Explanation
This hashed feature approach has the distinct advantage of requiring less memory and one less pass through the training data, but it can make it much harder to reverse engineer vectors to determine which original feature mapped to a vector location. This is because multiple features may hash to the same location. With large vectors or with multiple locations per feature, this isn't a problem for accuracy but it can make it hard to understand what a classifier is doing.
Models always have a coefficient per feature, which are stored in memory during model building. The hashing trick collapses a high number of features to a small number which reduces the number of coefficients and thus memory requirements. Noisy features are not removed; they are combined with other features and so still have an impact.
The validity of this approach depends a lot on the nature of the features and problem domain; knowledge of the domain is important to understand whether it is applicable or will likely produce poor results. While hashing features may produce a smaller model, it will be one built from odd combinations of real-world features, and so will be harder to interpret.
An additional benefit of feature hashing is that the unknown and unbounded vocabularies typical of word-like variables aren't a problem.

Explanation
Linear regression model are represented using the below equation

Where B(0) is intercept and B(1) is a slope. As B(0) and B(1) changes then fitted line also shifts accordingly on the plot. The purpose of the Ordinary Least Square method is to estimates these parameters B(0) and B(1).
And similarly it is a sum of squared distance between the observed point and the fitted line. Ordinary least squares (OLS) regression minimizes the sum of the squared residuals. A model fits the data well if the differences between the observed values and the model's predicted values are small and unbiased.

Explanation
Correlation measures the linear relationship (Pearson's correlation) or monotonic relationship (Spearman's correlation) between two variables, X and Y.
Mutual information is more general and measures the reduction of uncertainty in Y after observing X.
It is the KL distance between the joint density and the product of the individual densities. So Ml can measure non-monotonic relationships and other more complicated relationships Mutual information is a quantification of the dependency between random variables. It is sometimes contrasted with linear correlation since mutual information captures nonlinear dependence.
Features with high mutual information with the predicted value are good. However a feature may have high mutual information because it is highly correlated with another feature that has already been selected.
Choosing another feature with somewhat less mutual information with the predicted value, but low mutual information with other selected features, may be more beneficial. Hence it may help to also prefer features that are less redundant with other selected features.

Explanation
(Change the explanation even it is correct)A normalizing constant is positive, and multiplying or dividing a series of values by a positive number does not affect which of them is the largest. Maximum likelihood estimation is concerned only with finding a maximum value, so normalizing constants can be ignored.

Explanation
There is a tradeoff between cost and stability in unsupervised learning. The more tightly you fit the data, the less stable the model will be, and vice versa. The idea is to find a good balance with more weight given to the cost. Typically a good approach is to set a stability threshold and select the model that achieves the lowest cost above the stability threshold.