When is it ok to use "=" instead of ":="? I am having difficulty whilst writing my equations, deciding when I should be using the equals symbol "=" , and when I should be using the defined as symbol ":=". In particular, I am talking about equations for probabilities, when those probabilities are really approximations to the true probabilities. I am an engineer rather than a mathematician, so these approximations are common!
For example, if I was using Bayes rule in my equation, I would write $p(x|y) = \frac{p(y|x)p(x)}{p(y)}$, because the two are equivalent. On the contrary, suppose I was using a very naive and heuristic score, and then using that score as a probability in my future calculations. I might write, for example, $p(x) := 0.5 + 0.5 * (z/10)$ where $z$ are some observations I have made about $x$, and $0 < z < 10$. This is because $p(x)$ is clearly just a score being forced into a probability, and so it would not be accurate to use the "=" symbol.
However, what about the cases between these two examples? Should you only use the "=" symbol when the probability is derived purely and directly by maths? Or are there cases when the approximation is only small, and therefore "=" is appropriate? The reason I ask is that in engineering, almost all probabilities are approximations, but it would look odd to have every single equation using the ":=" symbol.
For example, suppose I decided to use a naive Bayesian model, where $x = \{a, b, c, d\}$. Would it be ok to write $p(x) = p(a)p(b|a)p(c|a)p(d|a)$, or would I have to write $p(x) := p(a)p(b|a)p(c|a)p(d|a)$? Is there a general rule as to how close to a pure approximation a probability definition has to be if you are going to use "=" rather than ":="?
 A: To answer one of your questions ("should = only be used when the probability is derived purely and directly by maths") the main take away is that, logically, $=$ and $:=$ mean the same thing. They both mean the thing on the left equals the thing on the right, no exceptions. The colon serves a purpose similar to the therefore or because sign - it has no actual mathematical meaning, it just clarifies the thoughts of the mathematician.
EDIT: And to that point, equals mean equals, always, and equals never means approximate. If you want approximate, then $\approx$ is the symbol for you (and notice that $:\approx$ wouldn't really make sense - you can't "define" something without being specific!)
EDITEDIT: I also want to clarify, thanks to Aernimund's comment, that although $=$ and $:=$ are logically equivalent, they are not equivalent, for instance, in the English language, or in the pidgin of English and formal logic that we use to write mathematics! They have different meanings, and I failed to underline the importance of that distinction, emphasizing instead that the distinction doesn't ever affect the truth value of a sentence.
A: I think you might be using the symbol $:=$ somewhat unnaturally. You are correct that this means to define something, for example I might say "let $Q$ be defined as the maximum of the numbers $q_1$ and $q_2$"
$$Q:=\max\{q_1,q_2\}$$
This is the context in which mathematicians would usually use the symbol, when they themselves are defining a quantity. In your question, writing $p(x):=p(a)p(b|a)p(c|a)p(d|a)$ looks odd as it looks like you are defining this quantity to be what is meant by the probability of $x$, but the probability of $x$ is usually understood as a predefined concept which is known (in your case) to be determined by $p(a)p(b|a)p(c|a)p(d|a)$. So really here the correct symbol is always =.
If you are conscious of being precise with what you are writing (and precision is something mathematicians take delight in!) you may wish to use the symbol $\simeq$ (/simeq in latex) when stating an approximate value. Otherwise you may wish to stick with the equals sign in order to prevent disconcerting repetition of the squiggly lines, but include in the text of your report lines such as "the quantity is approximately given by", or "the stated probablilties are approximated up to and error of (say) 0.001". 
Even better, if you know to what degree your measurements are approximate, you could write something like "$p(x)=0.5+0.5∗((z+\epsilon)/10)$, where $0.0005<\epsilon<0.0015$" and carry this quantity along the calculation to the final answer. 
However if the error is really always negligibly small then error terms could be cumbersome and not worth carrying around. You are then probably justified in stating the size of the error in your calculations at the offset for clean conscience whilst using = from then on.
Hope that helps :)
