Maximum likelihood estimation why multiply and least squares

Answer 1:

Maximum likelihood estimate: Now you have got a lot of samples (all the dependent variables in your dataset), these sample values have been implemented, and the maximum likelihood estimate is to find that (group) parameter estimates, so that the previously implemented sample value probability is the largest. Because the sample you have on hand has been implemented, the probability of its occurrence is the most logical. In this case, the joint probability maximization of all the observations of the sample is a continuous product, so long as the logarithm is taken, it becomes the linear plus total. By taking the derivative of the parameter and making the first derivative zero, the maximum likelihood estimate can be obtained by solving the equation (group).
Least squares: A (group) estimate is found so that the actual value is minimized from the estimated value. It is desirable to summarize and minimize the absolute value of the difference, but it is troublesome to find the minimum in mathematics, so the alternative is to look for a (group) estimate so that the sum of the squares of the difference between the actual and the estimated values is the smallest, called the least squares. "Least square" in English, in fact, the literal meaning of English is "the smallest square." At this point, the sum of the squared difference to the parameter derivative number, and take the first derivative of zero, is olse.

Answer 2:

In a more popular sense, the maximum likelihood estimate is that the value of the parameter, which is most likely to cause such a result, is reversed using the known sample results .
For example: A sack with a white ball and a black ball, but I do not know the ratio between them, then I have put back the extraction 10 times, and I found that I smoked 8 black ball 2 times the white ball, I asked the most likely between the black and white ball ratio when the maximum likelihood estimation method is taken: I suppose I'm the probability of a black ball being p, That's 8 black balls, 2 white balls. The probability of the result is:
P (Black =8) =p^8* (1-p) ^2, now I want to draw a p is how much ah, very simple, so that P (black =8) the largest p is the result of my request, then the process of derivation is the process of finding the extremum.
You may have questions about why you want ln, because ln turns multiplication into addition and does not change the position of the extremum (Monotonicity remains the same) so that the derivation will be much easier ~

Similarly, the probability density of setting the total X is
Known x1,x2: Xn is the sample observation value, and the maximum likelihood estimation of θ is obtained.

That's the same thing, to get x1,x2. The probability of xn such a set of sample observations is
P{x1=x1,x2=x2,... xn=xn}= F (x1,θ) F (x2,θ) ... f (xn,θ)
Then we ask to make P the largest θ is good, the same is the process of seeking extreme value, no longer repeat.

Answer 3:

Maximum likelihood estimation is a generic term for a class of methods, including least squares. For example, in the linear regression problem, assuming that the error obeys the Gaussian distribution, the maximum likelihood estimation of the model parameters is the least squares method.

http://www.fuzihao.org/blog/2014/06/13/%E4%B8%BA%E4%BB%80%E4%B9%88%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%98%E6%B3% 95%e5%af%b9%e8%af%af%e5%b7%ae%e7%9a%84%e4%bc%b0%e8%ae%a1%e8%a6%81%e7%94%a8%e5%b9%b3%e6%96%b9/

Answer 4:

The value of the maximum likelihood function may be small, but its value size is not useful. Things have happened, you have not given the greatest possible me.

Reference: http://www.zhihu.com/question/20447622

Maximum likelihood estimation why multiply and least squares