Help with Math notation: what is ψ & ϕ in this example?

Question

I am taking the Probabilistic Graphical Models class on Coursera. They post the question shown in the linked image ...

The course doesn't explain what the symbols "ψ" and "ϕ" mean. Can anyone help me understand this please?

My current best guess is that:

• ϕ means a function/product of X & Y and
• ψ means the result/return value of the function/product

Am I in the right area at all?

Answer 1

$\phi_1$ and $\phi_2$ are both functions of two variables. The functions appear to be defined by the first two tables. For example, $\phi_1(1,1) = 0.8$ from the first entry. The function $\psi(X,Y,Z)$ is defined as the product of the other two, according to the second paragraph. That is $$\psi(X,Y,Z) = \phi_1(X,Y)\phi_2(Y,Z)$$

We then find, for example, that

$$\psi(1,1,1) = \phi_1(1,1)\phi_2(1,1) = (0.8)(0.2)= 0.16$$

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The symbols themselves, $\phi$ and $\psi$, are simply names of the functions. These are the Greek lowercase letters phi and psi. The problem could have called them $f$ and $g$ or $bob$ and $alice$.

Answer 2

As pointed out in the comments, $\phi_1$, $\phi_2$ and $\psi$ are functions of $X$ and $Y$, $Y$ and $Z$ and $X$, $Y$ and $Z$ respectively.

Here, $\phi _1 $ and $\phi_2$ are defined by the tables given in the question. Also, $\psi $ is defined by $\psi (X,Y,Z) = \phi_1 (X,Y).\phi_2(Y,Z)$.

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You must be familiar with the concept of functions in probability. If not, then the gist is this -

A function in probability is a function which maps from the set of some events ( the sample space ), to the interval $[0,1]$.

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Here your $X$ and $Y$ take values $1$ and $2$. Corresponding to the ordered pair of $u(X,Y)$ of values taken by $X$ and $Y$ we get some value of $\phi_1 $ as given by the tables. Similarly, the other two functions also take si E values.

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In sum, $\psi $, $\phi_1 $ and $\phi_2 $ are just bands of functions and not some special notations.

Try to think of them as names of variables in the same spirit as $x$ , or more correctly ad names if functions in the same spirit as $f(x)$ or $g(x,y)$.