Descolonización perceptiva y variadas formas de saber

In solar physics the best-fitting parameters for a particular model of interest are often determined by a Levenberg-Marquardt least-squares fit to the relevant data, with each point weighted according to its error. Now a method based on Bayesian statistics will be discussed. Bayesian analysis allows robust estimation of how the output of the proposed model depends on the input parameters. It can be used to determine information about model parameters from data (inference) and to compare how well different models explain the observed data (model comparison) [see von Toussaint, 2011]. Bayesian inference is well used in many branches of physics, in particular astrophysics. Recently, it has been used to seismologically infer coronal loop parameters from observations of damped kink oscillations, as discussed several times in Section. 1.4 [e.g.,Arregui & Asensio Ramos,2011;Arregui et al., 2013a, 2015; Arregui, 2018]. In particular, Arregui et al. [2013b] used the Gaussian and exponential damping regimes described in Section 1.4.2, and describe an inversion procedure based on Bayesian analysis, conversely Arregui & Asensio Ramos [2014] apply Bayesian analysis to the ill-posed case, where only exponential

damping is considered.

The general methodology behind parameter inference and model comparison will now be described, in the next section the specific implementation used in this thesis will then be discussed. A parameter inference problem assumes that the observed data D can be fully interpreted by an assumed model M, which has a parameter set θ “ rθ1, θ2,¨ ¨ ¨, θNs. The aim is to obtain the values of the model

parameters θ that best described the observed data D, or to compare multiple different models and determine which is the most probable based on the data. The standard formulation of Bayesian parameter inference relies on three definitions:

1. The prior probability density function (PDF) Ppθq represents the knowledge about the model parameters θ before considering the observational data D. This is where knowledge from previous measurements or a model parameter being confined to a certain range may be included, and how the results are influenced by this prior information can be readily quantified.

2. The likelihood functionPpD|θqdescribes the conditional probability to obtain the observed data D for a set of values, θ, of the model parameters, i.e a function ofθ with fixed D.

3. The posterior PDF Ppθ|Dq describes the conditional probability that the model parameters are equal toθunder condition of observed data being equal toDand the assumed model, i.e a function ofDwith fixedθ. Computing this distribution is normally the main goal in Bayesian inference codes.

These three quantities are connected via the Bayes theorem

Ppθ|Dq “ PpD|θqPpθq

PpDq . (1.55)

where the normalisation constantPpDqin denominator is the Bayesian evidence or marginal likelihood, given by

PpDq “

ż

PpD|θqPpθqdθ. (1.56) This is an integral of the likelihood over the prior distribution, which normalises the likelihood such that it becomes a probability.

For the prior probabilityPpθqand likelihoodPpD|θqfunctions, the posterior probability distribution Ppθ|Dq can be calculated for any value of the parameter setθ using Eq. (1.55). For seismological applications the aim is to obtain the most probable value and corresponding uncertainties for each parameter θi, which are

obtain these values the full marginalised posterior is calculated for each parameter as

Ppθi|Dq “

ż

Ppθ1, θ2,¨ ¨ ¨, θN|Dqdθk‰i. (1.57)

This posterior distribution includes all the information for the given model param- eter, from both the prior and the data. The uncertainty from all of the model parameters is taken into account in the uncertainty of the parameter of interest for which the integral is computed. To define a particular value for model parameter

θi, the median value, or peak value of the distribution can be taken. Alternatively,

the maximum a posteriori value of the parameter can be estimated,θM AP_i , which is the value which maximises the posterior,Ppθ|Dq.

For low-parametric models the integrals in Equation (1.57) can be calculated directly using standard numerical integration methods. However, this is not possible for models with a large set of parameters due to the increase in the computation time. Therefore, sampling methods, such as Markov Chain Monte Carlo (MCMC), are often used for complex models. This is described further in Section 1.7.4.

For model comparison purposes the Bayes factor can be obtained from the ratio of the Bayesian evidence (1.56) for two models to be compared. This allows us to quantify how plausible one model is compared to the other. For two models

Mi and Mj the Bayes factor is defined as

Bij “

PpD|Miq

PpD|Mjq

, (1.58) where PpD|Mq are defined as above. To define evidence thresholds the natural logarithm of this factor,i.e.

Kij “2 lnBij, (1.59)

is often considered, where values of Kij greater than 2, 6 and 10 correspond to

“positive”, “strong”, and “very strong” evidence for model Mi over model Mj,

respectively [Kass & Raftery, 1995]. Negative values indicate evidence for model

Mj subject to the same thresholds.