Transcript ada09
Sample variance and sample error • We learned recently how to determine the sample variance using the sample mean. • How do we translate this to an unbiased estimate of the error on a single point? • We can’t just take the square root! This would introduce a bias: Y=sqrt(X) sqrt(S2) S2 , even though 2 S2 2 . Mean and variance of S2 • Like any other statistic, S2 has its own mean and variance. • Need to know these to compute bias in S: 2 N 1 2 2 S2 X X i N1 N 1 i1 N 1 S2 2 2 N1 2 (N 1) 2 N 1 N 1 2 2 2 2 Var(S ) Var( N 1 ) N 1 4 2 4 .2(N 1) 2 (N 1) N 1 Bias in sqrt(S2) • Define square root function g: • g(X) and its derivatives: g(S 2 ) S2 S 1 1/ 2 1 3 / 2 g(x) x , g' (x) x , g"(x) x 2 4 1/ 2 • Hence compute bias: S g S 2 g" S 2 2 Var S ... 2 2 4 g" 2 g 2 . ... 2 N 1 1 2 4 1 3. ... 1 ... 8 N 1 4(N 1) Unbiased estimator for • Re-define bias-corrected estimator for : S S 2 1 1 4(N 1) X X 2 1 / 2 i N 3 / 2 P(X, Y) P(X | Y)P(Y) P(Y | X)P(X) P(Y) P(Y|X) • Consider 2 random Conditional probabilities variables X and Y with a joint p.d.f. P(X,Y) that looks like: Y • To get P(X) or P(Y), project P(X,Y) on to X or Y axis and normalise. • Can also determine P(X|Y) (“probability of X given Y”) which is a normalised slice through P(X,Y) at a fixed P(X) value of Y or vice versa. • At any point along each slice, can get P(X,Y) from: P(X|Y) X Bayes’ Theorem and Bayesian inference P(X | Y)P(Y ) • Bayes’ Theorem: P(Y | X ) P(X) • This leads to the method of Bayesian inference: P(data | model)P(model) P(model | data) P(data) Inference Evidence Prior • We can determine the evidence P(data|model) using goodness-of-fit statistics. • We can often determine P(model) using prior knowledge about the models. • This allows us to make inferences about the relative probabilities of different models, given the data. Choice of prior X P(X|a) • Suppose our model of a set of data X is controlled by a parameter . • Our knowledge about before X is measured is quantified by the prior p.d.f. P(). • Choice of P() is arbitrary subject to common sense! • After measuring X get posterior p.d.f. P(|X) = P(X|).P() • Different priors P() lead to different inferences P(|X)! Uniform P(a) P(a|X) P(a)~1 / Log (a) P(a|X) P(X|a) • Suppose is the Doppler shift of a star. X • Adopting a search range –200 < < 200 km/sec in uniform velocity increments implicitly assumes a uniform prior. • Alternatively: scaling an emission-line profile of known shape. • If you know ≥ 0, can force > 0 by constructing the pdf in uniform increments of Log so P() ~ 1/Log(). Uniform P() • Posterior distributions are skewed differently according P( |X) to choice of prior. Examples P()~1 / Log () P( |X) Relative probabilities of models • Two models m1, m2 • Relative probabilities depend on – Ratio of prior probabilities – Relative ability to fit data P(m1 | X ) P(X | m1 ) P(m1 ) P(m2 | X ) P(X | m2 ) P(m2 ) • Note that P(data) cancels. Maximum likelihood fits • Suppose we try to fit a spectral line + continuum using a set of data points Xi, i=1...N 2 1 i 0 • Suppose our model is: 2 X i C Ae Var(Xi ) i2 • Parameters are C, A, 0, • i , i assumed known. i Likelihood of a model • Likelihood of a particular set of model parameters (i.e. probability of getting this set of data given model ), is: L() P(X | ) P(X1 | ) P(X2 | ) P(X N | ) N P(X i | ) i1 • If errors are gaussian, then: 1 P(Xi | ) e i 2 2 1 Xi i 2 i 1 N 2 1 L() (2 ) N / 2 e 2 i1 i To maximise L(), must minimise 2 2 ln i 2 ln L() 2 2 ln i constant i i Estimating • Data points Xi with no errors: Xi A i1 2 N 2 • To find A, minimise 2 2. Xi 120 110 100 90 80 70 60 0 5 10 15 20 350 300 250 • Can’t use 2 minimisation to estimate because : 2 0 as • Instead, minimise 2 ln L 2N ln 2 200 -2 ln L 150 2N ln 100 50 0 0 10 20 i