# What does $\propto$ mean?

$\ldots$a maximum-a-postiori "a maximum-a-posteriori $(MAP_{x,k}^{\,\,\,\,\,1})$ estimation, seeking a pair $(\hat{x}, \hat{k})$ maximizing: $$p(x, k\mid y) \propto p(y\mid x, k)p(x)p(k).$$

What does the symbol '∝' mean in this context?

• I think Seyhmus Güngören's answer is vague. It's just how this question is usually answered, and I think it bears elaboration. – Michael Hardy Mar 13 '13 at 2:26
• Bernoulli used it to mean "$=$". – Pedro Tamaroff Mar 13 '13 at 3:05
• Another question, are "x hat" and "k hat" the normalized version of vector x and k, having length 1? And why does this make sense? – user1095340 Mar 13 '13 at 10:26
• $\hat x$ and $\hat k$ are the values of $x$ and $k$ that make $p(x,k\mid y)$ as large as possible. – Michael Hardy Mar 13 '13 at 17:07

It means proportional as a function of two variables $x$ and $k$ with $y$ fixed.

You have a prior probability density as a function of $x$ and $k$, which is just a product of a function of $x$ and a function of $k$, so that the random variables involved are independent in the prior distribution. Then new data arrives: a random variable is observed to be equal to a number $y$. The conditional density of that random variable given that the first two were equal to $x$ and $k$, is the first factor in the expression on the right.

The expression on the left is the conditional density of the random variables corresponding to $x$ and $k$ given that observed value equal to $y$.

Since it means proportional as a function of $x$ and $k$ with $y$ fixed, you need to multiply it by a normalizing constant that may depend on $y$ but does not depend on $x$ and $k$, in order to make it a probability density function, as a function of $x$ and $k$. "Constant" in this case would mean not depending on $x$ and $k$.

"Constant" always means not depending on something. Usually it's clear from the context what the "something" is, but I think it should be stated explicitly more often than it is in present-day conventional practice. Here's a favorite example of mine, involving differentiation of exponential functions: \begin{align} \frac{d}{dx} 2^x & = \lim_{h\to0}\frac{2^{x+h}-2^x}{h} \\[10pt] & = 2^x\lim_{h\to0}\frac{2^h-1}{h}\tag{1} \\[10pt] & = (2^x\cdot\text{constant})\tag{2}. \end{align}

In $(1)$, the factor $2^x$ can be taken out of the limit because it's "constant" but "constant" means not depending on $h$.

In $(2)$, "constant" means not depending on $x$.

Some instructors in calculus classes actually present this proof without mentioning the contextual change in the meaning of "constant".

Later note in response to comments below: The linked paper uses a rather obnoxious notation, $p(x)$ and $p(k)$ for the probability density functions of two different random variables. One should distinguish between capital $X$ and lower-case $x$ in expressions like $\Pr(X=x)$, where capital $X$ is a random variable and lower-case $x$ is a particular value that $X$ might be equal to. Then, if one writes $p_X(x)$ for the value of the probability density function of a random variable (capital) $X$ at the point (lower-case) $x$, then one knows that $p_X(3)$ means something different from $p_Y(3)$.

But at any rate, $p(x)p(k)$ is the notation used in the linked paper for the joint density function of a pair of independent random variables, and $p(y\mid x,k)$ is the conditional density of another random variable given the values of those two. The idea is that if you multiply those, what you get is proportional, as a function of $x$ and $k$ with $y$ fixed, to the conditional density function of the random variables corresponding to $x$ and $k$, given an observed value of the random variable corresponding to $y$.

• But does it mean proportional as a function of $x$ with $k$ and $y$ fixed, or proportional as a function of $y$ with $k$ and $x$ fixed, or proportional as a function of $k$ with $x$ and $y$ fixed? My guess is none of the above: it means proportional as a function of two variables $x$ and $k$ with $y$ fixed. – Michael Hardy Mar 13 '13 at 2:15
• It means right and left hand side of the equation behaves similarly. Can happen that there is a simple constant factor. For each equality one can use this term. Additionally whenever this term is used, left hand side can be obtained from the right hand side by just a simple scaling. Examples: "If an object travels at a constant speed, then the distance traveled is directly proportional to the time spent traveling, with the speed being the constant of proportionality." "The circumference of a circle is directly proportional to its diameter, with the constant of proportionality equal to $\pi$. – Seyhmus Güngören Mar 13 '13 at 10:02
• It is MAP. So the difference between ML is the a priori probabilities, which are taken As some scaling factors in MAP. Therefore $p(x)$ and $p(k)$ in this context are jointly the scaling factor. – Seyhmus Güngören Mar 13 '13 at 18:09
• It might also happen that he is indicating the independecy between $T$ and $X$. One should see more about the content – Seyhmus Güngören Mar 13 '13 at 18:16