Point variable estimates for discriminative models

Predictions and residuals

Predictions

Our data \((\mathbf y, \mathbf X)\) is divided into \((y_i, \mathbf x_i)\).

We create a function \(\hat y_i = f(\mathbf x_i)\).

The best predictor of \(y\) given \(x\) is:

\(g(X)=E[Y|X]\)

The goal of regression is to find an approximation of this function.

Residuals

\(\epsilon_i = y_i- \hat y_i\)

Residual sum of squares (RSS)

\(RSS=\sum_i \epsilon_i^2\)

\(RSS=\sum_i (y_i-\hat y_i)^2\)

Explained sum of squares (ESS)

\(ESS=\sum_i (\bar y-\hat y_i)^2\)

Total sum of squares (TSS)

\(TSS=\sum_i (y_i-\bar y)^2\)

Relationship between prediction and probability distribution

\(P(y|X, \theta )\)

\(\hat y =f(\mathbf x)\)

Through integration?

\(E[y] = \int P(y|X, \theta ) dy\)

Coefficient of determination (\(R^2\))

\(R^2= 1-\dfrac{RSS}{TSS}\)

Classification

Binary classification

Classification models are a type of regression model, where \(y\) is discrete rather than continuous.

So we want to find a mapping from a vector \(X\) to probabilities across discrete \(y\) values.

A classifier takes \(X\) and returns a vector.

For a classifier we have \(K\) classes.

Classification

Confusion matrix. true positve, false positive, false negative, true negative

Can use this to get

Accuracy: percentage correct

Precision: percentage of positive predictions which are correct

Recall (sensitivity): percentage of poitive cases that were predicted as positive

Specificity: percentage of negative cases preicated as negative

Multiclass classification

What if can be email for work, friends, family, hobby?

Confusion matrix

Include error types here

Hard and soft classifiers

A hard classifier can return a sparce vector with \(1\) in the relevant classification.

A soft classifier returns probabilities for each entry in the vector.

The vector represents \(P(Y=k|X=x)\)

Transforming soft classifiers into hard classifiers

We can use a cutoff.

If there are more than two classes we can choose the one with the highest score.

Loss functions for point predictions

Minimum Mean Square Error (MMSE)

Mean estimate.

Can do for a parameter, or for a predicted estimate for \(y\).

Linear models

MLE is same as \(y^2\) loss

MAP is same as \(y^2\) loss with regularisation

Loss functions for soft classifiers

Hinge loss

Brier score

Loss functions for hard classifiers

Don’t want answers outside \(0\) and \(1\).

F score

F1 score

\(F_1\) score: \(\dfrac{2PR}{(P+R)}\)

may not just care about accuracy, eg breast cancer screening

high accurancy can result from v basic model (ie all died on titanic)

Receiver Operating Characteristic (ROC) Area Under Curve (AUC)

Other

Estimating other priors

Estimating \(P(k|T)\) - Which variables we split by, given the tree size

Estimating \(P(r|T, k)\) - The cutoff, given the tree size and the variables we are splitting by

Estimating \(P(\theta | T, k, r)\)

Maximum Likelihood Estimation (MLE) for generative and discriminative models

Maximum A-Priori estimation (MAP) for generative models

Bayesian regression for generative models

We know:

\(P(\theta |y,X)=\dfrac{P(y, \theta, X )}{P(y, X)}\)

\(P(\theta |y,X)=\dfrac{P(y, X |\theta )P(\theta )}{P(y, X)}\)

The bottom bit is a normalisation factor, and so we can use:

\(P(\theta |y,X)\propto P(y, X| \theta)P(\theta)\)

We have here:

Our prior - \(P(\theta )\)
Our posterior - \(P(\theta |y,X)\)
Our likelihood function - \(P(y, X| \theta )\)

Bayesian classifier

Classification risk

We can measure the risk of a classifier. This is the chance of misclassification.

\(R(C)=P(C(X)\ne Y)\)

The Bayesian classifier

This is the classifer \(C(X)\) which minimises the chance of misclassification.

It takes the output of the soft classifier and chooses the one with the highest chance.