Week 5 Least Square, Chi-Squared, Correlation

Question 1

what are the two ways to calculate joint probability

Answer 1

calculate directly if you know their analytical representation
require a full description of their individual probabilities so the product can be calculated numerically

Question 2

what does the prior probability from Bayesian statistics represent

Answer 2

degree of belief in a quantity/parameter that is to be estimated before some evidence is taken into account

Question 3

what are prior probabilities used for

Answer 3

constraining parameters

Question 4

define the two types of priors

Answer 4

flat priors - constant and independent of parameters

improper priors - priors can’t be normalised to a finite value

Question 5

what is the chi squared equation for a function f(xi, θ)

Answer 5

X^2(θ) = Σ[(yi - f(xi, θ))^2 / σi^2]

Question 6

what is chi squared used for

Answer 6

calculating the best fit parameters to a model given data

Question 7

what is the addition of two independent chi squared equation

Answer 7

X(k)^2 + X(j)^2 = X(k+j)^2

Question 8

when we have the function f(x; θ) that is a non linear function of θ what can be said

Answer 8

there is no general analytical solution to solve the M equations of each parameter. The MLE can only be found numerically

Question 9

when we have a distribution with N independent measures and a Gaussian error how do we get the best average of all elements

Answer 9

we minimize chi squared

Question 10

what is the chi squared equation for correlated measurements

Answer 10

X^2 = ΣiΣj[(yi - θ) V^-1(yj - θ)