EM Algorithm Essentials: Estimating standard errors using the empirical information matrix

Introduction

At the end of my latest blog post, I promised that I would talk about how to perform constrained maximization using unconstrained optimizers. This can be accomplished by employing clever transformation and nice properties of maximum likelihood estimators – see this fantastic post from the now defunct 🙁 Econometrics Beat. Eventually I will get around to discussing more of the statistical details behind this approach and its implementation from scratch in R.

RIGHT NOW, I want to talk about how to obtain standard errors for Gaussian mixture model parameters estimated using the EM algorithm. So far, I’ve written two posts with respect to Gaussian mixtures and the EM algorithm:

Embracing the EM algorithm: One continuous response, which motivates the theory behind the EM algorithm using a two component Gaussian mixture model; and
EM Algorithm Essentials: Maximizing objective functions using R’s optim, which demonstrates how to implement log-likelihood maximization from scratch using the optim function.

Both posts so far have focused only on POINT ESTIMATION! That is, obtaining estimates of our mixture model parameters.

Any estimate we obtain using any statistical method has some uncertainty associated with it. We quantify the uncertainty of a parameter estimate by its STANDARD ERROR. If we repeated the same experiment with comparable samples a large number of times, the standard error would reflect how much our estimates differ from experiment to experiment. Indeed, we would estimate the standard error as the standard deviation of the effect estimates across experiments. If we can understand how our estimates vary across experiments, i.e., estimate their standard errors, we can perform statistical inference by testing hypotheses or constructing confidence intervals, for example.

We usually do not need to conduct all of these experiments because theory tells us what form this distribution of effect estimates takes! We refer to this distribution as the SAMPLING DISTRIBUTION. When the form of the sampling distribution is not obvious, we may use resampling techniques to estimate standard errors. The EM algorithm, however, attempts to obtain maximum likelihood estimates (MLEs) which have theoretical sampling distributions. For well-behaved densities, MLEs can be shown to be asymptotically normally distributed and unbiased with a variance-covariance matrix equal to the inverse of the Fisher Information matrix, i.e., a function of the second derivatives of the log-likelihood or equivalently, the square of the first derivatives…

The tricky bit with the EM algorithm is we don’t maximize the observed log-likelihood directly to estimate Gaussian mixture parameters. Instead, we maximize a more convenient objective function. People much smarter than I have figured out this can give you the same answer. A question remains: if standard errors are estimated using the second derivative of the log-likelihood, but we used the objective function, how do we properly quantify uncertainty? Particularly as the derivatives of the log-likelihood are not straight-forward, for the same reasons that make optimization difficult.

Isaac Meilijson, another person smarter than me, provides us with a wealth of guidance in his 1989 paper titled A Fast Improvement to the EM Algorithm on its Own Terms (I love this title). In this blog post, we summarize and demonstrate how to construct simple estimators of the observed information matrix, and subsequently the standard errors of our EM estimates, when we have independent and identically distributed responses arising from a two-component Gaussian mixture model. Essentially, due to some nice properties, we can simply use the derivatives of our objective function, in lieu of our log-likelihood, to estimate the observed information. Standard errors computed using the observed information are then compared to those obtained using numerical differentiation of the observed log-likelihood.

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EM Algorithm Essentials: Maximizing objective functions using R’s optim

Overview

In my previous blogpost, I motivated the EM algorithm in the context of estimating the parameters of a two-component Gaussian mixture density. In this case, we can write the estimators of the mixing probability, means, and variance in a nice closed form, and I demonstrated how to implement the corresponding iterative estimation procedure from scratch. Results were then compared to those obtained from the very nice R package flexmix.

However, rarely do we get such nice closed form estimators! We usually need to use numerical methods to maximize our objective function directly. In this blog post, I demonstrate how we can specify our objective function, and use the optim function in R to obtain our parameter estimates. optim has lots of options, and we will cover how to change the optimization procedure and implement restrictions on our parameter spaces.

EM for two-component Gaussian mixture

Let’s quickly recap our motivation, previously discussed in Embracing the EM algorithm: One continuous response.

We randomly sample $N$ patients from a population, and examine the empirical density of their responses. We notice two modes, and based on prior knowledge, hypothesize that the density is actually a mixture of two Gaussian densities. For example, the density centered around greater responses may correspond to “healthy” patients and the other to “ill” patients. We would like to (1) estimate the probability of belonging to each subpopulation; and (2) estimate subpopulation parameters, e.g. mean response in among the healthy and ill. But, we have a problem: we don’t know who belongs to each subpopulation. In other words, subpopulation labels are unobserved or “latent.”

Figure: Observed two-component Gaussian mixture density (purple), and distribution of responses among latent healthy (blue) and ill (red) patient subpopulations.

We can represent the density of the observed responses as a mixture of the subpopulation densities:

$f(y) = \pi~ f_1(y) + (1-\pi)~ f_2(y).$

That is, individuals belong to the first subpopulation, or are distributed according to density $f_{1}(y)$ , according to probability $\pi$ and to the second, distributed per $f_2(y)$ , with probability $1-\pi$ . We assume both densities are Gaussian with respective means $\mu_1$ and $\mu_2$ and variances $\sigma_1^2$ and $\sigma_2^2$ .

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Embracing the EM algorithm: One continuous response

Overview

I’m currently working on a project that revolves around the EM algorithm, and am finally realizing the power of this machinery. It really is like that movie with Jim Carrey where he can’t stop seeing the number 23 everywhere, except for me it’s the EM algorithm. Apparently this is called THE BAADER-MEINHOF PHENOMENON, oooh that’s fancy. You’ve probably seen the EM algorithm around too – though perhaps you didn’t know it. It’s commonly used for estimation with missing data. A modified EM algorithm (EMis) is used by the Amelia library in R. The EM algorithm also underpins latent variable models, which makes sense because latent variables are really missing observations when you think about it, right?! The more I learn about statistics, the more I realize most things are really missing data problems… cough potential outcomes cough

Anyways, I was previously taught the EM algorithm using the classic multinomial example. This is a great teaching tool, but I’ve never run into a situation like this in my life (yet). But, I do run into mixture distributions a surprising amount – mostly when investigating heterogeneity within patient populations. There’s a whole textbook on this, see: Medical Applications of Finite Mixture Models. The EM algorithm makes a lot more sense to me in the context of mixture models:

We sample a group of patients and observe their response.
We notice a bimodal structure in the response distribution.
We hypothesize the observed distribution actually corresponds to two subpopulations or “classes.”
We don’t know who belongs to which subpopulation.
We estimate the probability of latent class membership using the EM algorithm.

Wouldn’t ya know it, this is unsupervised clustering.

In this blog post, I motivate the EM algorithm in the context of a two-component Gaussian mixture model. A thorough walkthrough of the underlying theory is provided. In this case, estimators take a nice closed form, but this is rarely the case for complex problems encountered in practice. R code for implementating the EM algorithm using the closed form estimators is provided. I also demonstrate how this model can be easily fit using the flexmix library.

Figure: A two-component Gaussian mixture density.

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Kernel Regression

Motivation

For observed pairs $(x_i, y_i)$ , $i = 1, …, n$ , the relationship between $x$ and $y$ can be defined generally as

$y_i = m(x_i) + \varepsilon_i$

where $f(x_i) = E[y_i | x = x_i]$ and $\varepsilon_i \stackrel{iid}{\sim} (0, \sigma^2)$ . If we are unsure about the form of $m(\cdot)$ , our objective may be to estimate $m(\cdot)$ without making too many assumptions about its shape. In other words, we aim to “let the data speak for itself”.

Simulated scatterplot of $y = f(x) + \epsilon$ . Here, $x \sim Uniform(0, 10)$ and $\epsilon \sim N(0, 1)$ . The true function $f(x) = sin(x)$ is displayed in green.

Non-parametric approaches require only that $m(\cdot)$ be smooth and continuous. These assumptions are far less restrictive than alternative parametric approaches, thereby increasing the number of potential fits and providing additional flexibility. This makes non-parametric models particularly appealing when prior knowledge about $m(\cdot)$ ‘s functional form is limited.

Estimating the Regression Function

If multiple values of $y$ were observed at each $x$ , $f(x)$ could be estimated by averaging the value of the response at each $x$ . However, since $x$ is often continuous, it can take on a wide range of values making this quite rare. Instead, a neighbourhood of $x$ is considered.

Result of averaging $y_i$ at each $x_i$ . The fit is extremely rough due to gaps in $x$ and low $y$ frequency at each $x$ .

Define the neighbourhood around $x$ as $x \pm \lambda$ for some bandwidth $\lambda > 0$ . Then, a simple non-parametric estimate of $m(x)$ can be constructed as average of the $y_i$ ‘s corresponding to the $x_i$ within this neighbourhood. That is,

(1) $\begin{equation*} \hat{f}_{\lambda}(x) = \frac{\sum_{n} \mathbb{I}(|x - x_i| \leq \lambda)~ y_i}{\sum_{n} \mathbb{I}(|x - x_i| \leq \lambda)} = \frac{\sum_n K\left( \frac{x - x_i}{\lambda} \right) y_i}{\sum_n K\left( \frac{x - x_i}{\lambda} \right) } \end{equation*}$

where

$K(u) = \begin{cases} \frac{1}{2} & |u| \leq 1 \\ 0 & \text{o.w.} \end{cases}$

is the uniform kernel. This estimator, referred to as the Nadaraya-Watson estimator, can be generalized to any kernel function $K(u)$ (see my previous blog bost). It is, however, convention to use kernel functions of degree $\nu = 2$ (e.g. the Gaussian and Epanechnikov kernels).

The red line is the result of estimating $f(x)$ with a Gaussian kernel and arbitrarily selected bandwidth of $\lambda = 1.25$ . The green line represents the true function $sin(x)$ .

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