# 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

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 hard classifiers

Don’t want answers outside $$0$$ and $$1$$.

#### 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)

## Other

### 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.