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# Naive Bayes Classifier From Scratch

Naive bayes is simple classifier known for doing well when only a small number of observations is available. In this tutorial we will create a gaussian naive bayes classifier from scratch and use it to predict the class of a previously unseen data point. This tutorial is based on an example on Wikipedia’s naive bayes classifier page, I have implemented it in Python and tweaked some notation to improve explanation.

## Preliminaries

```
import pandas as pd
import numpy as np
```

## Create Data

Our dataset is contains data on eight individuals. We will use the dataset to construct a classifier that takes in the height, weight, and foot size of an individual and outputs a prediction for their gender.

```
/* Create an empty dataframe */
data = pd.DataFrame()
/* Create our target variable */
data['Gender'] = ['male','male','male','male','female','female','female','female']
/* Create our feature variables */
data['Height'] = [6,5.92,5.58,5.92,5,5.5,5.42,5.75]
data['Weight'] = [180,190,170,165,100,150,130,150]
data['Foot_Size'] = [12,11,12,10,6,8,7,9]
/* View the data */
data
```

Gender | Height | Weight | Foot_Size | |
---|---|---|---|---|

0 | male | 6.00 | 180 | 12 |

1 | male | 5.92 | 190 | 11 |

2 | male | 5.58 | 170 | 12 |

3 | male | 5.92 | 165 | 10 |

4 | female | 5.00 | 100 | 6 |

5 | female | 5.50 | 150 | 8 |

6 | female | 5.42 | 130 | 7 |

7 | female | 5.75 | 150 | 9 |

The dataset above is used to construct our classifier. Below we will create a new person for whom we know their feature values but not their gender. Our goal is to predict their gender.

```
/* Create an empty dataframe */
person = pd.DataFrame()
/* Create some feature values for this single row */
person['Height'] = [6]
person['Weight'] = [130]
person['Foot_Size'] = [8]
/* View the data */
person
```

Height | Weight | Foot_Size | |
---|---|---|---|

0 | 6 | 130 | 8 |

## Calculate Priors

Priors can be either constants or probability distributions. In our example, this is simply the probability of being a gender. Calculating this is simple:

```
/* Number of males */
n_male = data['Gender'][data['Gender'] == 'male'].count()
/* Number of males */
n_female = data['Gender'][data['Gender'] == 'female'].count()
/* Total rows */
total_ppl = data['Gender'].count()
```

```
/* Number of males divided by the total rows */
P_male = n_male/total_ppl
/* Number of females divided by the total rows */
P_female = n_female/total_ppl
```

## Calculate Likelihood

This means that for each class (e.g. female) and feature (e.g. height) combination we need to calculate the variance and mean value from the data. Pandas makes this easy:

```
/* Group the data by gender and calculate the means of each feature */
data_means = data.groupby('Gender').mean()
/* View the values */
data_means
```

Height | Weight | Foot_Size | |
---|---|---|---|

Gender | |||

female | 5.4175 | 132.50 | 7.50 |

male | 5.8550 | 176.25 | 11.25 |

```
/* Group the data by gender and calculate the variance of each feature */
data_variance = data.groupby('Gender').var()
/* View the values */
data_variance
```

Height | Weight | Foot_Size | |
---|---|---|---|

Gender | |||

female | 0.097225 | 558.333333 | 1.666667 |

male | 0.035033 | 122.916667 | 0.916667 |

Now we can create all the variables we need. The code below might look complex but all we are doing is creating a variable out of each cell in both of the tables above.

```
/* Means for male */
male_height_mean = data_means['Height'][data_variance.index == 'male'].values[0]
male_weight_mean = data_means['Weight'][data_variance.index == 'male'].values[0]
male_footsize_mean = data_means['Foot_Size'][data_variance.index == 'male'].values[0]
/* Variance for male */
male_height_variance = data_variance['Height'][data_variance.index == 'male'].values[0]
male_weight_variance = data_variance['Weight'][data_variance.index == 'male'].values[0]
male_footsize_variance = data_variance['Foot_Size'][data_variance.index == 'male'].values[0]
/* Means for female */
female_height_mean = data_means['Height'][data_variance.index == 'female'].values[0]
female_weight_mean = data_means['Weight'][data_variance.index == 'female'].values[0]
female_footsize_mean = data_means['Foot_Size'][data_variance.index == 'female'].values[0]
/* Variance for female */
female_height_variance = data_variance['Height'][data_variance.index == 'female'].values[0]
female_weight_variance = data_variance['Weight'][data_variance.index == 'female'].values[0]
female_footsize_variance = data_variance['Foot_Size'][data_variance.index == 'female'].values[0]
```

```
/* Create a function that calculates p(x | y): */
def p_x_given_y(x, mean_y, variance_y):
/* Input the arguments into a probability density function */
p = 1/(np.sqrt(2*np.pi*variance_y)) * np.exp((-(x-mean_y)**2)/(2*variance_y))
/* return p */
return p
```

## Apply Bayes Classifier To New Data Point

To do this, we just need to plug in the values of the unclassified person (height = 6), the variables of the dataset (e.g. mean of female height), and the function (`p_x_given_y`

) we made above:

```
/* Numerator of the posterior if the unclassified observation is a male */
P_male *
p_x_given_y(person['Height'][0], male_height_mean, male_height_variance) *
p_x_given_y(person['Weight'][0], male_weight_mean, male_weight_variance) *
p_x_given_y(person['Foot_Size'][0], male_footsize_mean, male_footsize_variance)
```

```
6.197071843878078e-09
```

```
/* Numerator of the posterior if the unclassified observation is a female */
P_female *
p_x_given_y(person['Height'][0], female_height_mean, female_height_variance) *
p_x_given_y(person['Weight'][0], female_weight_mean, female_weight_variance) *
p_x_given_y(person['Foot_Size'][0], female_footsize_mean, female_footsize_variance)
```

```
0.0005377909183630018
```

Because the numerator of the posterior for female is greater than male, then we predict that the person is female.

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