# Derivation of an probability expansion used in Word2Vec classifier model

We are using the following notations, for this question, to calculate the probability values

$$\begin{array}{|c|c|} \hline \text{w} & \text{target word embedding vector} \\ \hline \text{c} & \text{context word embedding vector} \\ \hline \text{L} & \text{context window size} \\ \hline \text{c_i} & \text{i^{th} context word} \\ \hline \text{\sigma(x)} & \dfrac{1}{1+e^{-x}} \\ \hline \text{c.w} & \text{dot product of c and w} \\ \hline \end{array}$$

In Word2Vec, we learn embeddigs using logistic regression. In order to train logistic regression, there are two kinds of classes, one is positive (+) and other is negative (-).

The probability that the word $$c$$ is a real context word (positive class) for target word $$w$$ is

$$p(+/w, c) = \sigma(c.w) = \dfrac{1}{1+e^{-c.w}}$$

Since there are only two classes, we can imply the probability that the word $$c$$ is a not a real context word (negative class) for target word $$w$$ is

$$p(-/w, c) = \sigma(-c.w) = \dfrac{1}{1+e^{c.w}}$$

The first equation gives us the probability for one word, but there are many context words in the window. Skip-gram makes the simplifying assumption that all context words are independent, allowing us to just multiply their probabilities:

$$p(+/w, c_{1:L}) = \prod\limits_{i = 1}^{L} \sigma(-c.w)$$

My doubt is how does the RHS of the above equation is derived from the LHS?

I know that if all features $$\{ X_1, X_2, X_3, \cdots, X_n \}$$ are class conditionally independent with class $$Y$$, then

$$p(X_1, X_2, X_3, \cdots, X_n /Y) = p(X_1/Y)p(X_2/Y)p(X_3/Y) \cdots p(X_n/Y)$$

$$p(Y/ X_1, X_2, X_3, \cdots, X_n ) = \dfrac{p(X_1/Y)p(X_2/Y)p(X_3/Y) \cdots p(X_n/Y)}{p(X_1, X_2, X_3, \cdots, X_n)}$$

But I am not sure how to apply this and obtain the result I asked for.