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In the equation:

$f_\theta(x)=\theta_1x$

Is there a reason that $\theta$ might be a subscript of $f$ and not either a second parameter or left out of the left side of the equation altogether? Does it differ from the following?

$f(x,\theta)=\theta_1x$

(I've been following the Machine Learning class and the instructor uses this notation that I've not seen before)

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The notations are equivalent, but using a subscript sort of suggests that it's fixed for most of the discussion, and $x$ is the one that's changing. Leaving it out means for sure that it's fixed for the discussion (i.e., all the $f$'s you see should be taken with the same $\theta$.) –  Ted Oct 5 '11 at 5:38

1 Answer 1

up vote 4 down vote accepted

As you note, this is mostly notational choice. I might call the $\theta$ a parameter, rather than an independent variable. That is to say, you are meant to think of $\theta$ as being fixed, and $x$ as varying.

As an example (though I am not sure of the context you saw this notation), maybe you are interested in describing the collection of functions $$f(x) = x^2+c$$, where $c$ is a real number. I might call this function $f_c(x)$, so that I can later say that for $c\leq 0$, the function $f_c$ has two real roots, while for $c>0$ the roots are uniformly convex. I think these statements would be much more opaque if I made them about the function $f(x,c)$.

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