Prior Distributions

To sample the posterior of the PDF model parameters \(\boldsymbol{\theta}\), we must specify a prior distribution \(\pi(\boldsymbol{\theta})\).

Colibri’s release supports two prior types:

Uniform priors, with individually configurable bounds for each parameter.
Gaussian priors, defined by a mean vector and covariance matrix derived from the posterior samples of a previous fit.

The latter implements a Bayesian-update (posterior-factorisation): when you believe the earlier fit is a valid approximation (or exactly Gaussian), and the two datasets are uncorrelated, you can use the posterior of the first fit as a prior for the second fit.

Extra parameters of the forward map

Besides the PDF parameters, the forward map can have extra fit parameters, e.g. a normalisation or SMEFT Wilson coefficients. Their prior is independent of the PDF one:

prior_settings sets the prior of the PDF parameters only (the bayesian_prior provider, which takes the pdf_model). Bounds for parameters that are not in the PDF model are ignored, with a warning.
extra_prior_settings sets the prior of the extra parameters (the extra_parameters_prior provider). It has the same structure as prior_settings; by default only uniform_parameter_prior is supported, with per-parameter bounds or a global min_val and max_val. With the default provider, it is required as soon as the forward map has extra parameters.

Apps can override either provider with any prior, e.g. a non-uniform prior on the extra parameters, as long as it covers only its own part of the parameters.

prior_settings:                     # PDF parameters
  prior_distribution: uniform_parameter_prior
  prior_distribution_specs: {min_val: -1, max_val: 1}

extra_prior_settings:               # extra parameters
  prior_distribution: uniform_parameter_prior
  prior_distribution_specs:
    bounds: {norm: [0.5, 1.5]}      # or: {min_val: ..., max_val: ...}

The fit uses the product of the two priors, so any PDF prior, including the one of a PDF model app, can be combined with any extra parameters. The analytic fit takes its uniform prior bounds from the same two places.

Likewise, for the fits that need a starting point (Monte Carlo and Hessian), param_initialiser_settings initialises the PDF parameters, and extra_param_initialiser_settings the extra ones, with the same options (zeros, uniform, normal). The default for the extra parameters is zeros:

extra_param_initialiser_settings:
  type: uniform
  min_val: 0.9
  max_val: 1.1

In the next section we outline the basic idea and domain of validity of this posterior-factorisation approach.

Bayesian update

Let us suppose that experimental data comprising \(N_{\rm data}\) datapoints is distributed according to a multivariate normal distribution,

\[\begin{align} \mathbf{D} \sim N(FK(\boldsymbol{\theta}), \Sigma), \end{align}\]

where \(\Sigma\) is an \(N_{\rm data}\times N_{\rm data}\) experimental covariance matrix.

Since in Bayesian statistics, \(\boldsymbol{\theta}\) itself is assumed to be a random variable, it has some associated prior probability density \(\pi(\boldsymbol{\theta})\) which, in what follows, we will assume to be a “sufficiently” wide uniform probability density. Bayes’ theorem then tells us that after an observation \(\mathbf{D}_0\) of \(\mathbf{D}\), the probability density of \(\boldsymbol{\theta}\) is:

(1)\[p(\boldsymbol{\theta}\mid\mathbf{D}_0) = \frac{\pi(\boldsymbol{\theta})\,L(\mathbf{D}_0\mid\boldsymbol{\theta})}{Z} = \frac{\pi(\boldsymbol{\theta}) \exp\!\bigl(-\tfrac12 \|\mathbf{D}_0 - FK(\boldsymbol{\theta})\|^2_{\Sigma}\bigr)} {\displaystyle \int d\boldsymbol{\theta}\;\pi(\boldsymbol{\theta}) \exp\!\bigl(-\tfrac12 \|\mathbf{D}_0 - FK(\boldsymbol{\theta})\|^2_{\Sigma}\bigr)} ,\]

where we wrote the generalised \(L_2\) norm as:

\[\|\vec{x}\|^2_{\Sigma} = \vec{x}^T\,\Sigma^{-1}\,\vec{x} \quad\text{for}\quad\vec{x}\in\mathbb{R}^{N_{\rm data}}.\]

Now let’s assume that \(\mathbf{D}_0 = (\mathbf{D}_1, \mathbf{D}_2)^T\), with \(\mathbf{D}_1\in\mathbb{R}^{n_1}\), \(\mathbf{D}_2\in\mathbb{R}^{n_2}\) and \(n_1+n_2 = N_{\rm data}\), and that the two measurements are uncorrelated. That is, the covariance matrix factorises;

\[\Sigma = \Sigma_1 \oplus \Sigma_2, \quad \Sigma_1\in\mathbb{R}^{n_1\times n_1}, \;\; \Sigma_2\in\mathbb{R}^{n_2\times n_2}.\]

In this case, since the likelihood \(L(\mathbf{D}_0\mid\boldsymbol{\theta})\) factorises (block-diagonal \(\Sigma\)), we can write (1) as:

(2)\[p(\boldsymbol{\theta}\mid\mathbf{D}_0) = \frac{ \pi(\boldsymbol{\theta}) \exp\!\bigl(-\tfrac12\|\mathbf{D}_1-FK_1(\boldsymbol{\theta})\|^2_{\Sigma_1}\bigr) \exp\!\bigl(-\tfrac12\|\mathbf{D}_2-FK_2(\boldsymbol{\theta})\|^2_{\Sigma_2}\bigr) } { \displaystyle \int d\boldsymbol{\theta}\;\pi(\boldsymbol{\theta}) \exp\!\bigl(-\tfrac12\|\mathbf{D}_1-FK_1(\boldsymbol{\theta})\|^2_{\Sigma_1}\bigr) \exp\!\bigl(-\tfrac12\|\mathbf{D}_2-FK_2(\boldsymbol{\theta})\|^2_{\Sigma_2}\bigr) } ,\]

where we write \(FK(\boldsymbol{\theta}) = (FK_1(\boldsymbol{\theta}), FK_2(\boldsymbol{\theta}))^T\).

Now, by noticing that the posterior for parameters \(\boldsymbol{\theta}\) given only \(\mathbf{D}_1\) is:

(3)\[p_{\mathbf{D}_1}(\boldsymbol{\theta}\mid\mathbf{D}_1) = \frac{\pi(\boldsymbol{\theta}) \exp\!\bigl(-\tfrac12\|\mathbf{D}_1-FK_1(\boldsymbol{\theta})\|^2_{\Sigma_1}\bigr)} {\displaystyle \int d\boldsymbol{\theta}\;\pi(\boldsymbol{\theta}) \exp\!\bigl(-\tfrac12\|\mathbf{D}_1-FK_1(\boldsymbol{\theta})\|^2_{\Sigma_1}\bigr)} = \frac{\pi(\boldsymbol{\theta}) \exp\!\bigl(-\tfrac12\|\mathbf{D}_1-FK_1(\boldsymbol{\theta})\|^2_{\Sigma_1}\bigr)} {Z_1} ,\]

we can rewrite (2) as:

\[p(\boldsymbol{\theta}\mid\mathbf{D}_0) = \frac{ Z_1\,p_{\mathbf{D}_1}(\boldsymbol{\theta}\mid\mathbf{D}_1) \,\exp\!\bigl(-\tfrac12\|\mathbf{D}_2-FK_2(\boldsymbol{\theta})\|^2_{\Sigma_2}\bigr) } { \displaystyle \int d\boldsymbol{\theta}\;Z_1\,p_{\mathbf{D}_1}(\boldsymbol{\theta}\mid\mathbf{D}_1) \,\exp\!\bigl(-\tfrac12\|\mathbf{D}_2-FK_2(\boldsymbol{\theta})\|^2_{\Sigma_2}\bigr) } = \frac{ p_{\mathbf{D}_1}(\boldsymbol{\theta}\mid\mathbf{D}_1) \,\exp\!\bigl(-\tfrac12\|\mathbf{D}_2-FK_2(\boldsymbol{\theta})\|^2_{\Sigma_2}\bigr) } { \displaystyle \int d\boldsymbol{\theta}\;p_{\mathbf{D}_1}(\boldsymbol{\theta}\mid\mathbf{D}_1) \,\exp\!\bigl(-\tfrac12\|\mathbf{D}_2-FK_2(\boldsymbol{\theta})\|^2_{\Sigma_2}\bigr) }.\]

Note that, if we have a measurement \(\mathbf{D}\sim N(FK(\boldsymbol{\theta}),\Sigma)\) with covariance matrix given by the direct sum of individual contributions,

\[\Sigma = \Sigma_1 \oplus \Sigma_2 \oplus \dots \oplus \Sigma_n,\]

we can apply this recursively. The posterior after all \(n\) uncorrelated blocks is:

\[p(\boldsymbol{\theta}\mid\mathbf{D}_0) = \frac{ \prod_{i=1}^{n-1} p_{\mathbf{D}_i}(\boldsymbol{\theta}\mid\mathbf{D}_i) \,\exp\!\bigl(-\tfrac12\|\mathbf{D}_n-FK_n(\boldsymbol{\theta})\|^2_{\Sigma_n}\bigr) } { \displaystyle \int d\boldsymbol{\theta}\;\prod_{i=1}^{n-1} p_{\mathbf{D}_i}(\boldsymbol{\theta}\mid\mathbf{D}_i) \,\exp\!\bigl(-\tfrac12\|\mathbf{D}_n-FK_n(\boldsymbol{\theta})\|^2_{\Sigma_n}\bigr) } ,\]

with each intermediate posterior for \(k>1\) defined by:

\[p_{\mathbf{D}_k}(\boldsymbol{\theta}\mid\mathbf{D}_k) = \frac{ \displaystyle \prod_{i=1}^{k-1} p_{\mathbf{D}_i}(\boldsymbol{\theta}\mid\mathbf{D}_i) \,\exp\!\bigl(-\tfrac12\|\mathbf{D}_k-FK_k(\boldsymbol{\theta})\|^2_{\Sigma_k}\bigr) } { \displaystyle \int d\boldsymbol{\theta}\;\prod_{i=1}^{k-1} p_{\mathbf{D}_i}(\boldsymbol{\theta}\mid\mathbf{D}_i) \,\exp\!\bigl(-\tfrac12\|\mathbf{D}_k-FK_k(\boldsymbol{\theta})\|^2_{\Sigma_k}\bigr) }.\]