Lecture 6 Flashcards
The Likelihood function (L) is denoted by
L = n Π i=1 p(xi)
where n Π i=1 denotes a multiplication rather than a sum
Weighted Linear Least Squares fit
S = χ2(a,b) = n Σ i=1 [(yi-(a+bxi)/σi]^2
log least likelihood of a normal
l = ln(L) = ln[1/√(2πσ^2)]^n + ln[exp[ n Σ i=1 - ((xi-µ)^2/2σ^2 ]
l = - n/2 ln[2πσ^2] - n Σ i=1((xi-µ)^2/2σ^2 ]
l = C - [ n Σ i=1 ((xi-µ)^2/2σ^2 ]
log least likelihood of a Poisson
l = ln(L) = ln(µ^r) - µ ln(r!)
= rln(µ) - µ -ln(r!)
the maximum likelihood of a normal = sample mean
l ∝ (x1-2x1µ+µ^2) -(x2-2x2µ+µ^2) - … - (xn^2-2xnµ+µ^2)
l ∝ -nµ^2 2µ(x1+x2+ … +xn) - (x1^2+x2^2+…+xn^2)
-nµ^2+2µ(x1+x2+…+xn) - (x1^2 +x^2 +…+xn^2) = 0
quadratic pf ax^2+bx+c and xmax = -b/2a
What is the maximum likelihood of a parameter
a method to estimate the parameters of a distribution, which fit the observed data
- we then calculate the likelihood given the distribution we choose
the individual likelihoods, are then multiplied, and a minimization is used to maximise the likelihood
for an Analogue Digitial Convertor use
a uniform distribution
(xi-µ)
is the residual
the square is used in the least squares problem
the maximum likelihood estimator for the standard deviation
l = - n/2 ln[2πσ^2] - n Σ i=1((xi-µ)^2/2σ^2 ]
solve for dL/dσ = 0
0 = -n/2 σ + n Σ i=1(xi-µ)^2/σ^3
σ^2 = 1/n n Σ i=1(xi-µ)^2/n
in general, the maximum likelihood of function is
the derivative of the log least likelihood
