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: S2S2 sqrt(S 2 ) 22 Y=sqrt(X)
Mean and variance of S 2 Like any other statistic, S 2 has its own mean and variance. Need to know these to compute bias in S:
Bias in sqrt(S 2 ) Define square root function g: g(X) and its derivatives: Hence compute bias:
Unbiased estimator for Re-define bias-corrected estimator for :
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:
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.
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)! X P(X|a) P(a)~1 / Log (a) P(a|X) Uniform P(a) P(a|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. X P(X|a) P( )~1 / Log ( ) P( |X) Uniform P( ) P( |X)
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.
Maximum likelihood fits Suppose we try to fit a spectral line + continuum using a set of data points X i, i=1...N Suppose our model is: Parameters are C, A, 0, i, i assumed known.
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:
Estimating Data points X i with no errors: To find A, minimise 2 2. Can’t use 2 minimisation to estimate because : Instead, minimise i XiXi 2N ln -2 ln L