When should mutual information be used for feature selection over other feature selection methods like correlation, ANOVA , etc?
Pearson correlation and ANOVA will measure the linear variance between x,y.
A mutual information regression method is an entropy-based method which can find even non-linear correlation.
Let's see 2 pair of data (x, y_1) and (x, y_2).
##0. Data setup
x = np.array([1, 2, 3, 4, 5, 6, 7, 8, 9, 10])
y_1 = 3.39*x_1
y_2 = np.array([1, 1, -1, -1, 0, 0, -2, -2, 2, 2])
df_1 = pd.DataFrame([x,y_1]).T
df_2 = pd.DataFrame([x,y_2]).T

##1. Pearson corr
df_1.corr()
df_2.corr()

##2. f_test
from sklearn.feature_selection import f_regression, mutual_info_regression
f_test, _ = f_regression(x.reshape(10, 1),y_1)
f_test
f_test, _ = f_regression(x.reshape(10, 1),y_2)
f_test

##3. Mutual reg
mutual_info_regression(x.reshape(10, 1),y_1,n_neighbors=1,random_state=123)
mutual_info_regression(x.reshape(10, 1),y_2,n_neighbors=1,random_state=123)

##4. Tree-based feature importance is also based on entropy reduction 
##and is more reliable as  KNN needs a good k value 


 We can observe that Mutual importance is able to catch the relation of x with not just y_1 when it is simple Linear bust also with y_2 when it is not linear(but has a strong relation)
You should use either MI if you are only looking for a filter-based method. Otherwise, Tree feature importance would be best for most of the cases.
Check this link -
F_test Vs MI