Polynomial Fitting

We demonstrate in this tutorial how to use BayesBay to tackle a polynomial fitting problem. In the following, we utilise a third-degree polynomial, \(y(x) = m_0 + m_1 x + m_2 x^2 + m_3 x^3\), to generate data points \(y_1(x_1), y_2(x_2), \ldots, y_n(x_n)\). In matrix notation, these can be expressed as

\[\begin{split}\underbrace{\begin{pmatrix}y_0\\y_1\\\vdots\\y_N\end{pmatrix}}_{\text{Data}} = \underbrace{\begin{pmatrix}1&x_0&x_0^2&x_0^3\\1&x_1&x_1^2&x_1^3\\\vdots&\vdots&\vdots\\1&x_N&x_N^2&x_N^3\end{pmatrix}}_{\text{Forward operator or data kernel}} \underbrace{\begin{pmatrix}m_0\\m_1\\m_2\\m_3\end{pmatrix}}_{\text{Model}},\end{split}\]

or \(\mathbf{d}_{pred} = \mathbf{G m}\).

Having generated a synthetic data set using the model coefficients \(m_0\), \(m_1\), \(m_2\), and \(m_3\), we add noise to define the observed data \(\mathbf{d}_{obs} = \mathbf{d}_{pred} + \mathbf{e}\), where the \(i\)th entry of the observational error \(\mathbf{e}\) is randomly sampled from the normal distribution \(\mathcal{N}(0, \sigma)\). Finally, we use BayesBay to retrieve the true coefficients from the observations.

This tutorial comprises two parts:

• Part I: Known Data Noise
• Part II: Hierarchical Sampling