### Model Bias and Variance

**Over fitting**- Low Bias with High Variance
- Low training error – ‘Low Bias’
- High testing error
- Unstable model – ‘High Variance’
- The coefficients of the model change with small changes in the data

**Under fitting**- High Bias with low Variance
- High training error – ‘high Bias’
- testing error almost equal to training error
- Stable model – ‘Low Variance’
- The coefficients of the model doesn’t change with small changes in the data

### The Bias-Variance Decomposition

Y=f(X)+ϵ Var(ϵ)=σ2 SquaredError=E[(Y−f^(x0))2|X=x0]

= σ2+[Ef^(x0)−f(x0)]2+E[f^(x0)−Ef^(x0)]2

= σ2+(Bias)2(f^(x0))+Var(f^(x0))

**Overall Model Squared Error = Irreducible Error + \(Bias^2\) + Variance**

### Bias-Variance Decomposition

- Overall error is made by bias and variance together
- High bias low variance, Low bias and high variance, both are bad for the overall accuracy of the model
- A good model need to have low bias and low variance or at least an optimal where both of them are jointly low
- How to choose such optimal model. How to choose that optimal model complexity

### Choosing optimal model-Bias Variance Tradeoff

#### Bias Variance Tradeoff

### Test and Training Error

### Choosing Optimal Model

- Unfortunately
- There is no scientific method of choosing optimal model complexity that gives minimum test error.
- Training error is not a good estimate of the test error.
- There is always bias-variance tradeoff in choosing the appropriate complexity of the model.
- We can use cross validation methods, boot strapping and bagging to choose the optimal and consistent model.