text:
\section{Introduction}
\label{sec:intro}
Launched in April 2018, the Transiting Exoplanet Survey Satellite (\textit{TESS}) space mission \citep{Ricker2015} has discovered about 98 confirmed new exoplanets and more than 2450 candidates\footnote{Source: NASA Exoplanet Archive \citep{Akeson2013}, as of 20 January 2021.}. Among all the confirmed exoplanets, one class of particular interest are the small-sized planets (R\,$<$\,5\,R$_{\earth}$) orbiting bright stars (V\,$<$\,11 mag) with periods shorter than 10 days. Within this parameter space there is the so-called Neptunian desert \citep{SzaboKiss2011,Mazeh2016}, which is a region that presents a significant paucity of hot/highly irradiated planets with respect to the overall planet population. This deficit could be seen as evidence of photoevaporation and/or tidal disruptions \citep{OwenLai2018} or of core-powered atmospheric mass loss mechanism \citep{Ginzburg2018,Gupta2019}. Currently, of all the known planets in the Neptunian desert only a very small number have precisely measured masses \citep[e.g.][]{West2019NGTS4b,Armstrong2020TOI849b, Jenkins2020LTT9779b}, preventing a comprehensive understanding of the formation history of these objects. On the other side, there is a population of sub-Neptune size planets exposed to a milder stellar irradiation that usually present a gas-rich envelope with equilibrium temperatures below 1000\,K \citep[e.g.][]{Morley2015,Crossfield2017,Gao2020Nat}.  In this paper we present the discovery and confirmation of TOI-220\,$b$, a sub-Neptune orbiting TYC\,8897-01263-1, an old (10.1\,Gyr) K0\,V high proper motion southern star with a magnitude of 9.69 in the \textit{TESS} bandpass (V=10.47). The brightness of the star, therefore, enabled a precise high SNR radial velocity (RV) follow-up campaign with the High Accuracy Radial velocity Planet Searcher \citep[HARPS,][]{Pepe2002} spectrograph mounted at the ESO La Silla 3.6\,m telescope, in the framework of the NCORES \citep[e.g.,][]{Nielsen2020,Armstrong2020TOI849b} and KESPRINT \citep[e.g.,][]{Gandolfi2018,Carleo2020} collaborations. The confirmation of the planetary nature of TOI-220\,$b$ is based on the simultaneous analysis of the \textit{TESS} photometry and HARPS RV measurements.

The rest of the paper is organized as follows. The gathered observations of the TOI-220 system are described in Sect.~\ref{sec:obs}. The stellar analysis, the combined modelling and the timing analysis of the transits are presented in Sect.~\ref{sec:analysis}. In Sect.~\ref{sec:composition} we describe the Bayesian analysis of the internal structure of TOI-220\,$b$ and the final discussion is presented in Sect.~\ref{sec:discussion}.

\section{Observations}
\label{sec:obs}

\subsection{\textit{TESS} observations}
\label{subsec:tess}

During {\it TESS} Southern ecliptic hemisphere observations, TOI-220 was observed through sectors 1--12 with the exception of sector~3. By the end of the writing of this work, the sectors 27 and 28 light curves were released, as part of the {\it TESS} extended survey in the Southern hemisphere. These data were included in the timing analysis (Sect.~\ref{sec:ttv}) and in the search for transits of the hypothetical planet $c$ (Appendix~\ref{sec:bls_residuals}). The transit signals of TOI-220.01 (aka TOI-220\,$b$) were detected by the \textit{TESS} Science Processing Operations Center \citep[SPOC,][]{Jenkins2016} transit search pipeline \citep{Jenkins2002,Jenkins2010, Li2019}. The transit detection threshold crossing event, with a periodicity of 10.695\,d, a depth of $\sim$900\,ppm, and a SNR$\sim$28, subsequently passed all SPOC data validation diagnostic tests \citep{Twicken2018} and was promoted to {\it TESS} Object of Interest (TOI) status as a planet candidate (Guerrero et al., submitted).
For each {\it TESS} sector, the light curve of the target was retrieved from the Mikulski Archive for Space Telescopes (MAST) archive. In this work, the two-minute cadence Presearch Data Conditioning Simple Aperture Photometry (PDCSAP) light curves \citep{Stumpe2012, Stumpe2014, Smith2012} detrended with the SPOC algorithms were used. Two or three transits of TOI-220\,$b$ were identified using a simple Box Least Squares (BLS) analysis on per-Sector light curves. After a first inspection of the flux time series of each sector some transits of TOI-220\,$b$ were discarded due to SNR considerations and/or due to large systematics in the flux time series. This was the case for the full light curve of Sector 11. A few transits with poor coverage (incomplete transits) were also discarded. Therefore, only one or two transits per \textit{TESS} Sector were finally extracted. A total of 17 out of the 24 transits of TOI-220\,$b$ observed by \textit{TESS} were considered for further analysis. After a first estimation of the individual time of transits using the period obtained from the BLS analysis, we extracted the photometric light curves around 0.4 days before/after the transit central time, when possible. The {\it TESS} sector, central time, and the 2\,min RMS of each transit used in this work are reported in Table~\ref{tab:tess_transits} and the light curves are shown in Fig.~\ref{fig:lcs_all_vertical}.

\begin{figure*}
    \centering
    \includegraphics[scale=0.6]{Figure01.png}
    \caption{TOI-220\,$b$ transit light curves extracted from the \textit{TESS} data. The black solid line represents the best transit model (Sect.~\ref{sec:pastis}). The labels indicate the corresponding \textit{TESS} sector. }
    \label{fig:lcs_all_vertical}
\end{figure*}

    \begin{table}
        \centering

        \caption{Summary of TOI-220\,$b$ transits extracted from the \textit{TESS} data.}
        \label{tab:tess_transits}
        \begin{tabular}{cccl}
            \hline
            \textit{TESS} & transit mid-time & RMS(2\,min) \\
            Sector & [BJD\_TDB] & [ppm]  \\
            \hline

            S01 & 2458335.9024 & 890 \\
            S01 & 2458346.5976 & 941 \\
            S02 & 2458357.2928 & 897 \\
            S02 & 2458378.6832 & 903 \\
            S04 & 2458421.4640 & 979 \\
            S04 & 2458432.1592 & 909 \\
            S05 & 2458442.8544 & 942 \\
            S05 & 2458453.5496 & 949 \\
            S06 & 2458474.9400 & 952 \\
            S06 & 2458485.6352 & 882 \\
            S07 & 2458496.3304 & 908 \\
            S07 & 2458507.0257 & 929 \\
            S08 & 2458528.4161 & 924 \\
            S09 & 2458549.8065 & 911 \\
            S10 & 2458592.5873 & 833 \\
            S12 & 2458635.3681 & 961 \\
            S12 & 2458646.0633 & 938 \\

            \hline
        \end{tabular}
    \end{table}
\normalsize

\subsection{HARPS follow-up}
\label{subsec:harps}

We obtained radial velocity measurements of TOI-220 with the HARPS spectrograph \citep{Mayor2003} mounted on the 3.6\,m telescope at ESO's La Silla Observatory. HARPS is a stabilized high-resolution spectrograph with resolving power of R\,$\approx$\,115000, capable of sub m\,s$^{-1}$ RV precision. In total 101 observations were taken between 18 November 2018 and 11 October 2019 in high accuracy mode, of which 91 were used in the combined fit. Observations were performed under programmes 1102.C-0923 (PI: Gandolfi), 1102.C-0249 (PI: Armstrong), 0102.C-0584 (PI: De Medeiros), 60.A-9700 and 60.A-9709 (technical time). A typical exposure time of 1800\,s was used, on occasion adjusted between 1500\,s and 2100\,s depending on sky condition and observing schedule. We achieved a typical signal-to-noise per pixel of 60 and an RV precision of 1.0 m\,s$^{-1}$.

Eight observations acquired on the nights between 25 and 27 November 2018 were excluded. On these dates the ThAr lamp used for wavelength calibration of HARPS was deteriorating and was subsequently exchanged on 28 November 2018. The changing flux ratio between the thorium and argon emission lines of the dying ThAr lamp induced a 2\,m\,s$^{-1}\,$d$^{-1}$ drift in the wavelength solution of the instrument over 5\,d. The problematic data were confirmed by comparing unpublished data from the HARPS-N solar telescope \citep{Dumusque2015, CollierCameron2019} and those of the Helios program on HARPS, which also observes the Sun daily. Another observation done on the night of the 19 January 2019 was excluded due to an earthquake on that night\footnote{ \url{https://www.volcanodiscovery.com/earthquakes/2019/01/20/01h32/magnitude7-NearCoastofCentralChile-quake.html}.}. Finally, one observation on the night of the 17 April 2019 was excluded due to its very low signal-to-noise ratio. This gives 91 useful spectra.

The data were reduced with the HARPS data reduction software \citep[{\sc DRS};][]{Lovis2007}. Radial velocity measurements were derived using the weighted cross-correlation function (CCF) method using a K0 numerical mask \citep{Pepe2002}. For each spectrum, the {\sc DRS} provides also the contrast, the full width at half maximum (FWHM), and the bisector inverse slope (BIS) of the cross-correlation function, as well as the Ca\,{\sc ii} H\,\&\,K lines activity indicator ($\log$\,R$^{\prime}_\mathrm{HK}$). We also extracted additional activity indexes and spectral diagnostics, namely, H$\alpha$, Na D, chromaticity (CRX), and differential line width (dLW), using the code {\sc SERVAL} \citep{Zechmeister2018}. The 91 {\sc DRS} and {\sc SERVAL} RV measurements and activity indicators are listed in Tables~\ref{tab:rv_data} and \ref{tab:serval_data}. Time stamps are given in Barycentric Julian Dates in Barycentric Dynamical Time (BJD\_TDB).
The RV time series along with the best fitting model (Sect.~\ref{sec:pastis}) are shown in Fig.~\ref{fig:rvs_models}. We found an RV jitter below 2\,m\,s$^{-1}$, consistent with the low activity level of the star (see Sect.~\ref{sec:spectra_analysis} and \ref{sec:pastis}).

\subsection{High resolution imaging}
\label{subsec:hr}

To search for contaminating stars in the \textit{TESS} photometric aperture, TOI-220 was observed on 9 January 2020 with the Zorro speckle instrument at Gemini South telescope\footnote{\url{https://www.gemini.edu/sciops/instruments/alopeke-zorro/}.}. Zorro provides simultaneous speckle imaging in two bands, 562\,nm and 832\,nm, with output data products including a reconstructed image, and robust limits on companion detections \citep{howell2011}. The contrast curve for the 832\,nm is shown in Fig.~\ref{fig:bsc_zorro832}. We also calculated from our high-resolution images the probability of contamination from blended and undetectable sources in the \textit{TESS} aperture. We call this the blended source confidence (BSC) and the steps for estimating this probability are fully described in \cite{lillo-box14b}. We used a {\sc python} implementation of this technique ({\sc bsc}, by J. Lillo-Box) which uses the {\sc trilegal}\footnote{\url{http://stev.oapd.inaf.it/cgi-bin/trilegal}.} galactic model \citep[version 1.6;][]{girardi12} to retrieve a simulated source population of the region around the corresponding target\footnote{This is done in {\sc python} by using the {\sc astrobase} implementation by \cite{astrobase}.}. From this simulated population, the density of stars around the target position (radius r=1$^{\circ}$) was derived with the associated probability of chance-alignment at a given contrast magnitude and separation. We used the default parameters for the bulge, halo, thin/thick disks, and the lognormal initial mass function from \cite{chabrier01}. The Zorro contrast curve was then used to constrain this parameter space and estimate the final probability of undetected potentially contaminant sources. We considered as potentially contaminant sources those with a maximum contrast magnitude of $\Delta m_{\rm max} = -2.5\log{\delta}$, with $\delta$ being the transit depth of the candidate planet in the \textit{TESS} band. This represents the maximum magnitude that a blended star can have to mimic this transit depth. We converted the depth in the \textit{TESS} passband to the Zorro filters by using simple conversions using the TIC catalogue magnitudes and linking the 562\,nm filter to the SDSSr band and the 832\,nm filter to the SDSSz band. The corresponding conversions imply $\Delta m_{\rm 562\,nm} = 0.954\Delta m_{\rm TESS}$ and $\Delta m_{\rm 832\,nm} = 0.920\Delta m_{\rm TESS}$.

We applied this technique to TOI-220 using the two contrast curves available from the Gemini/Zorro instrument. The result provides a very low probability for an undetected source capable of mimicking the transit signal of 0.055\,\% for the 562\,nm image and 0.015\,\% for the 832\,nm image. We underline that using high-resolution images allowed us to significaly improve these probabilities by 26\,\% and 81\,\% for the 562\,nm and 832\,nm images, respectively, when comparing to the raw probabilities (i.e, without high-resolution images). As an example, we show in Fig.~\ref{fig:bsc_zorro832} the contrast curve and the results of the BSC analysis. In addition, by using the {\sc tpfplotter} code \citep{aller20} we ruled out the presence of contaminant sources within the \textit{TESS} photometric aperture down to a $\Delta {\rm mag} =8$\,mag in the Gaia DR2 catalogue (see Fig. \ref{fig:tpfplot}).

\begin{figure}
    \centering
    \includegraphics[scale=0.5]{Figure02.pdf}
    \caption{Contrast curve from the Gemini/Zorro instrument for the 832\,nm filter (solid black line). The color on each angular separation and contrast bin represents the probability of a chance-aligned source with these properties at the location of the target, based on TRILEGAL model (see Sect.~\ref{subsec:hr} within the main text). The maximum contrast of a blended binary capable of mimicking the planet transit depth is shown as a dotted horizontal line. The green-shaded region represents the non-explored regime by the high-spatial resolution image. The BSC corresponds to the integration of $P_{\rm aligned}$ over this shaded region.}
    \label{fig:bsc_zorro832}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.5]{Figure03.pdf}
    \caption{Target Pixel File of the \textit{TESS} frame for TOI-220 corresponding to Sector 1 and computed with {\sc tpfplotter} \citep{aller20}. The SPOC pipeline aperture is highlighted as red squares and the Gaia DR2 catalogue is over plotted with symbol sizes proportional to the magnitude contrast with the target.}
    \label{fig:tpfplot}
\end{figure}

\subsection{Ground based photometric follow-up}

We observed transit ingresses of TOI-220\,$b$ in Pan-STARRS $z$-short band on 13 February 2019 and 21 December 2019 from the LCOGT \citep{Brown:2013} 1.0\,m network node at South Africa Astronomical Observatory. We used the {\tt TESS Transit Finder}, which is a customized version of the {\tt Tapir} software package \citep{Jensen:2013}, to schedule our transit observations. The $4096\times4096$ LCOGT SINISTRO cameras have an image scale of 0.389$\arcsec$ per pixel, resulting in a $26\arcmin\times26\arcmin$ field of view. The images were calibrated by the standard LCOGT {\sc banzai} pipeline, and photometric data were extracted with {\tt AstroImageJ} \citep{Collins:2017}. The 13 February 2019 images were focused and have typical stellar point-spread-functions with a FWHM of $1\farcs9$, and circular apertures with radius $4\farcs3$ were used to extract the differential photometry. The 21 December 2019 images were defocused and have typical stellar point-spread-functions with a FWHM of $5\farcs2$, and circular apertures with radius $7\farcs8$ were used to extract the differential photometry. The light curves are presented in Appendix \ref{fig:lco_transits}.

\section{Data analysis}
\label{sec:analysis}

\subsection{Stellar parameters}
\label{sec:spectra_analysis}

We derived the spectroscopic parameters and chemical abundance of TOI-220 from the co-added spectrum, which has an S/N per pixel at 5500\,\AA{} of $\sim$600.  We used {\sc ares+moog}, and followed the methodology described in \citet{Santos-13} and \citet{Sousa-14} to derive the stellar atmospheric parameters (T$_{\mathrm{eff}}$, $\log g$, microturbulence, [Fe/H]), and their respective uncertainties. We first measured equivalent widths (EW) of iron lines on the combined HARPS spectrum of TOI-220 using the {\sc ares} v2 code\footnote{The {\sc ares} v2 code can be downloaded at \url{http://www.astro.up.pt/~sousasag/ares}.} \citep{Sousa-15}. Then we used a minimization process where we assume ionization and excitation equilibrium to converge in the best set of spectroscopic parameters. This process makes use of a grid of Kurucz model atmospheres \citep{Kurucz-93} and the radiative transfer code MOOG \citep{Sneden-73}. Stellar abundances of the elements were derived using the classical curve-of-growth analysis method assuming local thermodynamic equilibrium \citep[e.g.][]{Adibekyan-12, Adibekyan-15, Delgado-14, Delgado-17}. Abundances of the volatile elements, C and O, were derived following the method of \cite{Delgado-10, Bertrandelis-15}. Since the two analysed spectral lines of oxygen (6158.17\AA{} and 6300.3\AA{}) are usually weak and the 6300.3\AA{} line is blended with a Ni line \citep[see e.g.][]{Bertrandelis-15}, the EWs of these lines were manually measured with the task {\sc splot} in {\sc IRAF}. All the [X/H] ratios are obtained by doing a differential analysis with respect to a high S/N solar spectrum from HARPS, obtained from archival observations of the asteroid Vesta.

The stellar parameters and abundances of the elements are presented in Table \ref{tab:stellar_parameters}. These results are in agreement with the values obtained from the Bayesian analysis described in Sect.~\ref{sec:pastis}. We found that the [X/Fe] ratios of $\alpha$ elements are slightly above solar. This fact together with a metallicity of -0.22\,dex indicate that TOI-220 is probably an old star from the thin disk \citep[e.g.][]{Delgado-19}. Moreover, we used the chemical abundances of some elements to derive ages through the so-called chemical clocks (i.e. certain chemical abundance ratios which have a strong correlation for age). We applied the 3D formulas described in Table~10 of \citet{Delgado-19}, which also consider the variation in age produced by the effective temperature and iron abundance. We used the chemical clocks [Y/Mg], [Y/Zn], [Y/Ti], [Y/Si], [Y/Al], [Sr/Ti], [Sr/Mg] and [Sr/Si] from which we obtain a weighted average age of 10.1\,$\pm$\,1.4 Gyr. This age is in agreement (within their uncertainties) with the age obtained from the Bayesian analysis described in the Sect.~\ref{sec:pastis}.

Following \citet{Johnson-87} we calculated the space velocity components (UVW) with respect to the local standard of rest, adopting the standard solar motion (U$_{\odot}$, V$_{\odot}$, W$_{\odot}$) = (11.1, 12.24, 7.25) km\,s$^{-1}$ of \citet{Schonrich-10}. Then, following \citet{Bensby-03} we calculated the probability of the star belonging to different stellar populations. For more details about the calculations of the kinematic properties of the stars see \citet{Adibekyan-12}. Our calculations suggest that TOI-220 belongs to the galactic thin disk with a 98\,\% probability, in agreement with the chemical classification.

Finally, \citet{Martins2020a} studied TOI-220 in their recent study of 1000 TOIs and found no evidence of a coherent rotation period in the photometry, indicative of a low activity level. With a mean value of the Ca\,{\sc ii} H\,\&\,K lines activity index of $\log$\,R$^{\prime}_\mathrm{HK}$\,=\,$-5.07\pm0.05$, TOI-220 is magnetically less active than the Sun whose $\log$\,R$^{\prime}_\mathrm{HK}$ ranges from $-$4.86 to $-$4.95 from solar maximum to minimum \citep{Hall2007}.

\subsection{Periodogram analysis of the HARPS measurements}
\label{sec:periodograms}

We performed a frequency analysis of the HARPS RVs and spectral diagnostics to search for the Doppler reflex motion induced by TOI-220\,$b$ and unveil the presence of additional signals that might be associated to stellar activity and/or arise from the orbital motion of other planets in the system.

The generalized Lomb-Scargle (GLS) periodogram \citep{Zechmeister2009} of the HARPS DRS RVs (Fig.~\ref{fig:periodograms_activity}, upper panel) shows a significant peak at f$_\mathrm{b}$\,=\,0.093\,d$^{-1}$, i.e., the orbital frequency of the transiting planet detected in the \textit{TESS} light curve. We estimated its false-alarm probability (FAP) using the bootstrap method described in \cite{Murdoch1993}. Briefly, we computed the periodogram of 10$^6$ mock time-series obtained by shuffling the RV measurements and their uncertainties, while keeping the time-stamps fixed. We defined the FAP as the fraction of those periodograms whose highest power exceeds the observed power of f$_\mathrm{b}$ in the periodogram of the original HARPS data at any frequency. We found no false positives out of our 10$^6$ trials, implying that f$_\mathrm{b}$ has a FAP\,$<$\,10$^{-6}$. Moreover, the peak at f$_\mathrm{b}$ does not appear in any of the periodograms of the activity indicators\footnote{For the CCF's FWHM and BIS we arbitarily adopted uncertanties twice as large as those of the RV measurements.} (Fig.~\ref{fig:periodograms_activity}), confirming that the Doppler signal is induced by the presence of a \emph{bona fide} planet, namely, TOI-220\,$b$.

We subtracted the Doppler signal of the transiting planet TOI-220\,b from the HARPS RVs by fitting a circular model,fixing the period and T$_0$ to the values derived from the modelling of the \textit{TESS} light curves (with T$_0$ being equal to the mid-time of the first \textit{TESS} transit), while allowing for the systemic velocity and RV semi-amplitude to vary. The GLS periodogram of the RV residuals displays significant (FAP\,$\,<\,$\,0.1\,\%) power at frequencies lower than $\sim$0.008\,d$^{-1}$ (125 d), with a peak at about 0.0032\,d$^{-1}$  (309 d) (Fig.~\ref{fig:periodograms_activity}, second panel). Despite this peak being undetected in the periodograms of the activity indicators and line asymmetry diagnostics (Fig.~\ref{fig:periodograms_activity}) -- suggesting the presence of an additional outer companion -- we note that its period is poorly constrained, being comparable to the temporal baseline of our observations (327 d). Moreover, the position of this peak depends on the last poorly-sampled measurements. Therefore, as discussed in Sect.~\ref{sec:rv_1pl}, the exact nature of this long period signal can not be firmly constrained with the currently available RV data.

\begin{figure}
\centering
    \includegraphics[width=\linewidth, trim={0 1.9cm 0 0cm}, clip]{Figure04.pdf}
    \caption{From top to bottom: Generalized Lomb-Scargle periodograms of the HARPS RV measurements (upper panel); the RV residuals following the subtraction of the signal of the transiting planet (second panel); the activity indicators of TOI-220 (remaining panels). The 0.1\,\% false alarm probability (FAP) estimated using the bootstrap method is shown with horizontal dashed lines. The vertical red lines mark the orbital frequencies the transiting planet TOI-220\,$b$ (f$_\mathrm{b}$\,=\,0.093\,d$^{-1}$) and of the additional Doppler signal we found in the HARPS RVs ($f_\mathrm{c}$\,=\,0.0032\,d$^{-1}$). }
    \label{fig:periodograms_activity}
\end{figure}

\subsection{PASTIS analysis}
\label{sec:pastis}
The \textit{Planet Analysis and Small Transit Investigation Software} \citep{Diaz2014, Santerne2015}, {\sc pastis}, was used for the joint analysis of the transit light curves, the radial velocities and the spectral energy distribution of TOI-220. The {\sc pastis} package implements a fully Bayesian analysis of the data and uses the Markov Chain Monte Carlo (MCMC) method to sample the posterior distribution of the resulting parameters \citep[see e.g.][and references therein]{Santerne2019, Lopez2019}.  We draw the posterior distributions of all free parameters after merging the best MCMC chains, i.e. those with the largest model likelihood ($\log$\,{\it L}) from a sample of 30 chains with $10^6$ steps each. The convergence of these chains is checked by a Kologorov-Smirnov (K-S) test after the \textit{burn-in} period (BI) of each chain has been removed. Here, the BI phase corresponds to the initial portion of each chain with a $\log$\,{\it L} mean and variance 2\,$\sigma$ away from the corresponding values of the last 10\,\% of the chain. Then, after removing the BI phase, the selected chains correspond to those with a median $\log$\,{\it L} within 2\,$\sigma$ of the {\it best} chain (i.e. the chain with the largest median $\log$\,{\it L}) and with a K-S test p-value above 10$^{-30}$.

As mentioned in Sect.~\ref{sec:obs}, 17 individual transits of TOI-220\,$b$ were extracted from the \textit{TESS} light curves and input to {\sc pastis}.
Additionally, a nightly binning was applied to the RV data in order to minimize the jitter induced by stellar activity and/or granulation, ending up thus with 69 points out of the total 91 RV measurements. The transits were modelled using {\sc jktebop} software \citep{Southworth2008} and the RV data using keplerian orbits.

\subsubsection{Stellar SED analysis}
\label{sec:pastis_sed}

{\sc pastis} also models the host star by fitting the spectral energy distribution (SED) to the BT-Settl atmosphere models \citep{Allard2012}, Dartmouth evolutionary tracks \citep{Dotter2008} and the quadratic limb darkening coefficients for \textit{TESS} bandpass based on the tables from \cite{Claret2011}.

After a preliminary analysis using all the available magnitudes of TOI-220 (Table \ref{tab:star-ids}) we obtained a very poor fit of the visual magnitudes, in particular in the V band. We attributed this result to inconsistencies with the magnitude zero point used or with unreliable V magnitude estimations due to saturation of stars brighter than V=10\,mag in the case of the APASS catalogue \citep{Henden2015}. As {\sc pastis} does not implement the use of Gaia magnitudes by the time of this analysis, we decided to use only 2MASS \citep{2MASS} and AllWISE \citep{AllWISE2013} magnitudes for the bayesian analysis. For this, we took TOI-220's NIR magnitudes as reported in the \textit{TESS} Input Catalogue \citep[TICv8,][]{Stassun2019}. The host star was modelled for the effective temperature, T$_{\text{eff}}$; surface gravity, $\log$\,$g$; metallicity, [M/H]; distance, d; color excess, \textit{E(B$-$V)}; the systemic radial velocity, v$_{0}$, using as priors either normal or uniform distributions as presented in Table~\ref{tab:priors}. The SED and best stellar model obtained with {\sc pastis} are displayed in Fig.~\ref{fig:sed} and the fitted and derived stellar parameters are listed in Table~\ref{tab:stellar_parameters}. The resulting stellar parameters are consistent within their uncertainties with the values obtained independently from the spectral analysis described in Sect.~\ref{sec:spectra_analysis}.

\begin{figure}
    \centering
    \includegraphics[width=\linewidth]{Figure05.pdf}

    \caption{Top panel: spectral energy distribution of TOI-220 with the best stellar atmosphere model from the BT-SETTL library obtained with { \sc pastis} (blue solid line). Residuals are shown in the bottom panel (error bars include the fitted jitter from the analysis).  }
    \label{fig:sed}
\end{figure}

In order to corroborate the TOI-220's effective temperature obtained with {\sc pastis} and to discard any bias we could have introduced by using only NIR magnitudes, the observed and theoretical SEDs for TOI-220 were compared using the Virtual Observatory Spectral Analyzer \citep[VOSA,][]{Bayo2008}. The observed SED was constructed using the four IR bands {\it W1}-{\it W4} from allWISE, the {\it J}, {\it H}, and {\it Ks} bands from 2MASS \citep{2MASS}, and also including the 3 bands from Gaia \citep{Gaia2018}. As before, the theoretical fluxes were computed from the grids of theoretical stellar spectra BT-Settl \citep[AGSS2009,][]{Allard2012}. A best fit with the observed data was computed by $\chi^{2}$ minimization by testing effective temperature values ranging from 1000--7000\,K with steps of 100\,K. We also adopted the estimation of interstellar extinction provided in the \textit{TESS} input catalog \citep{Stassun2019}, then computing the reddening according to \cite{Fitzpatrick1999} and \cite{Schlafly2011}. The SED and the fitted stellar model are shown in Fig.~\ref{fig:SED2}. This independent analysis provides an effective temperature of 5300\,$\pm$\,50\,K, in very good agreement with the result from {\sc pastis}. This analysis also shows no IR-excess, suggesting the absence of circumstellar material around TOI-220, at least in the WISE infrared bands.

\begin{figure}
    \centering
    \includegraphics[width=\linewidth]{Figure06.pdf}
    \caption{Top: TOI-220\,$b$ transit light curve built by joining the 17 \textit{TESS} transits used in this work (blue points and 12\,min bins in red) together with the best model (solid curve) obtained with {\sc pastis} analysis. The residuals are shown in the bottom panel. }
    \label{fig:lc_all_joint}
\end{figure}

\subsubsection{Planetary analysis}
\label{sec:rv_1pl}

\begin{figure}
    \centering
    \includegraphics[width=\linewidth]{Figure07.pdf}
    \caption{Radial velocity time series with the best fitted model superimposed with the corresponding residuals are shown in the top and middle panel, respectively. The phased radial velocity measurements are shown in the bottom panel. The error bars include the fitted jitter for each case. Additional, a fitted quadratic long-term is represented by a solid red curve in the middle pannel (see Sect\,\ref{sec:rv_1pl} for details).}
    \label{fig:rvs_models}
\end{figure}

Based on the periodograms presented in Sect.~\ref{sec:periodograms}, the data analysis of the TOI-220 system was performed taking into account the presence of a single planet in the system to probe the 10.69\,d periodic signal. Additionally, we also included a quadratic, long-term RV trend in the modelling.

For the single planet scenario we modelled the planetary orbital period, P; the reference transit epoch, T$_{0}$; the star-planet radius ratio, $R_{\text{s}}/R_{\text{p}}$; the radial velocity semi-amplitude, K; and the inclination, eccentricity and argument of the periastron of the orbit: $i$, $e$ and $\omega$, respectively.
The prior distribution used for each parameter is reported in Table \ref{tab:priors}. The posterior distribution of all the free parameters was drawn from the best 15 MCMC chains after removing the \textit{burn-in} (BI) time (7--22\,\% of the chain). The full list of fitted and derived parameters of the system together with its respective 68.3\,\% confidence interval are given in Table \ref{tab:stellar_parameters} and \ref{tab:planet_results} for the star and planet, respectively.
Despite the good fit of the keplerian model, it seems that a structure is still present in the RV residuals (see e.g. Fig.~\ref{fig:periodograms_activity}, second panel). To better assess the significance of this trend, we applied the Anderson-Darling (A-D) test \citep{Stephens1974} to the RV residuals (Table~\ref{tab:stats_results}). The A-D test is a modification of the K-S test to evaluate whether a data sample comes from a specific distribution; in our case we tested the RV residuals against a Normal distribution. The test's result is compared then to a set of critical values in a range of significance levels. In particular, the A-D test statistic (see Table~\ref{tab:stats_results}) of the RV residuals is greater than the critical value at 15\,\% significance, 0.580 > 0.547 ($\alpha=0.15$), therefore the normality hypothesis of the RV residuals is rejected evidencing that they may contain astrophysical information not included in our single planet model.

Therefore, we also explored the effect of including a radial velocity quadratic trend in the RV modelling. For this, we used the same initial setup as before but adding two extra terms to account for a quadratic drift in the stellar radial velocity. We set a wide uniform distribution as a prior for the amplitude of the drift coefficients (Table~\ref{tab:priors}). Here, we selected the best 20 MCMC chains after removing the BI (7--27\,\% of the chain) to draw the posterior distributions of the fitted parameters. The fitted drift coefficients are reported in Table~\ref{tab:stellar_parameters} and the RV drift together with the 1-planet model are shown in Fig.\,\ref{fig:rvs_models}. Despite the amplitude of the residuals being smaller when compared to the single planet scenario, the structure in the first epochs (t < 2458440.20 BJD) in the RV residuals seems to remain. But in this case, the normality of the residuals is not rejected by the A-D test. We show in Table~\ref{tab:stats_results} the result of this and other statistical tests which favour the inclusion of the RV drift in the modelling. Notably, the TOI-220\,$b$ parameters are not affected by including this RV long term drift (Table\,\ref{tab:planet_results}). In Appendix\,\ref{sec:RVdrift-mass-constraint} we show the mass constraints of a hypothetical second orbital body based on the estimated RV drift.
Finally, we also checked the results when including a second planet in the modelling with a wide prior in the orbital period around the peak observed in the RV residual periodogram (300$\pm$100\,d). We found that the RMS of the RV residuals are reduced, however, the resulting parameters for the second planet are poorly constrained. Based on this together with the fact that all the obtained parameters of TOI-220\,$b$ remain fully compatible between models (see Table\,\ref{tab:planet_results}); and specially considering the baseline of the RV monitoring (only 327\,d compared with the periodogram peak of the RV residuals of $\sim$309\,d) and that the location of this peak strongly depends on the data points considered to calculate the periodogram; we adopt the values derived using the 1-planet with the quadratic RV long term drift shown in Tables~\ref{tab:stellar_parameters} and~\ref{tab:planet_results} for the star and planet, respectively. We obtained for TOI-220\,$b$ a planetary mass and radius of M$_{\text{p}}$=13.8$\pm$1.0\,M$_{\earth}$ and R$_{\text{p}}$=3.03$\pm$0.15\,R$_{\earth}$. The fitted orbital period and reference epoch are P=10.695264(86)\,d and T$_{0}$=2458335.9020(14) [BJD\_TDB], respectively. The best model for each individual transit is shown in Fig.~\ref{fig:lcs_all_vertical} and in Fig.~\ref{fig:lc_all_joint} for the complete folded light curve of the 17 \textit{TESS} transits. The best RV model including the RV quadratic drift and its residuals are displayed in Fig.~\ref{fig:rvs_models}.
As a sanity check, we performed an independent joint analysis of the transit photometry and HARPS Doppler measurements using the code \texttt{pyaneti} \citep{Barragan2019}. We adopted the same \texttt{PASTIS} RV models and found consistent results well within the nominal error bars, corroborating the results of our analysis.

\subsection{Transit Timing Variations}
\label{sec:ttv}

We performed the timing analysis of TOI-220\,$b$ using the 17 \textit{TESS} transits from sectors 1-12, the 4 transits of sectors 27-28; and the 2 transit observations obtained with the LCOGT telescopes. For this, we fixed all the transit parameters to the values obtained from {\sc pastis} analysis in Sect. \ref{sec:rv_1pl} except for the T$_{\text{c}}$, R$_{\text{p}}$/R$_{\text{s}}$ and the inclination which were let free to vary. As the two LCOGT transits have a relative lower quality in terms of photometric precision and sampling; and owing to the fact that they cover only the ingress of the transit (see Fig. \ref{fig:lco_transits}) we fixed also the inclination to i=87.88\,$\deg$ for the modelling of these light curves. For the mid-times of the \textit{TESS} transits, we used as priors a normal distribution with a 0.4\,d width centred in the values obtained from the BLS analysis (Table \ref{tab:tess_transits}). For each transit fit we used 20 MCMC chains of 10$^{5}$ iterations. The obtained values for R$_{\text{p}}$/R$_{\text{s}}$ and inclination are consistent within the uncertainties in all the transits. The resulting transit mid-times are reported in Table \ref{tab:o-c}. The average uncertainty of the transit mid-times is around 7\,min. The first LCOGT transit has an uncertainty larger than 30\,min in comparison with the 6\,min of second LCOGT transit. As this light curve has a very poor photometric quality and transit coverage, we removed it from the analysis. With the obtained mid-times and the ephemeris equation:
\begin{equation}
    \text{T}_{\text{c}}(n)=2458335.9021(14) [\text{BJD\_TBD}] +n\times10.695264(86),
    \label{eq:ephemeris}
\end{equation}
where P and T$_{0}$ are from  Sect. \ref{sec:rv_1pl}; we estimated the timing residuals of TOI-220\,$b$ as presented in Fig.~\ref{fig:ttv} and Table~\ref{tab:o-c}. All the transit times are consistent within the uncertainties with a constant orbital period; the RMS of the O$-$C points is around 4.7\,min. In addition, the statistical A-D test supports the hypothesis that the timing residuals are drawn from a normal distribution, discarding thus, orbital perturbations induced by a possible close companion.

\begin{figure}
    \centering
    \includegraphics[width=\linewidth]{Figure08.pdf}
    \caption{\textit{Observed minus Calculated} diagram of the transit mid-times of TOI-220\,$b$. Blue symbols correspond to \textit{TESS} transits while the red symbol correspond to the second LCOGT transit observation. The shaded region represents the projected 1$\sigma$ uncertainties of ephemeris equation of TOI-220\,$b$ (Sect. \ref{sec:ttv}).}
    \label{fig:ttv}
\end{figure}

    \begin{table}

        \caption{TOI-220 main IDs, celestial coordinates and magnitudes.}
        \label{tab:star-ids}
        \begin{tabular}{lll}
            \hline
            Parameter & value & reference \\
            \hline
            TIC & 150098860 & (1) \\
            TOI & 220 & (2) \\
            TYC & 8897-01263-1 & (1)\\
            APASS &29859752& (3)\\
            2MASS & J06071197-6159487 & (4) \\
            WISE &J060711.93-615949.5 & (5)\\
            Gaia DR2 & 5481210874877547904 & (6) \\
            RA (J2000) & 06:07:11.967 & (1) \\
            DEC (J2000)&-61:59:48.895 & (1) \\
            $\mu_{\text{RA}}$ [mas\,yr$^{-1}$] & -17.815 $\pm$ 0.046 &  (6)\\
            $\mu_{\text{DEC}}$ [mas\,yr$^{-1}$] & -68.361 $\pm$ 0.042 & (6)\\
            \hline
            Band & value & reference  \\
            \hline
            \textit{TESS} & $9.688 \pm 0.006$&   (1)    \\
            Johnson-{\it V} &	$10.416 \pm 0.263$ & (3) \\
            Johnson-{\it B} & $11.152 \pm 0.032$ & (3) \\
            2MASS-{\it J}	  & $9.030	\pm0.024$ &  (4) \\
            2MASS-{\it H}&	$8.631	\pm0.031$ & (4)  \\
            2MASS-{\it Ks}&	$8.542	\pm0.025$ & (4)  \\
            WISE-{\it W1}&	$8.476	\pm0.023$ & (5)  \\
            WISE-{\it W2}&	$8.525	\pm0.020$ & (5)  \\
            WISE-{\it W3} & $8.381	\pm0.021$ & (5)  \\
            WISE-{\it W4} & $8.526	\pm0.167$ & (5)  \\
            Gaia {\it G} & 10.1897 $\pm$ 0.003 &  (6) \\
            Gaia {\it B$_{p}$} & 10.6078 $\pm$ 0.0012 &  (6) \\
            Gaia {\it R$_{p}$} & 9.6408 $\pm$ 0.0007 &  (6) \\
            \hline
        \multicolumn{3}{l}{(1) TIC v8 \citep{Stassun2019}.} \\
        \multicolumn{3}{l}{(2) TOI Catalogue (Guerrero et al. Submitted).} \\
        \multicolumn{3}{l}{(3) AAVSO Photometric All-Sky Survey \citep{Henden2015}.} \\
        \multicolumn{3}{l}{(4) Two Micron All Sky survey \citep{2MASS}.}\\
        \multicolumn{3}{l}{(5) AllWISE Catalogue \citep{AllWISE2013}.}  \\
        \multicolumn{3}{l}{(6) Gaia DR2 \citep{Gaia2018}.}\\

        \end{tabular} \\
    \end{table}

\normalsize

    \begin{table}

        \caption{Stellar properties of TOI-220.}
        \label{tab:stellar_parameters}
        \begin{center}
        \begin{tabular}{lll}
            \hline
            Parameter & Value & Ref. \\
            \hline
            T$_{\text{eff}}$ [K] & 5273 $\pm$ 115 & (1)  \\
            T$_{\text{eff}}$ [K] ({\it adopted}) & 5298 $\pm$ 65 &  (2) \\
            T$_{\text{eff}}$ [K] &  $5384 \pm 77$ & (3) \\
            T$_{\text{eff}}$ [K] &  $5182 \pm 45$ & (4) \\
            $\log$\,$g$ [cgs] ({\it adopted}) & 4.37 $\pm$ 0.11 &  (2) \\
            $\log$\,$g$ [cgs] & 4.25 $\pm$ 0.07 &  (4) \\
            $\log$\,$g$ [cgs] & 4.49 $\pm$ 0.03 &  (3) \\
            V$_{\text{turb}}$ [km\,s$^{-1}$] & 0.694 $\pm$ 0.058 & (2) \\
            {}[Fe/H] [dex]& -0.217 $\pm$ 0.044 & (2) \\
            {}[M/H] [dex]& -0.19 $\pm$ 0.05 & (3)\\
            {}[Fe/H] [dex]& -0.2 $\pm$ 0.07 & (4) \\
            {}[Mg/H] [dex]& -0.10 $\pm$ 0.05 & (2) \\
            {}[Si/H] [dex]& -0.14 $\pm$ 0.04 & (2) \\
            {}[O/H]  [dex]& -0.031 $\pm$ 0.072 & (2) \\
            {}[C/H]  [dex]& -0.214 $\pm$ 0.045 & (2) \\
            {}[Cu/H] [dex]& -0.176 $\pm$ 0.044 & (2) \\
            {}[Zn/H] [dex]& -0.172 $\pm$ 0.027 & (2) \\
            {}[Sr/H] [dex]& -0.156 $\pm$ 0.077 & (2) \\
            {}[Y/H]  [dex]& -0.293 $\pm$ 0.094 & (2) \\
            {}[Zr/H] [dex]& -0.236 $\pm$ 0.071 & (2) \\
            {}[Ba/H] [dex]& -0.237 $\pm$ 0.025 & (2) \\
            {}[Ce/H] [dex]& -0.166 $\pm$ 0.045 & (2) \\
            {}[Nd/H] [dex]& -0.100 $\pm$ 0.039 & (2) \\
            $v$\,sin\,$i$ [km\,s$^{-1}$]& 2.90 $\pm$ 0.35 & (4) \\
            V$_{\text{mac}}$ [km\,s$^{-1}$]& 0.9 $\pm$ 0.4 & (4)\\
            V$_{\text{mic}}$ [km\,s$^{-1}$]& 0.85 $\pm$ 0.10& (4)\\
            Age [Gyr] ({\it adopted}) & 10.1 $\pm$ 1.4 & (2) \\
            \hline
            Age [Gyr] & 10.7$^{+2.1}_{-3.1}$& (3) \\
            Distance [pc]  & 89 $\pm$ 4 & (3)\\
            {}{\it E(B$-$V)} [mag] & 0.163 $\pm$ 0.087 &(3) \\
            Systemic RV, v$_{0}$ [m\,s$^{-1}$]  & 26.45827 $\pm$ 0.00037 & (3)\\
            RV drift lin. coeff. [m\,s$^{-1}$\,d$^{-1}$] & -0.0379(80) & (3) \\
            RV drift quad. coeff. [m\,s$^{-1}$\,d$^{-2}$]& 0.000149(31) & (3) \\
            Radius [R$_{\sun}$] & 0.858 $\pm$ 0.032 & (3)\\
            Mass [M$_{\sun}$] & 0.825 $\pm$ 0.028  & (3) \\
            Density [$\rho_{\sun}$] &1.31 $\pm$ 0.15  & (3) \\

            Limb darkening coeff. u$_{\text{a}}^{\text{\tiny{TESS}}}$ &0.352 $\pm$ 0.014 & (3)\\
            Limb darkening coeff. u$_{\text{b}}^{\text{\tiny{TESS}}}$ &0.2475 $\pm$ 0.0077& (3)\\
            \hline
            \multicolumn{3}{l}{(1) TIC v8 \citep{Stassun2019}.} \\
             \multicolumn{3}{l}{(2) From spectral analysis (Sect.\ref{sec:spectra_analysis}). Values correspond to weighted }\\
            \multicolumn{3}{l}{ average and the standard deviation.} \\
            \multicolumn{3}{l}{(3) From {\sc pastis} analysis (Sect. \ref{sec:pastis}). Derived values assume}\\
            \multicolumn{3}{l}{R$_{\sun}$=695\,508\,km and M$_{\sun}$= 1.98842 $\times$ 10$^{30}$\,kg.} \\

            \multicolumn{3}{l}{(4) From spectral analysis validation (Appendix \ref{sec:stellar_prop2})}.\\
            \end{tabular} \\
        \end{center}
    \end{table}

    \begin{table*}

        \caption{Results on the planetary parameters obtained from {\sc pastis} simultaneous analysis of the data assuming a single planet system and by including a long term quadratic drift in the RVs. Derived values assume M$_{\earth}$ = 5.9736$\times$10$^{24}$\,kg, R$_{\earth}$ = 6\,378\,137\,m, AU = 149\,597\,870.7\,km and zero albedo for the equilibrium temperature.}
        \label{tab:planet_results}
        \begin{center}

        \begin{tabular}{lcc}
            \hline
            Parameter (fitted) & 1 Planet & 1 Planet + RV drift ({\it adopted})  \\
            \hline
            Orbital period, P [d] & 10.695264(87) & 10.695264(86) \\
            Reference transit time,  T$_{0}$ [BJD$\_$TDB] & 2458335.9021(14) & 2458335.9020(14) \\
            Planet-to-star radius ratio, R$_{\text{p}}$/R$_{\text{s}}$& 0.03227 $\pm$ 0.00078 & 0.03235 $\pm$ 0.00076 \\
            Orbital inclination, $i$ [$\degr$] &$87.86 \pm 0.14 $&  $87.88 \pm 0.12$\\
            RV semi-amplitude, K [m\,s$^{-1}$] &  4.47 $\pm$ 0.36 & 4.56 $\pm$ 0.32\\
            Orbital eccentricity, $e$ &  $0.029^{\tiny{+0.034}}_{-0.021}$& $0.032^{+0.038}_{-0.023}$\\
            Argument of periastron, $\omega$ [$\degr$] & 159 $\pm$ 150 & $248^{+66}_{-190}$\\
            \hline
            Parameter (derived) && \\
            \hline
            Mass [M$_{\earth}$] & $13.6 \pm 1.2$ & $13.8 \pm 1.0$ \\
            Radius [R$_{\earth}$] & $3.02 \pm 0.17$  &  $3.03 \pm 0.15$ \\

            bulk density, $\rho_{\text{p}}$ [g\,cm$^{-3}$] & 2.69 $\pm$ 0.55 & 2.73 $\pm$ 0.47  \\
            System scale, a/R$_{\text{s}}$ & 22.3 $\pm$ 1.0 & 22.33 $\pm$ 0.80  \\
            Impact parameter, $b$ & $0.832 \pm 0.026$ & $0.836 \pm 0.022 $\\
            Transit duration, T$_{14}$ [h] & 2.231 $\pm$ 0.054 & 2.239 $\pm$ 0.055\\
            orbital semi-major axis, a [AU] &  0.08920 $\pm$ 0.0010 & 0.08911 $\pm$ 0.0010\\
            Mean equilibrium temperature, T$_{\text{eq}}$ [K]& 806 $\pm$ 25 & 805 $\pm$ 21\\
            \hline
            Instrument-related parameters &&\\
            \hline
            HARPS jitter [m\,s$^{-1}$] & 1.71000 $\pm$ 0.00021 &1.35000 $\pm$ 0.00024 \\
            SED jitter [mag] & 0.048 $\pm$ 0.025 &0.049 $\pm$ 0.026 \\
            \textit{TESS} jitter [ppm] & 111$\pm$74 & 111$\pm$ 74 \\
            \textit{TESS} out-of-transit flux & 1.000093(30) & 1.000093(30)  \\
            \hline

        \end{tabular}\\
        \end{center}
    \end{table*}
\normalsize

\section{Planetary Structure}
\label{sec:composition}

\begin{figure}
    \centering
    \includegraphics[scale=0.4]{Figure09.pdf}
    \caption{Mass-Radius diagram of the confirmed exoplanets with mass and radius measurements better than 20\,\% and 10\,\%, respectively. The symbol's color scales with the equilibrium temperature of the planet. TOI-220\,$b$ location in the diagram is highlighted with the red arrow and black circle. Solar System planets are also shown for reference. Composition models from \citet{Zeng2019} are displayed with dash brown and green lines. Those for a planet at 800\,K with an Earth-like composition topped with various proportion of a steam and supercritical water layer \citep{mousis20} are displayed with blue solid lines.}
    \label{fig:MRTemp}
\end{figure}

\begin{table}
        \centering

        \caption{Output parameters retrieved by the MCMC interior and atmosphere modeling.}
        \label{tab:internal_structure_results}
        \begin{tabular}{lc}
            \hline
            Parameter & Value \\
            \hline
            Total mass, M$_{\text{p}}$ [M$_{\oplus}$] & 13.8$\pm$0.7 \\
            Total radius, R$_{\text{p}}$ [R$_{\oplus}$] & 3.06$\pm$0.12 \\
            Total density, $\rho_{\text{p}}$ [g cm$^{-3}$] & 2.65$\pm$0.28 \\
            Core mass fraction, CMF & 0.08$\pm$0.03 \\
            Water mass fraction, WMF & 0.62$\pm$0.10 \\
            Fe/Si mole ratio & 0.64$\pm$0.11 \\
            Mg/Si mole ratio & 1.16$\pm$0.10 \\
            Mg number, \# Mg & 0.85-1.0 \\
            Temperature at 300 bar, T$_{\text{surf}}$ [K] & 3536$\pm$203 \\
            Planetary albedo, $a_{\text{p}}$ & 0.230$\pm$0.001 \\
            Atmospheric thickness, z [km] & 1111$\pm$99 \\
            Atmospheric mass, M$_{\text{atm}}$ [M$_{\oplus} \ 10^{-3}$] & 1.31$\pm$0.16 \\
            Core+Mantle radius, [R$_{\text{p}}$ units] & 0.47$\pm$0.05 \\

            \hline
        \end{tabular}
    \end{table}

\begin{figure}
    \centering
    \includegraphics[width=\linewidth]{Figure10.pdf}
    \caption{Sampled 2D marginal posterior distribution function for the CMF and WMF of TOI-220\,$b$ (blue region). The PDF mean and the 1$\sigma$ confidence interval is marked by the yellow cross and curve, respectively. TOI-220\,$b$ is consistent with a water rich planet (WMF=0.62$\pm$0.10) with a very small core (CMF=0.08$\pm$0.03). Earth and Mercury values are presented for comparison. }
    \label{fig:ternary_diagram}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[width=\linewidth]{Figure11.pdf}
    \caption{Mass-Period diagram of the confirmed exoplanets to date with masses and periods below 100\,M$_{\earth}$ and 50\,d, respectively (same planets sample as Fig.~\ref{fig:MRTemp}). The symbols size represents the radius of the planet while its equilibrium temperature is coded by the color of the symbol. TOI-220\,$b$ is outlined with a black circle. The dashed lines mark the edges of the Neptune-desert from \citet{Mazeh2016}.}
    \label{fig:MPdiagram}
\end{figure}

Figure \ref{fig:MRTemp} shows the mass-radius diagram of all the confirmed exoplanets to date\footnote{from \url{http://exoplanet.eu/} as of 27 January 2021.}, with a precision better than 20\,\% and 10\,\% in their mass and radius, respectively. TOI-220\,$b$ is well located in a domain of the sub-Neptune size planets. We investigate the interior of TOI-220\,$b$ in order to get some insights into its composition. To that purpose we used the model of internal structure developed by \cite{brugger17} and recently updated by \cite{mousis20} to include a steam and supercritical water layer for the special case of highly-irradiated planets. When the pressure and temperature are above the critical point of water, water is in supercritical phase. To describe this water layer, we use the Equation of State (EOS) of \cite{Mazevet2019}, which includes experimental and theoretical data of water at pressures and temperatures higher than the critical point of water. TOI-220\,$b$ being strongly irradiated with an equilibrium temperature of 806~K, the model allows to address how its composition could differ from that of a Neptune-like planet with a large gaseous envelope overlying a rocky core. We followed the method described in \cite{Lillo-Box2020} and \cite{Acuna2021} exploring the core mass fraction (CMF, the ratio of the mass of the core and the total planetary mass) and the water mass fraction (WMF, the ratio of the mass of the water layer -steam and supercritical- and the total mass) of the planet as free parameters in an MCMC Bayesian framework adapted from \cite{dorn15}.

We use as input parameters the planetary mass and radius, and the Fe/Si and Mg/Si mole ratios. The latter are calculated with the stellar abundances in Table \ref{tab:stellar_parameters} as Fe/Si = 0.65$\pm$0.09 and Mg/Si = 1.17$\pm$0.17. The code allows to explore the posterior distribution functions of three compositional parameters: CMF, WMF, and the Mg number ($\#$\,Mg), which is a proxy for the level of differentiation of the planet as described by \cite{sotin07} and \cite{brugger17}. If most of the Mg is located in the mantle and Fe in the core, the planet is highly differentiated and $\#$\,Mg is close to one (0.9 for the case of Earth). On the contrary, if Mg and Fe are mixed in the mantle, $\#$\,Mg will have lower values ($\simeq$ 0.6). In the MCMC algorithm, we assume uniform prior distributions of the CMF, WMF and the Mg number.

Table \ref{tab:internal_structure_results} shows the retrieved values of the three free compositional parameters, in addition to the observables (mass, radius and mole ratios), and the atmospheric variables. The posterior distributions of the CMF and WMF are also shown in Fig.~\ref{fig:ternary_diagram}. We used an MCMC algorithm to obtain samples of the posterior distribution of all our observable parameters: mass, radius, Fe/Si and Mg/Si. Convergence is achieved when the output parameters of the model reproduce the observed values within the 1$\sigma$ confidence intervals. Our results indicate that a planetary structure composed of a core rich in Fe, a silicate mantle and a supercritical water layer topped by a steam atmosphere in radiative-convective equilibrium is a likely scenario for TOI-220\,$b$.

Since its host star is less enriched in Fe than the Sun and Earth (Fe/Si = 0.93), its CMF is significantly lower ($0.08\pm0.03$) than the terrestrial value, 0.32. As can be seen in Table \ref{tab:internal_structure_results}, the large $\#$\,Mg obtained (0.85--1.0) is indicative of a highly-differentiated core and mantle configuration such as the Earth ($\#$\,Mg=0.9). Furthermore, as shown in Fig.~\ref{fig:ternary_diagram}, TOI-220\,$b$ could be a water-rich planet with a minimum WMF of 52\%, which is compatible with the composition derived for water-rich satellites in the Solar System, such as Titan, and a maximum WMF of 72\%, which is below the average water proportion found in comets \citep[see Fig. 12 of][]{McKay2019}. Moreover, most of the water mass would be under pressure and temperature conditions beyond the supercritical point, forming a supercritical and steam water layer that would constitute 47\% of the total radius.

\section{Discussion}
\label{sec:discussion}

\begin{figure}
    \centering
    \includegraphics[width=\linewidth]{Figure12.pdf}
    \caption{Histogram of the planetary radius for planets with periods below 100\,d as presented in Fulton \& Petigura (2018).  The location of the radius gap and the size of TOI-220\,$b$ ($\pm$1\,$\sigma$) are shown with the black-dashed and orange-shaded lines, respectively.}
    \label{fig:radiusgap}
\end{figure}

The combined analysis of the photometric and spectroscopic data has fully confirmed the planetary nature of the TOI-220\,$b$ transiting candidate.
We noted the presence of a structure in the RV residuals on the exact nature of which, instrumental or planetary, we could not conclude with the  data currently available. Therefore, any further investigation of the long-period signals would requires additional RVs. Thus, we adopt the values derived using a single planet model in the system and including a quadratic long term trend in the RVs.
In Fig.~\ref{fig:MPdiagram} we show the mass-period diagram of all the confirmed exoplanets including TOI-220\,$b$ (same sample as Fig.~\ref{fig:MRTemp}). The symbol size scales with the radii of planets while the color represents the planet's equilibrium temperature. TOI-220\,$b$ is located in the region of the \textit{warm sub-Neptunes} class of exoplanets (T$_{\text{eq}}$<1000\,K) with a relatively large size (3\,R$_{\earth}$).

A paucity of exoplanets has been reported in the planet size distribution around 1.5-2.0\,R$_{\earth}$ \citep[e.g.][]{Fulton2017, Fulton2018, VanEylen2018}, in particular for a M$_{\star} \sim$ 0.8\,M$_{\sun}$ the gap is around 1.6\,R$_{\earth}$. It is suggested that this gap is driven by photoevaporation mass-loss which acts shifting planets with gaseous envelopes towards a rocky Super-Earth population (R<1.7\,R$_{\earth}$). TOI-220\,$b$ lies above the gap in the {\it Fulton's diagram} (see Fig.\,\ref{fig:radiusgap}) in the not so populated high end of the size distribution of planets (R$\gtrsim$3\,R$_{\earth}$) orbiting low mass stars (M$_{\star}$<0.97\,M$_{\sun}$), as can be seen in Fig.\,8 and 9 in \cite{Fulton2018}. The atmospheric composition of TOI-220\,$b$, either with a thick H/He layer or water rich content, suggests a formation in the protoplanetary disk past the snow line followed by slow migration. The host's age of $\sim$10\,Gyr points to a very evolved system and the large planet's radius suggests the planet was massive enough to keep its atmosphere despite the large irradiation received by its host star.
To get a first estimate of the atmospheric escape of TOI-220\,$b$, we used equations of Aguichine et al. (Submitted). We found that the mass lost by Jeans' escape is negilible even for light species (H/He) due to the surface gravity of the planet. Only $\sim$0.1\,M$_{\earth}$ could be removed by XUV photo-evaporation along its 10.1\,Gyr of life. This picture is consistent with the idea that the fate of warm sub-Neptunes located above the gap in the {\it Fulton's diagram} (Fig.\,\ref{fig:radiusgap}) is not to end their evolution as naked cores \citep[see][and references therein]{Deleuil2020, Bean2021}. Precise measurements of physical parameters of planets and its host stars, like the ones presented in this work, are key to shape the knowledge of the formation and evolution processes that sculpt the planet population in this region of parameter space.

Complementarily, we have performed an interior structure and atmosphere analysis within a Bayesian framework to constrain the composition of TOI-220\,$b$ given its density and the stellar abundances of the host star. The low density of this strongly irradiated planet could be explained with a water-rich atmosphere that reaches the supercritical phase at its base, with a silicate mantle and an iron core as its bulk. As an alternative the planet might have a solid core surrounded by a thick H/He atmosphere. Future work should explore this possibility but also a composition with a well-mixed water and H/He atmosphere, since these are the most common volatiles that can form low-mass planets. The confirmation of the water-rich atmosphere scenario would also require conducting atmospheric characterisation to search for water spectral features. Following \cite{Kempton2018} we thus calculate TOI-220\,$b$ transmission spectroscopy metric to check whether the planet would be amenable to such a characterization by JWST or the ELT. We obtained a value of 45, which is well below the threshold of 84 for planets in this range of size, making of TOI-220\,$b$ atmosphere likely out of reach for the next decade.

\section*{Acknowledgements}
SH acknowledges CNES funding through the grant 837319. MD acknowledges the CNES support. V.A., E.D.M., S.G.S., N.C.S., S.C.C.B. and S.Hoj. acknowledge support by FCT - Funda\c{c}\~ao para a Ci\^encia e a Tecnologia (Portugal) through national funds and by FEDER through COMPETE2020 - Programa Operacional Competitividade e Internacionaliza\c{c}\~ao by these grants: UID/FIS/04434/2019; UIDB/04434/2020; UIDP/04434/2020; PTDC/FIS-AST/32113/2017 \& POCI-01-0145-FEDER-032113; PTDC/FIS-AST/28953/2017 \& POCI-01-0145-FEDER-028953. V.A., E.D.M., S.C.C.B., S.G.S. and O.D.S.D further acknowledge the support from FCT through Investigador FCT contracts IF/00650/2015/CP1273/CT0001, IF/00849/2015/CP1273/CT0003, IF/01312/2014/CP1215/CT0004, CEECIND/00826/2018--POPH/FSE(EC) and DL57/2016/CP1364/ CT0004, respectively. S.Hoj. further acknowledges the support by FCT through the fellowship PD/BD/128119/2016. DJA acknowledges support from the STFC via an Ernest Rutherford Fellowship (ST/R00384X/1). L.M.S. and D.G. gratefully acknowledge financial support from the CRT foundation under Grant No. 2018.2323 ``Gaseous or rocky? Unveiling the nature of small worlds". JRM, BLCM and ICL acknowledge continuous grants from the brazilian funding agencies CNPq, CAPES and FAPERN. This study was financed in part by the Coordena\c{c}\~{a}o de Aperfei\c{c}oamento de Pessoal de N{\'i}vel Superior-Brasil (CAPES)-Finance Code 001. PK and JS acknowledge the MSMT INTER-TRANSFER grant LTT20015. This paper includes data collected with the \textit{TESS} mission, which are publicly available from the Mikulski Archive for Space Telescopes (MAST). Funding for the {\it TESS} mission is provided by NASA's Science Mission directorate. We acknowledge the use of public \textit{TESS} Alert data from pipelines at the \textit{TESS} Science Office and at the \textit{TESS} Science Processing Operations Center. This research has made use of the Exoplanet Follow-up Observation Program website, which is operated by the California Institute of Technology, under contract with the National Aeronautics and Space Administration under the Exoplanet Exploration Program. Resources supporting this work were provided by the NASA High-End Computing (HEC) Program through the NASA Advanced Supercomputing (NAS) Division at Ames Research Center for the production of the SPOC data products. This research has made use of computing facilities operated by CeSAM data center at LAM, Marseille, France; and the Data \& Analysis Center for Exoplanets (DACE), which is a facility based at the University of Geneva (CH) dedicated to extrasolar planets data visualisation, exchange and analysis. DACE is a platform of the Swiss National Centre of Competence in Research (NCCR) PlanetS, federating the Swiss expertise in Exoplanet research. This research has made use of data obtained from the portal exoplanet.eu of The Extrasolar Planets Encyclopaedia. This research has made use of the NASA Exoplanet Archive, which is operated by the California Institute of Technology, under contract with the National Aeronautics and Space Administration under the Exoplanet Exploration Program. This work makes use of observations from the LCOGT network. D.G. warmly thanks Alessia Barbiero and Pietro Giordano for the inspiring conversations at the time the HARPS follow-up of TOI-220 was started in November 2018.\\

\textit{Software}: We gratefully acknowledge the open source software which made this work possible: \texttt{astropy} \citep{astropy2013, astropy2018}, \texttt{ipython} \citep{ipython}, \texttt{numpy} \citep{VanDerWalt2011}, \texttt{scipy} \citep{scipy}, \texttt{matplotlib} \citep{matplotlib}, \texttt{jupyter} \citep{jupyter2016}, \texttt{pandas} \citep{pandas-scipy-2010,pandas2020}.\\

\textit{Facilities}: \textit{TESS}, ESO:3.6m (HARPS spectrograph), Gemini:South (Zorro), LCOGT, Exoplanet Archive

\section{Data Availability}
The TESS photometric data used in this work is available via Mikulski Archive for Space Telescopes (MAST) archive and TFOP program.  Radial velocity measurements and the derived stellar activity indicators are provided in Tables \ref{tab:rv_data} and \ref{tab:serval_data}, respectively.

\bibliographystyle{mnras}

redpajama_set_name: RedPajamaArXiv


Grill A Chef: I'm always on the hunt for a simple meal, do you have any suggestions?
I'm always on the hunt for a simple meal, do you have any suggestions?
There is something, I was just futzing with a recipe this week. It's very easy.
It's a take on a Middle eastern way to prepare rice and chicken. The result is a crispy flavorful rice with aromatic chicken to boot. The method alone should broaden your horizons. I use chicken thighs, but breasts could be substituted, or even a fat piece of fish.
1. Boil the Rice for ten minutes in 3 qts. of rolling boiling water. Strain it well, return it to the pot, and mix in the chopped dill.
2. While the rice in cooking, brown the chicken thighs well in a ten inch skillet (that has a lid) over medium high heat. I did this in two batches.
3. Remove the brown chicken thighs and dump out the excess oil. Reduce the heat to medium and drop in the butter.
4. Once completely melted, add the dill and rice mixture to the skillet and spread it out evenly (no packing down) Lay the chicken on skin side up.
5. Using a non-terry cloth tea towel, wrap the lid and place it snugly over the skillet. Reduce the heat to medium low and set your timer for 25 minutes.
6. Once the thing is cooked, remove the thighs, and use a skinny edged metal spatula to loosen the rice. You can turn it out onto a platter and serve the chicken right on top.
Every Sunday from 11am to 4pm - You'll find me there.
Thanks to the Greenmarket for all of the hospitality and support. I couldn't do it without them.
Major thanks to Ross Connard who has been helping me with design stuff. Check out some of his other work.
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In the event I am admitted to a hospital for medical treatment and I am unable to speak, write, or perform any functions independently, I direct that I do not want to be kept alive by life-support systems in order to extend my life. I wish to die peacefully and with my dignity intact. I do not want and do not permit anyone to sustain my life by providing me water, food, or anything else by tubes inserted into my nostrils, mouth, or any part of my body (I allow use of a urinary catheter). I do not permit and do not want surgery or use of electronic equipment if my heart stops beating. Pain-relieving drugs and sleeping pills should be used to stop the pain only.
Injury to the brain, discovered during an x-ray, scan, or use of other medical equipment. The brain injury in this instance is minor and can be treated by surgery and the possibility of returning to normal is 90%. This exception is on condition that the surgery is done within 36 hours after I am admitted to a hospital.
I have been acquainted with this person for many years and appoint ………………………. with full authority to decide on any questions related to my medical treatment that may arise.
I release all person from any responsibility or legal proceedings from following my orders or the order of my representative. On the other hand, any persons not following my orders above cannot subsequently require payment for medical treatment from me or the administrator of my estate.
Chaninat & Leeds assists with the translations of Supreme Court decisions. Chaninat & Leeds specializes in Thailand real estate acquisition. Chaninat & Leeds is managed by an American attorney who specializes in immigration with a focus on K1 Visa Thailand.


text:
Victor Bogorad, The Moscow Times
Somewere I read that the number of students of Russian language in the USA is about a quarter of its peak during the cold war. It's certainly true that Russian has not been offered at my Chicago-area high school for a long time, and the Russian-language faculty at my university is roughly a third of the size I remember.
The theme of a decline in Russian expertise in the West (in language, economics, politics, and so on) has been around a while. Controversy swirled, for instance, around the defunding of the USA's "Title VIII" program of support for Russian studies last fall. And now this deficit of experts is blamed as a factor behind the USA's recent supposedly inept handling of relations with Russia.
Sean Guillory's Facebook page just posted a link to The Moscow Times's article "Getting Russia Wrong" by Peter Rutland. The article itself offers three purported examples of botched policy resulting from "shallow and schematic misunderstanding of Russian politics...." Comments on Sean's post, taken together, provide a compact review of the things area experts can and cannot provide. Given experts' fondness for "nuance" and lack of practical policy guidance, Mark Schrad confesses that he's "just a little cynical about the role of 'expert knowledge' these days and in these circumstances." Andrew Gentes responds, in part, "... But the idea that we have to have an all or nothing approach (either we really know what's in Putin's head or we're just playing a random guessing game) seems to be an expression of frustration with the realities of geopolitics. Oh, and I will continue to defend nuance over mere insight or (gasp!) Ultimate Truth."
What should we reasonably expect from experts?
Facts and thoughtful interpretations of the past
A professional vocabulary adequate to communicate the details and nuances of their field
Concrete data about the correlation of forces in a given situation
Balanced sampling of participants' voices in all the constituencies involved
Linguistic and cultural context
Honesty about sources and methodology
Recommendations for action.
What are we unlikely to get from experts? (Surprise me!)
Exact predictions
Perfect ability to communicate their findings to nonspecialists
Mind-reading
Perfect objectivity
Total honesty about past errors; complete disinterest in career advancement
Fair descriptions of competing or opposing points of view.
What work is required of the experts' audience members (the end users of their expertise)? (Can you add to this list?)
Respect experts and support their work; they're part of the team, perhaps not as important as they might think, but as essential as our own five senses
Don't depend on just one expert, and take special care to identify contrasting viewpoints; discount snarky descriptions of rivals!
Don't expect experts to reveal all their biases
Make the effort to get an adequate command of their jargon, but don't be afraid to request plain English
Take their recommendations into account, but don't allow or expect them to do your priority-setting work for you
Ask whether they are at all invested in the well-being of the people they study (it's much easier to be glib about people and places you don't care about)
Beware of fashions and trends in the field
Be at least as honest as you expect your experts to be; don't feign certainty; don't hide behind your experts when you fail
Above all, know your own enduring goals and values, and apply expertise accordingly.
Russian and Ukrainian Baptist leaders meet in Kiev on April 8.
Christian stewardship: Quaker "Options" (part one--excellent!) and Basil the Great, "Woe to the Rich."
"Faith for the Post-Christian Heart: A Conversation with Francis Spufford."
Irina Khrunova on the latest Pussy Riot court decision. Meanwhile, "Russia Pulls Voice of America Radio Off Air" and "American Councils Statement on NGO Status in Russia." (News story here.)
Martin E. Marty on the "End of Elite Denominational Headquarters." The article made me nostalgic for the "elite" offices of Friends United Meeting on Quaker Hill in Richmond, Indiana, USA. FUM's modest premises share a beautiful little campus with the Quaker Hill Conference Center, and now also serve the Right Sharing of World Resources program. In an odd coincidence, the Right Sharing program occupies the very same office I used in my time at FUM (1993-2000). My job before coming to FUM? Coordinator of Right Sharing! But then it was based in my home office in Wilmington, Ohio, from where it moved to Cincinnati for most of Roland Kreager's tenure.
Today's antidote to the scourge of false-witness-bearing: Big Daddy Wilson pleads, "Walk a Mile in My Shoes."
Posted by Johan Maurer at 10 April
Labels: blues, discipleship, education, evangelism, gospel, personal, politics, rhetoric, russia, stewardship, theology, ukraine
I would like to add something to this discussion. Research on prediction in the social sciences shows that there is a big difference between "experts" or pundits on the one hand, and social scientists who work with data on the other. The former are about as good as predicting as monkeys throwing darts, i.e. not very good. This is because they are repeating subjective opinions and biases and not systematically analyzing data. So I would not be surprised if some Russian "area experts" don't have a clear understanding of what is going on. Where we can find useful insights is in research that utilizes data in a systematic and scientific way and that does comparative research. Last year pundits made all kinds of statements about the presidential election outcome in the US, and many of them got it wrong. The people analyzing data, like Nate Silver, predicted every state right, and did it long before election day. So, my point is that I think political science has a lot to say about what is unfolding in Russia and Ukraine, but this is not the same as all the "experts" necessarily.
Erik Cleven
Nate Silver patiently explained his methodology to many bemused and skeptical commentators. We do have a right to know an expert's methodology. It turns out that pundits too often draw from a ready-made set of talking points. I think we'll find out that, in contrast to the social scientist, there is no methodology at all in some cases. They are either hired to speak persuasively in favor of some predetermined strategy or ideology, or their own identities are wrapped up in those predetermined approaches. Yet, they ooze familiarity with their subject and with the power players.
But even genuine social scientists often do not arrive at consensus recommendations--which is why long-term values and goals play such a crucial role. A leader is someone who keeps that perspective while remaining in a dynamic dialogue with those experts and thoughtfully assessing their methods.
George F. Kennan was such a great role model in those early postwar years in part because he remembered the anchoring power of values. He was neither seduced by his own expertise nor distorted by fear.
Thanks, Erik! I'd love to visit and talk some of this over with you. [To be continued by e-mail.]

redpajama_set_name: RedPajamaCommonCrawl


I have a data set where I have about 2000 observations of about 150 features or covariates. I'm trying to predict a 0/1 outcome. The outcome is a risk score for disease, R1. The assay generates all the feature values (F1, ..., F150) in a database, as well as a risk value for a related disease (R0). The thought is that the R0 is highly predictive for R1, and that if we use the features F1, ..., F150 we might have some sort of update rule we can apply to R0. Stated a bit differently, I'm thinking of using R0 as the Bernoulli parameter for a prior distribution of R1 and somehow training an update formula to use on R0 based on the values of F1, ..., F150.
The training algorithm would have to allow a 0/1 probability score model to be built that allows us to specify a prior value for the score. This sounds like a pretty straigthforward Bayesian problem. But I haven't done this before and I'm not sure how the feature information would be incorporated into the parameter update expression.
Would we generate a second probability score?
Would we use the linear predictor prior to going into the logit transform?
Would we modify R0 in some additive way, or by some multipicative factor?
Clearly I have more questions than answers. If anyone can point me in the right direction that would be great. If JMP allows this sort of thing, I'd love to hear about it.
I'm a bit confused - you say your model uses 100% of the features (150 of them!). Virtually all machine learning models I have run never use all the features. You may enter them all as possible, but rarely (never) do they all get used. In any case, if you think your model is too complex, you can drop out each feature (tedious, but someone can probably help script this) individually to see how much the performance goes down. An alternate version of this - which I have seen in other software but not in JMP yet - is to replace each feature (one at a time) with random data and compare the performance.
You also appear to want to see if you can get better predictions by adding something to the model you already have (what is left to add if all the features are in it already?). I think there are many ways to do this - but I think more about trying different models, and using some kind of ensemble method to combine them, rather than using additional features. Bayesian model averaging is one way to do this. But your original idea about using your first model as a prior and then adding features to produce a "Bayesian update" still doesn't make sense to me.
Yes, 100% of the features are used to calculate p0. The system actually generates thousands of features, but we needed to be able to validate the algorithm for computing p0 offline. So we found (through reading the code) every feature that was used to calculate p0 and saved them. Mind you, the "calculation" for p0 as a function of the features is too complex and implicit to ever let you write it down (hundreds of binary classifiers in parallel and serial configurations, I didn't build the system). But we do know that we have captured all the necessary and sufficient feratures to do an offline simulation and validate the computation for p0.
My goal is not to modify anything about the model for calculating p0. It is to see if, by chance, the very same features might hold contain some "magic" information that allows us to predict some other disease. My thought is that is unlikely, especially since the features have been generated specifically to, in some optimal and minimal manner, predict D0. There are probably some information theoretic ways to test this asssumption.
Again, I need to come up with a fair and principled wat to either show that the concept just will not work or that it will. It has been easy to generate plenty of red flags (covariate drift as an example). My attempt to use some sort of modifed "Bayesian" approach is my way of using the best statistic the features have been designed to generate, and then see if there is some independent, residual, information that might help predict D1. I'm a skeptic, but I'm among a group where others (management) have a dogmatic view that machine learning can magically solvce any problem imaginable. They don't appreciate that most of these models (LASSO, ridge, PLS) are just, at the end of the day, just simple main effects (generalized) linear models.
Once again, I appreciate your engagement. It has been very helpful.
I am by no means a Bayesian expert, but it seems to me that what you want to do is not correct. Bayesian updating usually refers to incorporating new information - such as additional data points. Why add additional features? Just put them in your original model. Or, if using something like boosted trees, the algorithm already will use the features to "update" its predictions based on the degree to which it is making incorrect classifications. In other words, I think you are suggesting an apparatus that would appear "Bayesian" but in fact is just examining how much each feature contributes to the model, given the other features in the model. That is essentially what all predictive models do as is - leaving some features out manually and then adding them in to see how predictions change seems like just manually doing what predictive models do automatically.
Please, if I'm wrong about this, somebody straighten me out.
I am probably quite in agreement with you. The whole intent of the project is to see if the features, already used to train on D0, might have some "orthogonal" information that can be used to predict D1. Restated, does the noise in the original model for p0, which is an incredibly complex model based 100% on the features, contain any information that can be used to train for p1. I want to keep p0 as the initial estimat since it has already proven to be quite useful for D0.
If I just go ahead and niavely use something like LASSO (or ridge or PLS or ...) to train a model for p1 based on the features , I get better performance than predicting D1 using only p0. But I'm skeptical that I might unknowingly be training on noise with all the CV going on. Moreover, I have evidence from independent experiments that the model for p1 suffers from covariate shift while p0 is quite resiliant to different data sets.
Of course it comes down to me to give either a thumbs down or up on the whoe concept and so I want to simultaneously challenge the cocept as well as give it the best chance. That's why I'm leaning toward this "Bayesian" approach. It's a bit of a stretch to call it Bayesian, I agree, because we don't get to observe a Bernoulli event, we only get to look at the predicates in a model for the prior parameter.


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2017 Paris Nice Arnaud Demare Geraint Thomas Sergio Henao World Tour
2017 Paris-Nice Video Race Highlights
Called the 'Race to the Sun' because the pro peloton winds its way from cold Northern France to (hopefully) sunny Nice and the Mediterranean, Paris-Nice was first contested back in 1933. Last year's winner was Geraint Thomas, continuing a long tradition of English speaking rider's success. Brad Wiggins, Floyd Landis, Bobby Julich, Stephen Roche and Tommy Simpson have all win Paris-Nice. Of course, the king is Sean Kelly. He won the race a record seven times, consecutively from 1981 to 1988.
PELOTON Magazine / Image: James Startt / Video: ASO
Running from Sunday March 5th to 12th 2017, the 75th Paris-Nice will be made up of eight stages and will cover a total distance of 1,233.5 kilometers. Enjoy all the 2017 Paris-Nice Highlights courtesy of the ASO right here. Each stage will be added as soon as it is available.
Highlights From the Entire Week of Racing at Paris-Nice 2017
Stage Eight
Stage Seven
RELATED: Check out the Race Highlights from an incredible Strada Bianchi on our Facebook page!
To the Summit
Italian Icons: Bartali, Coppi and Pantani
By John Wilcockson
STRADE BIANCHE: A TRUE CLASSIC
PELOTON Recall • Strade is a true classic


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Oliver Stone's Savages is a mild return to maniac form. For more than a decade, Stone has cooked up safe character studies (Wall Street: Money Never Sleeps, World Trade Center, W.), surefire duds (Alexander), and excellent documentaries that hardly anyone saw (Looking for Fidel, Comandante). Not since his excellent Any Given Sunday has Stone made a movie as charged as the drug drama Savages. And make no mistake, Savages is Stone flexing his madasscraziness, problem is, he's not flexing hard enough.
Chon (Taylor Kitsch) and Ben (Aaron Johnson) are two high level marijuana growers/dealers who both share the love and physical intimacy of O (Blake Lively), the free spirited woman who (pointlessly) narrates the film. Shortly into the movie, Chon and Ben are made an offer they can't refuse by Alex (Demián Bichir, perfect in his restrain), a suited thug who works for Mexico's strongest cartel, Elena (Salma Hayek, in her best role since Frida). Elena wants the boys to work for her, and in return, they will make a great deal more money. But Ben (a strict Buddhist) wants out of the game, so refuse the offer he does. In return, Elena has her go-to enforcer, Lado (Benicio Del Toro, stealing every scene) kidnap O as a means of convincing the men otherwise. They react by trying to get O back, and, you get it.
Savages is bogged down with plot to a fault. There is, frankly, a lot more going on here then there needs to be. But some of what is going on here, mind you, is pure Stoneian bliss.
Take, for instance, every single thing Benicio Del Toro say and does. Lado is a scruffy, even-keeled psychopath who always keeps us guessing. And laughing. In one scene, he and Alex examine a crime scene, during which Lado sips through the green straw of his large Starbucks beverage. What's perfect about this is that a.) We have a filthy drug lord drinking a posh Starbucks drink, and b.) We have to picture this filthy drug lord actually standing in line and ordering said posh beverage. It's brilliant character devices like that that make Lado one of the best roles Del Toro has ever had.
Equally as crazed (or is it panicked?) is John Travolta, who plays a federal agent willing to trade info for info with Chon and Ben. Without giving too much away, Del Toro and Travolta have a scene together that is by far the best moment in the movie. It's scary, funny, and completely unpredictable. It almost makes the film worth it.
Look, the acting across the board here is pretty solid. Those of you who have only been exposed to Kitsch's recent John Carter and Battleship disasters (and not his revelatory work in Friday Night Lights), can trust that his Savages performance is a step in the right direction. Chon is militant, precise, and exacting, and Kitsch fleshes him out well (not great, but well). I have slightly less positive praise for Johnson (best known for his lead roles in Kick-Ass and Nowhere Boy). His Doing-It-All-For-Love drug dealing philosophy isn't nearly as convincing as Chon's no hold's barred approach. It also doesn't help that Johnson's British accent is evident in nearly everything he says.
I've been a huge admirer or Blake Lively's since her career-changing work in The Town. She's good here, and I'm excited to see what she cooks up in the future. The rest of the supporting cast, as mentioned, are all utterly perfect.
Flaws with Savages are completely story based. At 131 minutes, it feels 20 minutes longer than it needs to be. It also switches from being wildly violent to flat out boring. Stone makes polarizing films, so if I say Savages is more like U-Turn and less like Natural Born Killers, some may take that as praise while others may balk. From where I'm sitting, Savages is an uneven film that I wish Stone would've made 15 years ago, when he wasn't afraid of anything. C+
thevoid99 July 11, 2012 at 5:02 PM
I don't know. Stone has been a total letdown for me for years. Plus, I have a very low opinion on Blake Lively who was one of few things in The Town I didn't like. Plus, hearing her narrate a film would be a chore.
Alex Withrow July 12, 2012 at 8:12 AM
Man, so I really am the only person I know who completely dug her work in The Town, huh? Maybe it was because I had literally never seen her in anything before, and I had this perception of her as some untalented kid actress trying to "make it real." Her performance in The Town definitely proved me wrong. Weird.
To be fair, her narration here isn't constant, it's only every now and again, but it is completely pointless and unnecessary. I did enjoy her acting work though. But I agree, Stone has been kind of a letdown as of late.
s. July 12, 2012 at 10:22 AM
I liked her in The Town, she was really great and for some inexplicable reason I did see season 1 of Gossip Girl :)
Alex Withrow July 12, 2012 at 10:58 AM
Nice, yeah I thought she was a perfectly sultry and trashy in The Town. Boston at its finest haha.
Dan O. July 11, 2012 at 5:16 PM
Lively's narrative takes this film down a lot but the rest of the performances make this film a lot better than you would expect from another Stone "drug movie". Good review Alex. Could have been a whole lot better but not terrible, and that's good considering what Stone has put out in the past decade.
Yup, completely agree. This is good not great (even thought it could've been great). Just a little too long and tame.
Candice Frederick July 11, 2012 at 9:14 PM
it's a very unorthodox film, but that's what i like most about it. i realy appreciated this film.
Oh Stone makes unorthodox flicks for sure, and I really dig that about him. I definitely appreciated it as well, I just wish there was... more, you know?
Excellent review, I'm definetly seeing this one - can't wait to see Hayek and Del Toro's work. I do love U Turn so is it safe to assume I'll like this one?
I think you'll like this one, yeah. It isn't as crazy as U-Turn, but it's definitely nuts. Del Toro makes the entire movie worth it. He is fantastically insane.
ruth July 12, 2012 at 5:58 PM
I have to admit, I really didn't know what to make of Blake Lively aside from her killer bod and great fashion sense. But she certainly displayed that she has potential in The Town. I think she's kinda like R-Patz where she has to constantly prove herself because of her looks.
In any case, I might rent this one for Aaron Johnson, it's just too brutal for me to see on the big screen. I had to close my eyes even during the trailer!
Alex Withrow July 12, 2012 at 6:12 PM
I definitely think Lively's looks work against her, in terms of people taking her seriously as an actress (which is really such a shame). Not sure about R-Patz... people really seem to think he's "got it" but honestly, he's been dog shit in everything I've seen him in, which is more than I'd like to admit haha. Hopefully Cosmopolis proves me wrong.
Yeah, if you aren't in the mood for brutal, I'd definitely skip this one for now. Parts of it are crazy brutal, for sure.
Eric July 13, 2012 at 12:01 AM
Great review, Alex. I might try to catch this one at the cheap theater this weekend, but I won't be going into it with high expectations. Now I want to know what this Travolta/Del Toro scene is all about, haha.
Oh that scene is priceless man. Just two guys completely going for it. Some may call it overacting, but I think they knew EXACTLY what they were doing and EXACTLY how far they could push it. One of the best acted scenes so far this year, in my opinion.
Diana July 13, 2012 at 10:11 AM
Great review Alex, I am really curious to see it, mostly for the supporting cast. I get what you're saying about Lively's performance in The Town, but she is in my top 2 I-can't-stand-this-woman-she-is-so-annoying-please-let-me-punch-her, alongside Jennifer Aniston, so I will probably suffer whilst seeing this movie, but I will endure for Hayek, Del Toro and others.
Hahhaa, see, you ARE funny.
If that's how you feel about Lively, Savages may not work for you. She's all over it, pointless narration included. But I definitely think the supporting cast makes up for it!
I can't think of a film with an ending as idiotic as the one this one had.
I thought the first ending was absurd, but the "real" one was pretty generic. I was apathetic about it, but I can see why people would not like it.
blahblahblah Toby November 8, 2012 at 2:04 AM
I haven't agreed with most of what you've said here Alex but I've enjoyed reading you as always. I hope when you read my review you forgive me for my comments on Blake Lively and it is counteracted by the link I throw your way. Also incase you have any interest in the brilliant source material I reviewed it yesterday.
blahblahblahgay.blogspot.com.au/2012/11/savages.html
Alex Withrow November 8, 2012 at 11:23 AM
I'm surprised you think we don't agree - just read your review and it seems like we agree a lot. I like Lively more than you, but I think our thoughts are pretty much aligned here. Aaaanyway. Christ, I hope Stone gets back to making, you know, good movies again.
blahblahblah Toby November 8, 2012 at 8:04 PM
I took an overall positive tone from your review then despite the c+ rating. I was just discssing the Stone problem with thevoid in my comments. I think he needs to go indie, learn to work with a much smaller budget. I used the Friedkin with Killer Joe example. To me that was the kind of film Stone should be making now.
Alex Withrow November 9, 2012 at 2:24 PM
Oh man, I couldn't agree more. I'd love to see a Killer Joe-esque Stone film.
But would he do it through choice? Looking back at U-Turn I think that was the last (only?) time he worked small and he achieved a slightly messy but very good film. Very Stone but small in scope. I get the feeling Hollywood needs to turn their back on him though, he seems addicted to the power.
Alex Withrow November 12, 2012 at 9:39 AM
I can't wait for his 10 part American History miniseries. That dude has such a batshit way of thinking, but he can communicate it so articulately. Be really interesting.
blahblahblah Toby November 12, 2012 at 7:49 PM
Despite not being American that does sound interesting. The kind of thing that people might actually be educated by rather than put to sleep. More talented film makers should direct educational videos.
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