Sample%20variance%20and%20sample%20error

1
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 cant 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
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(XY) (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(datamodel)
  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
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
Examples

• Suppose ? is the Doppler shift of a star.
• Adopting a search range 200 lt ? lt 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 ? gt 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, i1...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
Estimating ?
Xi

• Data points Xi with no errors
• To find A, minimise ?2 ?2.
• Cant use ?2 minimisation to estimate ? because
• Instead, minimise

i