Likelihoods, Priors and Posteriors

AIMS3 provides a Likelihood class for bookkeeping. User or pipeline scripts provides priors with which to construct a posterior distribution function.

The frequency function

The simplest way of using data for individual mode frequencies is to them directly but, for many reasons, we may wish to instead fit using some functions of the frequency. These may include averaged properties like the average large separation, fit by linear regression of the radial modes against their radial order, or frequency-insensitive quantities like separation ratios.

AIMS3’s Likelihood object uses a frequency function \(f(\nu)\), which maps the individual frequencies \(\nu\) onto the quantities we’ll actually fit. f is therefore a vector-valued function of the vector ν. AIMS3 is restricted to a mixture of functions that are either linear in ν (i.e. f(ν)=Aν for some matrix A) or a simple ratio of linear maps (i.e. f(ν)=Aν/Bν for matrices A and B). If we just want to use the mode frequencies directly, then we’d use a linear map A=I.

To propagate the uncertainties on the mode frequencies, which are defined by the covariance matrix C, we need the Jacobian matrix J, in which case the covariance of the frequency function is \(C' = JCJ^\mathrm{T}\). For a linear map, J=A. For a ratio of linear maps, the quotient rule gives

\[J = \frac{1}{B\nu}A-\frac{A\nu}{(B\nu)^2}B\]

AIMS3 constructs the frequency function and its Jacobian by stacking matrices for a sequence of options. The current options, given a set of mode frequencies \(\nu_{l,n}\), are

  • nu<l>: use the individual frequencies

  • nu_min<i>: use the i-th lowest radial mode frequency

  • r02: use the separation ratios

    \[r_{02,n} = \frac{\nu_{0,n}-\nu_{2,n-1}}{\nu_{1,n}-\nu_{1,n-1}}\]

  • r13: use the separation ratios

    \[r_{13,n} = \frac{\nu_{1,n}-\nu_{3,n-1}}{\nu_{0,n+1}-\nu_{0,n}}\]

  • r01: use the separation ratios

    \[r_{01,n}=\frac{\nu_{0,n-1}-4\nu_{1,n-1}+6\nu_{0,n}-4\nu_{1,n}+\nu_{0,n+1}}{8(\nu_{1,n}-\nu_{1,n-1})}\]

  • r10: use the separation ratios

    \[r_{10,n}=-\frac{\nu_{1,n-1}-4\nu_{0,n}+6\nu_{1,n}-4\nu_{0,n+1}+\nu_{1,n+1}}{8(\nu_{0,n+1}-\nu_{0,n})}\]

  • d2nu: use the second frequency differences

    \[\Delta_2\nu_{n,l} = \nu_{l,n-1}-2\nu_{l,n}+\nu_{l,n+1}\]

  • dnu<l>: use the mode-by-mode frequency differences \(\Delta\nu_{l,n}=\nu_{l,n}-\nu_{l,n-1}\)

  • avg_dnu<l>: use the average large separation \(\langle\Delta\nu_l\rangle\) (found by linear regression)

Where there is an optional angular degree <l>, it can be omitted to use all available modes. e.g., nu1 will add all the dipole mode frequencies and avg_dnu2 will add the average large separation of the quadrupole modes.

If a combination of options leads to an uninvertible covariance matrix C’, the AIMS3 will raise a LinAlgError.

Surface corrections

Functional forms:

  • Ball2014

    \[\delta\nu = a_3\left(\frac{\nu}{\nu_\mathrm{max}}\right)^3\mathcal{I}^{-1}\]

  • Ball2014_2

    \[\delta\nu = \left(a_{-1}\left(\frac{\nu}{\nu_\mathrm{max}}\right)^{-1} +a_3\left(\frac{\nu}{\nu_\mathrm{max}}\right)^3\right)\mathcal{I}^{-1}\]

  • Kjeldsen2008 or PowerLaw

    \[\delta\nu = a\left(\frac{\nu}{\nu_\mathrm{max}}\right)^b\]

  • Sonoi2015

    \[\delta\nu = a\left(1-\frac{1}{1+\left(\frac{\nu}{\nu_\mathrm{max}}\right)^b}\right)\]

Module options:

  • Ball2014: the one-term (or cubic) correction by Ball & Gizon (2014). The coefficient is free.

  • Ball2014_2: the two-term (or combined) correction by Ball & Gizon (2014). Both coefficients are free.

  • Kjeldsen2008: The linear coefficient a is free; the power-law index b is fixed.

  • Kjeldsen2008_scaling: The linear coefficient a is free; the power-law index b is determined by the scaling relation by Sonoi et al. (2015).

  • Kjeldsen2008_2: Both a and b are free.

  • PowerLaw: equivalent to Kjeldsen2008_2.

  • Sonoi2015: The linear coefficient a is free; the power-law index b is fixed.

  • Sonoi2015_scaling: The linear coefficient a is free; the power-law index b is determined by the scaling relation by Sonoi et al. (2015).

  • Sonoi2015_2 or PowerLaw: Both a and b are free.

Observational data format

The standard file format for observations is a mixture of comments, blank lines, lines for observed mode frequencies and lines for observed non-seismic data (e.g. effective temperature). These can be mixed in any order. The following is an example of a valid file:

# comments are OK
0 21 2866.79 0.50
M_H 0.09 0.09

0 3001.18 0.30
0 3135.44 0.50 # comments can follow a row of data, too
Teff 5761 84

Lines starting with an integer are presumed to be data for modes. If there are four numbers, the second is presumed to be the radial order, though it is ignored if the argument ignore_n=True is passed. If there is no radial order (or it is ignored), the data is given a null value defined by stats.unknown_n. The second last and last numbers are the frequency and its uncertainty.

Lines starting with a string are treated as a non-seismic constraint and added to the Likelihood object with that string as a key and the two floats as the mean and uncertainty.

The values and uncertainties need only be in the same units as each other and the same units as the grid, unless you’re getting very advanced in manipulating the grid or observations.