A very simple safe-Bayesian random forest

A Very Simple Safe-Bayesian Random Forest

Novi Quadrianto and Zoubin Ghahramani

Abstract—Random forests works by averaging several predictions of de-correlated trees. We show a conceptually radical approach to generate a random forest: random sampling of many trees from a prior distribution, and subsequently performing a weighted ensemble of predictive probabilities. Our approach uses priors that allow sampling of decision trees even before looking at the data, and a power likelihood that explores the space spanned by combination of decision trees. While each tree performs Bayesian inference to compute its predictions, our aggregation procedure uses the power likelihood rather than the likelihood and is therefore strictly speaking not Bayesian. Nonetheless, we refer to it as a Bayesian random forest but with a built-in safety. The safeness comes as it has good predictive performance even if the underlying probabilistic model is wrong. We demonstrate empirically that our Safe-Bayesian random forest outperforms MCMC or SMC based Bayesian decision trees in term of speed and accuracy, and achieves competitive performance to entropy or Gini optimised random forest, yet is very simple to construct. Index Terms—Bayesian methods, random forest, decision trees

F

1

INTRODUCTION

Decision trees [1] represent a classical induction method that is widely used. This is because decision trees exhibit many appealing properties: they are sim-ple to understand and interpret, applicable for both classification and regression tasks, and offer good predictive performance. Although defining a global objective to optimally learn decision trees is funda-mentally hard, there are several approaches to learn trees based on Gini impurity [1] and information-theoretic considerations [2], [3].

Breiman [4] used an ensemble of unpruned Gini-optimiseddecision trees to improve the performance of learning. This random forest framework has become a very popular and powerful tool with applications ranging from machine learning, computer vision, computer graphics, to medical image analysis, among others [5], [6]. For a recent comprehensive survey about random forests, refer to [7]. The randomness is introduced during training phase of the trees via: a) random sampling of the training dataset [4], and b) partitioning the data space with only randomised subsets of data features [8]. We will call trees trained with this procedure as randomly trained trees. An important ingredient of random forest is the compo-sition of trees that are randomly different from one another. This results in de-correlated individual tree predictions and, therefore, in an improved generalisa-tion. The notion of randomness helps the model to be robust with respect to noisy data.

Reflecting on how a random forest is built from • N. Quadrianto is with SMiLe CLiNiC, Department of Informatics,

University of Sussex, UK. E-mail: [email protected] • Z. Ghahramani is with Department of Engineering, University of

Cambridge, Cambridge, UK. Email: [email protected] Manuscript received May 2013.

randomly trained trees, we ask the following natural question: can we simplysample many treesfrom some prior distributions, and perform an ensembleof those randomly sampled trees? The Bayesian framework is a principled way to achieve this. We note that the usage of Bayesian statistics for learning decision trees has a long history of successful methods, among others, [9], [10], [11], [12], [13], [14].

Bayesian approaches start by defining a prior dis-tribution on the space of decision trees. Subsequently, a likelihood function, for example Gaussian (for re-gression tasks) or Bernoulli (for classification tasks), is evaluated within each data block induced by the decision tree. The likelihood describes the conditional distribution of the output data given input data falling into the corresponding block. Inall previous models using Bayesian statistics, the prior distribution of the decision tree is defined conditionally on the given input data. Sophisticated Markov Chain Monte Carlo (MCMC) methods [11] or sequential Monte Carlo [14] is then used to approximate intractable posterior computations. In this paper, we will show a novel usage of a prior that allows us to sample decision trees even before looking at the data.

As more data arrives, Bayesian decision trees will put a mass to a single tree. This is a desired be-haviour of Bayesian model averaging [15]. We are instead interested to explore ensemble of trees in the sense of model combination [16]. We achieve this by borrowing the concept of power likelihood [17], [18]. By utilising a data independent prior coupled with the power likelihood, we deliver fast yet safe Bayesian random forests that exceed the performance of Bayesian decision trees and are competitive with random forests and SVMs. We use the notion of Safe-Bayesian from [19] in a reference to procedures based on a so-calledβ-Bayesian posterior (power likelihood

several binary and multi-class classification problems. Finally, Section 6 concludes the paper.

2

THE

MODEL

Here we describe our model of Safe-Bayesian ran-dom forest. Assume that we are given input X =

{x1, . . . ,xN} with xi ∈ X = RD and output Y =

{y1, . . . , yN} with yi ∈ Y = {1, . . . , C}. The goal is

to infer a function F : X → Y that maps inputs to outputs. In this paper, we only focus on the clas-sification tasks. We are interested in the case where the latent function is anensembleof predictors, where each predictor partitions the input data space into axis-aligned blocks. Note that the partitioning can be represented graphically as adecision tree.

To elaborate the form of our Safe-Bayesian random forest model, we begin by establishing notation and model for a single tree. LetT be a rooted and strictly binary tree; a tree that has a single root node, internal nodes with exactly two outgoing edges (also known as children), and terminal (leaf) nodes. Each node of the tree v ∈ T corresponds to a block Bv ⊂ RD,

and has a splitting rule applied onBv. The rule splits

the block Bv into two halves: Bvl and Bvr associated

with the left and right child nodes, respectively. The splitting rule is characterised by the dimension of the split, κv ∈ {1, . . . , D}, and the location of the

split (also known as cut value), τv. Thus, we have

Bvl =Bv∩ {x∈RD :xκv ≤τv}, andBvr=Bv∩ {x∈

RD : xκv > τv}. Note that the root node has a

block B(root) = _RD_{, and terminal nodes do not}

have splitting rules associated to them. Each terminal node however describes conditional distribution of the labels given all input data falling into that node. We will denote a decision tree with tree structure, split dimensions, and split values as T = {T, κ, τ}. Our Bayesian analysis proceeds by specifying a prior probability distribution on the space of decision trees, that isp(T).

2.1 Tree Priorp(T)

In defining a prior on decision trees, we note that the dimension of the split κ, and the location of the split τ serve as an index for the splitting rule of each T. It is then more convenient to exploit the relationship that p(T, κ, τ) = p(κ, τ|T)p(T), and specify p(κ, τ|T)

and p(T) separately. We discuss the specifications of p(T)and p(κ, τ|T) in the subsequent sections.

A generative process of trees structurep(T)

codes are generated in depth-first order starting from

1 at the root node. Refer to Figure 1 for examples of tree coding, and to Figure 2 for samples of different trees structures.

A prior on cut dimensions and valuesp(κ, τ|T)

Given the structure of the tree, we will then need to specify the splitting rules for each of the internal nodes of the tree. For internal nodev, the dimension κv and location τv of the cut are chosen uniformly

from{1, . . . , D}and [0,1], respectively. That is κv ∼ U({1, . . . , D}) (1)

τv ∼ U([0,1]). (2)

We have assumed that the input features at each dimension lie in[0,1]. We will describe in Section 5 on how to enforce this.

Note that our prior on the space of decision trees defined above isindependentof the data X. Once we generate the structure of the tree, and fill up the in-ternal nodes with splitting rules, we have completely specified the decision tree. This is in contrast to exist-ing work on Bayesian decision trees, for example [11], [13], [14]. Under the Chipman et al. model that is the model used in these existing works, the probability that a nodevis split into two children isαs/(1 +|v|)βs

with the parameters αs ∈ (0,1) and βs ∈ [0,∞)

governing the shape of the resulting tree, and |v| is the depth of the node. For largerαsandβsthe typical

trees are larger, while deeperv is in the tree the less likely it will be split. This is in fact a data-independent tree structure prior. However, as noted in [14], given the data at a particular internal node, the split di-mension and location are sampled from the uniform distribution over the available data features and data values. Therefore, for all Bayesian decision trees, the probability of tree growing, and the distribution of the split dimension and location, depend on X so that every node in the tree will contain at least one data point. [14] put forward the theoretical need for data independent priors over the space of decision trees that include tree structures, cut dimensions and values to make the model coherent with respect to changing dataset sizes. On the practical side, the effect of data independent prior enables us to use exactly the same trees structures in all our experiments. We have just defined a way to partition the data, the next required thing is to define a conditional distribution of the labels given input data falling into corresponding blocks. This is described in the next section.

Fig. 1:Coding of a rooted and strictly binary tree. Internal nodes are denoted with circles and terminal nodes with squares. Each internal node has exactly two outgoing edges. The code for an internal node is1 and0 is for a terminal node. Codes are generated in depth-first order starting from1 at the root node.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 height = 2 1 2 3 4 5

Tree Samples withλ= 0.3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 height = 3 1 2 3 4 5 6 7 8 9

Tree Samples withλ= 0.3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 height = 4 1 2 3 4 5 6 7 8 9 10 11

Tree Samples withλ= 0.3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 height = 1 1 2 3

Tree Samples withλ= 0.3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 height = 3 1 2 3 4 5 6 7

Tree Samples withλ= 0.4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 height = 4 1 2 3 4 5 6 7 8 9 10 11 Tree Samples withλ= 0.4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 height = 9 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

Tree Samples withλ= 0.4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 height = 3 1 2 3 4 5 6 7

Tree Samples withλ= 0.4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 height = 14 1 2 3 4 5 6 7 8 9 10 11 12 1314 15 16 17 18 19 20 21 22 23 24 2526 27 2829303132 33 34 35 36 37 3839 40 41 4243 44 4546 47 48 49 50 515253545556 57 58 59 60 6162 63 64 65 66 67 68 6970 71 72 737475 Tree Samples withλ= 0.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 height = 8 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 Tree Samples withλ= 0.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 height = 3 1 2 3 4 5 6 7 8 9 Tree Samples withλ= 0.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 h ei ght = 161 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 8081828384 8586 87 888990919293949596979899 100 101 102 103 104 105 106 107119129128127108124123122120121118117109110111112113114115116143137142141140139138126136135134133132131130125144 145 146 147 148 149 150 151152 153 154 155 156 157 158 159171179178177176175174173172170160169168167166165164163162161180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199200201202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246247248249250251252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 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Fig. 2:Sampling trees structures from the prior atλ∈ {0.3,0.4,0.5}. Once we generate the structures, and fill

up the internal nodes with splitting rules, we will completely specify decision trees.

2.2 Dirichlet-Multinomial Tree Likelihood

p(Y|X,T)

To specify a predictive model within each block, we follow the Bayesian model of [9], [11]. We will repeat them here for the self consistency of the paper. For an input vector x that falls into the ν-th terminal node, we assign a parametric family f indexed by

θν, f(y|θν), that models the conditional distribution

y|x. For a classification problem where the output y belongs to one of C classes {1, . . . , C}, the natu-ral choice for f(y|θν) is a multinomial distribution

with conjugate Dirichlet prior on terminal node pa-rameters. Given a block partitioning and a predic-tive model within each partition, we assume that y values within a terminal node are i.i.d. given θν

and y values across terminal nodes are independent.

Thus the form of class conditional distribution at a particular node v for a specific decision tree T

is f(YNv|XNv,θ,T) = Qi∈Nv

QC

c=1θ

I[yi=c ]

c , where

θ = (θ1, . . . , θC), θ>1 = 1, and θc ≥ 0 for all

c∈ {1, . . . , C}. As well, for simplicity of presentation, we have denoted as Nv the set of indices of the

input vectors that fall into partition blockBv, that is

Nv = {i : xi ∈ Bv}. Thus, XNv and YNv are input

vectors and their labels in block Bv, respectively. In

the above, we make use of Iverson’s bracket notation:

I[P] = 1 for the condition P is true and it is 0

otherwise. We put a conjugate Dirichlet prior on θ

with parameterα= (α1, . . . , αC),αc>0, and assume

conditional independence of parameters across termi-nal nodes. Thus, we have the following likelihood: p(Y|X,T) = R Q_v∈Ω

Fig. 3:A visualisation of4trees with the highest (top row) and4trees with the lowest (bottom row) importance weights for a 3-class problem ‘iris’ dataset (best viewed in colour). The terminal nodes are colour encoded according to the label proportions of data points falling into the terminal nodes. The three colours, red, green, and blue, correspond to the three class labels. More homogeneous labels in the terminal node correspond to purer colour encoding. The terminal nodes with mixed class labels produce colour in RGB space. Some of the terminal nodes are coloured black as no data point falls into those particular nodes. The black internal nodes will not contribute to the importance weight of the tree as the associated terms in Equation (3) are1. Essentially, this allows us to have an in-built pruning of the trees structures.

have used ΩT to denote terminal nodes of decision

treeT. The above integration can be done analytically using standard properties of the Dirichlet distribu-tions (see for example [21]), we have

p(Y|X,T) = Y v∈ΩT Γ(PC c=1αc)Q C c=1Γ(mv,c+αc) QC c=1Γ(αc)Γ( PC c=1(mv,c+αc)) . (3) In the above, the symbol mv,c denotes the number

of labels yi = c among those i ∈ Nv. As noted

by [11], for a given tree, the value p(Y|X,T) will be larger whenever more homogeneous values of y are assigned to the terminal nodes. Further, a simple choice of α is the vector (1, . . . ,1) for which the Dirichlet prior is the uniform, unless further prior information is available.

2.3 Tree Posteriorp(T |X, Y)

We can compute the posterior distribution of the latent decision tree T given the observed data, up to a normalisation constant, by combining the like-lihood in (3) and the tree generating prior described in Section 2.1. We have the following: p(T |X, Y) ∝

p(Y|X,T)×p(T).Like all other Bayesian tree models, exact computation of the posterior is computationally intractable. To overcome this, several Markov Chain Monte Carlo methods have been proposed, particu-larly a Metropolis-Hastings (M-H) [11], [13], and a Sequential Monte Carlo (SMC) [14]. In this paper, we choose to use a very simple method based on importance sampling [22].

3

P

C

P

T

D

For a previously unseen test point x∗ ∈ RD, the

pre-dictive distribution over the latent output y∗ can be computed as follows: p(y∗=c|x∗, X, Y) = Z p(y∗=c|T,x∗, X, Y)p(T |x∗, X, Y)dT = Z p(y∗=c|T,x∗, X, Y)p(T |X, Y) p(T) p(T)dT (4) ≈X k p(y∗=c|T(k)_,x∗_{, X, Y)}p(T(k)|X, Y) p(T(k)₎ (5) ≈X k p(y∗=c|T(k)_,x∗_{, X, Y)p(Y|X,T}(k)₎ | {z } wk , (6) where p(y∗=c|T(k)_,x∗_{, X, Y) =} E[θc|T(k),x∗, X, Y] = mc+αc PC c=1(mc+αc) . (7) The importance weight wk in Equations (6) and

(3) will be larger for a decision tree T with more homogeneous values ofy. Equation (4) of the above is an importance sampling method with a prior pro-posal distributionp(T). We use importance sampling because of the following reasons: it is simple to im-plement and importantly it exploits the strength of our usage of data independent tree priors. However, we note that from a Bayesian perspective, importance sampling using the prior as the proposal is known

Algorithm 1 Training Phase of Simple Bayesian Random Forest

Inputnumber of trees K, parameter valueλ

GenerateK tree structure from the prior distributionp(T)

Inputnumber of dimensions D

Fill upthe internal nodes with splitting rules according to:κv∼ U({1, . . . , D}), andτv∼ U([0,1])

Inputparameter vectorα and paired input-output training data{(x1, y1), . . . ,(xN, yN)} ⊂RD× Y

fork= 1toK do

Computethe importance weight wk of the decision tree according top(Y|X, κ, τ)

end for

Generateoperating curve based on power likelihood [18] by varying the value ofβfrom1/N,2/N, . . . ,1

Return many randomly sampled decision trees {T(k)_{, κ}(k)_{, τ}(k)_}K

k=1 with associated importance weights

{wk}K

k=1 and a forest operating curve

to work not so well. Nevertheless, the prior proposal offers a better predictive accuracy versus computa-tional time tradeoff than any potentially more optimal proposal. The same observation is also made in [14]. And, Equation (7) is conditioned on the partition v that x∗ is in. To get a point estimate yˆ∗ from the predictive distributionp(y∗=c|x∗, X, Y), we perform

ˆ

y∗= arg maxcp(y∗=c|x∗, X, Y).

4

MODEL

COMBINATION VIA

POWER

LIKE-LIHOOD

It is important to note that in the limit of N → ∞, Bayesian model averaging (BMA) in (6) will put a mass to a single decision tree [15], [16]. Instead we want to explore the space spanned bycombination of several decision trees. We achieve this model combi-nation by borrowing the concept of power likelihood [17], [18], that is raising the likelihood function to a power between 0 and 1. Historically a motivation of introducing the power likelihood is to guarantee posterior consistency with simply a trivial Kullback-Leibler neighbourhoods condition on the prior [17], [18]. There are other reasons for undertaking inference with power likelihood. [23] use power likelihood to combine historical data from similar studies in con-structing the prior distributions for regression mod-els. It has also been used to do model selection via marginal likelihood as in [24]. Recently, [25] explore related ideas of tempering the distributions by power-ing them up to speed up the convergence of MCMC. Empirically we will also show that model combi-nation decision using the power likelihood greatly outperforms Bayesian model averaging of trees.

Specifically, when averaging, we use the marginal likelihood to the power β < 1. A sensible choice would be to use β =m/N where m is the ‘effective sample size’ used in the averaging. By dividing by the size of the dataset N, we ensure that we do not get domination by a single ensemble member as the size of the training set increases. For β = 1 we will recover BMA.

To summarise, our Bayesian random forest method involves two phases: training and test phases. The

trainingphase involves sampling trees structures, and their associated splitting rules from the prior distri-butions. This process can be performed offline. Once data arrive, we compute the importance weights via (marginal) likelihood in Equation (3). Subsequently, based on the power likelihood, we generate the oper-ating curve of Bayesian random forest by varying the value of β from 1/N,2/N, . . . , N/N [18] (see Figure 4 for examples of such curves). We generate the curve based on training data. We can then choose the operating value ofβ based on a small validation set. For this work, we choose the value of β based on the training set. In thetestphase, given a previously unseen data point, each decision tree hierarchically applies a number of splitting rules. Starting at the root, each internal node applies its associated split function to the new data point, and this process is repeated until the data point reaches a terminal node. At the terminal node, we compute the class predictive distribution using Equation (7). We repeat this procedure for all the trees in the forest, and the final forest prediction will be a weighted combination of individual trees prediction. Refer to Algorithm 1 for a training phase pseudo-code, and Algorithm 2 for a test phase.

5

EXPERIMENTS

Datasets We use the following nine UCI1 _datasets:

australian, breast-cancer, diabetes, heart, ionosphere, german.numer, iris, wine, and svmguide2, and two high dimensional image data from Israeli Image2

and Animals with Attributes (AwA)3 datasets. We use4,097 dimensional Fisher [26] representations for Israeli Image, and 2,000 dimensional SURF [27] rep-resentations for AwA.

Algorithms We compare the performance of our

Bayesian random forest with markov chain monte carlo (MCMC) and sequential monte carlo (SMC) based Bayesian decision trees4_{, random forest, kNN,}

1. http://archive.ics.uci.edu/ml/

2. http://people.cs.umass.edu/∼_{ronb/image clustering.html} 3. http://attributes.kyb.tuebingen.mpg.de

k=1 k

end for

Return forest predictive distributionp(y∗₁=c|x∗₁, X, Y), . . . , p(y_N∗0 =c|x∗_N0, X, Y)for allc={1, . . . , C}

and SVM. For the last three methods we use scikit-learn implementations5_{. For MCMC and SMC, we fix}

the hyperparameters to αs = 0.95 and βs = 0.5 as

suggested in [14]. We set the number of iterations to be 100,000 for MCMC. We consider two types of proposal distribution for SMC, that is prior proposal

(SMC: prior) and optimal proposal (SMC: post.).

ForRF, we set minimum number of samples required to split an internal node to be 1. We investigate two widely used criteria to measure the quality of the split: Gini impurity (RF: Gini) and entropy (RF: Ent.). We set the number of features to consider when looking for the best split to be 0.5D where D is the dimension of the data. ForkNN, we set the number of nearest neighbours to be3, and forSVMwe use a linear kernel with cross-validated regularisation parameter. For our S-Bayes RF, the parameterλ to be0.475 6_,

and we use a symmetric uniform Dirichlet prior. Our potential cut dimensions are chosen uniformly from

{1, . . . , D}. For our S-Bayes RF, we use exactly the same trees structures for all11datasets7_{. We use1,000}

trees for random forest, Bayesian random forest, and for number of particles in SMC.

PreprocessingWe use the probability integral

trans-form (PIT) to transtrans-form the input features at each dimension d ∈ {1, . . . , D} to lie in [0,1]. PIT allows us to generate uniform random variables out of any continuously distributed random variables by mak-ing use of the empirical estimate of the cumulative distribution function underlying our data. This trans-formation makes the distance between adjacent data points to be 1/(N+ 1), thus has the added potential to reduce the effect of noisy data points (outliers).

Results We use 80% of the available data to be

a training set and the remaining 20% as a test set. We assess the performance on the test set. The whole experiment was then repeated5times to obtain confi-dence bounds. In Figure 3, we provide a visualisation of 4trees with the highest importance weights wk in

Equation (6),(3) and 4 trees with the lowest impor-tance weights for a 3-class problem ‘iris’ dataset. We encode the terminal nodes with colour according to

5. http://scikit-learn.org/stable/ [28]

6. We recommend the use ofλ≤0.5for efficiency reason. 7. We will provide the tree structures and the source code.

the label proportions of data points falling into each of the terminal nodes. The more label homogeneous is the terminal node, the more red or green or blue is the colour encoding. Note also that some of the terminal nodes are coloured black due to no data point falling into those particular nodes (this is a property unique to our usage of data independent prior on decision trees). However, these black internal nodes will not contribute to the importance weight of the tree as the associated terms in Equation (3) equal to 1. We gen-erate, in Figure 4, the operating curve of our method by varying the value ofβ from 1/N,2/N, . . . ,1. It is clear from the curves that Bayesian model averaging (the point wherem/N = 1) is suboptimal, except for small ‘iris’ dataset. For all datasets, effective sample size m = 5 seems to be sufficient. The experimental results for UCI datasets are summarised in Table 1. Inall but twodatasets (‘diabetes’ and ‘iris’), proposed S-Bayes RF method outperforms Bayesian decision trees (MCMC and SMC variants), sometimes by a large margin. We credit this to our usage of power likelihood that does not assume a single decision tree as a good hypothesis. This fact is corroborated in Table 2 where our weighted ensemble of trees always outperforms a single tree with the highest importance weight and BMA (whenβ= 1). Even though Bayesian random forest consists of randomly sampled trees, it is competitive to the random forest, where each tree is built with respect to some goodness criteria, in this case Gini impurity and entropy. However, Table 3 shows the drawback of our usage of power likelihood that it does not produce a well-calibrated predictive probabilities. We also assess the sensitivity of S-Bayes RF with the choice of number of trees and the splitting probabilityλ. The results for4representative datasets are visualised in Figure 5. For completeness, we also provide results of Bayesian Additive Regression Trees (BART) [13] using BayesTree package for R in Table 1–second to last column. We note that BART can be thought of as a Bayesian version of boosted decision trees. BayesTree package uses a probit link function for handling a binary classification problem. Our re-sults are consistent with previous studies [29] that found calibrated boosted trees were the best learn-ing algorithm on several considered binary learnlearn-ing

T ABLE 1: Accuracy acr oss 5 random repeats. D : number of dimensions and N : number of observations, equal fr om each class. kNN : k-near est neighbours ( k = 3 ), SVM : Supp ort vector machine with linear kernels, RF: Gini : Random for est learned with Gini impurity , RF: Ent. : Random for est learne d with information gain, MCMC : Markov chain monte carlo of Bayesian decision tr ee, SMC: pr. : Sequential monte carlo of Bayesian decision tr ee with prior pr oposal, SMC: ps. : Sequential monte carlo of Bayesian decision tr ee with posterior pr oposal, BART : a Bayesian version of boosted decision tr ees, and S-Bayes RF (Ours) : our Safe-Bayesian random for est. D N kNN SVM RF: Gini RF: Ent. MCMC SMC: prior SMC: post. BART S-Bayes RF australian 14 618 83.04 ± 2.53 86.23 ± 0.89 86.23 ± 2.29 86.08 ± 2.82 86.09 ± 0.61 86.23 ± 1.02 86.23 ± 0.51 87.68 ± 0.88 87.39 ± 1.50 breast-cancer 10 517 98.39 ± 1.25 98.54 ± 1.15 98.10 ± 1.42 98.10 ± 1.42 96.93 ± 1.58 96.35 ± 2.42 96.49 ± 1.66 98.39 ± 1.40 97.95 ± 1.30 diabetes 8 574 72.59 ± 1.32 75.32 ± 4.68 74.80 ± 2.61 74.15 ± 3.13 74.42 ± 2.08 71.68 ± 1.63 73.76 ± 2.08 76.75 ± 5.08 74.28 ± 4.76 heart 13 270 81.11 ± 5.54 85.18 ± 2.62 82.96 ± 4.01 83.33 ± 4.34 80.37 ± 6.23 79.63 ± 5.86 81.48 ± 5.71 85.56 ± 3.56 87.41 ± 5.61 ionosphere 34 270 81.99 ± 3.33 81.99 ± 6.59 90.85 ± 3.58 91.14 ± 3.41 89.43 ± 3.13 88.85 ± 2.93 89.71 ± 3.41 92.00 ± 3.28 91.14 ± 2.11 german.numer 24 680 63.80 ± 5.81 72.09 ± 4.04 69.29 ± 3.47 68.99 ± 3.18 63.80 ± 4.80 63.59 ± 6.25 64.10 ± 5.97 71.80 ± 3.78 72.40 ± 2.88 iris 4 60 95.99 ± 5.33 98.00 ± 2.98 96.67 ± 3.33 96.67 ± 3.33 96.67 ± 3.33 97.33 ± 2.78 95.99 ± 3.65 NA 96.67 ± 3.33 wine 13 178 97.22 ± 1.76 97.22 ± 3.40 98.33 ± 2.48 98.33 ± 2.48 95.56 ± 1.52 86.67 ± 9.89 94.44 ± 1.96 NA 99.44 ± 1.24 svmguide2 20 159 60.61 ± 2.71 69.70 ± 7.72 69.09 ± 4.97 67.87 ± 7.90 63.64 ± 9.34 55.15 ± 11.81 62.42 ± 5.90 NA 67.27 ± 4.97 (a) australian (b) breast-cancer (c) diabetes (d) heart (e) ionosphere (f) german.numer (g) iris (h) wine (i) svmguide2 Fig. 4: Operating curve of our Bayesian random for est based on the concept of power likelihood. Accuracy ± STE.. The curves ar e generated based on training data. The point wher e m/ N = 1 corr esponds to Bayesian model averaging (BMA).

australian 86.23±0.79 86.23±0.89 87.39±1.50 breast-cancer 96.06±1.35 96.64±0.40 97.95±1.30 diabetes 71.42±2.53 70.39±3.45 74.28±4.76 heart 75.93±6.83 77.78±8.97 87.41±5.61 ionosphere 81.99±5.07 81.71±5.84 91.14±2.11 german.numer 63.20±5.71 63.70±6.02 72.40±2.88 iris 94.00±4.89 94.67±6.05 96.67±3.33 wine 86.67±5.93 88.33±4.56 99.44±1.24 svmguide2 51.51±5.42 53.94±8.41 67.27±4.97

TABLE 3: Log predictive probabilities of Markov

chain Monte Carlo of Bayesian decision tree (MCMC) and weighted ensemble of trees with β = 5/N

(S-Bayes RF). Log probability±STD. As expected, in

all cases MCMC provides better calibrated probabili-ties than S-Bayes RF with power likelihood.

MCMC S-Bayes RF australian -0.3330±0.0249 -0.5409±0.0043 breast-cancer -0.0999±0.0336 -0.3009±0.0129 diabetes -0.5277±0.0149 -0.6260±0.0061 heart -0.4464±0.0560 -0.5879±0.0053 ionosphere -0.3150±0.0605 -0.6097±0.0095 german.numer -0.5885±0.0281 -0.6759±0.0023 iris -0.2616±0.1284 -0.5473±0.0608 wine -0.2475±0.0581 -0.6625±0.0275 svmguide2 -0.8702±0.0858 -1.0283±0.0133

problems but random forests are close second. In 3

datasets, BayesTree outperforms S-Bayes RF by more than0.4%while only in2datasets it was the opposite. The results on high dimensional image data are summarised in Table 4. Our results are once again consistent with [29] that found it surprising that random forest variants perform well even for high dimensional data. For this high dimensional data, we use effective sample size of one. Among trees and forest variants, our S-Bayes RF is the fastest to train, while MCMC based Bayesian decision tree is the most time consuming. We also run the random forest variant where a random subset of candidate features is used plus cut values are drawn at random for each candidate feature and the best of these randomly-generated thresholds is picked as the cut value [30], [31]. SMC with optimal proposals does not finish in three days. Our S-Bayes RF is implemented in Mat-lab while others are in Python. Although in general S-Bayes RF is fast, getting a precise runtime com-parison is not straightforward since implementation languages differ. In terms of accuracy performance, Bayesian decision trees perform comparably to the baseline kNN (13.20±2.41) for Israeli, and perform rather poorly to kNN (36.53±2.69) for AwA. The random forest and our Bayes random forestoutperform

ers, machine learning, computer vision, computer graphics, and medical image analysis. It typically works by averaging several predictions of randomly trained trees. In this paper, we show a conceptually radical approach to generate a random forest based on Bayesian statistics: random sampling of many trees from a prior distribution, and subsequently perform-ing a weighted ensemble.

Unlike other Bayesian models of decision trees which require computationally intensive Markov Chain Monte Carlo procedures, our framework utilises a data independent tree prior that facilitates offline tree generation. This prior allows us to sample a collection of decision trees even before looking at the data. Furthermore with the use of power likelihood, our method is able to explore space spanned by com-bining decision trees. This is in contrast to Bayesian decision trees where in the infinite data limit will put a mass to a single tree.

Our experimental results are encouraging, our S-Bayes RF outperforms S-Bayesian decision trees in term of speed and predictive performance, and is compet-itive with state-of-the-art random forest algorithms on both speed and accuracy. In the future, we are interested in exploring the possibility of adapting the power likelihood exponentβ as new data points arrive using for example the method of [19].

A

The authors would like to thank two anonymous reviewers for constructive comments and Sara Wade, Wray L. Buntine, Viktoriia Sharmanska, Sebastian Nowozin, and James Lloyd for discussions.

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