distributions. A schematic illustration showing both the true and the reduced
posteriors can be found in Figure 17.

For $\beta_{\ce{H2O}}$ (slope of H2O abundance profile), we choose a broad
prior that allows for both vertically increasing and decreasing H2O profiles.

Figure 17: Reduction of selected retrieved posterior distributions with
respect to $P_{0}$ (left plot) and $R_{\text{pl}}$ (right plot). The blue-
filled histograms on the diagonal of the corner plots show the true retrieved
parameter posteriors. Red-outlined histograms indicate the reduced posteriors.
Below the diagonal, we show the 2D parameter posteriors in blue (true
posteriors) and red (reduced posteriors) scatter plots. Black dashed-dotted
lines represent the fitted linear correlation between parameters. Solid red
lines mark the $P_{0}{}=1$ bar and $R_{\text{pl}}{}=1R_{\oplus}{}$ position
corresponding to the reduced posteriors, red-dotted lines indicate the median
of the reduced posteriors.

In the following, let us consider one point in the true posterior
distribution. We denote the value of $P_{0}$ or $R_{\text{pl}}$ parameter
(i.e., the parameter we want to reduce over) as
$\theta_{\mathrm{reduce,true}}$ and the remaining parameter values as
$\theta_{\mathrm{param,true}}$. We assume a linear correlation between
$\theta_{\mathrm{red,true}}$ and $\theta_{\mathrm{param,true}}$. This means
that we can predict a value $\theta_{\mathrm{param,pred}}$ for a given value
$\theta_{\mathrm{red,true}}$ as follows:

$\theta_{\mathrm{param,pred}}=m\cdot\theta_{\mathrm{red,true}}+q.$ (F1)

Here, $m$ is the slope and $q$ the offset with respect to the origin of the
linear model. $\theta_{\mathrm{param,pred}}$ is the parameter value predicted
by the linear model. We search for the best fit linear model by minimizing the
square difference $\Delta$ between $\theta_{\mathrm{param,pred}}$ and
$\theta_{\mathrm{param,true}}$:

$\Delta=\sum_{\mathrm{Posterior}}\left(\theta_{\mathrm{param,pred}}-\theta_{\mathrm{param,true}}\right)^{2}=\sum_{\mathrm{Posterior}}\left(m\cdot\theta_{\mathrm{red,true}}+q-\theta_{\mathrm{param,true}}\right)^{2}.$
(F2)

The best fit linear models are indicated in Figure 17 as black dash-dotted
lines. From the figure, we see that the correlations between the parameters
considered here are well described by our linear model. In the next step, we
fix the value of $\theta_{\mathrm{red,true}}$ to $\theta_{\mathrm{red,fix}}$
and calculate the reduced posterior values $\theta_{\mathrm{param,red}}$ of
the other parameters as follows:

$\theta_{\mathrm{param,red}}=\theta_{\mathrm{param,true}}+m\cdot\left(\theta_{\mathrm{red,fix}}-\theta_{\mathrm{red,true}}\right).$
(F3)

This yields the reduced posterior distribution of a parameter, which we plot
in Figure 17. This reduction method allows us to remove the effect of one
parameter on the posterior distribution, and identify the origin of biases in
the retrieval results.