MACHINE LEARNING TECHNIQUES - LASA

187
The two most classical distribution probabilities are:
• The Uniform distribution. The simplest forms is when x is a constant p( x) = a, ∀ x
It can also be represented by a step function.
• The Gaussian or Normal distribution p( x)
1
= e
σ 2π
⎛⎛
⎜⎜−
⎜⎜
⎝⎝
( x−µ
) 2
To emphasize that the pdf depends on the definition of the mean and covariance, one
sometimes writes p( x µσ )
| , .
• The multi-dimensional Gaussian or Normal distribution has a pdf given by:
2σ
2
⎞⎞
⎟⎟
⎟⎟
⎠⎠
( | µ , )
p x
∑ =
( 2π
)
1
N 1
2 2
T −
( −( x−µ ) ∑ 1 ( x−µ
))
if x is N-dimensional, then
µ is a N − dimensional mean vector
∑ is a N×
N covariance matrix
∑
e
Figure 9-1: A 2-dimensional Gaussian distribution. The 2D oval-shape representation (left) is done by
projecting the density onto the 2D axes of the variate X. The color code represent surface of 1 standard
deviation (std) in red, 2 std in light blue, 3 std in darker blue, etc.
9.2.4 Marginal Probability Distribution or Marginal Density
Consider two random variables x and y with joint distribution p(x,y), then the marginal density of x
is givenby:
( ) ( , )
px
x = ∫ p x y dy
© A.G.Billard 2004 – Last Update March 2011