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