text:
ConU administration unfreezes funds
Concordia's administration has released $200,000 of students' fees it withheld from the CSU after November's executive byelection failed to produce an elected executive. Interim CSU President Patrice Blais announced the unfreezing of funds at the council of representatives meeting held on May 8.
Concordia's administration has released $200,000 of students' fees it
withheld from the CSU after November's executive byelection failed to
produce an elected executive.
Interim CSU President Patrice Blais announced the unfreezing of funds at
the council of representatives meeting held on May 8.
The fees, which were to be transferred to the CSU at the beginning of
the winter semester, have sat in an interest-earning account since the
judicial board's decision to annul the byelection results due to
election irregularities. Upon the announcement, the CSU council of
representatives appointed an interim executive headed by Blais, the VP
finance.
The administration issued a statement after the byelection saying it
would not deal with the interim executive because it was not elected and
therefore illegitimate. Furthermore, it announced it would withhold the
balance of the student transfer fees -$200,000- which was destined for
the CSU.
There was some hope that the administration might finally transfer the
fees following the March election of the CANDO slate, headed by Sabine
Friesinger. But CANDO is to only take office on June 1, and the
administration said it would only transfer the funds at that point.
But since that decision in early April, Rector Frederick Lowy has sat
down with Friesinger and negotiated a transfer of the fees to the CSU
before CANDO officially takes office. According to Concordia
Communications Officer Christine Mota, Lowy asked Friesinger if she
would be comfortable with the interim executive receiving the funds.
Friesinger, whose slate decried the administration's decision to
withhold the fees, agreed to the transfer.
Mota said the sole motivation for the unexpected decision was that a
new, representative executive slate had been elected. Blais, though,
feels there was more to it than that.
After making the announcement, he hinted that the administration's
decision was in part due to a court appearance to be held the following
Friday. The CSU had decided to file suit with the administration over
the withheld fees, as well as its refusal to deal with the interim
executive, both of which Blais said violated accreditation laws.
By transferring the fees back before the court case, said Blais, the CSU
went from a 98 per cent chance of winning to a fifty-fifty chance.
In fact, the case was thrown out on May 10 for exactly those reasons.
According to interim-vp internal Sameer Zuberi the judge decided that
any decision to transfer the fees to the CSU was in fact an acceptance
of its legitimacy.
The total sum of $203,500, including interest, has already been restored
to the CSU coffers.
Not a letter to the editor, but a request for general information
We're with you all summer long
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JMSB's MBA program second greenest in the world
On Sept. 23, results from Corporate Knights magazine's inaugural Global Green Survey showed that Concordia's John Molson School…
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Media fund may be indexed to consumer price index
One of the referendum questions that will be on the ballot will be about changing the media fund's collection and redistribution. Currently, the media fund is $0.15 per credit. The referendum question asks to change the media fund to an "ongoing media fund of $0.


Eat jail food to avoid being jailed!
Some 15 to 20 people knock on the gates of the Tihar Central Jail in New Delhi every month seeking rotis and dal. No, they are not poor, but those who believe that eating jail food ensures they will never get jailed.
Bizarre? May be, but the superstition is not going away.
The food prepared in the high-security but crowded prison, located in west Delhi, and given to the thousands of its inmates is very basic - rotis, lentils and, during festivals, an occasional sweet.
So who are the people who come seeking this food?
"They are mainly businessmen and traders," a jail official said. "They may have been told by their astrologers that there is a possibility of their getting jailed unless they seek out and eat the jail food.
"So we have people approaching us, either on their own or through their contacts, wanting a bit of jail food. It is embarrassing but we mostly accept their requests. We do have surplus food on occasions.
"Mostly the guards give them a roti or two and some dal. And the people are more than happy!"
Another official added: "We have come across so many who want dal and rotis at any cost."
There is yet another superstition related to the Tihar Jail and that is about the wooden plank from where convicts are hanged.
Pieces of wood from the plank are believed to possess the power to fight evil spirits.
"People say if you tie a piece around your neck, you can beat black magic," the official said.
Kamrun Nisha, 55, who has been selling snacks at the jail complex for 37 years, says this is more than a belief - she claims to have benefited from it.
"My daughter's three newborn babies died and I was depressed about it. One day a fakir came to my shop and advised me to get a small piece of wood from the plank and tie it around the baby's neck when he or she is born.
"After great difficulty I got a piece of wood. When my daughter gave birth to her fourth baby, I tied it around her neck. The girl not only survived but went on to have two brothers as well," said Nisha.
Not everyone believes in these superstitions.
"I have heard these stories but I doubt if they are true," said Shah Nawaz Khan, a 32-year-old who has been visiting the prison for the last three years.
Pankaj, 19, whose mother is a prisoner at the jail, said: "I have never heard of these things nor do I believe in them."


text:
"Breaking Bad"
Posts by kikachuck
kikachuck
yes yes and yes! Although I don't think this season has been as good as the first one was.
But I have to say, I've lived in New Mexico my whole life and I never heard the phrase "breaking bad" until this show. They say it's some sort of southwestern phrase :stu
My personal Blog!!! http://lostinthevalley.wordpress.com/
updated November 14th, 2010
UltraShy said:
I agree the first season was better, but last night's show wasn't bad.
I too had never before heard the phrase "breaking bad".
Yeah, last nights episode was pretty good. While I think that last season was better generally, I think Bryan Cranston has come to OWN this role this season. I wouldn't be surprised if he wins the Emmy next year too.
BeNice Apr 20, 2009

redpajama_set_name: RedPajamaCommonCrawl


Can I complete the painting if I have no artistic ability?
Yes! Our classes are for the inexperienced and experienced alike! The majority of our clients have never even picked up a paint brush so if that's you, you'll fit right in! We take you through step by step as you complete your masterpiece while having a blast doing it! And for those seasoned artists, you have the room to add your own artistic flair.
We provide all the supplies including acrylic paint, 16″ x 20″ canvasses (11″ x 14″ for senior, kids and other select events), easels, brushes, art smocks, art instruction and set-up/clean-up…you just bring yourself and enjoy! Food and beverages available for purchase.
You can dress up or dress down because we provide the apron for use during painting, but we are painting so you might not to wear that one-of-a-kind outfit.
Can I bring my own food/wine?
With the exception of a pre-arranged private party at a private residence we partner with venues that offer some of the best wine and food in San Diego! Please no byof or byow!
How do I start my own Paint and Palate?


What Better Way To Bring The Whole Family Together Than Loads Of Fun, Laughter, Dancing & Of Course Great Food! Brunch Kids Is The UK's First Themed Family Brunch, Specialising In Unique One-Off Experiences!
Heath & Safety Is Our Main Priority, All Necessary Precautions Have Been Put In Place To Make This The Most enjoyable experience for you and your children.
Buggies Should Be Kept In Buggy Park Areas And Not Taken Onto The Dance Floor As This Is A Tripping Hazard. Buggy's Parked At Owners Risk.
Music Levels Will Be Kept Low But Guests Are Welcome To Carry/Bring Bring Ear Defenders.
By Attending Brunch Kids You Give Permission For Our Photographer To Take Photos & Videos Of The Event (For Press Purposes Only).


Swift's Wireless Gas Sensors are used for measuring the concentration of ammonia, carbon dioxide, methane and benzene. These units are designed for in leak detection, air quality monitoring environmental monitoring in hazardous environments such as industrial plants, manufacturing facilities and warehouses. They allow user to set warning and critical thresholds for sensor measurements and the data is stored on Swift Sensors Cloud via computer, tablet, or smartphone.
AUSTIN, TEXAS (PRWEB) AUGUST 16, 2018 - Swift Sensors, Inc., a supplier of plug-and-play cloud-based wireless sensor systems for Industrial IoT, has released four gas sensors for air quality and safety monitoring. The wireless gas sensors simplify and reduce the cost of monitoring dangerous and hazardous gasses in industrial plants, manufacturing facilities, warehouses. Common applications for the new sensors include leak detection, air quality monitoring, OSHA compliance monitoring and reporting, environmental monitoring in poultry and livestock farms, and facility monitoring.
The new sensors measure concentration of ammonia, carbon monoxide, methane, and benzene. Each sensor is AC powered and packed in a small ABS enclosure. Warning and critical thresholds can be set for the sensor measurements, and notifications sent if thresholds are exceeded. All data from the sensors are stored on the Swift Sensors Cloud and can be managed and viewed via the Swift Sensors Dashboard from any computer, tablet, or smartphone.
We've added Swift Sensors' ammonia sensor to our new mobile monitoring platform for poultry farms," said Mark Moshier, President, Manufacturing Division of Arrowsight, the leading supplier of remote video auditing for manufacturers, food service companies, and hospitals. The breadth of measurements Swift Sensors offers gives us the ability to add monitoring of critical environmental conditions in our customer's operations.
Swift Sensors, Inc. is a cloud wireless sensor IoT company, providing a low-cost sensor system for industrial and commercial applications. Its flagship product is the Swift Sensors Cloud Wireless Sensor System, comprised of low-power wireless sensors and cloud-based monitoring, notifications, analytics, and reporting. The sensor system proactively protects and monitors a wide range of equipment and processes. Swift Sensors applications include manufacturing, food service, facility management, cold chain, transportation, research & development, and power. The headquarters is located at 4611 Bee Caves Road, Suite 100, Austin, Texas, 78746. For more information, please visit http://www.swiftsensors.com or call +1-866-308-1340.


Our regional workforce is the fuel that drives our economy.
We have the resources and the drive to help your business thrive.
Entrepreneurship opens up a world of opportunity here at home.
Our region is an ideal place to live, and build or grow a business.
More jobs, more people, and a more attractive business environment.
We are dedicated to achieving consistent, long-term economic growth for the communities of Grand Bay-Westfield, Quispamsis, Rothesay, Saint John and St. Martins.
live in the greater Saint John area, and our historic Uptown core is the Maritimes' most densely populated mixed-use neighbourhood.
Saint John is located in the Atlantic Time zone, perfectly positioned to support customers in the UK and west coast of North America, and within easy reach of Halifax, Montreal, and the US northeast.
In Greater Saint John, 15% of the workforce is bilingual.
Saint John is home to the University of New Brunswick, Dalhousie Medicine New Brunswick, the New Brunswick Community College, and other private options.
Our health cluster includes the Saint John Regional Hospital, Dalhousie Medicine New Brunswick, Nursing and Health Sciences at UNB Saint John, and Allied Health programs at NBCC.
Greater Saint John's variety of logistics and manufacturing advantages will ensure you have the tools you need to get your product to market affordably and efficiently.
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Greater Saint John is the largest metropolitan area on the Bay of Fundy – a body of water that is an enormous economic asset to the region.


Tricia Kagan Bettman was born on July 8, 1950 and passed away on June 19, 2018.
When Tricia walked into a room she lit it up with her smile. She was always unselfish in all her work with her synagogue, Hadassah and all her friends.
To the Bettman and Kagan Family…my heartfelt sympathy and condolences.
Tricia was a wonderful person and a valuable member of Hadassah. I still remember her leadership trainings from 10 years ago. She will be missed. Love and prayers for her family.
Dear Bettman Family,I recall Mrs.Bettman when i worked at Publix,she was a very kind lady.I am shocked by her sudden passing.To you all i pray G-d's comfort and healing.My deepest sympathy to you dear folks.


text:
The Friends of the Grand Western Canal exists to enhance the experiences of all users of the Grand Western Canal, the historic water way that connected Tiverton with Taunton 150 years ago.
The Grand Western runs for 24 miles through the beautiful unspoilt countryside of Mid Devon and western Somerset. 11 miles remain in water and form one of Devon County Council's most prized Country Parks The remainder, although back in private ownership, has a public footpath running along much of its former towing path.
Residents and visitors of all ages can find much to enjoy and great satisfaction in the peace and tranquility that exist along this linear haven. Whether walking, cycling, jogging, boating, fishing, wild-life watching or just enjoying the stunning views, there's something for everyone.
The Friends, and its predecessor, the Grand Western Canal Trust, have been supporting the DCC rangers and the officers of Taunton Deane DC since 1988 in working to preserve and enhance this unique heritage.
Since the devastating flood on 21st Nov 2012 that washed out an embankment at Halberton, we are focusing on reinstatement of the waterway in time for our bi-centenary in 2014.
Boats on the Grand Western Canal.jpg
The Grand Western Canal hosted the National Trailboat Festival in 2008
Grand Western.jpg
The Grand Western Canal forms part of national cycle route 3 - passing beneath on of Rennies classic arched bridges.
Friends Grand Western launch 2lr.jpg
The Friends of the Grand Western Canal were launched on April 15 2012
Friends of the Grand Western Canal
'Grand Western Canal Friends' for up-to-date news & comment
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Join the Friends - click here


text:
\section{Introduction}

The great success of the  Dirichlet process within the Bayesian nonparametric framework has paved the way for the definition and investigation of a large variety of random probability measures. Indeed, since  the seminal paper by \cite{Fer73}, several discrete nonparametric priors have been proposed to accommodate for exchangeable observations, among these we mention: the Pitman-Yor process or two parameter Poisson-Dirichlet process \citep{Per92, Pit96}; species sampling processes \citep{Pit96}; priors based on normalization of completely random measures \citep{regz_03,Lij10}. Gibbs-type priors are another important class of Bayesian nonparametric models early introduced by \citep{Gne05} and recently investigated in \citep{Deb15}. The Gibbs-type family has the advantage to balance modelling flexibility and mathematical tractability. These  processes  have been successfully used in several frameworks, just to mention a few examples: to face prediction within species sampling framework \citep[e.g.][]{Lij07}, to define mixture models \citep[e.g.][]{Ish01,Lij07b}, for survival analysis \citep[e.g.][]{Jar10}, and for applications in linguistic and information retrieval \citep[e.g.][]{Teh06, Teh10}. \cite{Roy20} have recently discussed a class of feature allocation models  parametrized by Gibbs-type random probability measures.

We introduce a new family of Bayesian nonparametric models where a Gibbs-type prior is contaminated with an exogenous diffuse probability measure, called \textit{contaminant measure}. More precisely, we define a new random probability measure as  a convex linear combination of a Gibbs-type prior $\tilde{q}$ and a diffuse probability $P_0$, i.e. we deal with $\tilde{p}= \beta \tilde{q}+ (1-\beta) P_0$, where $\beta \in [0,1]$ is a weight which tunes the impact of the contaminant measure. We refer to $\tilde{p}$ as a contaminated Gibbs-type prior (see Definition \ref{def:cGP}) and its distribution is then used  as a nonparametric prior in a Bayesian context.
We will show that the advantage of this process in the Bayesian nonparametric setting is twofold: i)  $\tilde{p}$ is a tractable prior outside the Gibbs-type family which allows to enrich the predictive structure of exchangeable models, through the inclusion of the additional sampling information on the number of observations with frequency one out of the observed sample; ii) the contaminant measure $P_0$ accounts naturally for the presence of anomalies in the data (observations which are under some respects singular), thus resulting particularly suited for several applied problems.
With regard to point ii), contaminated Gibbs-type prior can be exploited for modelling discrete data directly, when one needs to inflate the observations with frequency one. As an example, in Section \ref{sec:discrete_app}, we consider species detection data from the Global Biodiversity Information Facility project \citep{GBI21} with a high number of species detected only once. Within this framework we show the advantage of our model with respect to the traditional Pitman-Yor process to model the excess of ones.
Besides, the proposed construction turns out to be useful in other contexts, such as in language modelling, disclosure risk assessment and operational taxonomic unit data analysis. See Section \ref{sec:discussion} for a thoughtful discussion on these applications. Furthermore, the new construction can be exploited as a mixing measure in mixture models to account for the presence of outliers in a dataset. In particular we are motivated by an astronomical dataset \citep{Iba11} composed by $n = 139$ stars, and we aim to understand which stars belong to a globular cluster and which stars are contaminants, i.e., outliers. Outlier detection is a crucial problem in Statistics and similar convex constructions are available also in classical setting, see, e.g., \cite{clustering_19} for an account. In the Bayesian framework, some contributions are available and rely on the use of traditional Dirichlet process. \cite{Qui_03,Qui_06} focus on product partition models, and they develop a decision-theoretic approach  that allows selecting a partition
with the purpose of outlier detection in regression problems.
\cite{Sho_11} identify an outlier detection criterion based on the Bayes factor, where they compare a
partition containing outliers against a partition with fewer or no outliers. As a remarkable addition with respect to the current literature, our prior process contemplates a specific component in the model, i.e. the contaminant measure $P_0$, to account for the presence of outliers, which thus follow a different generating process with respect to observations. This modeling strategy results in smoothed density estimates with respect to the ones obtained without the presence of the contaminant measure in the model.
Motivated by all these applications, we introduce and deeply investigate the random partition structure induced by contaminated Gibbs-type priors. All the stated results are available in a closed form, they are simple and with a natural interpretation. The induced prediction rule can be easily explained in terms of a new Chinese restaurant with a social and a non-social room. As a concrete example, throughout the paper we focus on the contaminated version of the Pitman-Yor process, which exhibits more tractable expressions for all the quantity of interest and to face predictive inference. In particular, a simple extended P\'olya urn representation of the updating mechanism implied by this process can also be obtained.

To the best of our knowledge, the Bayesian nonparametric literature has never focused on Gibbs type priors contaminated with the inclusion of a diffuse measure to model anomalies or the excess of singular observations. Experiments with simulated and real data will show their advantage to properly model the presence of such data with respect to standard Gibbs-type priors, which result in severe bias to estimate model parameters and in poor predictive performances.

\section{Contaminated Gibbs-type priors and their properties} \label{sec:model}

Let  $\{X_i\}_{i \geq 1}$  be a sequence of observations
taking values in a Polish space $\mathds{X}$, equipped with its Borel $\sigma$-field $\mathscr{X}$.
In the Bayesian nonparametric setting $X_1, X_2, \ldots$ are typically supposed to be exchangeable \citep{defin_37}, which is tantamount to saying that there exists a random probability measure
$\tilde{p}$ such that $X_i | \tilde{p} \stackrel{\mbox{\scriptsize{iid}}}{\sim} \tilde{p}$, where the distribution of $\tilde{p}$ works as a prior in the Bayesian nonparametric framework.
The distribution of $\tilde{p}$, indicated by $\mathscr{Q}$, is called the de Finetti measure of the sequence $X_1, X_2, \ldots$, and several prior specifications
$\mathscr{Q}$ are available in the Bayesian nonparametric literature.
Among these we mention the remarkable class of species sampling models \citep{Pit96}. We recall that  an exchangeable sequence of observations $\{ X_i \}_{i \geq 1}$  is called a \textit{species sampling sequence}  if and only if it is governed by a distribution of the following type
\begin{equation}\label{eq:tilde_p_gen}
\tilde{p} = \sum_{j \geq 1} p_j \delta_{Z_j} +\Big( 1-\sum_{j \geq 1} p_j\Big) P_0,
\end{equation}
for a sequence of random weights $\{ p_j  \}_{j \geq 1}$ with $p_j \geq 0$ and $\sum_{j \geq 1}p_j \leq 1$ almost surely, and a sequence of random atoms $\{ Z_j \}_{j \geq 1}$ i.i.d. from $P_0$ independent of  $\{ p_j  \}_{j \geq 1}$, where  $P_0$ is assumed to be a  diffuse probability measure on  $(\mathds{X}, \mathscr{X})$. A random distribution $\tilde{p} $ of the form in \eqref{eq:tilde_p_gen} is called a \textit{species sampling model}. Further, a species sampling model is termed \textit{proper} if and only if $\sum_{j \geq 1} p_j = 1$ almost surely, and most of the current Bayesian nonparametric literature focuses on the proper species sampling models. In this paper we discuss the case in which $\sum_{j \geq 1} p_j < 1$ with positive probability, and we show how non-proper models are particularly suited to take into account contaminated observations or more generally observations with frequency one.

Among the very general class of species sampling models we recover special subclasses of priors, which have been duly investigated in the literature, e.g., homogeneous normalized random measures with independent increments \citep{regz_03} and Gibbs-type priors \citep{Gne05,Deb15}. Here we focus on a contaminated version of Gibbs-type priors. For this reason, it is worth recalling that Gibbs-type random probability measures are typically characterized in terms of the exchangeable random partition \citep{Pit06} induced by the data.
More precisely, given a sample $X_{1:n}:= (X_1, \ldots , X_n)$ from a species sampling model governed by a random probability measure $\tilde{p}$, the $n$ observations are naturally partitioned into $K_n=k$ groups of distinct values, denoted here as
$X_1^*, \ldots , X_k^*$, with corresponding frequencies $(N_{n,1}, \ldots , N_{n, K_n})= (n_1, \ldots , n_k)$. The exchangeable partition probability function (EPPF) corresponds to the probability of observing a specific partition of the data into clusters of distinct values, and it can be formalized as
\begin{equation}
\label{eq:EPPF_def}
\Pi_k^{(n) } (n_1, \ldots , n_k)  := \int_{\mathds{X}^k } \mathds{E}  \prod_{j=1}^k  \tilde{p}^{n_j} ({\rm d}  x_j^*).
\end{equation}
Gibbs-type priors are proper species sampling models  $\tilde{p}$ characterized by means of their sequence of EPPFs $\{ \Pi_k^{(n)} : \; n \geq 1, \, 1 \leq k \leq n  \}$, which can be expressed in the following form
\begin{equation}\label{eq:EPPF_def_gibbs}
\Pi_k^{(n) } (n_1, \ldots , n_k) = V_{n,k} \prod_{i=1}^k (1-\sigma)_{n_i-1},
\end{equation}
for all $n \geq 1$, $k \leq n$ and positive integers $n_1, \ldots , n_k$ with $\sum_{i=1}^k n_i =n$,  where $(a)_b= \Gamma (a+b)/\Gamma (a)$ in \eqref{eq:EPPF_def_gibbs}, for $a, b >0$, denotes the Pochhammer symbol. The discount parameter $\sigma <1$ and the non-negative weights $\{ V_{n,k} : \; n \geq 1, \, 1 \leq k \leq n \}$ must satisfy the recurrence  relation $V_{n,k} = (n -\sigma k) V_{n+1,k} +V_{n+1, k+1}$ for all $k =1, \ldots , n$, $n \geq 1$, with the proviso $V_{1,1}=1$ and $V_{0,0}=1$. The sequence of weights $V_{n,k}$s can be specified to recover prior processes commonly used in literature, such as the Dirichlet process \citep{Fer73}, the Pitman-Yor process \citep{Pit97}, the normalized inverse Gaussian process \citep{Lij05} and the normalized generalized gamma process \citep[see e.g.][and references therein]{Lij07b}.
Building upon Gibbs-type priors, we now introduce a new family of prior processes which account for the possibility of contaminated observations.
\begin{definition} \label{def:cGP}
	Let $\tilde{q}$ be a Gibbs-type prior, specified by the sequence of weights $\{ V_{n,k} : \; n \geq 1, \, 1 \leq k \leq n \}$ and $\sigma<1$.
	A \textit{contaminated Gibbs-type prior} is a random probability measure  on $(\mathds{X}, \mathscr{X})$ defined as
	\begin{equation}\label{eq:HGT_def}
	\tilde p = \beta \tilde q + (1 - \beta) P_0, \qquad \beta \in (0, 1),
	\end{equation}
	where $Q_0(\, \cdot \,) = \mathds{E}[\tilde{q}(\, \cdot \,)]$ is the base measure of $\tilde{q}$, and $Q_0,P_0$ are diffuse probability measures.
\end{definition}
The prior $\tilde{p}$ in \eqref{eq:HGT_def} is a convex linear combination of two components: an almost surely discrete component $\tilde{q}$ which generates the data, and a diffuse probability measure $P_0$ which accounts for contaminated observations. In the sequel we refer to $P_0$ as the \textit{contaminant measure}.
Sampling from $\tilde p$ can be interpreted as sampling from a population formed by two parts: the first one, representing  a $\beta$ fraction of the entire population, is composed by a countable number of species each appearing with positive probability. The second part ($1-\beta$ fraction) can be thought of as composed by a continuum of individuals each belonging to a different species. Therefore any time we sample from this second part a new species is obtained that cannot be re-observed. As stated above, for simplicity we shall call  \textit{contaminant} this second part and \textit{contaminated} the relative observations. However, the diffuse part can be used more generally to account for any population which displays unique elements (see Section \ref{sec:discussion}) and/or to model a high number of generic singletons in the observations. Finally, note that in Definition \ref{def:cGP}, the contaminant measure $P_0$ may be different from the base measure $Q_0$, thus $\tilde{p}$ in \eqref{eq:HGT_def} may not be a species sampling model. This additional flexibility is introduced because it was found useful in some applied contexts to distinguish the distribution of contaminated observations from the others.

We first derive the expectation and the covariance structure of a contaminated Gibbs-type prior in order to understand  how
the contaminant measure affects the distribution of $\tilde{p}$.
\begin{proposition}\label{thm:prior_quant}
	Let $\tilde p$ be a contaminated Gibbs-type prior as in Definition \ref{def:cGP}. Let $A, B \in \mathscr{X}$, then
	\[
	\begin{split}
	\mathds{E}[\tilde p(A)] &= \beta Q_0(A) + (1-\beta)P_0(A),\\
	\mathrm{Cov}(\tilde p(A), \tilde p(B)) &= \beta^2 (1-\sigma) \frac{V_{2,1}}{V_{1,1}} [Q_0(A \cap B) - Q_0(A)Q_0(B)].
	\end{split}
	\]

\end{proposition}
As consequence of Proposition \ref{thm:prior_quant},
one has $\mathrm{Var}(\tilde p(A)) = \beta^2 \mathrm{Var}(\tilde q (A))$, therefore the diffuse probability measure $P_0$ in \eqref{eq:HGT_def} has the effect to shrink $\tilde q (A)$ towards its expected value. See Section \ref{sup:proof_prior} for a proof of Proposition \ref{thm:prior_quant}.
\begin{example}[Contaminated Pitman-Yor process]  \label{ex:cPY1}
	Among the class of Gibbs-type priors, the Pitman-Yor process represents a noteworthy example, widely used in numerous applications. The contaminated Pitman-Yor process  can be constructed by selecting $\tilde q$ in \eqref{eq:HGT_def} to be a Pitman-Yor process. In such a case we recall that
	\begin{equation}\label{eq:w_PY}
	V_{n,k} = \frac{\prod_{i=1}^{k-1}(\vartheta + i \sigma)}{(\vartheta + 1)_{n-1}}
	\end{equation}
	with $\sigma \in [0,1)$ and $\vartheta > -\sigma$. Moreover, if  $\sigma = 0$ we recover the Dirichlet process.
\end{example}

\section{Random partition, prediction and asymptotic properties} \label{sec:inference}
Having introduced all the modeling assumptions in Section \ref{sec:model}, we now study the partition structure induced by a sample of observations from
the random probability measure in \eqref{eq:HGT_def}, we further derive a closed form expression for predictive distributions and asymptotic properties for the number of clusters.
We first focus on the random partition induced by a sequence of exchangeable observations governed by a contaminated Gibbs-type prior, deriving  the EPPF.
\begin{theorem}\label{thm:EPPF_Gibbs}
	Let $\tilde p$ be a contaminated Gibbs-type prior  as in \eqref{eq:HGT_def}, with  $P_0$ and  $Q_0$ two diffuse probability measures
	on $(\mathds{X}, \mathscr{X})$. Suppose that  $X_i | \tilde{p} \stackrel{\mbox{\scriptsize{iid}}}{\sim} \tilde{p}$, as $i \geq 1$, then the probability that $n$ observations $X_{1:n}$ are partitioned into $K_n=k$ clusters of distinct values
	$X_1^*, \ldots , X_{k}^*$  with corresponding frequencies $(N_{n,1}, \ldots , N_{n, K_n})= (n_1, \ldots , n_k)$ equals
	\begin{equation}\label{eq:EPPF_convex_comb}
	\Pi_k^{(n)} (n_1, \ldots , n_k) =  \mathds{E}_{\bar{M}_{m_1}} [V_{n-\bar{M}_{m_1}, k- \bar{M}_{m_1}}] \beta^{n-m_{1}} \prod_{i=1}^{k} (1-\sigma)_{n_i-1}
	\end{equation}
	where $\bar{M}_{m_1} \sim {\mathrm{Binom}} (m_{1}, 1 - \beta)$  and $m_{1}= \# \{ i : \; n_i=1 \}$ denotes the number of singletons (i.e. observations with frequency one) out of the sample of size $n$.
\end{theorem}

See Section \ref{prof:th1} of the Appendix for a proof of Theorem \ref{thm:EPPF_Gibbs}.
From the expected value in \eqref{eq:EPPF_convex_comb}, it is apparent that the use of the contaminant measure $P_0$ in \eqref{eq:HGT_def} acts on observations with frequency one and, as expected, they play a central role in the expression of the EPPF. In order to fix the terminology we call \textit{singletons} the observations with frequency one, while the \textit{structural singletons} are those values generated from the contaminant measure $P_0$, whose number equals the latent quantity $\bar{M}_{m_1}$. Note that the term \textit{structural} refers to the fact that these values cannot be observed twice and this statistic could be of potential interest in certain applied problems, as it will be discussed in Section \ref{sec:discussion}.

We now get a glimpse of the probabilistic implications of the random partition \eqref{eq:EPPF_convex_comb} induced by contaminated Gibbs-type priors
as compared to the pure Gibbs-type priors.
In order to do this, we
denote by  $(n_1, \ldots , n_k)$  and $(n_1', \ldots , n_k')$ two distinct compositions having the same number of distinct  values $k$ and corresponding to two samples with the same size $n$; the probability ratio between the EPPFs corresponding to the two compositions will be denoted by
$\mathrm{R} (n_1, \ldots  n_k ;n_1', \ldots  n_k '; n,k): =\Pi_k^{(n)} (n_1, \ldots , n_k)/ \Pi_k^{(n)} (n_1', \ldots , n_k')$.
In Proposition \ref{prop:compare_EPPF} (Section \ref{sec:proof_compare_EPPF}) we compare the probability ratio $\mathrm{R}$ when $\Pi_k^{(n)}$ is a Gibbs-type EPPF \eqref{eq:EPPF_def_gibbs} and when it equals the EPPF of a contaminated Gibbs-type prior \eqref{eq:EPPF_convex_comb}.
Proposition \ref{prop:compare_EPPF} of the Appendix clarifies that if the two compositions have the same number of singletons, the ratio is the same for the contaminated and non-contaminated model. On the other side, if the number of singletons out of the composition $(n_1, \ldots , n_k)$ is bigger w.r.t. the number of singletons out of $(n_1', \ldots , n_k')$, the relative ratio increases in the contaminated model. See Section \ref{sec:proof_compare_EPPF} for details.
Thus, in relative terms and given the number $k$ of distinct values, the contaminated Gibbs model modifies the probabilities of compositions only when a different number of singletons is involved, favouring compositions with a higher number of these elements.

For computational convenience, we can equivalently describe the EPPF \eqref{eq:EPPF_convex_comb} introducing a set of suitable latent variables on an augmented probability space. Indeed, we can denote by $J_1, \ldots, J_{n}$ Bernoulli random variables, where the generic $J_i$ indicates if the $i$th observation is generated from the contaminant measure $P_0$ ($J_i=0$), or from the a.s. discrete component $\tilde{q}$ ($J_i = 1$).
Thus, the introduction of latent elements $J_1, \ldots , J_n$ leads us to deal with the following augmented model
\begin{equation}
\begin{aligned}\label{eq:hgibbs_aug}
X_i | \tilde{p}, J_i  & \stackrel{\mbox{\scriptsize{iid}}}{\sim} J_i \tilde{q} + (1-J_i) P_0 \\
J_i  &\stackrel{\mbox{\scriptsize{iid}}}{\sim} \mathrm{Bern} (\beta),
\end{aligned}
\end{equation}
from which we may recover the marginal model by integrating  \eqref{eq:hgibbs_aug} with respect to $J_i$. Furthermore, $J_i =1 $ in \eqref{eq:hgibbs_aug} if the corresponding observation $X_i$ has been recorded at least twice in the sample: indeed
if an observation $X_i$ is generated from $P_0$, it does not appear again in the sample with probability $1$.
Thus, the non-degenerate $J_i$s are those values referring to singletons out of the sample $X_{1:n}$. Without loss of generality
we can assume that the elements appearing once in the sample are the first $m_{1}$ observations $X_1, \dots, X_{m_{1}}$. Based upon this augmentation, the random variable $\bar{M}_{m_1}$ in \eqref{eq:EPPF_convex_comb} equals $\sum_{i=1}^{m_1} (1-J_i)$ which represents the number of structural singletons among the observations recorded only once and it could be of potential interest in many application areas, as discussed in Section \ref{sec:discussion}.
We now  describe the predictive distribution of the next observation $X_{n+1}$, conditionally given $X_{1:n}$  and the latent variables $J_{1:m_1}= (J_1, \dots, J_{m_{1}})$.
\begin{proposition}\label{prp:predictive}
	Let $\tilde p$ be a contaminated Gibbs-type prior  as in \eqref{eq:HGT_def}, with  $P_0$ and  $Q_0$ two diffuse probability measures
	on $(\mathds{X}, \mathscr{X})$.  Assume that  $X_i | \tilde{p} \stackrel{\mbox{\scriptsize{iid}}}{\sim} \tilde{p}$, as $i \geq 1$, and consider a  sample  $X_{1:n}$ which displays $K_n=k$ distinct values, denoted as $X_1^*, \ldots , X_k^*$, with respective frequencies $(N_{n,1}, \ldots , N_{n, K_n})= (n_1, \ldots , n_k)$, and the first $m_{1}$ values $X_1^*, \ldots , X_{m_{1}}^*$ are singletons. Then
	\begin{equation}
	\label{eq:pred_cond}
	\begin{split}
	&\P (X_{n+1} \in {\rm d} x^* | X_{1:n}, J_{1:m_1} ) = (1-\beta) P_0({\rm d} x^*) + \beta \frac{V_{n-\bar{M}_{m_1}+1, k-\bar{M}_{m_1}+1}}{V_{n-\bar{M}_{m_1}, k-\bar{M}_{m_1}}} Q_0({\rm d} x^*)\\
	&\qquad+ \beta\frac{V_{n-\bar{M}_{m_1}+1, k-\bar{M}_{m_1}}}{V_{n- \bar{M}_{m_1}, k- \bar{M}_{m_1}}} \left( \sum_{i=1}^{m_1} J_i (1-\sigma)  \delta_{X_i^*} ({\rm d} x^*)+ \sum_{i=m_1 + 1}^{k} (n_i-\sigma)  \delta_{X_i^*} ({\rm d} x^*) \right),
	\end{split}
	\end{equation}
	where $\bar{M}_{m_1} = \sum_{i=1}^{m_1} (1 - J_i)$ represents the latent number of structural singletons.
\end{proposition}
From the sampling mechanism dictated by the predictive distribution \eqref{eq:pred_cond}, it is apparent that those values sampled from the contaminant measure $P_0$ cannot be observed twice; moreover, at each sampling step, the probability of sampling a contaminated observation equals $1-\beta$ and does not depend on $n$.
Note also that the sample without the $\bar{M}_{m_1}$ structural singletons is characterized by the usual predictive mechanism of  Gibbs-type priors. Finally, it is worth mentioning that the prediction rule has a nice interpretation in terms of a modified Chinese restaurant metaphor. Consider a restaurant with two rooms: a social and a non-social room.
The first customer arrives and she chooses a table either in the social room with probability $\beta$ or in the non-social room with probability $(1-\beta)$, she also chooses a dish which is shared by all the customers that will join the same table.
When the $n$th customer arrives, she first selects either the social room with probability $\beta$ or the non-social room with probability $1-\beta$. In the former case she can either sits at new table or at an occupied table according to the traditional Chinese restaurant metaphor in the social room, while in the latter case she sits alone at a new table eating a new dish.

If we further assume that $P_0 = Q_0$, which corresponds to a proper species sampling model, we can derive an explicit form of the predictive distribution integrating over $J_{1:m_1}$ as shown in the following result.
\begin{proposition} \label{prp:predictive_mar}
	Let $\tilde p$ be a contaminated Gibbs-type prior  as in \eqref{eq:HGT_def}, with  $P_0=Q_0$ a diffuse probability measure
	on $(\mathds{X}, \mathscr{X})$. Suppose that  $X_i | \tilde{p} \stackrel{\mbox{\scriptsize{iid}}}{\sim} \tilde{p}$, as $i \geq 1$, and consider a  sample  $X_{1:n}$ which displays $K_n=k$ distinct values, denoted as $X_1^*, \ldots , X_k^*$, with respective frequencies $(N_{n,1}, \ldots , N_{n, K_n})= (n_1, \ldots , n_k)$, and the first $m_{1}$ values $X_1^*, \ldots , X_{m_{1}}^*$ are singletons. Then
	\begin{align}
	\label{eq:predictiveG}
	&\P (X_{n+1} \in {\rm d} x^* | X_{1:n} ) =  \left\{ (1-\beta) +\beta \frac{\mathds{E}_{\bar M_{m_1}}[V_{n-\bar M_{m_1} + 1, k-\bar M_{m_1} + 1}]}{
		\mathds{E}_{\bar M_{m_1}}[V_{n-\bar M_{m_1}, k-\bar M_{m_1}}]} \right\} P_0 ({\rm d} x^*)\notag \\
	&\qquad+ \frac{1}{m_1}\sum_{i=1}^{m_1} \beta (1-\sigma) \frac{\mathds{E}_{\bar M_{m_1}}[(m_1 - \bar M_{m_1}) V_{n-\bar M_{m_1} + 1, k-\bar M_{m_1}}]}{
		\mathds{E}_{\bar M_{m_1}}[V_{n-\bar M_{m_1}, k-\bar M_{m_1}}]} \delta_{X_i^*} ({\rm d} x^*)\\
	&\qquad\qquad+ \sum_{i=m_1+1}^{k} \beta (n_i-\sigma) \frac{\mathds{E}_{\bar M_{m_1}}[ V_{n-\bar M_{m_1} + 1, k-\bar M_{m_1}}]}{
		\mathds{E}_{\bar M_{m_1}}[V_{n-\bar M_{m_1}, k-\bar M_{m_1}}]} \delta_{X_i^*} ({\rm d} x^*)\notag
	\end{align}
	where $\bar M_{m_1}\sim \mathrm{Binom}(m_1, 1 - \beta)$.
\end{proposition}
We refer to Section \ref{sup:predictive} of the Appendix for a proof of Proposition \ref{prp:predictive_mar}.
The predictive distribution \eqref{eq:predictiveG} clearly shows that the probability that $X_{n+1}$ does not belong to $\{X_1^*, \ldots , X_k^*\}$ depends on the initial sample through the sample size $n$, the number of distinct values $k$ and the number of singletons  $m_1$. This is a remarkable addition w.r.t. the Gibbs-type family, in which such a probability does not depend on $m_1$ \citep{Bat_17}. Moreover  the probability that $X_{n+1}$ equals a previously observed value $X_i^*$, with $i=1,\ldots,k$, not only depends on $n$, $n_i$ and $k$, as in the  Gibbs-type framework, but also on $m_1$. As a consequence contaminated models allows to enrich the predictive structure of an exchangeable model, though the inclusion of the additional sampling information on the number of singletons out of the observable sample. On the other side analytical tractability is still preserved.
In Section \ref{sup:prob_new_comparison} we study the re-sampling mechanism induced by contaminated Gibbs-type prior in comparison with standard Gibbs-type priors. More precisely we show that the contaminant measure mainly acts on singletons by decreasing their re-sampling probabilities w.r.t. observations with higher frequencies. On the other side, for observations with frequency larger than one, we  preserve   the same reinforcement as the discrete term of the model, and the parameter $\sigma$ exhibits the same behavior as in the Gibbs-type case.   We now specialize all the results   for the contaminated Pitman-Yor process of Example \ref{ex:cPY1}.
\begin{example}[contaminated Pitman-Yor (continued)]\label{ex:PYp2}
	Consider the contaminated Pitman-Yor process of Example \ref{ex:cPY1}. We may recover an explicit expression  for  its EPPF starting from \eqref{eq:EPPF_convex_comb} and  by observing that  the weights $V_{n,k}$s equal \eqref{eq:w_PY}. Thus, we obtain

	\begin{equation}
	\label{eq:EPPF_convex_comb_PY}
	\Pi_k^{(n)} (n_1, \ldots , n_k) =  \prod_{i=1}^k  (1-\sigma)_{n_i-1}   \sum_{\bar m_1=0}^{m_1} \binom{m_1}{\bar m_1} \beta^{n - \bar m_1} (1-\beta)^{\bar m_1}  \frac{\sigma^{k - \bar m_1} (\vartheta/\sigma)_{k - \bar m_1}}{(\vartheta)_{n - \bar m_1}}.
	\end{equation}

	See Section \ref{sup:eppf_PY} of the Appendix for the derivation of \eqref{eq:EPPF_convex_comb_PY}. The expression of the EPPF \eqref{eq:EPPF_convex_comb_PY} plays a central role to carry out posterior inference in our applications,  indeed all the algorithms we have developed (see Section \ref{sup:disc} of the Appendix) are based on this expression.
	We conclude the example specializing the predictive distribution \eqref{eq:pred_cond} for the contaminated Pitman-Yor model:
	\begin{equation}
	\label{eq:predictive}
	\begin{split}
	&\P (X_{n+1} \in {\rm d} x | X_{1:n}, J_{1:m_1}) =  (1 - \beta)P_0({\rm d} x) + \beta \frac{\vartheta + (k - \bar{M}_{m_1}) \sigma}{\vartheta + n - \bar{M}_{m_1}} Q_0({\rm d} x)\\
	&\qquad\qquad + \sum_{i=1}^{m_1}J_i \beta\frac{1 - \sigma}{\vartheta + n- \bar{M}_{m_1}} \delta_{X_i^*}({\rm d} x)+ \sum_{i=m_1 + 1}^{k}\beta\frac{n_i - \sigma}{\vartheta + n - \bar{M}_{m_1}} \delta_{X_i^*}({\rm d} x).
	\end{split}
	\end{equation}
	In Section \ref{sec:lemma} we show that the probability of sampling a new value is monotone as a function of the number of distinct values $m_1$, which  results in a richer predictive structure w.r.t. the Pitman-Yor case, where $m_1$ does not appear in the probability of sampling a new value. For example, the dependence on $m_1$ is always increasing in the Dirichlet process case ($\theta=0$), whereas it is always decreasing in the stable process one ($\sigma=0$). Some numerical experiments are presented in Section \ref{sup:prob_new}. Finally  the predictive distribution \eqref{eq:predictive} can be described in terms of an urn model, with solid and strip balls, when the prior for $\beta$ is a beta with parameters $\alpha$ and $\vartheta$, which correspond to the initial weight of strip colored balls and of black solid balls, respectively. At the first sampling step, if a strip colored ball is drawn from the urn, then we return the ball in the urn with an additional strip colored ball of a new color. On the other side if we draw a black solid ball, then we return a black ball in the urn with an additional black ball of weight $\sigma$ and a solid ball of a new color with weight $1-\sigma$. At the generic $i$th step, one can sample three different kinds of balls: a strip ball of an arbitrary color, a black solid ball or a  colored solid ball, where once that we draw a colored solid ball, we replace that ball in the urn with an additional one of the same color. See Section \ref{sup:urn} for a detailed description of the updating mechanism.
\end{example}

We conclude this section with some considerations on distributional properties of the number of clusters with a given frequency in a sample of size $n$:
this helps us to better understand the advantage of contaminated Gibbs-type priors. To fix the notation, we consider a sample $X_{1:n}$ from a  contaminated Gibbs-type prior, and we denote by  $M_{n,r}$ the random number of elements observed $r$ times out of the sample. In the sequel, if $V$ is a statistic depending on the sample $X_{1:n}$, we write $V(\beta)$ to make explicit the dependence on the parameter $\beta$ of the contaminated prior \eqref{eq:HGT_def}.
The following proposition clarifies the effect of the contaminant component with respect to the Gibbs-type model in terms of stochastic dominance and asymptotic properties.
\begin{proposition}\label{prp:asymptotic}
	If $\beta_1 < \beta_2$, then $K_n (\beta_1)$ (resp. $M_{n,1} (\beta_1)$) stochastically dominates $K_n (\beta_2)$ (resp. $M_{n,1} (\beta_2)$).
	Moreover, as $n \to +\infty$, we have
	\[
	\frac{K_n}{n}  \stackrel{a.s.}{\to} 1-\beta, \quad  \frac{M_{n,1}}{n}  \stackrel{a.s.}{\to} 1-\beta \quad \text{and} \quad
	\frac{M_{n,r}}{n^\sigma} \stackrel{a.s.}{\to} \frac{\sigma (1-\sigma)_{r-1}}{r!}  S_{\sigma} \beta^\sigma,
	\]
	where $S_\sigma$ denotes the $\sigma$-diversity random variable \citep{Pit06}.
\end{proposition}
The first part of Proposition \ref{prp:asymptotic} is a result of first order stochastic dominance and it clarifies the effect of the contaminant measure in the model \eqref{eq:HGT_def}. As $\beta$ decreases, the number of distinct values and the number of singletons out of $X_{1:n}$ increases.
By noticing that the case $\beta=1$ corresponds to a Gibbs-type prior, it is now apparent that our model has the advantage to increase (in mean) the number of distinct values and the number of singletons: the smaller beta, the higher $\mathds{E} [K_n (\beta)]$ and $\mathds{E} [M_{n,1} (\beta)]$.
The second part of Proposition \ref{prp:asymptotic} tells us that the number of distinct values $K_n$ and the number of unique values scale linearly with $n$: this is a remarkable difference with respect to Gibbs-type priors. Indeed, as $n \to +\infty$, for Gibbs-type priors both $K_n$ and $M_{n,1}$ grows as $n^\sigma$ \citep{Pit06}. Also, the asymptotic behavior of $M_{n,r}$ remains unchanged with respect to Gibbs-type priors, apart for the presence of the factor $\beta^\sigma$. This asymptotic behavior clarifies the role of the contaminant measure $P_0$ in \eqref{eq:HGT_def}, which produces an inflation of the number of singletons, and consequentially of the number of unique elements, but it is not acting on higher frequencies values.
We now specialize the results for the contaminated Pitman-Yor process.

\begin{example}[contaminated Pitman-Yor (continued)]  \label{ex:PY3}
	As for contaminated Pitman-Yor priors of Example \ref{ex:cPY1}, it is possible to evaluate the expected value of $M_{n,r}$ and $K_n$, in particular we have obtained:
	\begin{align*}
	\mathds{E} [M_{n,1}] &=  n (1-\beta) +  n \beta  \mathds{E} [(\beta B_1 +(1-\beta))^{n-1}] \\
	\mathds{E} [M_{n,r} ] &= \frac{(1-\sigma)_{r-1}}{(\vartheta+1)_{r-1}} \binom{n}{r} \beta^r \mathds{E}[(B_r \beta +1-\beta)^{n-r}], \quad \text{if } r \geq 2 \\
	\mathds{E} [K_n]& = \frac{\vartheta}{\sigma} \mathds{E} [ (B_1\beta +1-\beta)^n ] +\frac{n \beta}{\sigma} \mathds{E} [B_1  (B_1\beta+1-\beta)^{n-1}]-\frac{\vartheta}{\sigma} +n (1-\beta)
	\end{align*}
	where $B_r $ is a Beta random variable with parameters $(\vartheta+\sigma, r-\sigma)$, as $r \geq 1$.
	See Section \ref{sec:app_cPY} of the Appendix for further details.  In Figure \ref{fig:prior_plots}, we compare the behavior of the expected values of the statistics $K_n$, $M_{n,1}$ and $M_{n,2}$  in the Pitman-Yor case with the same quantities for the contaminated model. It is apparent that for the latter model the two curves of $\mathds{E}[K_n]$ and $\mathds{E}[M_{n,1}]$ grow faster as function of $n$, with respect to the Pitman-Yor model.
	We finally underline that, resorting to the results by \cite{Fav13}, one may face prediction for a large number of statistics arising in species sampling models. Indeed, in Section \ref{sec:app_cPY} of the Appendix, we evaluated the posterior expected value of the following meaningful statistics: i)  $K_m^{(n)}$, which denotes
	the number of distinct species out a future sample  $X_{n+1:n+m}= (X_{n+1}, \ldots , X_{n+m})$ not yet observed in the initial sample $X_{1:n}$; ii) $N_{m,r}^{(n)}$, which denotes the number of new and distinct observations recorded with frequency $r$ out of the additional sample $X_{n+1:n+m}$, hitherto unobserved in the initial sample of size $n$.
	All these posterior expected values display closed form expressions (see Section \ref{sec:app_cPY} for details), which depend  not only on $n$ and $k$, as for all the class of Gibbs-type priors \citep{Bat_17}, but also on the number of singletons $m_1$, thus improving the flexibility of the Pitman-Yor process. Building upon the results of \cite{Fav13}, one may derive formulas also for  contaminated Gibbs-type random partitions.
	\begin{figure}
		\centering
		\includegraphics[width=0.9\textwidth]{exp_prior_plots}
		\caption{Curves of $\mathds{E}[K_n]$, $\mathds{E}[M_{n,1}]$  and $\mathds{E}[M_{n,2}]$ as $n$ increases, for the contaminated Pitman-Yor model with $\beta=0.9$ (red), the contaminated Pitman-Yor model with $\beta=0.95$ (green), and the Pitman-Yor model (blue). The parameters are $\vartheta=50$ and $\sigma=0.2$.}\label{fig:prior_plots}
	\end{figure}
\end{example}

\section{Mixtures of contaminated Gibbs-type priors} \label{sec:mixture}

Contaminated Gibbs-type priors are not restricted only to species sampling models, but they can be used also as mixing measures to build contaminated mixture models.
Mixture models in Bayesian nonparametrics were early introduced by \citep{Lo84} for the Dirichlet process mixtures of univariate Gaussian distribution case, and later extended in several directions, by considering different kernel functions or mixing measures.
The standard general framework can be described as follows. It is assumed that  observations are $\mathds{R}^d$-valued random elements generated from a random density $\tilde f(y) = \int_\Xi \mathcal K(y;\xi)  p({\rm d} \xi)$, where $\mathcal K(y; \xi):  \mathds{R}^d \times \Xi  \to \mathds{R}^+$ is a kernel and $\Xi$ is  a general Polish space.
Furthermore, the mixing measure $ p$ is usually assumed to belong to a specific class of discrete random probability measures.
If one denotes by $\xi_1, \ldots , \xi_n$ the latent variables corresponding to a sample of size $n$ from  $p$, the standard mixture model may be expressed in the following hierarchical form
\begin{equation}
\label{eq:mix_model}
\begin{split}
Y_{i} | \xi_i  \stackrel{\mbox{\scriptsize{ind}}}{\sim}   \mathcal K(\, \cdot \, ;\xi_i) , \quad \xi_i | {p}  \stackrel{\mbox{\scriptsize{iid}}}{\sim} {p} \\
\end{split}
\end{equation}
for any $i =1, \ldots , n$.
We remark that the model \eqref{eq:mix_model} describes a general formulation of a mixture models. Indeed a realization of $\tilde{f}$ can be a continuous distribution, a discrete distribution \citep[e.g.][]{Krn08}, or a distribution defined on more abstract spaces, depending on the kernel function $\mathcal K$. Nowadays it is an established opinion in the applied statistics framework that mixture models are flexible tools for density estimation and model-based clustering analysis \citep{Fru19}.

Here we propose to extend such framework by choosing as mixing measure $p$ the contaminated Gibbs-type prior $\tilde p$.
Thanks to the definition of $\tilde p$, which is a linear convex combination of two elements, we can decompose the mixture in two terms, a first term corresponding to the discrete part of $\tilde p$ and a second term which corresponds to the diffuse component,
\begin{equation}\label{eq:model2}
\tilde f(y) = \beta \sum_{j=1}^{\infty} p_j \mathcal K(y;Z_j) + (1 - \beta) \int_\Xi \mathcal K(y;\xi) P_0({\rm d} \xi)
\end{equation}
where the last equality holds in force of the almost sure discreteness of $\tilde q= \sum_{j \geq 1} p_j \delta_{Z_j}$. The first term on the r.h.s. of equation \eqref{eq:model2} describes the standard random mixture components of the model, while the second term corresponds to a different probabilistic mechanism contaminating the mixture. A noteworthy application of this model is to the cases where outliers are possibly present in the data. Indeed, according to well developed classical theory \citep{Fru19} they can be interpreted as generated by a different random process with respect to the other observations.

If one considers $Q_0 \neq P_0$, she can specify the contaminant measure $P_0$ depending on a specific scenario of interest: if our prior opinion is translated into contaminant observations on a particular subset of $\mathds{R}^d$, we can force $P_0$ to shrink its mass on such subset. On the other hand, if we aim to model possible contaminant observations spreading over the entire support, we can specify $P_0$ over-disperse with respect to $Q_0$.

From the hierarchical formulation \eqref{eq:mix_model}, it is apparent that the random probability measure $\tilde p$ governs the distribution of the latent parameters $\xi_i$s. Thus, posterior inference for mixture models may  be performed by exploiting the results described in the previous sections deriving a marginal sampling strategy in the spirit of the seminal works of \citet{Esc88} and \citet{Esc95}.
See Section \ref{sup:mix} of the Appendix for a description of a possible sampling strategy to perform posterior inference with mixtures of contaminated Pitman-Yor processes.

\section{Illustrations}  \label{sec:exp}
\subsection{Simulation studies}  \label{sec:simulations}

In the Appendix we carried out some simulation studies to illustrate the use of the contaminated Pitman-Yor process. We first tested the proposed model in discrete scenarios by simulating  observations from the contaminated Pitman-Yor process of Example \ref{ex:cPY1}, with different values of the parameters $\beta$, $\vartheta$ and $\sigma$. See Section \ref{sup:sim_disc}.
We faced posterior inference on the main parameters of the model, on $\beta$ and the number of structural singletons $\bar{M}_{m_1}$ by exploiting Algorithm \ref{algo:discrete}. Our strategy provides good results in terms of parameters' estimation, also in comparison with the Pitman-Yor process, which, for example, overestimates the discount parameter of the model in presence of contamination of the data. With the proposed model, we also  obtain reliable estimates of the weight $\beta$ and the number of structural singletons.

We then moved to a simulation study within the framework of mixture models in Section \ref{sec:cont_simu}. We tested the model on different simulated scenarios, where observations are generated from a mixture of Gaussian distributions, with the inclusion of some outliers in the sample. See Section \ref{sup:mix} of the Appendix for further details on the data generating process. We compare posterior inference faced with three different Gaussian mixture models, where the random mixing measure is specified as: i) a contaminated Pitman-Yor with $P_0 = Q_0$; ii) a contaminated Pitman-Yor with $P_0 \neq Q_0$, forcing an over-dispersion of the contaminant measure; iii) a Pitman-Yor.
Posterior inference is carried out on the basis of a marginal sampling scheme with the goal of outlier detection (see Algorithm \ref{algo:mixture} in Section \ref{sup:mix} of the Appendix). Note that  we consider an observation to be an outlier iff it is clustered as a singleton in the posterior point estimate of the latent partition dictated by the data. From Table \ref{tab:sing} in the Appendix, one may realize that the Pitman-Yor mixture model is not appropriate to perform outlier detection, indeed only few clusters with frequency one are detected. On the other hand, the two specifications of the contaminated Pitman-Yor mixture models produce reasonable estimates of the number of contaminants in the data, and the model with  $P_0 \neq Q_0$ displays appreciable superior performances.

\subsection{The North America Ranidae dataset}  \label{sec:discrete_app}

We consider a set of species detection data from the Global Biodiversity Information Facility project \citep{GBI21}. The project is an extensive database consisting in record of species found across the world, where for each individual is reported the taxonomy, location and possibly other relevant information. Our sample consists of $n = 131\,204$ observations belonging to $k = 619$ distinct species of the Ranidae family observed in North America, and identified by their scientific name.
Among the $k=619$ species, $m_1 = 296$ species were observed only once in the sample, creating a possible inflation of the number of elements with frequency one. Such inflation might be caused by miss reported scientific name of the observed animals. We aim to investigate the benefit of including a contaminant measure in the prior model specification by comparing posterior inference when we use a contaminated and a standard Pitman-Yor process.  We choose non-informative prior specifications  for the parameters, namely $\vartheta \sim \mathrm{Gamma}(2, 0.02)$ and $\sigma , \beta \sim \mathrm{Unif}(0,1)$. We carried out posterior inference by  exploiting Algorithm \ref{algo:discrete} described in Section \ref{sup:disc} of the Appendix, and similarly for the standard model. Refer to Section \ref{sup:disc_app} for diagnostic summaries and algorithmic details.

\begin{figure}
	\centering
	\includegraphics[width = 0.9 \textwidth]{posterior_plots2.pdf}
	\caption{Posterior summaries for contaminated Pitman-Yor model (green) and Pitman-Yor model (orange). Left panel: frequency spectrum of the first non-empty frequencies, with the posterior expectation of the discrete part of the two models, shaded bands represent $90\%$ posterior credible intervals; the dashed line corresponds to the inflation of the diffuse component;. Right-top panel: posterior probability of sampling a new species. Right-bottom: posterior distribution of $\sigma$.}
	\label{fig:disc_app}
\end{figure}

Figure \ref{fig:disc_app} clarifies how the presence of a large number of species observed only once leverages the estimation of the parameters in the Pitman-Yor model, while the use of a  contamination component helps to obtain a much more suitable modeling of the data. Indeed, in the latter case, some of the observations with frequency $1$ are assigned to the diffuse component. As consequence of the excessive number of singletons, the estimated posterior distribution of the frequency spectrum is remarkably different on small values of the support, as emphasized in the left panel of Figure \ref{fig:disc_app}. Furthermore, both the probability of sampling a new species and the posterior distribution of $\sigma$ in  the Pitman-Yor case are translated with respect to the contaminated model. Additional posteriors summaries are reported in the Appendix: the posterior distributions of $\vartheta, \beta$ and $\bar{M}_{m_1}$.

We finally consider the task  of predicting the number of new species and the number of new species observed with a given frequency in a follow-up sample,  given an initial training sample. We have retained the $80\%$ of the $n$ data for purposes of training, and the remaining $m$ data points are used as a test set. We focused on estimation of: i) $K_{m}^{(n-m)}$, the distinct number of new species in a follow-up sample hitherto unobserved in the initial training sample of size $n-m$; ii) $N_{m,1}^{(n-m)}$, the number of new species observed with frequency one in an additional sample of size $m$, hitherto unobserved in the training dataset. The posterior expectations of $K_{m}^{(n-m)}$  and $N_{m,1}^{(n-m)}$ are evaluated using the corresponding closed-form expressions, reported in Equations \eqref{eq:Kmn_posterior} and \eqref{eq:Nmr1} respectively, for the contaminated Pitman-Yor model.
The predicted values are compared with the true ones, obtained by extrapolating to the remaining
$m$ data. We repeated the experiment  $1\,000$ times in order to asses variability.
Figure \ref{fig:add_sample} shows the cross-validated distributions of $K_{m}^{(n-m)}$ and $N_{m,1}^{(n-m)}$ when we exploit the contaminated Pitman-Yor model in comparison with the predicted values obtained by using the Pitman-Yor process. The average true value is represented with a dashed black line.

\begin{figure}
	\centering
	\includegraphics[width = 0.45 \textwidth]{cv_dist.pdf}\centering
	\includegraphics[width = 0.45 \textwidth]{cv_dist3.pdf}
	\caption{Cross-validated distributions of $K_{m}^{(n-m)}$  and $N_{m,1}^{(n-m)}$, for the contaminated Pitman-Yor model (green) and the Pitman-Yor model (orange). The black dashed lines correspond to the true values.

	}\label{fig:add_sample}
\end{figure}

From Figure \ref{fig:add_sample}, it is apparent how the contaminant measure in the model specification can be crucial also for its predictive properties. Indeed the cross-validated distributions of $N_{m,1}^{(n-m)}$ and $K_{m}^{(n-m)}$ for the contaminated model, conditionally on an observed sample, shrink to the corresponding average of the observed values (black dashed line), while the distributions for the model without a contaminant term provides a systematic error in prediction. Such behavior is also confirmed by the mean squared error ($\mathrm{MSE}$) of the predictions, which is bigger for the Pitman-Yor (PY) model with respect to the contaminated Pitman-Yor (CPY), indeed: $\mathrm{MSE}(N_{m,1}^{(n-m)}) = 74.12$ for CPY and $\mathrm{MSE}(N_{m,1}^{(n-m)}) = 423.36$ for PY; $\mathrm{MSE}(K_{m}^{(n-m)}) = 75.48$ under CPY and $\mathrm{MSE}(K_{m}^{(n-m)}) = 370.67$ under PY. See Section \ref{sec:inf_new_samp} of the Appendix for further details on the cross-validation study.
Finally we stress that this example highlights the strong lack of robustness of the pure Pitman-Yor model. Indeed, drastically erroneous inferential conclusions are caused by relatively few singletons ($m_1=296 $) compared to the total number of observations ($n = 131\,204 $).

\subsection{Analysis of the NGC 2419 data}\label{sec:cont_app}
We consider a set of data composed by $n = 139$ stars, possibly belonging to the globular cluster NG 2419 and sharing the same galactic center. The data were early introduced and studied by \citet{Iba11}.
For each observation we have measurements of $d = 4$ different variables: the two-dimensional projection on the plane of the position of the star $(D_1, D_2)$, the line of sight velocity $V$ on a logarithmic scale, and the metallicity of the star $[Fe/H]$ on a logarithmic scale, which is a measure of the abundance of iron relative to hydrogen. We denote by $Y_i = (D_{1,i}, D_{2,i}, V_i, [Fe/H]_i)$ the
$i$th observed vector.

A crucial problem for the astronomical community is to identify which stars belong to the globular cluster, and which star are contaminants (or outliers), to properly study the dynamic of a group of stars. To this aim, we consider a contaminated Pitman-Yor mixture model, specified with a multivariate Gaussian kernel function $\mathcal K(\, \cdot \, ; (\mu, \Sigma))$, with expectation $\mu$ and covariance matrix $\Sigma$. We further assume a base measure conjugate to the kernel function, i.e., $Q_0 \sim \mathrm{NIW}(\mu_0, \kappa_0, \nu_0, S_0)$ is a Normal-Inverse-Wishart distribution. We consider the same distributional law also for the diffuse component:  $P_0 \sim \mathrm{NIW}(\mu_1, \kappa_1, \nu_1, S_1)$.
We specify the base measure of the discrete component $Q_0$ by setting $\mu_0$ equal to the sample mean of the data, $\kappa_0 = 1$, $\nu_0 = d + 3 = 7$, and $S_0$ equals to the diagonal of the sample variance of the data. We further specify the parameters of the contaminant measure as follows: $\mu_1$ equals the sample mean of the data, $\kappa_1 = 0.1$, $\nu_1 = d + 3 = 7$, and $S_1$ matches the sample variance of the data, in order to force an over-disperse contaminant measure with respect to the base measure. We complete the model specification assuming vague priors for the parameters of the mixing measure, $\vartheta \sim \mathrm{Gamma}(2,0.02)$ and  $\sigma , \beta\sim \mathrm{Unif}(0,1)$. Posterior inference is carried out by exploiting the sampling scheme described in Section \ref{sup:mix} of the Appendix, see also Section \ref{sup:mixt_app} for diagnostics. We exploit the decisional  approach based on the variation of information loss function \citep{Wad18,Ras18}
to provide an optimal posterior point estimate of the latent partition induced by the data.

The results are summarized in Table \ref{tab:vsIBA} in comparison with the previous clusters identified by \cite{Iba11}. Within the $16$ stars identified as contaminants, $4$ belongs to the globular cluster identified by \citet{Iba11}, $5$ stars to the likely globular cluster group, and $7$ to the contaminants. Most of the stars of the main estimated cluster, denoted by $A$ in Table \ref{tab:vsIBA}, belong to the globular cluster of \citet{Iba11}. We have also recovered two additional clusters: a cluster of stars mainly belonging to the globular cluster in \citet{Iba11}, and a cluster with two contaminant stars. Our findings are coherent with the previous literature, but providing a more conservative detection of the contaminant stars. Figure \ref{fig:application_plot} of the Appendix shows the optimal partition with the mean of the estimated posterior random density. From the contour lines in Figure \ref{fig:application_plot}, we can see how the inclusion of a diffuse component is producing a smoothed estimate of the expectation of the posterior random density, while for the standard Pitman-Yor mixture model the expectation of the posterior random density shows some peaks in correspondence of the contaminants, as shown in Figure \ref{fig:application_plot_PY}. We further compare our findings with the latent partition obtained using a Pitman-Yor mixture model, as described in Section \ref{sec:mixt_comparison} of the Appendix: the optimal latent partition recovered with the contaminated Pitman-Yor prior is characterized by a larger main globular cluster and a higher number of singletons.

\begin{table}
	\centering
	\begin{tabular}{llc*{5}{c}}
		\\
		&       && \multicolumn{5}{c}{CPY partition} \\
		&       && \textit{Singletons}    & \textit{A}   & \textit{B}    & \textit{C}  \vspace{5pt}\\
		&&\multicolumn{1}{c}{\textit{total}}&\textit{16}&\textit{115}&\textit{4}&\textit{4}\\
		\multirow{3}{*}{\citet{Iba11}} & \textit{globular cluster}   &\multicolumn{1}{c}{\textit{118}}& 4  & 109 & 3 &2   \\
		& \textit{likely globular cluster} &\multicolumn{1}{c}{\textit{12}}& 5 & 6 & 1 &0 \\
		& \textit{contaminants} & \multicolumn{1}{c}{\textit{9}}& 7 &0 &0&2    \\
	\end{tabular}
	\caption{Comparison between the partition described in \citet{Iba11} and optimal partition estimated using a contaminated Pitman-Yor mixture model.}
	\label{tab:vsIBA}
\end{table}

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

We introduced a new family of priors outside the Gibbs-type one which are still tractable from an analytical viewpoint.  According to the characterization by \cite{Bat_17}, the predictive probability weights of Gibbs-type priors cannot depend on the number of observations recorder with frequency one  $M_{n,1}$ in  the initial sample.
With the inclusion of a contaminant component, we have enriched the predictive structure  of Gibbs-type priors by including the additional sampling information on $M_{n,1}$.
Moreover we discussed the usage of contaminated Gibbs-type priors in two situations: i) for discrete data in presence of and excess of ones; ii) in mixture models to account for outliers. Nevertheless, the use of contaminated Gibbs-type priors is not restricted to the scenarios presented in this manuscript, but they can be relevant in other applications, where the presence of elements with frequency one is a key inferential interest.

Firstly, contaminated Gibbs-type priors could be of potential interest in the analysis of disclosure risk for microdata. Microdata files typically contains two types of categorical information about individuals: identifying and sensitive information. Before releasing a dataset, statistical agencies estimate different measures of disclosure risk, which are typically based on the number of sample records which have a unique combination of the categorical variables and that are not shared with any element of the entire population. See, e.g., \cite{Bet_90,Ski_02,Ski_94} for possible definitions and estimators of disclosure indexes. In the disclosure risk framework, the random variable $\bar{M}_{m_1}$ appearing in our model represents a measure of disclosure, i.e., the number of records that contain a unique element both in the sample and in the whole population.

Language modeling constitutes another application area when one is interested to estimate the number of \textit{hapax legomena} in a corpus of documents. An hapax legomena is indeed a word that occurs only once in the entire production of an author.
These unique words are particularly important since they have been recognized as peculiar
usage of words by specific authors, and they represent an interesting problem to study
from a statistical perspective. See e.g. \citet{Baa01} for further details on word frequency distributions. In such a framework, one may use a contaminated Gibbs-type prior to estimate the number of hapax legomena on the basis of the latent quantity $\bar{M}_{m_1}$.

Finally a contaminated Gibbs-type model may be exploited to test the presence of contaminant observations in a set of data by selecting a spike and slab prior \citep{Mit88} for the parameter $\beta$ in \eqref{eq:HGT_def}. More precisely one may specify  a prior for $\beta$ which assigns positive mass
to the point $\beta =1$. A prior specification of this type may  be exploited to perform posterior inference on the presence of a contaminant term in the model, looking at the posterior probability of $\{ \beta =1 \}$.
Work on these points is ongoing.

\section*{Acknowledgement}
The authors gratefully acknowledge the financial support from the Italian Ministry of Education, University and Research (MIUR), ``Dipartimenti di Eccellenza" grant 2018-2022, and the DEMS Data Science Lab for supporting this work through computational resources. Federico Camerlenghi received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme under grant agreement No 817257.

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The Hoxton, Portland officially opened its doors on November 12. The much anticipated opening was the second U.S location for the Hoxton brand. The Hoxton, based in London is planning two additional openings in the United States later in 2019—The Hoxton Chicago and The Hoxton Downtown L.A. The Hoxton Portland, located in historic Old Town Chinatown in a building dating back to 1906, features 119 rooms and three restaurants including a cocktail bar.
The Hoxton Portland also offers multiple meeting and event opportunities in The Apartment which is scheduled to open later this year. By day, this space is perfect for meetings, brainstorming sessions, photo and video shoots, and presentations. By night, it is ideal for private events and parties. The Apartment includes separate meeting rooms in a variety of sizes. This space will also host workshops and master classes available to guests.
A defining quality of The Hoxton is its incorporation of the diverse neighborhood culture. The Hoxton Portland invites local culture into its space by offering local artists and start-ups the opportunity to showcase their work. Hox Friends is a program to initiate engagement between hotel guests and locals by inviting neighborhood members to handpick 10 reads and insights about the area to share with guests. This program establishes a relationship between hotel guests and locals, immersing visitors in the rich cultural aspects of the surrounding neighborhood. For more information, view the full story here.


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Amy Chang made me sweat and groan for every ounce lost and/or redistributed - and I would still highly recommend her to anyone looking for a personal fitness trainer!

I became Amy's reluctant client several years ago at the urging of my family.  I was overweight, lethargic and eating really badly. With a lot of patience, Amy brushed aside all of my excuses and worked to make me stronger. She emphasized correct form (to avoid injuries) and set a pace which was definitely hard but not punishing.

After I started feeling stronger and less lethargic, Amy steered me towards group exercise classes to add a cardio component to my workouts. She also started making suggestions about my diet. Overall, she inspired me feel that I had to be more accountable to myself for my over all health and well-being.

Today I'm two dress sizes smaller, more energetic and much more conscious of my eating habits. My improvements encouraged my husband to sign on as Amy's client. We have such confidence in Amy that we asked her to help our 14 year old daughter stay in shape when our daughter was recovering from foot surgery.

The best thing about Amy - even if she doesn't say it - you know that she sincerely wants you to succeed in reaching your health goals and that she's with you every step of the (difficult) way.

- Louella T.

Amy is the most amazing trainer I have ever worked with. She helped me reach my weight and strength goals through her extensive knowledge and, most importantly, her encouragement. She is motivating, and she pushed me to levels I never thought I could reach. She is also a wonderful person who cares about her clients, and she is passionate about what she does.

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I've been training with Thao for several years now and am in the best shape of my life. She has helped me through several injuries and has also provided special training for different activities. Thao has increased my core strength and my flexibility which has helped me in golf, skiing, triathlons and half marathons. She motivates me and pushes me to my limits and makes each and every workout unique so I do not get bored with coming to the gym. I really like the facilities at BMI Fitness as they have all the equipment I need for a complete workout, TRX, cardio machines, free weights, cable machines, Pilates equipment and Thao incorporates all of these into my workouts. Thanks to Thao I have been able to maintain my health and fitness.

- Jeannette

For the last 3 years, Amy Chang has been my personal trainer and she has been fabulous!! Our weekly sessions have substantially elevated my fitness to a new level: my "body" age is almost 20 years younger than my chronological age. I have always been quite athletic, but now, and in great part thanks to Amy, I am the fittest I have been in my entire life.  Amy is intelligent, engaging and very informed; she makes each and every session unique and properly tailored to the individual client. She is also an amazing group fitness instructor who has the ability to engage and attract a huge crowd (including my two daughters) to many of her very challenging and fun classes.

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Amy Chang is a dedicated personal trainer and a friend. Her skills, knowledge and experiences are far beyond her profession. A year ago, before we signed up with her for one-on-one training, we were a bit hesitant. It's a big investment after all. We're happy to say that this journey has turned out to be one of our most successful investments.

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The most dramatic impact of training with Amy has been on our health. Jenny's rheumatoid arthritis has tormented her for years. While all the other remedies, medical or none-medical, have been tried and failed, personal training with Amy has proven to be a huge success. With Amy's thoughtful workout plan, her ailment has been effectively contained. A year ago, Jenny's joints were so stiff, especially during the winter, that she even had a hard time getting up every morning. Now her joints are much more flexible and the pains are substantially subdued. She's active everywhere in the gym now, spin class, U-Jam, kickboxing etc.

The treatment of Charles' chronic back pain is another success story. For years, Charles has been beset by the back pain that goes back to his athletic young adulthood. Invariably, all the medical approaches have been tried and proven ineffective. Through the structured core training with Amy, by systematically working on the relevant muscle groups, Charles' back pain is literally gone.

Thanks to Amy, we are able to live our active lives again. Now we have confidently signed up our son and daughter, Lue and Charlene, to train with Amy.

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Any personal trainer can teach exercises and suggest diet changes to help a client lose a few pounds. What is special and rare, is a personal trainer who can inspire a client to make fitness an integral part of their life. This is what Amy Chang has done for me. I have been working with Amy for several years and she has inspired and supported me beyond measure. She has taught me success doesn't necessarily mean a decreasing number on a scale, but rather how setting a goal like getting to the gym 5 days a week without exception can and will make you successful. She has encouraged me to think outside my comfort zone and push to achieve goals like training for and running a half marathon. More importantly, she has taught me that it's not about being accountable to my trainer every week, but rather to myself. Thanks to Amy, I have and continue to achieve my fitness goals.

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Thao is extremely effective. During the last two years, she has reduced my back pain, increased my strength training, and increased my flexibility. She is also good at working around my busy schedule. Finally, the facilities at BMI are comprehensive, clean, and well maintained.

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I have been working with Lynn Christensen-Aden for many years now and feel she is a friend, as well as a wonderful trainer.  She helped me through my original 35 pound weight loss and strengthening regimen and made physical exercise, as well as rolling, a part of my life.  We identify goals to work towards and the next ones are being fit enough to achieve:  The Millford Walk in New Zealand, Machu Pichu and Everest Base Camp.

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There are many trainers out there, but they truly are not all the same. I have tried dozens of others over the years and have been sorely disappointed each time. Unfortunately, I no longer live in the area or else I would still be working out with Vince. Vince takes the time to understand your goal(s), develops a plan, and helps you execute the plan. Even though you may think you are exhausted, somehow he knows when you have more in you to complete an exercise. If you are committed to putting in the work, Vince will support, inspire, and motivate you to accomplishing your goal(s). If you are looking for a personal trainer and Vince is in your area, put your money to good use and call him. Seriously, you won't find a better personal trainer.

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When I first met David over 8 years ago, my doctor had told me I had to lose weight or face serious medical issues. Ever since that day, David has been my personal trainer.  He created workouts to help me lose over 75 pounds, has trained me through various athletic events (that I never thought I would do) and also helped rehab me through major knee surgery (a result of carrying too much weight for years).  He has spent countless hours reviewing meal plans and providing endless pep talks when I was ready to give up. He teaches his clients how to develop healthy habits that lead to a healthy lifestyle.  It may sound cliche but he helped "save my life". The road to healthy living is not always easy and I know I had to put in the work but having a personal trainer like David in my corner is a key part to my health successes past, present and future.

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Angelina has been my partner in my journey to lose weight for 2 years. She transformed me to adopt a healthy lifestyle. I have realized that the hard part of this journey isn't getting your body in shape, it is about getting your mind in shape. Angelina has figured out the algorithm to work with the person, emotionally and physically. The results I have achieved to date are truly a reflection of her experience and excellent training techniques. Her knowledge of human body anatomy and how human muscle groups are connected truly reflects in her plans.

She individualizes the training by an introductory session in which she evaluates muscle groups that one need to work on to achieve maximum results. One of her strengths is posture correction and working with small muscle groups which is mostly neglected by many trainers.  My day to day job requires working on a computer and, I used to suffer from neck and lower back pain due to tight and week muscles. She helped me to condition and strengthen those muscles around my neck and lower back. She consistently takes feedback and is very respectful of pain and would always accommodate to make changes to the routine. She quickly transitioned me into handling a rigorous workout regime.

One of my biggest challenges is, I get bored easily on a cardio machine or having to do the same exercise. Angelina would plan my routine around different gym props and do various partner workouts. She incorporates variation into the routine. Going to the gym, now, is fun and something I eagerly look forward to, which is a huge personal milestone. As I kept building on my strength and endurance, she would push my limits and challenge my abilities in the same session length. Sometimes we don't know what we can handle. She understood and motivated me to do better.  As I look at myself now, I am genuinely amazed at how strong she helped me become. I lost 45 lbs and gained both mental and physical stamina. This is what made me stick with her for two years, and more importantly, I see myself continuing for the long term. Angelina deserves all the best compliments.

P.S - Make an appointment soon before her calendar fills up.

- Sravya

Dave's spin classes are challenging, inspiring, and entertaining.  Dave has helped me gain the strength, endurance, and confidence to complete a 50 mile mountain bike endurance ride in the high Sierra mountains, and now I'm usually the first one to reach the top of the ascents in my mountain biking group!  I've just started road biking, and the 3500 foot climb to the top of Mt. Umunhum was a snap, after all the "max-hill climb" drills in spin class.  His fun and challenging workouts keep you coming back for more.

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Beth Pagano has been my personal trainer for years. I have recommended her services to several friends. I credit her training for the fact that my body, now in extremely late middle age, still has the original knees and hips, a straight back, and full range of motion. I credit her personality for the fact that our sessions together are so enjoyable.

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I had a great experience with Ai as may health coach for over 6 months. Here are some highlights of her coaching: 1) Active and involve when explaining an exercise, providing a demonstration and then watching her client closely to make sure they are doing it the right way (to avoid injury and to get the most from the exercise) 2) knowledgeable of many different exercise routines so she can tailor programs to meet your needs 3) Takes video of the workout and provides it to clients for reinforcing the learning (ie, form) 4) Discusses diet and other healthy habits as needed to build upon the workout experience More important than all of the above is that I found Ai to be dedicated to her profession and to me as a client. She is the type of person who expects a lot of herself and pours that energy and enthusiasm into her clients. I recommend Ai Nguyen as a health coach without any reservations. You will benefit from the experience in many ways, starting with your health.

- Chris R.

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Nykaa, an online cosmetics retailer, was started by former investment banker Falguni Nayar in 2012.
Nykaa in talks to raise Rs75 crore at a valuation of Rs3,000 crore
2 min read . Updated: 13 Apr 2018, 03:55 PM IST Swaraj Singh Dhanjal
Nykaa investor TVS Capital is looking to sell part of its stake in this new funding round
mint-india-wire NykaaNykaa fundingNykaa valuationFalguni NayarNykaa investorsTVS CapitalNykaa productscosmetics onlineonline retaile-commercestart-ups
Mumbai: Nykaa, a leading player in the online Indian beauty and cosmetics space, is in talks to raise Rs75 crore in a new round of funding that could see the company being valued at almost Rs3,000 crore, two people aware of the development said.
Nykaa was started by former investment banker Falguni Nayar in 2012. Before starting Nykaa, Nayar was managing director at domestic investment bank Kotak Mahindra Capital Co. Ltd. The company offers makeup, skincare, hair care, fragrances, bath and body, luxury and wellness products for women and men.
"Nykaa is in talks with various people to raise its latest round of capital, its fourth round so far. They are talking to both institutional investors such as PE funds, as well as with HNIs and family offices. The company is seeking a valuation of around Rs3,000 crore," said one of the persons cited above, requesting anonymity as the talks are private.
The company has been growing at a fast clip, he added, and recorded a topline of over Rs200 crore in 2016-17.
According to the second person cited above, TVS Capital an investor in the company, is looking to sell part of its stake in this new round.
"TVS Capital invested in the company in 2015 and they are now looking to exit a large part of their stake in this round. TVS Capital is looking to dilute its stake as its current fund is coming close to the end of its life cycle," he said, also requesting anonymity.
"We don't comment on these matters," said Gopal Srinivasan, founder chairman of TVS Capital Funds in an email response.
Calls and text messages sent to Falguni Nayar too went unanswered.
What stands out about Nykaa is that unlike other e-commerce ventures, the company has until now relied upon home-grown capital to build up the franchise.
Nykaa had last raised funds in 2016, when it raised a little over Rs100 crore from Max Ventures and Industries Ltd, the Munjal Family Office, Yogesh Mahansaria and Harsh Mariwala. Earlier in 2015, the company had raised Rs60 crore in a round led by TVS Capital. In its first round of fundraising in 2014, Nykaa had raised Rs20 crore from various investors.
Having started as an online retail firm, Nykaa has in the last couple of years evolved into an omni-channel retailer and now has around 11 offline stores. The company plans to reach 30 offline stores by 2020. Nykaa had opened its first offline store at Delhi airport's Terminal 3.
Nykaa claims to offer over 80,000 products across 700 brands on its online portal.


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WWE ring announcer Lillian Garcia responds to rumor that she's been released
By Nick Schwartz | FoxSports
Lillian Garcia has been absent from WWE programming for weeks, and on Friday a report surfaced that the longtime ring announcer had been released from the company. Garcia has worked for WWE for 15 years, and returned in 2011 two-year absence.
Garcia explained in a Facebook post that she was forced to miss work to be with her father, who is battling cancer. Garcia writes that her father's condition has fortunately improved, and that she will be returning to Monday Night Raw this week.
Via Facebook:
"Imagine my surprise today when I woke up to find out that I was among the top trends in the world on twitter! What for, I said to myself....wait, does that say "Lilian Garcia released from WWE"?? That's when I knew I had to set the story straight.
.... I mentioned on twitter that it was due to a family member being ill. Well, that family member is my father. He has been battling bladder cancer and multiple Myeloma (blood cancer) and due to a reaction to the chemotherapy, he became very ill these past two weeks. On top of that, he had to go into two emergency procedures on his kidneys. To say he has had it tough is an understatement. However, he is a retired Lieutenant Colonel from the US Army, being a Veteran from the Korean and Vietnam Wars, so his spirit is that of a fighter. I am so inspired by how he has taken on this disease and continues to battle it with all his might. He has allowed me to share this story as he hopes his journey will inspire anyone else who is battling cancer. Luckily he is doing much better now and gearing up for a new treatment next week. As you can imagine, my entire family has been through a rollercoaster of emotions and cherishing all the time we have with him.
So, on to Monday, I will be back on RAW and couldn't be happier as I have missed you all very much!! Thank you and I love you!!!
Lilian"