learn_params: Learn the parameters of a Bayesian network structure.

Description

Learn parameters with maximum likelihood or Bayesian estimation, the weighting attributes to alleviate naive bayes' independence assumption (WANBIA), attribute weighted naive Bayes (AWNB), or the model averaged naive Bayes (MANB) methods. Returns a bnc_bn.

Usage

lp(
  x,
  dataset,
  smooth,
  awnb_trees = NULL,
  awnb_bootstrap = NULL,
  manb_prior = NULL,
  wanbia = NULL
)

Value

A bnc_bn object.

Arguments

x: The bnc_dag object. The Bayesian network classifier structure.
dataset: The data frame from which to learn network parameters.
smooth: A numeric. The smoothing value ($\alpha$) for Bayesian parameter estimation. Nonnegative.
awnb_trees: An integer. The number ($M$) of bootstrap samples to generate.
awnb_bootstrap: A numeric. The size of the bootstrap subsample, relative to the size of dataset (given in [0,1]).
manb_prior: A numeric. The prior probability for an arc between the class and any feature.
wanbia: A logical. If TRUE, WANBIA feature weighting is performed.

Details

lp learns the parameters of each local distribution $\theta_{ijk} = P(X_i = k \mid \mathbf{Pa}(X_i) = j)$ as $$\theta_{ijk} = \frac{N_{ijk} + \alpha}{N_{ ij \cdot } + r_i \alpha},$$ where $N_{ijk}$ is the number of instances in dataset in which $X_i = k$ and $\mathbf{Pa}(X_i) = j$, $N_{ ij \cdot} = \sum_{k=1}^{r_i} N_{ijk}$, $r_i$ is the cardinality of $X_i$, and all hyperparameters of the Dirichlet prior equal to $\alpha$. $\alpha = 0$ corresponds to maximum likelihood estimation. Returns a uniform distribution when $N_{ i j \cdot } + r_i \alpha = 0$. With partially observed data, the above amounts to available case analysis.

WANBIA learns a unique exponent 'weight' per feature. They are computed by optimizing conditional log-likelihood, and are bounded with all $w_i \in [0, 1]$. For WANBIA estimates, set wanbia to TRUE.

In order to get the AWNB parameter estimate, provide either the awnb_bootstrap and/or the awnb_trees argument. The estimate is: $$\theta_{ijk}^{AWNB} = \frac{\theta_{ijk}^{w_i}}{\sum_{k=1}^{r_i} \theta_{ijk}^{w_i}},$$ while the weights $w_i$ are computed as $$w_i = \frac{1}{M}\sum_{t=1}^M \sqrt{\frac{1}{d_{ti}}},$$ where $M$ is the number of bootstrap samples from dataset and $d_{ti}$ the minimum testing depth of $X_i$ in an unpruned classification tree learned from the $t$-th subsample ($d_{ti} = 0$ if $X_i$ is omitted from $t$-th tree).

The MANB parameters correspond to Bayesian model averaging over the naive Bayes models obtained from all $2^n$ subsets over the $n$ features. To get MANB parameters, provide the manb_prior argument.

References

Hall M (2004). A decision tree-based attribute weighting filter for naive Bayes. Knowledge-based Systems, 20(2), 120-126.

Dash D and Cooper GF (2002). Exact model averaging with naive Bayesian classifiers. 19th International Conference on Machine Learning (ICML-2002), 91-98.

Pigott T D (2001) A review of methods for missing data. Educational research and evaluation, 7(4), 353-383.

Examples

Run this code

data(car)
nb <- nb('class', car)
# Maximum likelihood estimation
mle <- lp(nb, car, smooth = 0)
# Bayesian estimaion
bayes <- lp(nb, car, smooth = 0.5)
# MANB
manb <- lp(nb, car, smooth = 0.5, manb_prior = 0.5)
# AWNB
awnb <- lp(nb, car, smooth = 0.5, awnb_trees = 10)

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