Inference for hidden Markov models (HMMs) is numerically unstable. A standard approach to resolving this instability is to use scaling factors. I discuss this idea in detail.

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Conjugate gradient descent (CGD) is an iterative algorithm for minimizing quadratic functions. CGD uses a kind of orthogonality (conjugacy) to efficiently search for the minimum. I present CGD by building it up from gradient descent.

2

I derive the evidence lower bound (ELBO) in variational inference and explore its relationship to the objective in expectation–maximization and the variational autoencoder.

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Expectation–maximization for hidden Markov models is called the Baum–Welch algorithm, and it relies on the forward–backward algorithm for efficient computation. I review HMMs and then present these algorithms in detail.

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The unscented transform, most commonly associated with the nonlinear Kalman filter, was proposed by Jeffrey Uhlmann to estimate a nonlinear transformation of a Gaussian. I illustrate the main idea.

5

In Bayesian optimization, a popular acquisition function is predictive entropy search, which is a clever reframing of another acquisition function, entropy search. I rederive the connection and explain why this reframing is useful.

6

The expectation–maximization (EM) updates for several Gaussian latent factor models (factor analysis, probabilistic principal component analysis, probabilistic canonical correlation analysis, and inter-battery factor analysis) are closely related. I explore these relationships in detail.

7

I annotate my Python implementation of the framework in Adams and MacKay's 2007 paper, "Bayesian Online Changepoint Detection".

8

Wang and Fleet's 2008 paper, "Gaussian Process Dynamical Models for Human Motion", introduces a Gaussian process latent variable model with Gaussian process latent dynamics. I discuss this paper in detail.

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A Gaussian process latent variable model (GPLVM) can be viewed as a generalization of probabilistic principal component analysis (PCA) in which the latent maps are Gaussian-process distributed. I discuss this relationship.

10

Linderman, Johnson, and Adam's 2015 paper, "Dependent multinomial models made easy: Stick-breaking with the Pólya-gamma augmentation", introduces a Gibbs sampler for Gaussian processes with multinomial observations. I discuss this model in detail.

11

Following Linderman, Johnson, and Adam's 2015 paper, "Dependent multinomial models made easy: Stick-breaking with the Pólya-gamma augmentation", I show that a multinomial density can be represented as a product of binomial densities.

12

The posterior mean from a Gaussian process regressor is related to the prediction of a kernel ridge regressor. I explore this connection in detail.

13

Rahimi and Recht's 2007 paper, "Random Features for Large-Scale Kernel Machines", introduces a framework for randomized, low-dimensional approximations of kernel functions. I discuss this paper in detail with a focus on random Fourier features.

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I disentangle the what I call the "lifting trick" from the kernel trick as a way of clarifying what the kernel trick is and does.

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Robert Morris's algorithm for counting large numbers using 8-bit registers is an early example of a sketch or data structure for efficiently processing a data stream. I introduce the algorithm and analyze its probabilistic behavior.

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Bayesian inference for models with binomial likelihoods is hard, but in a 2013 paper, Nicholas Polson and his coauthors introduced a new method fast Bayesian inference using Gibbs sampling. I discuss their main results in detail.

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I discuss Rasmussen and Williams's Algorithm 2.1 for an efficient implementation of Gaussian process regression.

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Adams and MacKay's 2007 paper, "Bayesian Online Changepoint Detection", introduces a modular Bayesian framework for online estimation of changes in the generative parameters of sequential data. I discuss this paper in detail.

19

The definition of a Gaussian process is fairly abstract: it is an infinite collection of random variables, any finite number of which are jointly Gaussian. I work through this definition with an example and provide several complete code snippets.

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Determinantal point process are point processes characterized by the determinant of a positive semi-definite matrix, but what this means is not necessarily obvious. I explain how such a process can model repulsive systems.

21

Probabilistic machine learning is a useful framework for handling uncertainty and modeling generative processes. I explore this approach by comparing two models, one with and one without a clear probabilistic interpretation.

22

A common explanation for the reparameterization trick with variational autoencoders is that we cannot backpropagate through a stochastic node. I provide a more formal justification.

23

Backprogation is an algorithm that computes the gradient of a neural network, but it may not be obvious why the algorithm uses a backward pass. The answer allows us to reconstruct backprop from first principles.

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Most explanations of CNNs assume the reader understands the convolution operation and how it relates to image processing. I explore convolutions in detail and explain how they are implemented as layers in a neural network.

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