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Section 10.5 Fitting Using Probabilistic Models 307

log-likelihood of the data under this model as

L(a, b, c, σ) = log P(x i ,y i |a, b, c, σ)
i∈data

= log P(ξ i |σ) + log P(u i ,v i |a, b, c).
i∈data

But P(u i ,v i |a, b, c) is some constant, because this point is distributed uniformly
along the line. Since ξ i is the perpendicular distance from (x i ,y i ) to the line (which
2
2
is ||(ax i + by i + c)|| as long as a + b = 1), we must maximize
ξ 2 i 1 2
log P(ξ i |σ) = − − log 2πσ
2σ 2 2
i∈data i∈data
(ax i + by i + c) 1 2
2
= − − log 2πσ
2σ 2 2
i∈data
2
2
(again, subject to a + b = 1). For fixed (but perhaps unknown) σ this yields the
problem we were working with in Section 10.2.1. So far, generative models have
just reproduced what we know already, but a powerful trick makes them much more
interesting.
10.5.1 Missing Data Problems
A number of important vision problems can be phrased as problems that happen to
be missing useful elements of the data. For example, we can think of segmentation
as the problem of determining from which of a number of sources a measurement
came. This is a general view. More specifically, fitting a line to a set of tokens
involves segmenting the tokens into outliers and inliers, then fitting the line to
the inliers; segmenting an image into regions involves determining which source of
color and texture pixels generated the image pixels; fitting a set of lines to a set
of tokens involves determining which tokens lie on which line; and segmenting a
motion sequence into moving regions involves allocating moving pixels to motion
models. Each of these problems would be easy if we happened to possess some data
that is currently missing (respectively, whether a point is an inlier or an outlier,
which region a pixel comes from, which line a token comes from, and which motion
model a pixel comes from).
A missing data problem is a statistical problem where some data is missing.
There are two natural contexts in which missing data are important: In the first,
some terms in a data vector are missing for some instances and present for others
(perhaps someone responding to a survey was embarrassed by a question). In the
second, which is far more common in our applications, an inference problem can be
made much simpler by rewriting it using some variables whose values are unknown.
Fortunately, there is an effective algorithm for dealing with missing data problems;
in essence, we take an expectation over the missing data. We demonstrate this
method and appropriate algorithms with two examples.