Sample variance and sample error

1. Y=sqrt(X) sqrt(S2)  2 S2 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:

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:

3. Bias in sqrt(S2) • Define square root function g: • g(X) and its derivatives: • Hence compute bias:

4. Unbiased estimator for  • Re-define bias-corrected estimator for :

5. Y P(Y|X) P(Y) X P(X) P(X|Y) Conditional probabilities • Consider 2 random variables X and Y with a joint p.d.f. P(X,Y) that looks like: • 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 value of Y or vice versa. • At any point along each slice, can get P(X,Y) from:

6. Bayes’ Theorem and Bayesian inference • Bayes’ Theorem: • This leads to the method of Bayesian inference: • 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.

7. X P(X|a)  Uniform P(a) P(a)~1 / Log (a) P(a|X) P(a|X) Choice of prior • 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)!

8. X P(X|a)  Uniform P() P()~1 / Log () P( |X) P( |X) Examples • Suppose  is the Doppler shift of a star. • 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(). • Posterior distributions are skewed differently according to choice of prior.

9. Relative probabilities of models • Two models m1, m2 • Relative probabilities depend on • Ratio of prior probabilities • Relative ability to fit data • Note that P(data) cancels.

10. Maximum likelihood fits • Suppose we try to fit a spectral line + continuum using a set of data points Xi, i=1...N • Suppose our model is: • Parameters are C, A, 0,  • i , i assumed known.

11. Likelihood of a model • Likelihood of a particular set  of model parameters (i.e. probability of getting this set of data given model  ), is: • If errors are gaussian, then:

12. 350 300 250 200 -2 ln L 150 100 2N ln  50  0  0 10 20  Estimating  Xi • Data points Xi with no errors: • To find A, minimise 22. • Can’t use 2 minimisation to estimate  because : • Instead, minimise i