Writing sigmoid function with input as (X * theta)

Question

In Machine learning course, I am unable to visualize below input.

we have below equation in Logistic Regression :

We can write it in octave as below in sigmoid.m :

g = (1 ./ ( 1 + e.^(-z)));

Now, to calculate costFUnction.m , we are getting probability as:

h = sigmoid(X*theta);

From, the above picture, shouldn't it be:

h = sigmoid(theta'*X);

What am I missing here. I am newbie to ML so forgive me if I am missing something here.

Answer 1

The most important thing is to understand what every vector means. In most courses they talk about

    h = theta'* x

But here they use colum vectors so h is a scalar for one training example. The vectorized notation tells you

    h = X * theta

Where X is a matrix all your training examples, where each example is a row and the features are colums. So mxn with m number of training examples and n number of features. You want h to give an output for each training example so you want amx 1 matrix. You know that theta will be anx 1 matrix as it is a theta for each feature and you have 1 model. If you do the second formula i wrote down at the top you will get as hamx 1 matrix which is prefered.

Answer 2

If you'll refer to the material shared here , you can see that

and what we want from h(x) is:

to visualize it:

X =  [ 1 x1 ; 1 x2 ; 1 x3;]
theta = [ t0 t1;]
X * theta
% will give  [ t0+(x1*t1) ; t0+(x2*t1) ; t0+(x3*t1) ; ] 

where each row of above matrices represent separate hypothesis.