24 sept 2018 log x. A more refined answer: it looks like a certain integral called ... Proof (Ben): The derivative of logx is 1 x and the derivative of.
LS
1 mar 2016 Strongly convex if ∃α > 0 such that f(x) − α
ORF S Lec gh
Exercise 4 Prove the Laws of Exponents. Hint: make use of the fact that log x is a 1-1 function. Derivatives of the Exponential Function. We already
chapter
New sharp bounds for the logarithmic function
5 mar 2019 In this paper we present new sharp bounds for log(1 + x). We prove that our upper bound is sharper than all the upper bounds presented ...
13 feb 2019 x log x for all x ≥ 17. (1). January 2014]. PROOF OF THE SHELDON CONJECTURE ... Letting z = log x this derivative is z + log z + 1/z.
sheldon
Lemma 1. If x> 1 then. 0<log (r(x))-{(x-$) log (x)-x+i log (2«). Proof. Proof. We prove that for JC^6 the second derivative of h(x) is negative.
div class title some inequalities involving span class italic r span span class sup span class italic r span span div
log p ps . Thus without justification of proving that the derivative of a series Denote the sum over the logarithm of primes by θ (x) = ∑ p≤x log p.
Notes
log. = x b. 6. logb b = 1 and logb 1 = 0. (iv) The derivative of ex Example 9 If ex + ey = ex+y prove that ... Example 13 If xy = ex–y
leep
6 abr 2016 log f(X
Fisher
zk) (from Cauchy-Schwarz inequality) geometric mean: f(x)=(∏ n k=1 xk). 1/n on R n. ++ is concave. (similar proof as for log-sum-exp). Convex functions.
functions
213578
Fisher Information
April 6, 2016 Debdeep Pati
1 Fisher Information
AssumeXf(xj) (pdf or pmf) with2R. Dene
I
X() =E@@
logf(Xj) 2 where logf(Xj) is the derivative of the log-likelihood function evaluated at the true value. Fisher information is meaningful for families of distribution which are regular: 1.
Fixed supp ort:fx:f(xj)>0gis the same for all.
2. logf(xj) must exist and be nite for allxand. 3.
If EjW(X)j<1for all, then
k E
W(X) =@@
kZ
W(x)f(xj)dx=Z
W(x)@@
k f(xj)dx
1.1 Regular families
One parameter exponential families: Cauchy location or scale family: f(xj) =1(1 + (x)2) f(xj) =1(1 + (x=)2) and lots more. (Most families of distributions used in applications are regular).
1.2 Non-regular families
Uniform(0;)
Uniform(;+ 1):
1
1.3 Facts about Fisher Information
Assume a regular family.
1. E logf(Xj) = 0: Here logf(Xj) is called the \score" functionS().
Proof.
E logf(Xj) =Z logf(xj) f(xj)dx Z f(xj)f(xj)f(xj)dx Z@@ f(xj)dx Z f(xj)dx= 0 since
Rf(xj)dx= 1 for all.2.IX() = Var
logf(Xj)
Proof.SinceE
logf(Xj) = 0 Var logf(Xj) =E@@ logf(Xj) 2 =IX():3.If X= (X1;X2;:::;Xn) andX1;X2;:::;Xnare independent random variables, then I
X() =IX1() +IX2() +IXn().
Proof.Note that
f(xj) =nY i=1f i(xij) 2 wherefi( j) is the pdf (pmf) ofXi. Observe that logf(Xj) =nX i=1@@ logfi(Xij) and the random variables in the sum are independent. This Var logf(Xj) =nX i=1Var@@ logfi(Xij) so thatIX() =Pn i=1IXi() by 2.4.If X1;X2;:::;Xnare i.i.d andX= (X1;X2;:::;Xn), thenIXi() =IX1() for alli so thatIX() =nIX1(). 5.
An alternate form ulafor Fisher infor mationis
I
X() =E
@2@
2logf(Xj)
Proof.AbbreviateRf(xj)dxasRf, etc. Since 1 =Rf, applying@@ to both sides, 0 = Z f=Z@f@ =Z f f
Fisher Information
April 6, 2016 Debdeep Pati
1 Fisher Information
AssumeXf(xj) (pdf or pmf) with2R. Dene
I
X() =E@@
logf(Xj) 2 where logf(Xj) is the derivative of the log-likelihood function evaluated at the true value. Fisher information is meaningful for families of distribution which are regular: 1.
Fixed supp ort:fx:f(xj)>0gis the same for all.
2. logf(xj) must exist and be nite for allxand. 3.
If EjW(X)j<1for all, then
k E
W(X) =@@
kZ
W(x)f(xj)dx=Z
W(x)@@
k f(xj)dx
1.1 Regular families
One parameter exponential families: Cauchy location or scale family: f(xj) =1(1 + (x)2) f(xj) =1(1 + (x=)2) and lots more. (Most families of distributions used in applications are regular).
1.2 Non-regular families
Uniform(0;)
Uniform(;+ 1):
1
1.3 Facts about Fisher Information
Assume a regular family.
1. E logf(Xj) = 0: Here logf(Xj) is called the \score" functionS().
Proof.
E logf(Xj) =Z logf(xj) f(xj)dx Z f(xj)f(xj)f(xj)dx Z@@ f(xj)dx Z f(xj)dx= 0 since
Rf(xj)dx= 1 for all.2.IX() = Var
logf(Xj)
Proof.SinceE
logf(Xj) = 0 Var logf(Xj) =E@@ logf(Xj) 2 =IX():3.If X= (X1;X2;:::;Xn) andX1;X2;:::;Xnare independent random variables, then I
X() =IX1() +IX2() +IXn().
Proof.Note that
f(xj) =nY i=1f i(xij) 2 wherefi( j) is the pdf (pmf) ofXi. Observe that logf(Xj) =nX i=1@@ logfi(Xij) and the random variables in the sum are independent. This Var logf(Xj) =nX i=1Var@@ logfi(Xij) so thatIX() =Pn i=1IXi() by 2.4.If X1;X2;:::;Xnare i.i.d andX= (X1;X2;:::;Xn), thenIXi() =IX1() for alli so thatIX() =nIX1(). 5.
An alternate form ulafor Fisher infor mationis
I
X() =E
@2@
2logf(Xj)
Proof.AbbreviateRf(xj)dxasRf, etc. Since 1 =Rf, applying@@ to both sides, 0 = Z f=Z@f@ =Z f f