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Density estimation Typical approaches Hardy Haar
Hardy Haar
Non-linear density estimation using a sparse Haar prior
Arthur Breitman
April 3, 2016
Arthur Breitman
Hardy Haar

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Table of contents
Density estimation
Problem statement
Applications
Typical approaches
Parametric density
Kernel density estimation
Hardy Haar
Principles
Recipe
Linear transforms
How to use for data mining
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Hardy Haar

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Problem statement
What is density estimation?
▶ Given an i.i.d sample xn
1 from unknown distribution P,
estimate P(x) for arbitrary x
▶ For instance P belongs to some parametric family, but non
Bayesian treatment possible
▶ Examples:
▶ Model P is a multivariate gaussian with unknown mean and
covariance.
▶ Kernel density estimation is non paremetric (but morally ∼ to
P as a uniform mixture of n distributions, ﬁt with maximum
likelihood)
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Hardy Haar

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Applications
Why is density estimation useful?
▶ Allows unsupervized learning by recovering the latent
parameters of a distribution describing the data
▶ But can also be used for supervized learning.
▶ Learning P(x, y) is more general than learning y = f (x).
▶ For instance, to minimize quadratic error, use
f (x) =
∫
y P(x, y) dx
∫
P(x, y) dx
▶ Knowledge of the full density permits the use of any∗ loss
function
∗
oﬀer void under fat tails
Arthur Breitman
Hardy Haar

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Applications
Mutual information
Of particular is the ability to compute “mutual information”.
Mutual information is the principled way to measure what is
loosely referred to as “correlation”
I(X; Y ) =
∫
Y
∫
X
p(x, y) log
(
p(x, y)
p(x) p(y)
)
dx dy
Measures the amount of information one variable gives us about
another.
Arthur Breitman
Hardy Haar

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Applications
Correlation doesn’t always capture this relation
In the case of a bivariate gaussian,
I = −
1
2
log
(
1 − ρ2
)
We can get a correlation equivalent by using
ˆρ =
√
1 − e−2I
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Bivariate normal
Assume that the data observed was drawn from a bivariate normal
distribution
▶ Latent parameters: mean and covariance matrix
▶ Unsupervized view: learn the relationship between two
random variables (mean, variance, correlation)
▶ Supervized view: equivalent to simple linear regression
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Parametric density
Bivariate normal density
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Hardy Haar

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Parametric density
Kernel density estimation is a non-parametric density estimator
deﬁned as
ˆfh(x) =
1
nh
n∑
i=1
K
(x − xi
h
)
▶ K is a non negative function that integrates to 1 and has
mean 0 (typically gaussian)
▶ h is the scale or bandwith.
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Gaussian kernel density estimation
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Bandwith selection
Picking h can be tricky
▶ h too small =⇒ overﬁt the data
▶ h too large =⇒ underﬁt the data
▶ There are rules of thumbs to pick h from variance of data and
number of points
▶ Can be picked by cross-validation
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Gaussian kernel density estimation
Under and over-ﬁtting with diﬀerent bandwiths (the correlation of
the kernel is estimated from the data)
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Hardy Haar

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Issues with kernel density estimation
Naive Kernel density estimation has several drawbacks
▶ Kernel covariance is ﬁxed for the entire space
▶ Does not adjust bandwith to local density
▶ No distributed representation =⇒ poor generalization
▶ Performs poorly in high dimensions
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Hardy Haar

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Adaptive kernel density estimation
One approach is to use a diﬀerent kernel for every point, varying
the scale based on local features.
Ballon estimators make the kernel width inversely proportional to
density at the test point
h =
k
(nP(x))1/D
Pointwise estimator try to vary the kernel at each sample point
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Hardy Haar

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Local bandwith choice
▶ If latent distribution has peaks, tradeoﬀ between accuracy
around the peaks and in regions of low density.
▶ This is reminiscent of time-frequency tradeoﬀs in fourier
analysis (hint: h is called bandwith)
▶ Suggests using wavelets which have good localization in time
and frequency
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Hardy Haar

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Principles
Introducing Haardy Haar
Hardy Haar attempts to address some of these shortcomings
▶ Full Bayesian treatment of density estimation
▶ (Somewhat) distributed representation
▶ Fast!
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Principles
Examples of Haardy Haar
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Principles
Goals
Coming up with a prior?
▶ In principle, any distribution whose support contains the
sample is potential.
▶ Some distributions are more likely than others, but why not
just take the empirical distribution?
▶ May work ﬁne for integration problems for instance
▶ Doesn’t help with regression or to understand the data
▶ There should be some sort of spatial coherence to the
distribution
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Principles
Sparse wavelet decomposition prior
To express the spatial coherence constraint, we take a L0 sparsity
prior over the coefficient of the decomposition of the PDF in a
suitable wavelet basis.
▶ This creates a coefficient ”budget” to describe the distribution
▶ Large scale wavelet describe coarse features of the distribution
▶ Sparse areas can be described with few coefficients
▶ Areas with a lot of sample points are described in more detail
▶ Closely adheres to the minimum description length principle
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Principles
Haar basis
The Haar wavelet is the simplest wavelet
Not very smooth, but no overlap between wavelets at the same
scale =⇒ tractability
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Principles
Model
▶ The minimum length principle suggests penalizing the log
likelihood of producing the sample with the number of non
zero coefficients.
▶ We can put a weight on the penalty, which will enforce more
or less sparsity
▶ We can “cheat” with an improper prior: use an infinite
number of coefficient, but favor models with many zeros
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Recipe
Sampling
To sample from this distribution over distributions, conditional on
the observed sample, we interpret the data as generated by a
recursive generative model.
As n is held ﬁxed, the number of datapoints in each orthant is
described by a multinomial distribution. We put a non-informative
dirichlet prior on the probabilities of each quadrant. This processus
is repeated for each orthant.
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Hardy Haar

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Recipe
Orthant tree
Place the data points in an orthant tree. The structure is built in
time O(n log n)
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Recipe
Probabily of each orthant
Conditional on the number of datapoints falling in each orthant,
the distribution of the probability mass over each orthant is given
by the Dirichlet distribution:
Γ
(∑2d
i=1 ni
)
∏2d
i=1 Γ(1 + ni )
2d
∏
i=1
pni
i
d is the dimension of the space, thus there are 2d orthants.
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Recipe
What about our prior?
Having a zero coefficient in the Haar decomposition translate itself
in certain symmetries. In two dimension there are eight cases
▶ 1 non-zero coeffs: each quadrant has 1
4 of the mass
▶ 2 non-zero coeff
▶ Left vs. right, top and bottom weight independent of side
▶ Top vs. bottom
▶ Diagonal vs. other diagonal
▶ 3 non-zero coeffs
▶ Shared equally between left and right, but each side has its
own distribution between top and bottom
▶ Same for top and bottom
▶ Same for diagonals
▶ 4 non-zero coeffs: each quadrant is independent
N.B. probabilities must sum to 1
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Recipe
Single point example
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Recipe
Marginalizing
▶ The distribution of weight over orthants is independent
between the “levels” of the tree
▶ We can marginalize to eﬃciently compute the mean density at
each point.
▶ The cost is then O(2d n log n)
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Hardy Haar

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Linear transforms
Orthant artifacts
▶ The model converges to the true distribution
▶ But the choice of orthants is arbitrary
▶ Introduces unnecessary variance
▶ We’d like to remove that sensitivity
Solution: integrate over all aﬃne transforms of the data
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Linear transforms
Integrating over linear transforms
Fortunately, the Haar model gives us the evidence
▶ Assume the data comes from a gaussian Copula
▶ Adjust by the Jacobian of the transform
▶ To sample from linear distributions
▶ perform PCA
▶ translate by ( ux√
n
, uy√
n
)
▶ rotate randomly
▶ scale variances by 1√
2n
▶ ... then weight by evidence from the model
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Hardy Haar

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How to use for data mining
Application to data-mining
▶ select relevant variables to use as regressors
▶ evaluate the quality of hand-crafted features
▶ explore unknown relationships in the data
▶ in time series, mutual information between time and data
detects non stationarity
Arthur Breitman
Hardy Haar

�ݺ�ߣ

Non-linear density estimation using a sparse Haar prior

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Non-linear density estimation using a sparse Haar prior