One prominent application of the variational inference methodology of Chapter 36 arises in the context of topic modeling. In this application, the objective is to discover similarities between texts or documents such as news articles. For example, given a large library of articles, running perhaps into the millions, such as a database of newspaper articles written over 100 years, it would be useful to be able to discover in an automated manner the multitude of topics that are covered in the database and to cluster together articles dealing with similar topics such as sports or health or politics. In another example, when a user is browsing an article online, it would be useful to be able to identify automatically the subject matter of the article in order to recommend to the reader other articles of similar content. Latent Dirichlet allocation (or LDA) refers to the procedure that results from applying variational inference techniques to topic modeling in order to address questions of this type.

The maximum-likelihood (ML) formulation is one of the most formidable tools for the solution of inference problems in modern statistical analysis. It allows the estimation of unknown parameters in order to fit probability density functions (pdfs) onto data measurements. We introduce the ML approach in this chapter and limit our discussions to properties that will be relevant for the future developments in the text. The presentation is not meant to be exhaustive, but targets key concepts that will be revisited in later chapters. We also avoid anomalous situations and focus on the main features of ML inference that are generally valid under some reasonable regularity conditions.

The temporal learning algorithms TD(0) and TD($\lambda$) of the previous chapter are useful procedures for state value evaluation; i.e., they permit the estimation of the state value function $v^{\pi }\left(s\right)$ for a given target policy $\pi \left(a\right|s)$ by observing actions and rewards arising from this policy (on‐policy learning) or another behavior policy (off‐policy learning).In most situations, however, we are not interested in state values but rather in determining optimal policies, denoted by ${\pi }^{\star }\left(a\right|s)$ (i.e., in selecting what optimal actions an agent should follow in a Markov decision process (MDP)).

We have described a range of supervised learning algorithms in the previous chapters, including several neural network implementations and their training by means of the backpropagation algorithm. The performance of some of these algorithms has been demonstrated in practice to match or even exceed human performance in important applications. At the same time, it has also been observed that the algorithms are susceptible to adversarial attacks that can drive them to erroneous decisions under minimal perturbations to the data. For instance, adding small perturbations to an image that may not even be perceptible to the human eye has been shown to cause learning algorithms to classify the image incorrectly.

We expectation-maximization (EM) algorithm can be used to estimate the underlying parameters of the conditional probability density functions (pdfs) by approximating the maximum-likelihood (ML) solution. We found that the algorithm operates on a collection of independent observations, where each observation is generated independently from one of the mixture components. In this chapter and the next, we extend this construction and consider hidden Markov models (HMMs), where the mixture component for one observation is now dependent on the component used to generate the most recent past observation.

Oftentimes, the dimension of the feature space, $h_n\in {\mathbb{\text{R}}}^M$, is prohibitively large either for computational or visualization purposes. In these situations, it becomes necessary to perform an initial dimensionality reduction step where each $h_n$ is replaced by a lower-dimensional vector $h_n^{\prime }\in {\mathbb{R}}^{M^{\prime }}$ with $M^{\prime }\ll M$.

Maximum likelihood (ML) is a powerful statistical tool that determines model parameters $\theta$ in order to fit probability density functions (pdfs) onto data measurements. The estimated pdfs can then be used for at least two purposes. First, they can help construct optimal estimators or classifiers (such as the conditional mean estimator, the maximum a-posteriori (MAP) estimator, or the Bayes classifier) since, as we already know from previous chapters, these optimal constructions require knowledge of the conditional or joint probability distributions of the variables involved in the inference problem. Second, once a pdf is learned, we can sample from it to generate additional observations. For example, consider a database consisting of images of cats and assume we are able to characterize (or learn) the pdf distribution of the pixel values in these images. Then, we could use the learned pdf to generate “fake” cat-like images (i.e., ones that look like real cats). We will learn later in this text that this construction is possible and some machine-learning architectures are based on this principle: They use data to learn what we call a “generative model,” and then use the model to generate “similar” data. We provide a brief explanation to this effect in the next section, where we explain the significance of posterior distributions.

In Chapters 33 and 34 we described three methods for approximating posterior distributions: the Laplace method, the Markov chain Monte Carlo (MCMC) method, and the expectation-propagation (EP) method. Given an observable $\mathit{y}$ and a latent variable $\mathit{z}$, the Laplace method approximates $f_{\mathit{z}\mid \mathit{y}}\left(z\mid y\right)$ by a Gaussian distribution and was seen to be suitable for problems with small-dimensional latent spaces because its implementation involves a matrix inversion. The Gaussian approximation, however, is not sufficient in many instances and can perform poorly. The MCMC method is more powerful, and also more popular, and relies on elegant sampling techniques and the Metropolis–Hastings algorithm. However, MCMC requires a large number of samples, does not perform well on complex models, and does not scale well to higher dimensions and large datasets. The EP method, on the other hand, limits the class of distributions from which the posterior is approximated to the Gaussian or exponential families, and can be analytically demanding. In this chapter, we develop a fourth powerful method for posterior approximation known as variational inference. One of its advantages is that it usually scales better to large datasets and large dimensions.

We formulated the maximum-likelihood (ML) approach in the previous chapter, where an unknown parameter $\theta$ is estimated by maximizing the log-likelihood function. We showed there that in some cases of interest this problem can be solved analytically in closed form and an expression for the parameter estimate can be determined in terms of the observations. However, there are many important scenarios where the ML solution cannot be pursued in closed form, either due to mathematical intractability or due to missing data or hidden variables that are unobservable. In this chapter, we motivate and describe the expectation maximization (EM) procedure as a useful tool for constructing ML estimates under these more challenging conditions. We also illustrate how EM can be used to fit mixture models onto data.

In this chapter we illustrate one important application of the linear meansquare-error (MSE) theory to the derivation of the famed Kalman filter. The filter is a powerful recursive technique for updating the estimates of the state (hidden) variables of a state-space model from noisy observations. The state evolution satisfies a Markovian property in the sense that the distribution of the state ${\mathit{x}}_n$ at time $n$ is only dependent on the most recent past state, ${\mathit{x}}_{n-1}$. Likewise, the distribution of the observation ${\mathit{y}}_n$ at the same time instant is only dependent on the state $x_n$. The state and observation variables are represented by a linear state-space model, which will be shown to enable a powerful recursive solution. One key step in the argument is the introduction of the innovations process and the exploitation to great effect of the principle of orthogonality. In Chapter 35 we will allow for nonlinear state-space models and derive the class of particle filters by relying instead on the concept of sequential importance sampling.