In the equation:


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?


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

  • 2
    $\begingroup$ 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 $\theta$.) $\endgroup$
    – Ted
    Oct 5, 2011 at 5:38

2 Answers 2


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 $f(x,c)$.


I personally dislike the notation. And I believe it means different things in different contexts. For example, it means the opposite of what the other answers say in probability, see:


It is shown that $P_{\theta}(X = x)$ and $p_{\theta}(x)$ both mean that $X/x$ is the constant variable in this context and we are plotting with respect to $\theta$.

  • $\begingroup$ I think it makes sense in probability. $p_{\theta}(x)$ denotes a probability density function on $x$, which is parametrised by some "model parameters" $\theta$. It's a probability distribution on $x$, not on $\theta$ ($\int_{-\inf}^{\inf} p_{\theta}(x d\theta \neq 1$). Treated as a probability distribution it makes sense to think of $\theta$ as parameters and $x$ as the argument on which the distribution is defined. But in inference, you are interested in finding the value of $\theta$ that maximises the distrbution for some $x$, which is why you often vary and plot $\theta$ instead. $\endgroup$
    – Marses
    Aug 25, 2020 at 17:12
  • $\begingroup$ That's why in inference you often write it as a likelihood, $L(\theta) = p_{\theta}(x)$, because in inference the result of your random variable ($x$) is often "measured" and fixed, and you want to find $p_{\theta}(x)$ at different values of $\theta$. If you write $p(x, \theta)$, unless you normalise it again, you won't have a probability distribution anymore (if you can even normalise it). Take the standard normal distribution: if you treat the mean as an argument, you can't say $\int\int\frac{1}{\sqrt{2\pi}}e^\frac{(x-\mu)^2}{2} dxd\mu = 1$. In fact you can't even normalise that. $\endgroup$
    – Marses
    Aug 25, 2020 at 17:22

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