What is the purpose of weights and biases in tensorflow word2vec example?

The idea of skip-gramm is comparing words by their contexts. So we consider words equal if they appear in equal contexts. The first layer of NN represents words vector encodings (basically what is called embeddings). The second layer represents context. Every time we take just one row (Ri) of first layer (because input vector always looks like 0, ..., 0, 1, 0, ..., 0) and multiply it by all columns of second layer (Cj , j = 1..num of words) and that product will be the output of NN. We train neural network to have maximum output components Ri * Cj if word i and j appear nearby (in the same context) often. During each cycle of training we tune only one Ri (again because of the way input vectors are chosen) and all Cj, j = 1..w. When training ends we toss the matrix of the second layer because it represents context. We use only matrix of the first layer which represents vector encoding of the words.

In word2vec what you want is a vector representation of words. In order to do that you can use, among other things, a neural network. So you have inputs neurons, outputs and hidden layers. What you do to learn the vector representation is to have a hidden layer which number of neurons is the same as the dimension you want in your vectors. There is one input per word and one output per word. And then you train the network to learn the input from the output but in the middle you have a smaller layer which you can see it as a codification of the input in a vector. So here are the weights and biases. But you don't need them later, what you use for testing is a dictionary which contains the word and the vector which represents that word. This is faster than running the neural network to get the representation. That is why you don't see it later.

The last code you write about the cosine distance is to know which vectors are closed to your calculated vector. You have some words (vectors) you make some operations (like: king - man + woman) and then you have a vector that you want to convert in the result. This is the cosine function run among all the vectors (queen would have the minimum distance with the result vector of the operation).

To sum up, you don't see the weight and bias in the validation phase because you don't need them. You use the dictionary you have created in the training.

UPDATE s0urcer has explained better how the vector representation is created.

The input layer and the output layer of the networks represents words. It means the value is 0 if the word is not there and 1 if the word is there. First position is one word, second another one, etc. You have as input/output neurons as words.

The middle layer is the context, or you vector representation of the words.

Now you train the network with sentences or group of consecutive words. From this group you take one word and set it in the inputs and the other words are the outputs of the network. So basically the network learns how a word is related with other words in its context.

To get the vector representation of each word you set the input neuron of that word to 1 and see the values of the context layer (the middle layer). Those values are the values of the vector. As all the inputs are 0 except the word that is 1, those values are the weights of the connections of the input neuron with the context.

You don't use the network later because you don't need to calculate all the values of the context layer, that will be slower. You only need to check in your dictionary what are those values for the word.