# laplace¶

• Available in: Naïve-Bayes
• Hyperparameter: yes

## Description¶

This option specifies a value for the Laplace smoothing factor, which sets the conditional probability of a predictor. If the Laplace smoothing parameter is disabled (laplace = 0), then Naive Bayes will predict a probability of 0 for any row in the test set that contains a previously unseen categorical level. However, if the Laplace smoothing parameter is used (e.g. laplace = 1), then the model can make predictions for rows that include previously unseen categorical level.

Laplace smoothing adjusts the maximum likelihood estimates by adding 1 to the numerator and $$k$$ to the denominator to allow for new categorical levels in the training set:

$$\phi_{j|y=1}= \frac{\Sigma_{i=1}^m 1(x_{j}^{(i)} \ = \ 1 \ \bigcap y^{(i)} \ = \ 1) \ + \ 1}{\Sigma_{i=1}^{m}1(y^{(i)} \ = \ 1) \ + \ k}$$

$$\phi_{j|y=0}= \frac{\Sigma_{i=1}^m 1(x_{j}^{(i)} \ = \ 1 \ \bigcap y^{(i)} \ = \ 0) \ + \ 1}{\Sigma_{i \ = \ 1}^{m}1(y^{(i)} \ = \ 0) \ + \ k}$$

$$x^{(i)}$$ represents features, $$y^{(i)}$$ represents the response column, and $$k$$ represents the addition of each new categorical level. ($$k$$ functions to balance the added 1 in the numerator.)

Laplace smoothing should be used with care; it is generally intended to allow for predictions in rare events. As prediction data becomes increasingly distinct from training data, new models should be trained when possible to account for a broader set of possible feature values.

This value must be >=0 and defaults to 0.

## Example¶

library(h2o)
h2o.init()

# import the cars dataset:
prostate <- h2o.importFile("http://s3.amazonaws.com/h2o-public-test-data/smalldata/prostate/prostate.csv.zip")

# Converting CAPSULE, RACE, DCAPS, and DPROS to categorical
prostate$CAPSULE <- as.factor(prostate$CAPSULE)
prostate$RACE <- as.factor(prostate$RACE)
prostate$DCAPS <- as.factor(prostate$DCAPS)
prostate$DPROS <- as.factor(prostate$DPROS)

# Compare with Naive Bayes when x = 3:9, y = 2, and use laplace smoothing
prostate.nb <- h2o.naiveBayes(x = 3:9, y = 2, training_frame = prostate, laplace = 1)
print(prostate.nb)

# Predict on training data
prostate.pred <- predict(prostate.nb, prostate)

import h2o
h2o.init()
from h2o.estimators.naive_bayes import H2ONaiveBayesEstimator

# import prostate dataset:
prostate = h2o.import_file("http://s3.amazonaws.com/h2o-public-test-data/smalldata/prostate/prostate.csv.zip")

# Converting CAPSULE, RACE, DCAPS, and DPROS to categorical, and set the response column
prostate['CAPSULE'] = prostate['CAPSULE'].asfactor()
prostate['RACE'] = prostate['RACE'].asfactor()
prostate['DCAPS'] = prostate['DCAPS'].asfactor()
prostate['DPROS'] = prostate['DPROS'].asfactor()
response_col = 'CAPSULE'

# Compare with Naive Bayes when x = 3:9, y = 2, and use laplace smoothing
prostate_nb = H2ONaiveBayesEstimator(laplace = 1)
prostate_nb.train(x=list(range(3,9)), y=response_col, training_frame=prostate)
prostate_nb.show()

# Predict on training data
prostate_pred = prostate_nb.predict(prostate)