Cfrgtkky

Can someone please help me understand below notation I encountered while studying logistic regression? I am pretty sure I lack the mathematical maturity but would like to give it a try.

$l(beta_0,beta_1) = prod_{i:y_{i=1}} p(x_i) prod_{i':y_{i'=0}} (1-p(x_i'))$

where $p(X)$ = $frac{e^{beta_0+beta_1X}}{1+e^{beta_0+beta_1X}}$

1. What does $prod$ notation stand for?


2. What does the index $i$ comprise of?


3. What does the lower bound $i:y_{i=1}$ mean?


4. What does the first equation mean on the whole?

Please let me know if I missed providing any context information.

It is not the notation I would choose, but essentially you seem have some observations $x_a,x_b,ldots$ which produce successful responses, which seem to be shown as $y_a=1, y_b=1, ldots$, and some other observations $x^prime_j,x^prime_k,ldots$ which produce unsuccessful responses, which seem to be shown as $y^prime_j=0, y^prime_k=0, ldots$

The model is based of the probability that $x$ produces a successful response $y=1$ being expressed as $p(x) = frac{e^{beta_0+beta_1x}}{1+e^{beta_0+beta_1x}}$ and so the probability that $x$ produces an unsuccessful response $y=0$ being $1-p(x)$

In answer to your questions:

1. 
   $prod$ means take the product over the index, in a similar way to $sum$ meaning the sum


2. In the first product $i$ seem to represent the index of the successful cases, while the second product $i^prime$ seem to represent the index of the unsuccessful cases


3. 
   $i:y_i=1$ seems to mean taking the index over the successful cases, $i^prime:y_{i^prime}=0$ seems to mean taking the index over the unsuccessful cases,


4. The whole expression is saying that the likelihood of the observations is (proportional to) the product of the probabilities of the successful cases being successful and the unsuccessful cases being unsuccessful assuming the probabilities follow the model family. Presumably the next step is to find the $beta_0$, $beta_1$ which maximise this likelihood