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Cross val predict expects as input an already fitted model?

I am reading Geron's Hands-on Machine Learning. In page 90, there is a section about Confusion Matrix. He says that we need some predictions, so he does the following:
from sklearn.model_selection import cross_val_predict
y_train_pred = cross_val_predict(sgd_clf, X_train, y_train5, cv=3)
This object sgd_clf is a stochastic gradient descent classifier which was previously fitted with the train data in the previous section. My question is: why, if already fitted, it is better to split the train set in three parts and retrain (?) the sgd_clf in two of them, then make a prediction and so on, if sgd_clf is already trained? Why not just let it predict on full X_train? Or just take a new not-fitted classifier as imput? Why put sgd_clf already trained as imput to retrain? I am a bit confused.

I see your confusion and I think Geron doesn't mean you should use the fitted model for cross-validation. He just wants to compare the naive fitting method with cross-validation.
The complete code should be as follows:
from sklearn.linear_model import SGDClassifier
# No cross-validation
sgd_clf1 = SGDClassifier(random_state=42)
sgd_clf1.fit(X_train, y_train)
# With cross-validation
sgd_clf2 = SGDClassifier(random_state=42)
cross_val_score(sgd_clf2, X_train, y_train, cv=3, scoring='accuracy')

How to get predicted labels during training of any classifier?

Is there any way that I can track my model's performance in terms of it's classified labels, during the training phase? Any classifier from sklearn would work as an example.
To be more specific, I want to get something like a list of Confusion Matrices here:
clf = LinearSVC(random_state=42).fit(X_train, y_train)
# ... here ...
y_pred = clf.predict(X_test)
My objective here is to see how well the model is learning (during training). This is similar to analyzing the training loss, that is a common practice in DNN's, and libraries such as pyTorch, Keras, and Tensorflow have such capability already implemented.
I thought a quick browsing of the web would give me what I want, but apparently not. I still believe this should be fairly simple though.

Some ML practitioners like to work with three folds of data: training, validation and testing sets. The latter should not be seen in any training at all, but the middle could. For example, cross-validation uses K different folds of validation sets "during the training phase" to get a less biased performance estimation when training with different parts of the data.
But you can do this on a single validation fold for the purpose of what you asked.
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)
X_train2, X_valid, y_train2, y_valid = train_test_split(X_train, y_train, test_size=0.2)
# Fit a classifier with train data only
clf = LinearSVC(random_state=42).fit(X_train2, y_train2)
y_valid_pred = clf.predict(X_valid)
confusionm_valid = confusion_matrix(y_valid, y_valid_pred) # ... here ...
# Refit with all your training data
clf = LinearSVC(random_state=42).fit(X_train, y_train)
y_pred = clf.predict(X_valid)

Using cross_val_predict against test data set

I'm confused about using cross_val_predict in a test data set.
I created a simple Random Forest model and used cross_val_predict to make predictions:
from sklearn.ensemble import RandomForestClassifier
from sklearn.cross_validation import cross_val_predict, KFold
lr = RandomForestClassifier(random_state=1, class_weight="balanced", n_estimators=25, max_depth=6)
kf = KFold(train_df.shape[0], random_state=1)
predictions = cross_val_predict(lr,train_df[features_columns], train_df["target"], cv=kf)
predictions = pd.Series(predictions)
I'm confused on the next step here. How do I use what is learnt above to make predictions on the test data set?

I don't think cross_val_score or cross_val_predict uses fit before predicting. It does it on the fly. If you look at the documentation (section 3.1.1.1), you'll see that they never mention fit anywhere.

As #DmitryPolonskiy commented, the model has to be trained (with the fit method) before it can be used to predict.
# Train the model (a.k.a. `fit` training data to it).
lr.fit(train_df[features_columns], train_df["target"])
# Use the model to make predictions based on testing data.
y_pred = lr.predict(test_df[feature_columns])
# Compare the predicted y values to actual y values.
accuracy = (y_pred == test_df["target"]).mean()
cross_val_predict is a method of cross validation, which lets you determine the accuracy of your model. Take a look at sklearn's cross-validation page.

I am not sure the question was answered. I had a similar thought. I want compare the results (Accuracy for example) with the method that does not apply CV. The CV valiadte accuracy is on the X_train and y_train. The other method fit the model using X_trian and y_train, tested on the X_test and y_test. So the comparison is not fair since they are on different datasets.
What you can do is using the estimator returned by the cross_validate
lr_fit = cross_validate(lr, train_df[features_columns], train_df["target"], cv=kf, return_estimator=Ture)
y_pred = lr_fit.predict(test_df[feature_columns])
accuracy = (y_pred == test_df["target"]).mean()

Cross validation and model selection

I am using sklearn for SVM training. I am using the cross-validation to evaluate the estimator and avoid the overfitting model.
I split the data into two parts. Train data and test data. Here is the code:
import numpy as np
from sklearn import cross_validation
from sklearn import datasets
from sklearn import svm
X_train, X_test, y_train, y_test = cross_validation.train_test_split(
iris.data, iris.target, test_size=0.4, random_state=0
)
clf = svm.SVC(kernel='linear', C=1)
scores = cross_validation.cross_val_score(clf, X_train, y_train, cv=5)
print scores
Now I need to evaluate the estimator clf on X_test.
clf.score(X_test, y_test)
here, I get an error saying that the model is not fitted using fit(), but normally, in cross_val_score function the model is fitted? What is the problem?

cross_val_score is basically a convenience wrapper for the sklearn cross-validation iterators. You give it a classifier and your whole (training + validation) dataset and it automatically performs one or more rounds of cross-validation by splitting your data into random training/validation sets, fitting the training set, and computing the score on the validation set. See the documentation here for an example and more explanation.
The reason why clf.score(X_test, y_test) raises an exception is because cross_val_score performs the fitting on a copy of the estimator rather than the original (see the use of clone(estimator) in the source code here). Because of this, clf remains unchanged outside of the function call, and is therefore not properly initialized when you call clf.fit.

how to properly use sklearn to predict the error of a fit

I'm using sklearn to fit a linear regression model to some data. In particular, my response variable is stored in an array y and my features in a matrix X.
I train a linear regression model with the following piece of code
from sklearn.linear_model import LinearRegression
model = LinearRegression()
model.fit(X,y)
and everything seems to be fine.
Then let's say I have some new data X_new and I want to predict the response variable for them. This can easily done by doing
predictions = model.predict(X_new)
My question is, what is this the error associated to this prediction?
From my understanding I should compute the mean squared error of the model:
from sklearn.metrics import mean_squared_error
model_mse = mean_squared_error(model.predict(X),y)
And basically my real predictions for the new data should be a random number computed from a gaussian distribution with mean predictions and sigma^2 = model_mse. Do you agree with this and do you know if there's a faster way to do this in sklearn?

You probably want to validate your model on your training data set. I would suggest exploring the cross-validation submodule sklearn.cross_validation.
The most basic usage is:
from sklearn.cross_validation import train_test_split
X_train, X_test, y_train, y_test = train_test_split(X, y)

It depends on you training data-
If it's distribution is a good representation of the "real world" and of a sufficient size (see learning theories, as PAC), then I would generally agree.
That said- if you are looking for a practical way to evaluate your model, why won't you use the test set as Kris has suggested?
I usually use grid search for optimizing parameters:
#split to training and test sets
X_train, X_test, y_train, y_test =train_test_split(
X_data[indices], y_data[indices], test_size=0.25)
#cross validation gridsearch
params = dict(logistic__C=[0.1,0.3,1,3, 10,30, 100])
grid_search = GridSearchCV(clf, param_grid=params,cv=5)
grid_search.fit(X_train, y_train)
#print scores and best estimator
print 'best param: ', grid_search.best_params_
print 'best train score: ', grid_search.best_score_
print 'Test score: ', grid_search.best_estimator_.score(X_test,y_test)
The Idea is hiding the test set from your learning algorithm (and yourself)- Don't train and don't optimize parameters using this data.
Finally you should use the test set for performance evaluation (error) only, it should provide an unbiased mse.