I was just going through Neural-Network tutorials and I have some question regarding Bias neuron :-  1.] Is the act of introducing bias neuron same as introducing the X(index 0)=1 in the logistic regression i.e. [1, X1, X2]  2.] Is bias neuron an additional degree of freedom ?  3.] Can bias neuron be analogous to the constant term in  the linear equation :-  Y= mX+C   because without C i can only rotate the line x around the origin, but if I take into consideration C also then I can translate this line up or down also.  For example consider the figure below :-    Then the blue line (hypothesis) is simply Y=m*X (blue line passes through origin), so even if you keep fiddling with the m paramater but you wont be able to seperate the green and blue dots ever with just this hypothesis. So with just m to fiddle with, my hypothesis will always have to pass through the origin, i.e. I can choose from the infinite numbers of lines through the origin by tweaking the parameter 'm'.  Whereas if I include C/constant term (which Im calling the bias) in my hypothesis then my equation becomes Y=m*X+C then a line like Y=.4*X+3  (magenta line ) easily classifies the data. So now that I have an additional parameter "C" , I can translate(shift up/shift down) apart from the rotation.  Is my understanding correct ?
I'm curious to know how important the bias node is for the effectiveness of modern neural networks. I can easily understand that it can be important in a shallow network with only a few input variables. However, modern neural nets such as in deep learning often have a large number of input variables to decide whether a certain neuron is triggered. Would simply removing them from, e.g., LeNet5 or ImageNet have any real impact at all?
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