Probability distributions for ml

Introduction
Binary Variables
Multinomial Variables
The Gaussian Distribution
The Exponential Family
Nonparametric Methods
Probability Distributions for ML
Sung-Yub Kim
Dept of IE, Seoul National University
January 29, 2017
Sung-Yub Kim Probability Distributions for ML

Introduction
Binary Variables
Bishop, C. M. Pattern Recognition and Machine Learning Information Science and Statistics, Springer, 2006.
Kevin P. Murphy. Machine Learning - A Probabilistic Perspective Adaptive Computation and Machine
Learning, MIT press, 2012.
Ian Goodfellow and Yoshua Bengio and Aaron Courville. Deep Learning Computer Science and Intelligent
Systems, MIT Press, 2016.

Introduction
Binary Variables
Purpose: Density Estimation
Assumption: Data Points are independent and identically distributed.(i.i.d)
Parametric and Nonparametric
Parametric estimations are more intuitive but has very strong assumption.
Nonparametric estimation also has some parameters, but they control
model complexity.

Introduction
Binary Variables
Bernouli and Binomial Distribution
MLE of Bernouli parameter
The Beta Distribution
Bayesian Inference on binary variables
Diﬀerence between prior and posterior
Bernouli Distribution(Ber(θ))
Bernouli Distribution has only one parameter θ which means the success
probability of the trial. PMF of bernouli dist is shown like
Ber(x|θ) = θI(x=1)
(1 − θ)I(x=0)
Binomial Distribution(Bin(n,θ))
Binomial Distribution has two parameters n for number of trials, θ for
success prob. PMF of binomial dist is shown like
Bin(k|n, θ) =
n
k
θk
(1 − θ)n−k

Introduction
Binary Variables
Likelihood of Data
By i.i.d assumption, we get
p(D|µ) =
N
n=1
p(xn|µ) =
N
n=1
µxn
(1 − µ)1−xn
(1)
Log-likelihood of Data
Take logarithm, we get
ln p(D|µ) =
N
n=1
ln p(xn|µ) =
N
n=1
{xn ln µ + (1 − xn) ln(1 − µ)} (2)
MLE
Since maximizer is stationary point, we get
µML := ˆµ =
1
N
N
n=1
xn (3)

Introduction
Binary Variables
Prior Distribution
The weak point of MLE is you can be overfitted to data. To overcome this
deficiency, we need to make some prior distribution.
But same time our prior distribution need to has a simple interpretation
and useful analytical properties.
Conjugate Prior
Conjugate prior for a likelihood is a prior distribution which your prior and
posterior distribution are same given your likelihood.
In this case, we need to make our prior proportional to powers of µ and
(1 − µ). Therefore, we choose Beta Distribution
Beta(µ|a, b) =
Γ(a + b)
Γ(a)Γ(b)
µa−1
(1 − µ)b−1
(4)
Beta Distribution has two parameters a,b each counts how many occurs
each classes(effective number of observations). Also we can easily valid
that posterior is also beta distribution.

Introduction
Binary Variables
Posterior Distribution
By some calculation,
p(µ|m, l, a, b) =
Γ(m + l + a + b)
Γ(m + a)Γ(l + b)
µm+a−1
(1 − µ)l+b−1
(5)
where m,l are observed data.
Bayesian Inference
Now we can make some bayesian inference on binary variables. We want
to know
p(x = 1|D) =
1
0
p(x = 1|µ)p(µ|D)dµ =
1
0
µp(µ|D)dµ = E[µ|D] (6)
Therefore we get
p(x = 1|D) =
m + a
m + a + l + b
(7)
If observed data(m,l) are suﬃciently big, its asymptotic property is
identical to MLE, and this property is very general.

Introduction
Binary Variables
Since
Eθ[θ] = ED[Eθ[θ|D]] (8)
we know that poseterior mean of θ, averaged over the distribution generating
the data, is equal to the prior mean of θ.
Also since
Varθ[θ] = ED[Varθ[θ|D]] + VarD[Eθ[θ|D]] (9)
We know that on average, the posterior variance of θ is smaller than the prior
variance.

Introduction
Binary Variables
Multinomials and Multinouli Distributions
MLE of Multinouli parameters
The Dirichlet Distribution and Bayesian Inference
Multinomial Distribution(Mu(x|n, θ))
Multinomial distribution is diﬀerent from binomial with respect to
dimension of ouput and θ. In binomial, k means the number of success. In
multinomial each index of x means the number of state. Therefore we can
see binomial as multinomial when the dimension of x and θ is 2.
Mu(x|n, θ) =
n
x0, . . . , xK−1
K−1
j=0
θ
xj
j
Multinouli Distribution(Mu(x|1, θ))
Sometimes we are intersted in the special case of Multinomial when the n
is 1 that is called Multinouli distribution:
Mu(x|1, θ) =
K−1
j=0
θ
I(xj =1)
j

Introduction
Binary Variables
Likelihood of Data
By i.i.d assumption, we get
p(D|µ) =
N
n=1
K
k=1
µ
xnk
k =
K
k=1
µ n xnk
k =
K
k=1
µ
mk
k (10)
where mk = n xnk (suﬃcient statistics)
Log-likelihood of Data
Take logarithm, we get
ln p(D|µ) =
K
k=1
mk ln µk (11)
MLE
Therefore, we need to solve following optimization problem for MLE
max{
K
k=1
mk ln µk |
K
k=1
µk = 1} (12)

Introduction
Binary Variables
MLE(cont.)
We already know that Lagrangian stationaty point is a necessary condition
for constrained optimization problem. Therefore,
µL(µ; λ) = 0, λL(µ; λ) = 0 (13)
where
L(µ; λ) =
K
k=1
mk ln µk + λ(
K
k=1
µk − 1) (14)
Therefore, we get
µML
k =
mk
N
(15)

Introduction
Binary Variables
Dirichlet Distribution
By the same intuition in Beta distribution, we can get conjugate prior for
Multinouli
Dir(µ|α) =
Γ(α0)
Γ(α1) · · · Γ(αK )
K
k=1
µ
αk −1
k (16)
where α0 = k αk
Bayesian Inference
By the same argument in binomial, we can get posterior probability
p(µ|D, α) = Dir(µ|α + m) =
Γ(α0 + N)
Γ(α1 + m1) · · · Γ(αK + mK )
K
k=1
µ
αk +mk −1
k
(17)

Introduction
Binary Variables
Uni and Multi variate Gaussian
Basic Property
Conditional and Marginal Distributions
Inference for Gaussian
Student’s t-distribution
Univariate Gaussian Distribution(N(x|µ, σ2
) = N(x|µ, β−1
))
N(x|µ, σ2
) =
1
√
2πσ2
exp(−
1
2σ2
(x − µ)2
) (18)
N(x|µ, β−1
) =
β
2π
exp(−
β
2
(x − µ)2
) (19)
Multivariate Gaussian Distribution(N(x|µ, Σ) = N(x|µ, β−1
))
N(x|µ, Σ) =
1
(2π)
D
2 det(Σ)
1
2
exp(−
1
2
(x − µ) Σ−1
(x − µ)) (20)
N(x|µ, β−1
) =
1
(2π)
D
2 det(Σ)
1
2
exp(−
1
2
(x − µ) β(x − µ)) (21)

Introduction
Binary Variables
Basic Property
Mahalanobis Distance
By EVD, we can get
∆2
= (x − µ) Σ−1
(x − µ) =
D
i=1
y2
i
λi
(22)
where yi = ui (x − µ)
Change of Variable in Gaussian
By above, we can get
p(y) = p(x)|Jy→x | =
D
j=1
1
(2πλj )
1
2
exp{−
y2
j
2λj
} (23)
which means product of D independent univariate Gaussian Distribution.
First and Second Moment of Gaussian
By using above, we can get
E[x] = µ, E[xx ] = µµ + Σ (24)

Introduction
Binary Variables
Basic Property
Limitations of Gaussian and Solutions
There are two main limitations for Gaussian.
First, we have to infer so many covariance parameters.
Second, we cannot represent multi-modal ditriubtions. Therefore, we
deﬁne some auxilarily concepts.
Diagonal Covariance
Σ = diag(s2
) (25)
Isotropic Covariance
Σ = σ2
I (26)
Mixture Model
p(x) =
K
k=1
πk p(x|πk ) (27)

Introduction
Binary Variables
Basic Property
Partitions of Mahalanobis distance
First, partition the covariance matrix and precision matrix.
Σ =
Σaa Σab
Σba Σbb
, Σ−1
= Λ =
Λaa Λab
Λba Λbb
(28)
where aa, bb are symmetric and ab and ba are conjugate transpose.
Now, partition the Mahalanobis distance.
(x − µ) Σ−1
(x − µ)
= (xa − µ) Σ−1
aa (xa − µ) + (xa − µ) Σ−1
ab (xb − µ)
+(xb − µ) Σ−1
ba (xa − µ) + (xb − µ) Σ−1
bb (xb − µ)(29)
Schur Complement
Like gaussian elimination, we can use some block matrix elimination by
Schur Complement
A B
C D
−1
=
M −MBD−1
−D−1
CM D−1
+ D−1
CMBD−1 (30)
where M = (A − BD−1
C)−1

Introduction
Binary Variables
Basic Property
Schur Complement(cont.)
Therefore, we get
Λaa = (Σaa − ΣabΣ−1
bb Σba)−1
(31)
Λab = −(Σaa − ΣabΣ−1
bb Σba)−1
ΣabΣ−1
bb (32)
Conditional Distribution
Therefore, we get
xa|xb ∼ N(x|µa|b, Σa|b) (33)
where
µa|b = µa + ΣabΣ−1
bb (xb − xa) (34)
Σa|b = Σaa − ΣabΣ−1
bb Σba (35)
Marginal Distribution
Removing xb by integrating, we can get marginal distribution of xa
p(xa) = −
1
2
xa (Λaa − ΛabΛbbΛba)xa + xa (Λaa − ΛabΛbbΛba)µa + const (36)
Therefore, we get
xa ∼ N(x|µa, Σaa) (37)

Introduction
Binary Variables
Basic Property
Given a marginal Gaussian for x and a conditional Gaussian for y given x in the
form
x ∼ N(x|µ, Λ−1
) (38)
y|x ∼ N(y|Ax + b, L−1
) (39)
Then we can get marginal distribution of y and the conditional distribution of x
given y are given by
y ∼ N(y|Aµ + b, L−1
+ AΛ−1
A ) (40)
x|y ∼ N(x|Σ{A L(y − b) + Aµ}, Σ) (41)
where
Σ = (Λ + A LA)−1
(42)

Introduction
Binary Variables
Basic Property
Log-likelihood for data
By same argument in categorical data, we can get log-likelihood for
Gaussian
ln p(D|µ, Σ) = −
ND
2
ln 2π −
N
2
ln |Σ| −
1
2
N
n=1
(xn − µ) Σ−1
(xn − µ) (43)
and this log-likelihood depends only on these quantities called Suﬃcient
Statistics
N
n=1
xn,
N
n=1
xnxn (44)
MLE for Gaussian
Since MLE is a maximizer for log-likelihood, we can get
µML =
1
N
N
n=1
xn (45)
ΣML =
1
N
N
n=1
(xn − µML)(xn − µML) (46)

Introduction
Binary Variables
Basic Property
Sequential estimation
Since we get MLE for gaussian analytically, we can do this sequentially like
µN
ML = µN−1
ML +
1
N
(xN − µN−1
ML ) (47)
Robbins-Monro Algorithm
By same intuition, we can generalize sequential learning. Robbins-Monro
algorithm gives us root θ such that f (θ) = E[z|θ] = 0. The iterate process
of RM algorithm can be represented by
θN
= θN−1
− aN−1z(θN−1
) (48)
where z(θN−1
) means observed value of z when θ takes the value θN−1
and aN is an sequence satisfy
lim
N→∞
aN = 0,
∞
N=1
aN = ∞,
∞
N=1
aN < ∞ (49)

Introduction
Binary Variables
Basic Property
Generalized Sequential Learning
We can apply RM algorithm for sequential learning. In this case, our f (θ)
is a gradient of log-likelihood function. Therefore, we can get
z(θ) = −
∂
∂θ
ln p(x|θ) (50)
In Gaussian case, we put aN to σ2
/N.
Bayesian Inference for mean given variance
Since gaussian likelihood takes the form of the exponential of a quadratic
form in µ, we can choose a prior also Gaussian. Therefore, if we choose
µ ∼ N(µ|µ0, σ2
0) (51)
for prior, we get following for posterior
µ|D ∼ N(µ|µN , σ2
N ) (52)
where
µN =
σ2
Nσ2
0 + σ2
µ0 +
Nσ2
0
Nσ2
0 + σ2
µML,
1
σ2
N
=
1
σ2
0
+
N
σ2
(53)

Introduction
Binary Variables
Basic Property
Bayesian Inference for mean given variance(cont.)
1. Posterior mean compromises between the priot and the MLE.
2. Precision is given by the precision of the prior plus one contribution of
the data precision from each of the observed data.
3. If we take σ2
0 → ∞ then the posterior mean reduces to the MLE.
Bayesian Inference for variance given mean
Since gaussian likelihood takes the form of proportional to the product of
a power of precision and the exponential of a linear function of precision.
We choose gamma distribution which is deﬁned by
Gam(λ|a0, b0) =
1
Γ(a0)
ba
00λa0−1
exp(−b0λ) (54)
Then we can get posterior
λ|D ∼ Gam(λ|aN , bN ) (55)
where
aN = a0 +
N
2
, bN = b0 +
N
2
σ2
ML (56)

Introduction
Binary Variables
Basic Property
Bayesian Inference for variance given mean(cont.)
1. We can interpret the parameter 2a0 eﬀective prior observations for
number of data. 2. We can interpret the parameter b0/a0 eﬀective prior
observations for variance.
Bayesian Inference for no data
By apply same argument on mean and variance, we can get prior
p(µ, λ) ∼ N(µ|µ0, (βλ)−1
)Gam(λ|a, b) (57)
where
µ0 = c/β, a = 1 + β/2, b = d − c2
/2β (58)
Note that precision of µ is a linear function of λ
For Multivariate case, we can similarly get prior
p(µ, Λ|µ0, β, W , ν) = N(µ|µ0, (βΛ)−1
)W(Λ|W , ν) (59)
where W is Wishart distribution.

Introduction
Binary Variables
Basic Property
Univariate t-distribution
If we integrate out the precision given that our prior for precision is
Gamma, we get t-distribution.
St(x|µ, λ, ν) =
Γ(ν/2 + 1/2)
Γ(ν/2)
(
λ
πν
)1/2
[1 +
λ(x − µ)2
ν
]−ν/2−1/2
(60)
where ν = 2a(degrees of freedom) and λ = a/b.
We can think t-dstribution as an inﬁnite mixture of Gaussians.
Since t-distribution has fat tail(than Gaussian), we can obtain more robust
model when we estimate.
Multivariate t-distribution
We also can get multivariate case of inﬁnite mixture of Gaussians, then we
get multivariate t-distribution
St(x|µ, Λ, ν) =
Γ(ν/2 + D/2)
Γ(ν/2)
(
Λ1/2
(πν)D/2
)[1 +
∆2
ν
]−ν/2−D/2
(61)

Introduction
Binary Variables
Distribution for the exponential family
Sigmoid and Softmax
MLE for the exponential family
Conjugate priors for exponential family
Noninformative priors
The exponential family of distributions over x, given parameters η, is
deﬁned to be the set of distributions of the form
p(x|η) = g(η)h(x) exp{η u(x)} (62)
where η is natural parameters of the distribution, and u(x) is a function
of x.
The fnuction g(η) can be interpereted as the normalization factor.

Introduction
Binary Variables
Sigmoid and Softmax
Logistic Sigmoid
In case of bernouli distribution, our parameter is µ, although our natural
parameter is η. Those two parameter can be connected by following
η = ln(
µ
1 − µ
), µ := σ(η) =
exp(µ)
1 + exp(µ)
(63)
And we call this σ(η) sigmoid function.
Softmax function
By same argument, we can ﬁnd some realtionship between our parameter
and natural parameter. That is Softmax function.
µk =
exp(ηk )
K
j=1 exp(ηj )
(64)
Note that in this case, u(x) = 1, h(x) = 1, g(x) = ( K
j=1 exp(ηj ))−1

Introduction
Binary Variables
Sigmoid and Softmax
Gaussian
Gaussian also can be interpreted as the exponential family by
u(x) =
x
x2 (65)
η =
µ/σ2
−1/2σ2 (66)
g(η) = (−2η2)1/2
exp(
η2
1
4η2
) (67)

Introduction
Binary Variables
Sigmoid and Softmax
Problem of estimating the natural parameter
We can generalize the argument in MLE in other cases.
First, we consider the log-likelihood of the data.
ln p(D|η) =
N
n=1
h(xn) + N ln g(η) + η
N
n=1
u(xn) (68)
Next, we need to ﬁnd the stationary point of the log-likelihood.
N η ln g(η) +
N
n=1
u(xn) = 0 (69)
Therfore, we get MLE
− η ln g(η) =
1
N
N
n=1
u(xn) (70)
We see that the solution for the MLE depedns on the data only through
σnu(xn), which is therefore called the suﬃcient statistic of the
exponential family.

Introduction
Binary Variables
Sigmoid and Softmax
Conjugate prior
For any member of the exponential family, there exists a conjugate prior
that can be written in the form
p(η|χ, ν) = f (χ, ν)g(η)ν
exp{νη χ} (71)
where f (χ, ν) is a normalization factor, and g(η) is the same function as
the exponential family.
Posterior distribution
If we choose prior as conjugate prior, we get
p(η|D, χ, ν) ∝ g(η)ν+N
exp{η (
N
n=1
u(xn) + νχ)} (72)
Therefore, we see that the parameter ν can be interpreted as the eﬀective
number of pseudo-observations in the prior, each of which has a value
for the suﬃcient statistics u(x) given by χ.

Introduction
Binary Variables
Sigmoid and Softmax
Noninformative Priors
We may seek a form of prior distribution, called a noninformative prior,
which is intended to have as little inﬂuence on the posterior distribution as
possible.
Generalizations of Noninformative priors
It leads to two generalizations, namely the principle of transformation
groups as in the Jeﬀreys prior, and the principle of maximum entropy.

Introduction
Binary Variables
Histogram Technique
Kernel Density Estimation
Nearest-Neighbour methods
Histogram Technique
Standard histograms simply partition x into distinct bins of width ∆i and
then count the number ni of observations of x falling in bin i. In order to
turn this count into a normalized probability density, we simply divide by
the total number N of observations and by the width ∆i of the bins to
obtain probability values for each bin given by
pi =
ni
N∆i
(73)
Limitations of Hitogram
The estimated density has discontinuities that are due to the bin edges
rather than any property of the underlying distribution that generated the
data.
Histogram approach also sacling with dimensionality.
Lessons of Histogram
First, to estimate the probability density at a particular location, we should
consider the data points that lie within some local neighbourhood of that
point.
Second, the value of the smoothing parameter should be neither too large
nor too small in order to obtain good results.

Introduction
Binary Variables
Histogram Technique
Motivation
For large N, the bernouli trial that data point fall within small region
mathcalR will be sharply peaked around the mean and so
K NP (74)
If, however, we also assume that the region R is sufficiently small that the
probability density p(x) is roughlt over the region, then we have
P p(x)V (75)
where V is the volume of R. Therefore,
p(x) =
K
NV
(76)
Note that in our assumption, R is sufficiently small tha the density is
approximately constant over the region and the yet sufficiently large that
the number K of points falling inside the region is sufficient for the
binomial distribution to be sharply peaked.

Introduction
Binary Variables
Histogram Technique
Kernel Density Estimation(KDE)
If we ﬁx V and determine K from the data, we use kernel approach. For
instance, we ﬁx V to 1 and count the data point by following function
k(u) =
1, if |ui | ≤ 1/2, i = 1, · · · , D,
0, otherwise
(77)
which called Parzen window In this case, we can use this by
K =
N
n=1
k(
x − xn
h
) (78)
and it leads density function
p(x) =
1
N
N
n=1
1
hD
k(
x − xn
h
) (79)
We can also use another kernel like Gaussian kernel. If we do so, then we
get
p(x) =
1
N
N
n=1
1
(2πh2)D/2
exp{−
x − xn
2h2
} (80)

Introduction
Binary Variables
Histogram Technique
Limitation of KDE
One of the difficulties with the kernel approach to density estimation is
that the parameter h governing the kernel width is fixed for all kernels. In
regions of high data density, a large value of h may lead to over-smoothing
and in lower data density, a small value of h may lead to overfitting. Thus
the optimal choice for h may be dependent on location within data space.
Nereat-Neighbor(NN)
Therefore we consider a fixing K and use the data to find an appropriate V
and we call this method K-NN methods.
In this case, the value of K governs the degree of smoothing and we need
to optimizae(hyper-parameter optimize) K.
Erro of KNN
Note that for sufficiently big N, the error rate is never more than twice the
minimum achievable error rate of an optimal classifier.

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