MOTIVATION
Figure 1 illustrates a typical example of a prediction problem: given some noisy observations
of a dependent variable at certain values of the independent variable
, what is
our best estimate of the dependent variable at a new value, ✂✁
?
If we expect the underlying function ✄✆☎✞✝
to be linear, and can make some assumptions
about the input data, we might use a least-squares method to fit a straight
line (linear regression). Moreover, if we suspect ✄✆☎✂✝
may also be quadratic, cubic, or
even nonpolynomial, we can use the principles of model selection to choose among the
various possibilities.
Gaussian process regression (GPR) is an even finer approach than this. Rather
than claiming ✄✆☎✞✝
relates to some specific models (e.g. ✄✆☎✞✝✠✟☛✡☞✍✌✏✎),
a Gaussian
process can represent ✄✆☎✞✝
obliquely, but rigorously, by letting the data ‘speak’ more
clearly for themselves. GPR is still a form of supervised learning, but the training data
are harnessed in a subtler way.
As such, GPR is a less ‘parametric’ tool. However, it’s not completely free-form,
and if we’re unwilling to make even basic assumptions about ✄✆☎✂✝
, then more general
techniques should be considered, including those underpinned by the principle of
maximum entropy; Chapter 6 of Sivia and Skilling (2006) offers an introduction.
MOTIVATIONFigure 1 illustrates a typical example of a prediction problem: given some noisy observationsof a dependent variable at certain values of the independent variable, what isour best estimate of the dependent variable at a new value, ✂✁?If we expect the underlying function ✄✆☎✞✝to be linear, and can make some assumptionsabout the input data, we might use a least-squares method to fit a straightline (linear regression). Moreover, if we suspect ✄✆☎✂✝may also be quadratic, cubic, oreven nonpolynomial, we can use the principles of model selection to choose among thevarious possibilities.Gaussian process regression (GPR) is an even finer approach than this. Ratherthan claiming ✄✆☎✞✝relates to some specific models (e.g. ✄✆☎✞✝✠✟☛✡☞✍✌✏✎),a Gaussianprocess can represent ✄✆☎✞✝obliquely, but rigorously, by letting the data ‘speak’ moreclearly for themselves. GPR is still a form of supervised learning, but the training dataare harnessed in a subtler way.As such, GPR is a less ‘parametric’ tool. However, it’s not completely free-form,and if we’re unwilling to make even basic assumptions about ✄✆☎✂✝, then more generaltechniques should be considered, including those underpinned by the principle ofmaximum entropy; Chapter 6 of Sivia and Skilling (2006) offers an introduction.
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