary decision rule for estimating the mean of a multivariate Gaussian distribution (with dimensions, $$n \geq 3$$) is inadmissible under mean squared error risk.
• Proof of Stein’s example
• Probability in high dimensions29
• High-Dimensional Probability:An introduction with applications in data science30

## Statistical models

### Parametric models

• Data: $$x_i$$
• Parameters: $$\theta_j$$
• Model: $$f(\vec{x} ; \vec{\theta})$$

### Canonical distributions

#### Bernoulli distribution

$\mathrm{Ber}(k; p) = \begin{cases} p & \mathrm{if}\ k = 1 \\ 1-p & \mathrm{if}\ k = 0 \end{cases} \label{eq:bernoulli}$

which can also be written as

$\mathrm{Ber}(k; p) = p^k \: (1-p)^{(1-k)} \quad \mathrm{for}\ k \in \{0, 1\}$

or

$\mathrm{Ber}(k; p) = p k + (1-p)(1-k) \quad \mathrm{for}\ k \in \{0, 1\}$

• Binomial distribution
• Poisson distribution

TODO: explain, another important relationship is

#### Normal/Gaussian distribution

$N(x \,|\, \mu, \sigma^2) = \frac{1}{\sqrt{2\,\pi\:\sigma^2}} \: \exp\left(\frac{-(x-\mu)^2}{2\,\sigma^2}\right) \label{eq:gaussian}$

and in $$k$$ dimensions:

$N(\vec{x} \,|\, \vec{\mu}, \boldsymbol{\Sigma}) = (2 \pi)^{-k/2}\:\left|\boldsymbol{\Sigma}\right|^{-1/2} \: \exp\left(\frac{-1}{2}\:(\vec{x}-\vec{\mu})^{\mathsf{T}}\:\boldsymbol{\Sigma}^{-1}\:(\vec{x}-\vec{\mu})\right) \label{eq:gaussian_k_dim}$

where $$\boldsymbol{\Sigma}$$ is the covariance matrix (defined in eq. $$\eqref{eq:covariance_matrix_indexed}$$) of the distribution.

• Central limit theorem
• $$\chi^2$$ distribution
• Univariate distribution relationships
• The exponential family of distributions are maximum entropy distributions.

## Point estimation and confidence intervals

### Inverse problems

Recall that in the context of parametric models of data, $$x_i$$ the pdf of which is modeled by a function, $$f(x_i ; \theta_j)$$ with parameters, $$\theta_j$$. In a statistical inverse problem, the goal is to infer values of the model parameters, $$\theta_j$$ given some finite set of data, $$\{x_i\}$$ sampled from a probability density, $$f(x_i; \theta_j)$$ that models the data reasonably well.33

### Bias and variance

The bias of an estimator, $$\hat\theta$$, is defined as

$\mathrm{Bias}(\hat{\theta}) \equiv \mathbb{E}(\hat{\theta} - \theta) = \int dx \: P(x|\theta) \: (\hat{\theta} - \theta) \label{eq:bias}$

The mean squared error (MSE) of an estimator has a similar formula to variance (eq. $$\eqref{eq:variance}$$) except that instead of quantifying the square of the difference of the estimator and its expected value, the MSE uses the square of the difference of the estimator and the true parameter:

$\mathrm{MSE}(\hat{\theta}) \equiv \mathbb{E}((\hat{\theta} - \theta)^2) \label{eq:mse}$

The MSE of an estimator can be related to its bias and its variance by the following proof:

\begin{align} \mathrm{MSE}(\hat{\theta}) &= \mathbb{E}(\hat{\theta}^2 - 2 \: \hat{\theta} \: \theta + \theta^2) \nonumber \\ &= \mathbb{E}(\hat{\theta}^2) - 2 \: \mathbb{E}(\hat{\theta}) \: \theta + \theta^2 \end{align}

noting that

$\mathrm{Var}(\hat{\theta}) = \mathbb{E}(\hat{\theta}^2) - \mathbb{E}(\hat{\theta})^2$

and

\begin{align} \mathrm{Bias}(\hat{\theta})^2 &= \mathbb{E}(\hat{\theta} - \theta)^2 \nonumber \\ &= \mathbb{E}(\hat{\theta})^2 - 2 \: \mathbb{E}(\hat{\theta}) \: \theta + \theta^2 \end{align}

we see that MSE is equivalent to

$\mathrm{MSE}(\hat{\theta}) = \mathrm{Var}(\hat{\theta}) + \mathrm{Bias}(\hat{\theta})^2 \label{eq:mse_variance_bias}$

For an unbiased estimator, the MSE is the variance of the estimator.

TODO:

• Note the discussion of the bias-variance tradeoff by Cranmer.
• Note the new deep learning view. See Deep learning.

### Maximum likelihood estimation

A maximum likelihood estimator (MLE) was first used by Fisher.35

$\hat{\theta} \equiv \underset{\theta}{\mathrm{argmax}} \: \mathrm{log} \: L(\theta) \label{eq:mle}$

Maximizing $$\mathrm{log} \: L(\theta)$$ is equivalent to maximizing $$L(\theta)$$, and the former is more convenient because for data that are independent and identically distributed (i.i.d.) the joint likelihood can be factored into a product of individual measurements:

$L(\theta) = \prod_i L(\theta|x_i) = \prod_i P(x_i|\theta)$

and taking the log of the product makes it a sum:

$\mathrm{log} \: L(\theta) = \sum_i \mathrm{log} \: L(\theta|x_i) = \sum_i \mathrm{log} \: P(x_i|\theta)$

Maximizing $$\mathrm{log} \: L(\theta)$$ is also equivalent to minimizing $$-\mathrm{log} \: L(\theta)$$, the negative log-likelihood (NLL). For distributions that are i.i.d.,

$\mathrm{NLL} \equiv - \log L = - \log \prod_i L_i = - \sum_i \log L_i = \sum_i \mathrm{NLL}_i$

#### Invariance of likelihoods under reparametrization

• Likelihoods are invariant under reparametrization.36
• Bayesian posteriors are not invariant in general.

#### Ordinary least squares

• Least squares from MLE of gaussian models: $$\chi^2$$
• Ordinary Least Squares (OLS)
• Geometric interpretation

### Bayesian credibility intervals

• Inverse problem to find a posterior probability distribution.
• Maximum a posteriori estimation (MAP)
• Prior sensitivity
• Not invariant to reparametrization in general
• Jeffreys priors are
• TODO: James

## Statistical hypothesis testing

### Null hypothesis significance testing

• Karl Pearson observing how rare sequences of roulette spins are
• Null hypothesis significance testing (NHST)
• goodness of fit
• Fisher

Fisher:

[T]he null hypothesis is never proved or established, but is possibly disproved, in the course of experimentation.50

### Neyman-Pearson theory

#### Neyman-Pearson lemma

Neyman-Pearson lemma:54

For a fixed signal efficiency, $$1-\alpha$$, the selection that corresponds to the lowest possible misidentification probability, $$\beta$$, is given by

$\frac{L(H_1)}{L(H_0)} > k_{\alpha} \,, \label{eq:np-lemma}$

where $$k_{\alpha}$$ is the cut value required to achieve a type-1 error rate of $$\alpha$$.

Neyman-Pearson test statistic:

$q_\mathrm{NP} = - 2 \ln \frac{L(H_1)}{L(H_0)} \label{eq:qnp-test-stat}$

Profile likelihood ratio:

$\lambda(\mu) = \frac{ L(\mu, \hat{\theta}_\mu) }{ L(\hat{\mu}, \hat{\theta}) } \label{eq:profile-llh-ratio}$

where $$\hat{\theta}$$ is the (unconditional) maximum-likelihood estimator that maximizes $$L$$, while $$\hat{\theta}_\mu$$ is the conditional maximum-likelihood estimator that maximizes $$L$$ for a specified signal strength, $$\mu$$, and $$\theta$$ as a vector includes all other parameters of interest and nuisance parameters.

#### Neyman construction

Cranmer: Neyman construction.

TODO: fix

$q = - 2 \ln \frac{L(\mu\,s + b)}{L(b)} \label{eq:q0-test-stat}$

#### Flip-flopping

• Flip-flopping and Feldman-Cousins confidence intervals55

### p-values and significance

• $$p$$-values and significance56
• Coverage
• Fisherian vs Neyman-Pearson $$p$$-values

Cowan et al. define a $$p$$-value as

a probability, under assumption of $$H$$, of finding data of equal or greater incompatibility with the predictions of $$H$$.57

Also:

It should be emphasized that in an actual scientific context, rejecting the background-only hypothesis in a statistical sense is only part of discovering a new phenomenon. One’s degree of belief that a new process is present will depend in general on other factors as well, such as the plausibility of the new signal hypothesis and the degree to which it can describe the data. Here, however, we only consider the task of determining the $$p$$-value of the background-only hypothesis; if it is found below a specified threshold, we regard this as “discovery.”58

#### CLs method

• Conservative coverage; used in particle physics
• Junk60
• ATLAS62

### Asymptotics

• Analytic variance of the likelihood-ratio of gaussians: $$\chi^2$$
• Wilks63
• Under the null hypothesis, $$-2 \ln(\lambda) \sim \chi^{2}_{k}$$, where $$k$$, the degrees of freedom for the $$\chi^{2}$$ distribution is the number of parameters of interest (including signal strength) in the signal model but not in the null hypothesis background model.
• Wald64
• Wald generalized the work of Wilks for the case of testing some nonzero signal for exclusion, showing $$-2 \ln(\lambda) \approx (\hat{\theta} - \theta)^{\mathsf{T}}V^{-1} (\hat{\theta} - \theta) \sim \mathrm{noncentral}\:\chi^{2}_{k}$$.
• In the simplest case where there is only one parameter of interest (the signal strength, $$\mu$$), then $$-2 \ln(\lambda) \approx \frac{ (\hat{\mu} - \mu)^{2} }{ \sigma^2 } \sim \mathrm{noncentral}\:\chi^{2}_{1}$$.
• Pearson $$\chi^2$$-test
• Cowan et al.65
• Criteria for projected discovery and exclusion sensitivities of counting experiments66

### Student’s t-test

• Student’s t-test
• ANOVA
• A/B-testing

### Frequentist vs bayesian decision theory

Support for using Bayes factors:

which properly separates issues of long-run behavior from evidential strength and allows the integration of background knowledge with statistical findings.69

### Examples

• Difference of two means: $$t$$-test
• A/B-testing
• New physics

## Uncertainty quantification

### Sinervo classification of systematic uncertainties

• Class-1, class-2, and class-3 systematic uncertanties (good, bad, ugly), Classification by Pekka Sinervo (PhyStat2003)70
• Not to be confused with type-1 and type-2 errors in Neyman-Pearson theory
• Heinrich, J. & Lyons, L. (2007). Systematic errors.71
• Caldeira & Nord72

Lyons:

In analyses involving enough data to achieve reasonable statistical accuracy, considerably more effort is devoted to assessing the systematic error than to determining the parameter of interest and its statistical error.73

• Poincaré’s three levels of ignorance

### Profile likelihoods

• Profiling and the profile likelihood
• Importance of Wald and Cowan et al.
• hybrid Bayesian-frequentist method

### Examples of poor estimates of systematic uncertanties

• OPERA. (2011). Faster-than-light neutrinos.
• BICEP2 claimed evidence of B-modes in the CMB as evidence of cosmic inflation without accounting for cosmic dust.

## Statistical classification

### Introduction

• Precision vs recall
• Recall is sensitivity
• Sensitivity vs specificity
• Accuracy

• TODO

## Causal inference

### Causal models

• Structural Causal Model (SCM)
• Pearl, J. (2009). Causal inference in statistics: An overview.76
• Robins, J.M. & Wasserman, L. (1999). On the impossibility of inferring causation from association without background knowledge.77
• Peters, J., Janzing, D., & Schölkopf, B. (2017). Elements of Causal Inference.78

• TODO

## Exploratory data analysis

### Look-elsewhere effect

• Look-elsewhere effect (LEE)
• AKA File-drawer effect
• Stopping rules
• validation dataset
• statistical issues, violates the likelihood principle

## “Statistics Wars”

### Introduction

• Kruschke
• Carnap
• “The two concepts of probability”80
• Royall
• “What do these data say?”81

Cranmer:

Bayes’s theorem is a theorem, so there’s no debating it. It is not the case that Frequentists dispute whether Bayes’s theorem is true. The debate is whether the necessary probabilities exist in the first place. If one can define the joint probability $$P (A, B)$$ in a frequentist way, then a Frequentist is perfectly happy using Bayes theorem. Thus, the debate starts at the very definition of probability.82

Neyman:

Without hoping to know whether each separate hypothesis is true or false, we may search for rules to govern our behaviour with regard to them, in following which we insure that, in the long run of experience, we shall not be too often wrong.83

### Likelihood principle

• Likelihood principle
• The likelihood principle is the proposition that, given a statistical model and a data sample, all the evidence relevant to model parameters is contained in the likelihood function.
• The history of likelihood85
• Berger & Wolpert. (1988). The Likelihood Principle.88

O’Hagan:

The first key argument in favour of the Bayesian approach can be called the axiomatic argument. We can formulate systems of axioms of good inference, and under some persuasive axiom systems it can be proved that Bayesian inference is a consequence of adopting any of these systems… If one adopts two principles known as ancillarity and sufficiency principles, then under some statement of these principles it follows that one must adopt another known as the likelihood principle. Bayesian inference conforms to the likelihood principle whereas classical inference does not. Classical procedures regularly violate the likelihood principle or one or more of the other axioms of good inference. There are no such arguments in favour of classical inference.89

Mayo:

Likelihoods are vital to all statistical accounts, but they are often misunderstood because the data are fixed and the hypothesis varies. Likelihoods of hypotheses should not be confused with their probabilities. … [T]he same phenomenon may be perfectly predicted or explained by two rival theories; so both theories are equally likely on the data, even though they cannot both be true.94

### Discussion

Lyons:

Particle Physicists tend to favor a frequentist method. This is because we really do consider that our data are representative as samples drawn according to the model we are using (decay time distributions often are exponential; the counts in repeated time intervals do follow a Poisson distribution, etc.), and hence we want to use a statistical approach that allows the data “to speak for themselves,” rather than our analysis being dominated by our assumptions and beliefs, as embodied in Bayesian priors.95

Goodman:

The idea that the $$P$$ value can play both of these roles is based on a fallacy: that an event can be viewed simultaneously both from a long-run and a short-run perspective. In the long-run perspective, which is error-based and deductive, we group the observed result together with other outcomes that might have occurred in hypothetical repetitions of the experiment. In the “short run” perspective, which is evidential and inductive, we try to evaluate the meaning of the observed result from a single experiment. If we could combine these perspectives, it would mean that inductive ends (drawing scientific conclusions) could be served with purely deductive methods (objective probability calculations).115

## Replication crisis

### Introduction

• Ioannidis, J.P. (2005). Why most published research findings are false.116

### p-value controversy

[N]o isolated experiment, however significant in itself, can suffice for the experimental demonstration of any natural phenomenon; for the “one chance in a million” will undoubtedly occur, with no less and no more than its appropriate frequency, however surprised we may be that it should occur to us. In order to assert that a natural phenomenon is experimentally demonstrable we need, not an isolated record, but a reliable method of procedure. In relation to the test of significance, we may say that a phenomenon is experimentally demonstrable when we know how to conduct an experiment which will rarely fail to give us a statistically significant result.120

From “The ASA president’s task force statement on statistical significance and replicability”:

P-values are valid statistical measures that provide convenient conventions for communicating the uncertainty inherent in quantitative results. Indeed, P-values and significance tests are among the most studied and best understood statistical procedures in the statistics literature. They are important tools that have advanced science through their proper application.123

## Classical machine learning

### Introduction

• Classification vs regression
• Supervised and unsupervised learning
• Classification = supervised; clustering = unsupervised
• Hastie, Tibshirani, & Friedman124
• Information Theory, Inference, and Learning125
• Murphy, K.P. (2012). Machine Learning: A probabilistic perspective. MIT Press.126
• Murphy, K.P. (2022). Probabilistic Machine Learning: An introduction. MIT Press.127
• Shalev-Shwarz, S. & Ben-David, S. (2014). Understanding Machine Learning: From Theory to Algorithms.128
• VC-dimension
• Vapnik (1994)129
• Shalev-Shwarz, S. & Ben-David, S. (2014).130

### Logistic regression

From a probabilistic point of view,134 logistic regression can be derived from doing maximum likelihood estimation of a vector of model parameters, $$\vec{w}$$, in a dot product with the input features, $$\vec{x}$$, and squashed with a logistic function that yields the probability, $$\mu$$, of a Bernoulli random variable, $$y \in \{0, 1\}$$.

$p(y | \vec{x}, \vec{w}) = \mathrm{Ber}(y | \mu(\vec{x}, \vec{w})) = \mu(\vec{x}, \vec{w})^y \: (1-\mu(\vec{x}, \vec{w}))^{(1-y)}$

The negative log-likelihood of multiple trials is

\begin{align} \mathrm{NLL} &= - \sum_i \log p(y_i | \vec{x}_i, \vec{w}) \nonumber \\ &= - \sum_i \log\left( \mu(\vec{x}_i, \vec{w})^{y_i} \: (1-\mu(\vec{x}_i, \vec{w}))^{(1-y_i)} \right) \nonumber \\ &= - \sum_i \log\left( \mu_i^{y_i} \: (1-\mu_i)^{(1-y_i)} \right) \nonumber \\ &= - \sum_i \big( y_i \, \log \mu_i + (1-y_i) \log(1-\mu_i) \big) \label{eq:cross_entropy_loss0} \end{align}

which is the cross entropy loss. Note that the first term is non-zero only when the true target is $$y_i=1$$, and similarly the second term is non-zero only when $$y_i=0$$.135 Therefore, we can reparametrize the target $$y_i$$ in favor of $$t_{ki}$$ that is one-hot in an index $$k$$ over classes.

$\mathrm{CEL} = \mathrm{NLL} = - \sum_i \sum_k \big( t_{ki} \, \log \mu_{ki} \big) \label{eq:cross_entropy_loss1}$

where

$t_{ki} = \begin{cases} 1 & \mathrm{if}\ (k = y_i = 0)\ \mathrm{or}\ (k = y_i = 1) \\ 0 & \mathrm{otherwise} \end{cases}$

and

$\mu_{ki} = \begin{cases} 1-\mu_i & \mathrm{if}\ k = 0 \\ \mu_i & \mathrm{if}\ k =1 \end{cases}$

This readily generalizes from binary classification to classification over many classes as we will discuss more below. Note that in the sum over classes, $$k$$, only one term for the true class contributes.

$\mathrm{CEL} = - \left. \sum_i \log \mu_{ki} \right|_{k\ \mathrm{is\ such\ that}\ y_k=1} \label{eq:cross_entropy_loss2}$

Logistic regression uses the logit function,136 which is the logarithm of the odds—the ratio of the chance of success to failure. Let $$\mu$$ be the probability of success in a Bernoulli trial, then the logit function is defined as

$\mathrm{logit}(\mu) \equiv \log\left(\frac{\mu}{1-\mu}\right) \label{eq:logit}$

Logistic regression assumes that the logit function is a linear function of the explanatory variable, $$x$$.

$\log\left(\frac{\mu}{1-\mu}\right) = \beta_0 + \beta_1 x$

where $$\beta_0$$ and $$\beta_1$$ are trainable parameters. (TODO: Why would we assume this?) This can be generalized to a vector of multiple input variables, $$\vec{x}$$, where the input vector has a 1 prepended to be its zeroth component in order to conveniently include the bias, $$\beta_0$$, in a dot product.

$\vec{x} = (1, x_1, x_2, \ldots, x_n)^{\mathsf{T}}$

$\vec{w} = (\beta_0, \beta_1, \beta_2, \ldots, \beta_n)^{\mathsf{T}}$

$\log\left(\frac{\mu}{1-\mu}\right) = \vec{w}^{\mathsf{T}}\vec{x}$

For the moment, let $$z \equiv \vec{w}^{\mathsf{T}}\vec{x}$$. Exponentiating and solving for $$\mu$$ gives

$\mu = \frac{ e^z }{ 1 + e^z } = \frac{ 1 }{ 1 + e^{-z} }$

This function is called the logistic or sigmoid function.

$\mathrm{logistic}(z) \equiv \mathrm{sigm}(z) \equiv \frac{ 1 }{ 1 + e^{-z} } \label{eq:logistic}$

Since we inverted the logit function by solving for $$\mu$$, the inverse of the logit function is the logistic or sigmoid.

$\mathrm{logit}^{-1}(z) = \mathrm{logistic}(z) = \mathrm{sigm}(z)$

And therefore,

$\mu = \mathrm{sigm}(z) = \mathrm{sigm}(\vec{w}^{\mathsf{T}}\vec{x})$

### Softmax regression

Again, from a probabilistic point of view, we can derive the use of multi-class cross entropy loss by starting with the Bernoulli distribution, generalizing it to multiple classes (indexed by $$k$$) as

$p(y_k | \mu) = \mathrm{Cat}(y_k | \mu_k) = \prod_k {\mu_k}^{y_k} \label{eq:categorical_distribution}$

which is the categorical or multinoulli distribution. The negative-log likelihood of multiple independent trials is

$\mathrm{NLL} = - \sum_i \log \left(\prod_k {\mu_{ki}}^{y_{ki}}\right) = - \sum_i \sum_k y_{ki} \: \log \mu_{ki} \label{eq:nll_multinomial}$

Noting again that $$y_{ki} = 1$$ only when $$k$$ is the true class, and is 0 otherwise, this simplifies to eq. $$\eqref{eq:cross_entropy_loss2}$$.

• Multinomial logistic regression
• Softmax is really a soft argmax. TODO: find ref.
• Softmax is not unique. There are other squashing functions.138
• Roelants, P. (2019). Softmax classification with cross-entropy.
• Gradients from backprop through a softmax
• Goodfellow et al. point out that any negative log-likelihood is a cross entropy between the training data and the probability distribution predicted by the model.139

## Deep learning

### Regularization

Regularization = any change we make to the training algorithm in order to reduce the generalization error but not the training error.165

Most common regularizations:

• L2 Regularization
• L1 Regularization
• Data Augmentation
• Dropout
• Early Stopping

Papers:

### Batch size vs learning rate

Papers:

1. Keskar, N.S. et al. (2016). On large-batch training for deep learning: Generalization gap and sharp minima.

[L]arge-batch methods tend to converge to sharp minimizers of the training and testing functions—and as is well known—sharp minima lead to poorer generalization. In contrast, small-batch methods consistently converge to flat minimizers, and our experiments support a commonly held view that this is due to the inherent noise in the gradient estimation.

• $$\eta \propto \sqrt{m}$$
1. Goyal, P. et al. (2017). Accurate large minibatch SGD: Training ImageNet in 1 hour.

• $$\eta \propto m$$
2. You, Y. et al. (2017). Large batch training of convolutional networks.

• Layer-wise Adaptive Rate Scaling (LARS)
3. You, Y. et al. (2017). ImageNet training in minutes.

• Layer-wise Adaptive Rate Scaling (LARS)
4. Jastrzebski, S. (2018). Three factors influencing minima in SGD.

• $$\eta \propto m$$
5. Smith, S.L. & Le, Q.V. (2018). A Bayesian Perspective on Generalization and Stochastic Gradient Descent.

6. Smith, S.L. et al. (2018). Don’t decay the learning rate, increase the batch size.

• $$m \propto \eta$$
7. Masters, D. & Luschi, C. (2018). Revisiting small batch training for deep neural networks.

This linear scaling rule has been widely adopted, e.g., in Krizhevsky (2014), Chen et al. (2016), Bottou et al. (2016), Smith et al. (2017) and Jastrzebski et al. (2017).

On the other hand, as shown in Hoffer et al. (2017), when $$m \ll M$$, the covariance matrix of the weight update $$\mathrm{Cov(\eta \Delta\theta)}$$ scales linearly with the quantity $$\eta^2/m$$.

This implies that, adopting the linear scaling rule, an increase in the batch size would also result in a linear increase in the covariance matrix of the weight update $$\eta \Delta\theta$$. Conversely, to keep the scaling of the covariance of the weight update vector $$\eta \Delta\theta$$ constant would require scaling $$\eta$$ with the square root of the batch size $$m$$ (Krizhevsky, 2014; Hoffer et al., 2017).

1. Lin, T. et al. (2020). Don’t use large mini-batches, use local SGD.
- Post-local SGD.

2. Golmant, N. et al. (2018). On the computational inefficiency of large batch sizes for stochastic gradient descent.

Scaling the learning rate as $$\eta \propto \sqrt{m}$$ attempts to keep the weight increment length statistics constant, but the distance between SGD iterates is governed more by properties of the objective function than the ratio of learning rate to batch size. This rule has also been found to be empirically sub-optimal in various problem domains. … There does not seem to be a simple training heuristic to improve large batch performance in general.

1. McCandlish, S. et al. (2018). An empirical model of large-batch training.
• Critical batch size
2. Shallue, C.J. et al. (2018). Measuring the effects of data parallelism on neural network training.

In all cases, as the batch size grows, there is an initial period of perfect scaling ($$b$$-fold benefit, indicated with a dashed line on the plots) where the steps needed to achieve the error goal halves for each doubling of the batch size. However, for all problems, this is followed by a region of diminishing returns that eventually leads to a regime of maximal data parallelism where additional parallelism provides no benefit whatsoever.

1. Jastrzebski, S. et al. (2018). Width of minima reached by stochastic gradient descent is influenced by learning rate to batch size ratio.
• $$\eta \propto m$$

We show this experimentally in Fig. 5, where similar learning dynamics and final performance can be observed when simultaneously multiplying the learning rate and batch size by a factor up to a certain limit.

1. You, Y. et al. (2019). Large-batch training for LSTM and beyond.
• Warmup and use $$\eta \propto m$$

[W]e propose linear-epoch gradual-warmup approach in this paper. We call this approach Leg-Warmup (LEGW). LEGW enables a Sqrt Scaling scheme in practice and as a result we achieve much better performance than the previous Linear Scaling learning rate scheme. For the GNMT application (Seq2Seq) with LSTM, we are able to scale the batch size by a factor of 16 without losing accuracy and without tuning the hyper-parameters mentioned above.

1. You, Y. et al. (2019). Large batch optimization for deep learning: Training BERT in 76 minutes.
• LARS and LAMB
2. Zhang, G. et al. (2019). Which algorithmic choices matter at which batch sizes? Insights from a Noisy Quadratic Model.

Consistent with the empirical results of Shallue et al. (2018), each optimizer shows two distinct regimes: a small-batch (stochastic) regime with perfect linear scaling, and a large-batch (deterministic) regime insensitive to batch size. We call the phase transition between these regimes the critical batch size.

1. Li, Y., Wei, C., & Ma, T. (2019). Towards explaining the regularization effect of initial large learning rate in training neural networks.

Our analysis reveals that more SGD noise, or larger learning rate, biases the model towards learning “generalizing” kernels rather than “memorizing” kernels.

1. Kaplan, J. et al. (2020). Scaling laws for neural language models.

2. Jastrzebski, S. et al. (2020). The break-even point on optimization trajectories of deep neural networks.

Blogs:

Resources:

### Natural language processing

Chain rule of language modeling (chain rule of probability):

$P(x_1, \ldots, x_T) = P(x_1, \ldots, x_{n-1}) \prod_{t=n}^{T} P(x_t | x_1 \ldots x_{t-1}) \label{eq:chain_rule_of_lm}$

or for the whole sentence:

$P(x_1, \ldots, x_T) = \prod_{t=1}^{T} P(x_t | x_1 \ldots x_{t-1}) \label{eq:chain_rule_of_lm_2}$

$= P(x_1) \: P(x_2 | x_1) \: P(x_3 | x_1 x_2) \: P(x_4 | x_1 x_2 x_3) \ldots$

Auto-regressive inference follows this chain rule. If done with greedy search:

$\hat{x}_t = \underset{x_t \in V}{\mathrm{argmax}} \: P(x_t | x_1 \ldots x_{t-1}) \label{eq:greedy_search}$

$\mathrm{attention}(Q, K, V) = \mathrm{softmax}\left(\frac{Q\, K^{\mathsf{T}}}{\sqrt{d_k}}\right) V \label{eq:attention}$

### Reinforcement learning

Pedagogy:

Tutorials:

More:

#### Q-learning

• Q-learning and DQN
• Uses the Markov Decision Process (MDP) framework
• The Bellman equation211
• Q-learning is a values-based learning algorithm. Value based algorithms updates the value function based on an equation (particularly Bellman equation). Whereas the other type, policy-based estimates the value function with a greedy policy obtained from the last policy improvement (source: towardsdatascience.com).
• DQN masters Atari212

## Theoretical machine learning

### Algorithmic information theory

• Ray Solomonoff (1926-2009)
• Solomonoff induction
• Naturally formalizes Occam’s razor
• Incomputable
• Rathmanner, S. & Hutter, M. (2011). A philosophical treatise of universal induction.233

### No free lunch theorems

Raissi et al.:

encoding such structured information into a learning algorithm results in amplifying the information content of the data that the algorithm sees, enabling it to quickly steer itself towards the right solution and generalize well even when only a few training examples are available.248

Roberts:

From an algorithmic complexity standpoint it is somewhat miraculous that we can compress our huge look-up table of experiment/outcome into such an efficient description. In many senses, this type of compression is precisely what we mean when we say that physics enables us to understand a given phenomenon.249

## Automation

### AutoML

• Neural Architecture Search (NAS)
• AutoML frameworks
• RL-driven NAS
• learned sparsity

Lectures:

## Implications for the realism debate

• Nope: Hennig

### Word meanings

Wittgenstein in PI:

The meaning of a word is its use in the language.291

and

One cannot guess how a word functions. One has to look at its use, and learn from that.292

My docs:

My talks:

## Annotated bibliography

• Mayo (1996)

• TODO

### Cowan, G. (1998). Statistical Data Analysis.

• Cowan (1998) and Cowan (2016)

• TODO

• James (2006)

• TODO

### Cowan, G. et al. (2011). Asymptotic formulae for likelihood-based tests of new physics.

• Cowan et al. (2011)
• Glen Cowan, Kyle Cranmer, Eilam Gross, Ofer Vitells

• TODO

• TODO

### Cranmer, K. (2015). Practical statistics for the LHC.

• Cranmer (2015)

#### My thoughts

• TODO

• All of Statistics293
• The Foundations of Statistics294

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1. Edwards (1974), p. 9.↩︎

2. Hacking (1971).↩︎

3. Bernoulli, J. (1713). Ars Conjectandi, Chapter II, Part IV, defining the art of conjecture [wikiquote].↩︎

4. Venn (1888).↩︎

5. Peirce (1883), p. 126–181.↩︎

6. Pearson (1900).↩︎

7. Fisher (1912).↩︎

8. Fisher (1915).↩︎

9. Fisher (1921).↩︎

10. Fisher (1955).↩︎

11. Salsburg (2001).↩︎

12. Reid (1998).↩︎

13. Neyman (1955).↩︎

14. Stuart, Ord, & Arnold (2010).↩︎

15. James (2006).↩︎

16. Cowan (1998) and Cowan (2016).↩︎

17. Cranmer (2015).↩︎

18. Lista (2016b).↩︎

19. Lista (2016a).↩︎

20. Cox (2006).↩︎

21. Cousins (2018).↩︎

22. Weisberg (2019).↩︎

23. Carnap (1947).↩︎

24. Goodfellow, Bengio, & Courville (2016), p. 72-73.↩︎

25. Cowan (1998), p. 20-22.↩︎

26. Fienberg (2006).↩︎

27. Weisberg (2019), ch. 15.↩︎

28. Fisher (1921), p. 15.↩︎

29. van Handel (2016).↩︎

30. Vershynin (2018).↩︎

31. Leemis & McQueston (2008).↩︎

32. Cranmer, K. et al. (2012).↩︎

33. This assumption that the model models the data “reasonably” well reflects that to the degree required by your analysis, the important features of the data match well within the systematic uncertainties parametrized within the model. If the model is incomplete because it is missing an important feature of the data, then this is the “ugly” (class-3) error in the Sinervo classification of systematic uncertainties.↩︎

34. Cowan (1998) and Cowan (2016), p. TODO.↩︎

35. Aldrich (1997).↩︎

36. James (2006), p. 234.↩︎

37. Cox (2006), p. 11.↩︎

38. Murphy (2012), p. 222.↩︎

39. Fréchet (1943), Cramér (1946), Rao (1945), and Rao (1947).↩︎

40. Rice (2007), p. 300–2.↩︎

41. Cowan (1998), p. 130-5.↩︎

42. James (2006), p. 234.↩︎

43. James & Roos (1975).↩︎

44. Cowan, Cranmer, Gross, & Vitells (2012).↩︎

45. Wainer (2007).↩︎

46. Tegmark, Taylor, & Heavens (1997).↩︎

47. Clopper & Pearson (1934).↩︎

48. Hanley & Lippman-Hand (1983).↩︎

49. L. D. Brown, Cai, & DasGupta (2001).↩︎

50. Fisher (1935), p. 16.↩︎

51. Goodman (1999a). p. 998.↩︎

52. ATLAS and CMS Collaborations (2011).↩︎

53. Cowan, Cranmer, Gross, & Vitells (2011).↩︎

54. Neyman & Pearson (1933).↩︎

55. Feldman & Cousins (1998).↩︎

56. Sinervo (2002) and Cowan (2012).↩︎

57. Cowan et al. (2011), p. 2–3.↩︎

58. Cowan et al. (2011), p. 3.↩︎

59. Cousins & Highland (1992).↩︎

60. Junk (1999).↩︎

62. ATLAS Statistics Forum (2011).↩︎

63. Wilks (1938).↩︎

64. Wald (1943).↩︎

65. Cowan et al. (2011).↩︎

66. Bhattiprolu, Martin, & Wells (2020).↩︎

67. Murphy (2012), p. 197.↩︎

68. Goodman (1999b).↩︎

69. Goodman (1999a). p. 995.↩︎

70. Sinervo (2003).↩︎

71. Heinrich & Lyons (2007).↩︎

72. Caldeira & Nord (2020).↩︎

73. Lyons (2008), p. 890.↩︎

74. Pearl (2018).↩︎

75. Lewis (1981).↩︎

76. Pearl (2009).↩︎

77. Robins & Wasserman (1999).↩︎

78. Peters, Janzing, & Scholkopf (2017).↩︎

79. Tukey (1977).↩︎

80. Carnap (1945).↩︎

81. Royall (1997), p. 171–2.↩︎

82. Cranmer (2015), p. 6.↩︎

83. Neyman & Pearson (1933).↩︎

84. Kruschke & Liddell (2018).↩︎

85. Edwards (1974).↩︎

86. Birnbaum (1962).↩︎

87. Hacking (1965).↩︎

88. Berger & Wolpert (1988).↩︎

89. O’Hagan (2010), p. 17–18.↩︎

90. Gandenberger (2015).↩︎

91. Evans (2013).↩︎

92. Mayo (2014).↩︎

93. Mayo (2019).↩︎

94. Mayo (2019).↩︎

95. Lyons (2008), p. 891.↩︎

96. Sznajder (2018).↩︎

97. Hacking (1965).↩︎

98. Neyman (1977).↩︎

99. Zech (1995).↩︎

100. Royall (1997).↩︎

101. Berger (2003).↩︎

102. Mayo (1981).↩︎

103. Mayo (1996).↩︎

104. Mayo & Spanos (2006).↩︎

105. Mayo & Spanos (2011).↩︎

106. Mayo (2018).↩︎

107. Gelman & Hennig (2017).↩︎

108. Murphy (2012), ch. 6.6.↩︎

109. Murphy (2022), p. 195–198.↩︎

110. Gandenberger (2016).↩︎

111. Wakefield (2013), ch. 4.↩︎

112. Efron & Hastie (2016), p. 30–36.↩︎

113. Kruschke & Liddell (2018).↩︎

114. Steinhardt (2012).↩︎

115. Goodman (1999a). p. 999.↩︎

116. Ioannidis (2005).↩︎

117. Wasserstein & Lazar (2016).↩︎

118. Wasserstein, Allen, & Lazar (2019).↩︎

119. Benjamin, D.J. et al. (2017).↩︎

120. Fisher (1935), p. 13–14.↩︎

121. Mayo (2021).↩︎

122. Gorard & Gorard (2016).↩︎

123. Benjamini, Y. et al. (2021), p. 1.↩︎

124. Hastie, Tibshirani, & Friedman (2009).↩︎

125. MacKay (2003).↩︎

126. Murphy (2012).↩︎

127. Murphy (2022), p. 195–198.↩︎

128. Shalev-Shwarz & Ben-David (2014).↩︎

129. Vapnik, Levin, & LeCun (1994).↩︎

130. Shalev-Shwarz & Ben-David (2014), p. 67–82.↩︎

131. McCarthy, Minsky, Rochester, & Shannon (1955).↩︎

132. Solomonoff (2016).↩︎

133. Kardum (2020).↩︎

134. Murphy (2012), p. 21.↩︎

135. Note: Label smoothing is a regularization technique that smears the activation over other labels, but we don’t do that here.↩︎

136. “Logit” was coined by Joseph Berkson (1899-1982).↩︎

138. Blondel, Martins, & Niculae (2020).↩︎

139. Goodfellow et al. (2016), p. 129.↩︎

140. T. Chen & Guestrin (2016).↩︎

141. Slonim, Atwal, Tkacik, & Bialek (2005).↩︎

142. Batson, Haaf, Kahn, & Roberts (2021).↩︎

143. Hennig (2015).↩︎

144. Lauc (2020), p. 103–4.↩︎

145. Ronen, Finder, & Freifeld (2022).↩︎

146. Bengio (2009).↩︎

147. LeCun, Bengio, & Hinton (2015).↩︎

148. Sutskever (2015).↩︎

149. Goodfellow et al. (2016).↩︎

150. Kaplan, J. et al. (2019).↩︎

151. Rumelhart, Hinton, & Williams (1986).↩︎

152. LeCun & Bottou (1998).↩︎

153. Bottou (1998).↩︎

154. Sutton (2019).↩︎

155. Watson & Floridi (2019).↩︎

156. Bengio (2009).↩︎

157. Belkin, Hsu, Ma, & Mandal (2019).↩︎

158. Nakkiran, P. et al. (2019).↩︎

159. Dar, Muthukumar, & Baraniuk (2021).↩︎

160. Balestriero, Pesenti, & LeCun (2021).↩︎

161. Nagarajan (2021).↩︎

162. Bubeck & Sellke (2021).↩︎

163. Bach (2022), p. 225–230.↩︎

164. Steinhardt (2022).↩︎

165. Mishra, D. (2020). Weight Decay == L2 Regularization?↩︎

166. S. Chen, Dobriban, & Lee (2020).↩︎

167. Chiley, V. et al. (2019).↩︎

168. Kiani, Balestriero, Lecun, & Lloyd (2022).↩︎

169. Fukushima & Miyake (1982).↩︎

170. LeCun, Y. et al. (1989).↩︎

171. LeCun, Bottou, Bengio, & Haffner (1998).↩︎

172. Ciresan, Meier, Masci, & Schmidhuber (2012).↩︎

173. Krizhevsky, Sutskever, & Hinton (2012).↩︎

174. Simonyan & Zisserman (2014).↩︎

175. He, Zhang, Ren, & Sun (2015).↩︎

176. Haber & Ruthotto (2017).↩︎

177. Howard, A.G. et al. (2017).↩︎

178. R. T. Q. Chen, Rubanova, Bettencourt, & Duvenaud (2018).↩︎

179. Tan & Le (2019).↩︎

180. Dosovitskiy, A. et al. (2020).↩︎

181. Tan & Le (2021).↩︎

182. H. Liu, Dai, So, & Le (2021).↩︎

183. Ingrosso & Goldt (2022).↩︎

184. Park & Kim (2022).↩︎

185. Firth (1957).↩︎

186. Nirenburg (1996).↩︎

187. Hutchins (2000).↩︎

188. Mikolov, Chen, Corrado, & Dean (2013), Mikolov, Yih, & Zweig (2013), and Mikolov, T. et al. (2013).↩︎

189. Hochreiter & Schmidhuber (1997).↩︎

190. Werbos (1990).↩︎

191. Sutskever, Vinyals, & Le (2014).↩︎

192. Bahdanau, Cho, & Bengio (2015).↩︎

193. Wu, Y. et al. (2016).↩︎

194. Stahlberg (2019).↩︎

195. Church & Hestness (2019).↩︎

196. Kaplan, J. et al. (2020).↩︎

197. Vaswani, A. et al. (2017).↩︎

198. Devlin, Chang, Lee, & Toutanova (2018).↩︎

199. Lan, Z. et al. (2019).↩︎

200. Radford, Narasimhan, Salimans, & Sutskever (2018).↩︎

201. Radford, A. et al. (2019).↩︎

202. Brown, T.B. et al. (2020).↩︎

203. Yang, Z. et al. (2019).↩︎

204. Zaheer, M. et al. (2020).↩︎

205. Tay, Dehghani, Bahri, & Metzler (2022).↩︎

206. Phuong & Hutter (2022).↩︎

207. Jurafsky & Martin (2022).↩︎

208. Sutton & Barto (2018).↩︎

209. Arulkumaran, Deisenroth, Brundage, & Bharath (2017).↩︎

210. Cesa-Bianchi & Lugosi (2006).↩︎

211. Bellman (1952).↩︎

212. Mnih, V. et al. (2013) and Mnih, V. et al. (2015).↩︎

213. Silver, D. et al. (2016).↩︎

214. Silver, D. et al. (2017b).↩︎

215. Silver, D. et al. (2017a).↩︎

216. Hart & Mas‐Colell (2000).↩︎

217. Zinkevich, Johanson, Bowling, & Piccione (2007).↩︎

218. Lanctot, Waugh, Zinkevich, & Bowling (2009).↩︎

219. Neller & Lanctot (2013).↩︎

220. Gibson (2014).↩︎

221. Burch (2018).↩︎

222. Bowling, Burch, Johanson, & Tammelin (2015).↩︎

223. Heinrich & Silver (2016).↩︎

224. Moravcik, M. et al. (2017).↩︎

225. N. Brown & Sandholm (2018).↩︎

226. N. Brown & Sandholm (2019a).↩︎

227. N. Brown, Lerer, Gross, & Sandholm (2019).↩︎

228. N. Brown & Sandholm (2019b).↩︎

229. N. Brown, Bakhtin, Lerer, & Gong (2020).↩︎

230. N. Brown (2020).↩︎

231. Spears, B.K. et al. (2018).↩︎

232. Cranmer, Seljak, & Terao (2021).↩︎

233. Rathmanner & Hutter (2011).↩︎

235. Wolpert (1996).↩︎

237. Shalev-Shwarz & Ben-David (2014), p. 60–66.↩︎

238. McDermott (2019).↩︎

239. Wolpert (2007).↩︎

240. Wolpert & Kinney (2020).↩︎

241. Mitchell (1980).↩︎

242. Roberts (2021).↩︎

243. Goldreich & Ron (1997).↩︎

244. Joyce & Herrmann (2017).↩︎

245. Lauc (2020).↩︎

246. Nakkiran (2021).↩︎

247. Bousquet, O. et al. (2021).↩︎

248. Raissi, Perdikaris, & Karniadakis (2017a), p. 2.↩︎

249. Roberts (2021), p. 7.↩︎

250. Minsky & Papert (1969).↩︎

251. Hornik, Stinchcombe, & White (1989).↩︎

252. Lu, Z. et al. (2017).↩︎

253. Ismailov (2020).↩︎

254. Bishop (2006), p. 230.↩︎

255. Bahri, Y. et al. (2020).↩︎

256. Halverson, Maiti, & Stoner (2020).↩︎

257. Canatar, Bordelon, & Pehlevan (2020).↩︎

258. Roberts, Yaida, & Hanin (2021).↩︎

259. Cohen & Welling (2016).↩︎

260. Cohen, Weiler, Kicanaoglu, & Welling (2019).↩︎

261. Fuchs, Worrall, Fischer, & Welling (2020).↩︎

262. Smith (2019).↩︎

263. Nielsen (2018).↩︎

264. Amari (2016).↩︎

265. Balasubramanian (1996a).↩︎

266. Balasubramanian (1996b).↩︎

267. Calin & Udriste (2014).↩︎

268. Lei, Luo, Yau, & Gu (2018).↩︎

269. Gao & Chaudhari (2020).↩︎

270. Bronstein, Bruna, Cohen, & Velickovic (2021).↩︎

271. Fefferman, Mitter, & Narayanan (2016).↩︎

272. Raissi et al. (2017a) and Raissi, Perdikaris, & Karniadakis (2017b).↩︎

273. Karniadakis, G.E. et al. (2021).↩︎

274. Howard, Mandt, Whiteson, & Yang (2021).↩︎

275. Thuerey, N. et al. (2021).↩︎

276. Cranmer, Brehmer, & Louppe (2019).↩︎

277. Baydin, A.G. et al. (2019).↩︎

278. Anderson (2008).↩︎

279. Asch, M. et al. (2018).↩︎

280. D’Agnolo & Wulzer (2019).↩︎

281. Udrescu & Tegmark (2020).↩︎

282. Cranmer, M. et al. (2020).↩︎

283. Z. Liu, Madhavan, & Tegmark (2022).↩︎

284. Krenn, M. et al. (2022).↩︎

285. Asch, M. et al. (2018).↩︎

286. Korb (2001).↩︎

287. Williamson (2009).↩︎

288. Bensusan (2000).↩︎

289. Perone (2018).↩︎

290. Skelac & Jandric (2020).↩︎

291. Wittgenstein (2009), §43.↩︎

292. Wittgenstein (2009), §340.↩︎

293. Wasserman (2003).↩︎

294. Savage (1954).↩︎