small docs fixes

docs/src/api.md

@@ -1,14 +1,10 @@
 # API
 
-## Functions and macros
-
 ```@index
-Pages = ["internal_api.md"]
-Order = [:macro, :function]
+Pages = ["api.md"]
+Order = [:function]
 ```
 
-## Documentation
-
 ### analysis.jl
 ```@docs
 ZeroNuFit.Analysis.retrieve_real_fit_results

docs/src/config.md

@@ -1,3 +1,5 @@
+# Configuration file
+
 Table of contents:
 
 ```@contents

docs/src/index.md

@@ -20,6 +20,7 @@ Pages = [
     "inputs.md",
     "toys.md",
     "tutorial.md",
+    "api.md",
 ]
 Depth = 1
 ```

docs/src/inputs.md

@@ -1,3 +1,5 @@
+# Partitions and events
+
 The fit takes in inputs two files in JSON format (for a full customization of the fit), which paths have to be specified in the `config.json` file.
 
 Table of contents:

docs/src/likelihood.md

@@ -1,3 +1,5 @@
+# Likelihood implementation
+
 Table of contents:
 
 ```@contents
@@ -10,12 +12,12 @@ The implemented unbinned Likelihood function reads as:
 
 ```math
 \begin{aligned}
-    \mathcal{L}(\Gamma) = \prod_k \bigg[ \textrm{Pois}(s_k+b_k) \bigg[ \prod_{i_k=1}^{N_k} \frac{1}{s_k + b_k} \left( b_k\cdot p_{\rm b}(E) + s_{\rm k}\cdot p_{\rm s}(E) \right)  \bigg] \bigg]
+    \mathcal{L}(\Gamma,\, \boldsymbol{BI},\,\boldsymbol{\theta}|D) = \prod_k \bigg[ \textrm{Pois}(s_k+b_k) \bigg[ \prod_{i_k=1}^{N_k} \frac{1}{s_k + b_k} \left( b_k\cdot p_{\rm b}(E) + s_{\rm k}\cdot p_{\rm s}(E) \right)  \bigg] \bigg]
 \end{aligned}
 ```
 
+where $\Gamma$ is the signal rate, BI is the background index, $\boldsymbol{\theta}$ are the nuisance parameters, and $D$ are the observed data.
 Here, the first product runs over the number of partitions _k_ ($N_{\rm p}$ partitions in total) and the second over the events _i_ in a given partition ($N_{\rm k}$ events in total).
-
 In case no events are found in a given partition _k_, the above Likelihood expression simplifies into
 
 ```math
@@ -46,7 +48,7 @@ Taking $x=Q_{\beta\beta} - \Delta_{\rm k}$, the signal energy distribution for e
 \end{aligned}
 ```
 
-Alternatively, the signal energy distribution can also be shaped as a Gaussian with a tail at low energies (e.g. for MAJORANA DEMONSTRATOR), 
+Alternatively, the signal energy distribution can also be shaped as a Gaussian with a tail at low energies (e.g. for MAJORANA DEMONSTRATOR data), 
 
 ```math
 \begin{aligned}
@@ -68,7 +70,7 @@ and
 
 ```math
 \begin{aligned}
-s_{\rm k} = \frac{\text{ln}\,2\,\mathcal{N}_{\rm A}}{m_{\rm 76}} \cdot (\varepsilon_{\rm k} + \alpha \cdot \sigma_{\varepsilon_{\rm k}}) \cdot \mathcal{E}_{\rm k} \cdot \Gamma
+s_{\rm k} = \frac{\text{ln}\,2\cdot \mathcal{N}_{\rm A}}{m_{\rm 76}} \cdot (\varepsilon_{\rm k} + \alpha \cdot \sigma_{\varepsilon_{\rm k}}) \cdot \mathcal{E}_{\rm k} \cdot \Gamma
 \end{aligned}
 ```
 
@@ -86,10 +88,10 @@ We defined our "log Likelihood" ($LL$) as:
 \end{aligned}
 ```
 
-The sum over all partitions $k$ was separated in a sum over partitions containing an event $i$ ($j$) and in a sum over partitions with no events ($l$).
+The sum over all partitions $k$ was separated in a sum over partitions containing an event $i$ with energy $E_{\rm i}$ (sum with index $j$) and in a sum over partitions with no events (sum with index $l$).
 
 
-## Free parameter priors
+## Prior terms
 
 Different free prameters can be identified within the framework:
 - signal, $\Gamma$
@@ -140,7 +142,7 @@ The above products, then, can be expressed again as
 \end{aligned}
 ```
 
-### Marginalization and posterior distributions
+### Posterior distributions and marginalization
 
 The combined posterior probability density function is calculated according to Bayes’ theorem as:
 

docs/src/toys.md

@@ -1,3 +1,5 @@
+# Generating toys
+
 Table of contents:
 
 ```@contents

docs/src/tutorial.md

@@ -1,3 +1,5 @@
+# Tutorial
+
 The aim of this tutorial consists in building proper config JSON files in order to run a neutrinoless double-beta decay analysis over GERDA and MAJORANA DEMONSTRATO (MJD) published data.
 Additional info on the meaning of input parameters can be found under the "Configuration file" section, and for input files under the "Partitions and events" section.