I. Basics terms in statistical learning


First of all let’s get familiar with the notation:
X =  inputs, features (all of the possible x)

X = [x1, x2, ... , xm]

Y = outputs, target variables (all of the possible y)

Y = [y1, y2, ..., ym]

let’s call f(X) the true fonction of X that gives Y => what we are looking for but is unknown. We write:

 Y = f(X) + 𝓔 

meaning that f(x1) + 𝓔 = y1, f(x2) + 𝓔 = y2, …, f(xm) +𝓔 = ym where 𝓔 is the error term (independent of x and has mean zero) also called noise.
Usually we don’t have access to all the possible inputs-outputs but we got a dataset D of input-output examples. This data set is used to learn the model g(X) that approximates f, the true (unknown) target function. 

Y' = g(X) + 𝓔


Prediction and inference
The main goal in statistical learning is to estimate f. Depending on your goal, there could be two reasons to estimate f, you might want to predict or to infer. Prediction means that we want to approximate f as well as possible. Inference means that we want to understand the relationship between X and Y (f is not the main focus). If your goal is to predict, then you will try to find a model g close to f that gives a good approximation of Y. If your goal is to infer, then you will try to make g as readable as possible which will allow you to interpret the model. 

Parametric/non parametric methods
Depending on your priors on the problem you want to solve, you might use a parametric or a non-parametric method. The first one involves (1)making an assumption about the shape of f (ex: linear) and (2)fit or train the model (ex: gradient descent). The non-parametric method however, doesn’t make any explicit assumption about the shape of f, it fits several possible shapes for f. The main problem of the parametric approach is that the true f might be completely different from our assumption (ex: we try to fit a linear regression whereas f is highly non-linear). For the non-parametric, we don’t have this problem since no assumption is made but it needs a very large number of data points.

Supervised/unsupervised learning
There are three principal types of learning:
Supervised, reinforcement and unsupervised learning. In the supervised learning, we got a dataset with inputs and outputs (x1 -> y1, x2 -> y2, …, xm -> ym). In the reinforcement learning, we don’t get explicitly the correct outputs. In the unsupervised learning, we don’t have any output.

Regression/classification problem
Depending on the nature of the outcomes, there could be two kind of problem: Regression and classification. In regression, Y is composed of continuous values. In classification, Y is composed of class label (for example 0 and 1). In regression, we usually aim to estimate or predict a response. In classification, we aim to identify group membership.


Trade-off between model interpretability and prediction accuracy
A model could be more or less restrictive, leading to more or less flexibility. A very restrictive model is relatively inflexible, it can produce only a small range of shapes to estimate f. A flexible model will allow a broader range of shapes to estimate f. Choosing which one is more adapted depends on your problem: If you are interested in making prediction, you will care more about prediction accuracy than model complexity (meaning you want a good approximate of f which will give you a prediction y’ close to y), if you want to infer, then you want a simple model, highly interpretable (g(X) = 3X + 2 is more interpretable than g(X) = 2X³ + 3X²). However in some cases, complexity is not synonymous of high prediction accuracy and can lead to overfitting (see below).
Assessing a model and the no free lunch theorem  
The no free lunch theorem states that no one method is better than the others over all data sets. Meaning that there is a need to assess the model accuracy. In order to evaluate your model on a given dataset, the most commonly-used measure is the mean squared error (MSE) (see LinearRegression for more details). Basically, it is the total difference between the real Y (from f) and the predicted Y’ (from g). The bigger the MSE, the bigger the bias (g moves away from f). A model with a low MSE might be considered as having a high prediction accuracy because the Y’ are really close to the Y. Be careful though, it can also be because of overfitting (see below).

The bias-variance trade off
As seen before, the bias is the error introduced by trying to approximate f. The higher the bias, the more different g is from f. The variance on the other hand, is the amount by which g would change if we estimated it using a different data set. We used one particular data set to fit the model, so using a new one would have an influence on g. In general, more flexible model got higher variance and less bias. In other words, As a flexible model increases the variance, it decreases the bias until a certain point where there is only a small effect on bias but an important one on the variance. If you want to infer, you will focus on minimising the variance. In prediction, the focus will be on minimising the bias.

Training and test set
We often only have a limited dataset to determinate g. If we use the whole dataset to fit our model g, we won’t be able to assess its prediction accuracy (if we don’t have any new data). One strategy to remedy this problem is the validation set approach. It involves to randomly split the dataset into a training and a validation set (test set). We use the training set to find or fit the model, and the validation set to assess its accuracy. There are different methods to get a training and a test set. The most common are the k-fold cross-validation and the leave-one-out cross validation. The first one divides the data set into k fold and each will play the role of training and test set. In the second approach, each data point will play the role of test-set. 

After fitting our model using the training set, we will assess it using the test set. If our model fit our training set too well (very low MSE) then the generalisation won’t work well and lead to overfitting meaning that we also fitted the noise present in our training set.
Statistical learning and classical statistics
In statistical learning, the word “learning” stands for the fact that we learn from the data to get or to fit a model (or several). basically, there are two big trends in the statistics used in neuroscience: the statistical learning and the classical statistics. Read this article if you want to go further (Bzdok, 2016, Classical statistics and statistical learning in imaging neuroscience. PDF). In sum:

Screen Shot 2016-04-01 at 15.22.05


Next: Univariate Linear Regression and gradient descent


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