How do I calculate the maximum likelihood (machine learning statistics) of this table of data? I’m taking a machine learning course and I’m stuck. I've already tried to ask around in the forums of the class but I don't understand how to implement the formulas they give me. Can you please help me understand how to solve this problem from the ground up?


  “Calculate the likelihood of this data.”
  
  X  | Y
  
  2.5 | +1
  
  0.3 | -1
  
  2.8 | +1
  
  0.5 | +1
  
  the current estimates of the weights are $w_0 = 0$ and $w_1 = 1$. ($w_0$: the intercept, $w_1$: the weight for $x$).


Now Here's what I've already done in case that matters:

When I went on to the forums to try to understand I was told this:


  You can use the sigmoid function to find the likelihood of any given
  one, and then once you have the probabilities of any given one
  occurring you should be able to find the probability of the whole
  thing


Which is nice; now I know it has something to do with the sigmoid function: $$\frac{1} {1 + e^{-x}}$$

sigmoid(2.5) = 0.9241418


I hope I'm on the right track, I really have no way of knowing. So then I calculated the sigmoid for all of them and multiplied them together hoping that would give me the probability of the whole thing because that's how you combine probabilities, right? (The probability of a dice role hitting 2 three times is $1/6 \times 1/6 \times 1/6$).

But that apparently isn't the right answer, and I didn't take into account the $y$ column or the weights because I don't know how. How sad for me.

Can you please explain it to me, I feel like I'm a blind man looking for the right colored paper.
You might have mistaken the equation for maximum likelihood , and it should be like this :


  $p^y*(1-p)^{1-y}$


while y is your label , it should be 0 or 1 , 0 for negative and 1 for positive , and p is your sigmoid function value.

so , in your case it should be :


  p(2.5)*(1 - p(0.3))*p(2.8)*p(0.5)


Hopes this will help you , good luck !