text:
\section{\label{sec:Introduction}Introduction}
Many real-world phenomena can be modelled by random graphs, or more
generally, by dynamically changing random graphs. Specifically,
host-pathogen biological systems that may combine primary and
nearest-neighbour or long-range secondary infection processes can be
efficiently described by spatio-temporal models based on random
graphs evolving in time~\cite{Gibson&et&al&2006}.

Although a continuous observation of an epidemic is not always
possible, a spatial `snapshot' may provide one with some, albeit
highly incomplete, knowledge about the epidemic. In terms of the
model this knowledge results in a random graph realised in some
metric space. Moreover, under some circumstances it is not even
possible to observe some or all of the edges of such a random
graph---all one would know then are the vertices which correspond to
the infected sites, i.e.\ to those sites which interacted as a
result of the evolution of the process under consideration.

One particular application refers to the colonisation of susceptible
sites, such as seeds or plants grown on a lattice, by virus, fungal,
or bacterial pathogens with limited dispersal abilities. A typical
example is the spread of infections through populations of seedlings
by the fungal pathogen, \textit{Rhizoctonia solani} K\"{u}hn. This
economically-important pathogen is wide spread with a remarkably
wide host range \cite{Chase1998}. In addition to its intrinsic
economic importance, it has been extensively used as an experimental
model system to test epidemiological hypotheses in replicated
microcosms
\cite{Gibson&Kleczkowski&Gilligan2004,Otten&Bailey&Gilligan2004} and
to study biological control of pathozone behaviour by an
antagonistic fungus and disease dynamics~\cite{Bailey&Gilligan1997}.
Transmission of infection between plants occurs by mycelial growth
from an infected host, with preferential spread along soil
surfaces---hence the missing information about the structure of
interactions.

The spread of infections with limited dispersal abilities among
plants can be viewed as a spatial \textit{SIR}
(susceptible$\thinspace\rightarrow\thinspace$infective$\thinspace\rightarrow\thinspace$removed)
epidemic with nearest-neighbour secondary infections and removals,
and can be related to a percolation process on a finite static
network~\cite{Gibson&et&al&2006,Bailey&Otten&Gilligan2000}.
Percolation has also been widely used to model various phenomena of
disordered media in physics, chemistry, biology, engineering and
materials science~\cite{Sahimi1994,Frisch&Hammersley}, and also
forest fires~\cite{Zhang,Cox&Durrett}.

Bayesian estimation for percolation models of disease spread in
plant populations in the context of the spread of
\textit{Rhizoctonia solani} has been presented in
\cite{Gibson&et&al&2006}. In \cite{Bailey&Otten&Gilligan2000} the
spread of this soil-borne fungal plant pathogen among discrete sites
of nutrient resource is studied using simple concepts of percolation
theory; a distinction is made between invasive and noninvasive
saprotrophic spread. The authors of
\cite{Gibson&et&al&2006,Bailey&Otten&Gilligan2000} formulated
statistical methods for fitting and testing percolation-based,
spatio-temporal models that are generally applicable to biological
or physical processes that evolve in time in spatially distributed
populations. Estimation of spatial parameters from a single snapshot
of an epidemic evolving on a discretised grid under the assumption
that fundamental spatial statistics are near equilibrium is studied
in~\cite{Keeling&Brooks&Gilligan2004}.

The difficulties in performing inference for these models in the
presence of observational uncertainty or incomplete observations can
be overcome to an extent by employing a Bayesian approach and modern
powerful computational techniques---mainly Markov chain Monte Carlo
(McMC), e.g.~\cite{Gibson}. McMC methods often offer important
advantages over existing methods of analysis. Particularly, they
allow a much greater degree of modelling flexibility, although the
implementation of McMC methods can be problematic because of
convergence and mixing difficulties which arise due to the amount
and nature of missing data.

An aspect which has received little attention in the context of the
described models is that of experimental design. Statisticians have
investigated the question of experimental design in the Bayesian
framework (see \cite{Chaloner_Verdinelli1995} for a review). The
work of M\"{u}ller and others~\cite{Muller1999,Verdinelli1992}
examined the ways of identifying designs that maximise the
expectation of a utility function.

The goal of this paper is to study inference and optimal design
problems for a model which is obtained by taking a final snapshot of
nearest-neighbour disease spread spatio-temporal dynamics and
relating it to the percolation process on a regular grid. We use the
utility-based Bayesian framework for this purpose and discuss
generic issues that arise in this context.

The paper is arranged as follows. In Section 2 we describe the
percolation model considered and describe the partial sampling
scenarios that form the focus of the paper. In Section 3 we describe
how, by embedding the problem in a framework of Bayesian inference,
it is possible to estimate likelihood functions for percolation
models under the respective sampling scenarios using Markov chain
Monte carlo (McMC) methods.  Moreover we prove some theoretical
results regarding the limiting properties of the likelihood for the
case when only cluster size is observed, and elicit interesting
connections with percolation thresholds. In Sections 5 and 6 we turn
our attention to the question of experimental design and, in
particular, the identification of experimental lattices that can
maximise the information on the percolation probability yielded by
an experimental process observed on it, when only the vertex set of
the connected cluster is observed.  We demonstrate that the optimal
lattice may be one which includes regions which have been sparsified
by removal of certain vertices and edges.  These results have
important implications for the design of experiments on, for
example, fungal pathogen spread in populations of agricultural
plants, where hosts are typically arranged on a lattice.

\section{\label{sec:Main}Nearest-neighbour interaction model and percolation}

\subsection{Model description}
Disease spread as a result of (typically) short-range contact
between, for example, plants can be modelled as a transmission
process on an undirected graph.  Nodes, or vertices, of the graph
correspond to possible locations of plants, and edges of the graph
link locations which are considered to be \emph{neighbours}.  In a
classical \textit{SIR} model each node, or vertex, of the graph is
in one of three states: either it is occupied by a \emph{healthy},
but susceptible, plant (state~\textit{S}), or it is occupied by an
\emph{infected} and infectious plant (state~\textit{I}), or finally
it is \emph{empty}, any plant at that location having died and thus
being considered removed (state~\textit{R}).  A plant at node $i$,
once infected (or from time $0$ if initially infected), remains in
the infected state~\textit{I} for some random lifetime $\tau_i$
after which it dies, so that node $i$ then remains in the empty
state~\textit{R} ever thereafter. During its infected lifetime the
plant at node $i$ sends further infections to each of its
neighbouring nodes $j$ as a Poisson process with rate $\lambda_{ij}$
(so that the probability that an infection travels from $i$ to $j$
in any small time interval of length $h$ is $\lambda_{ij}h+o(h)$ as
$h\to0$ while the probability that two or more infections travel in
the same interval is $o(h)$ as $h\to0$); any infection arriving at
node $j$ changes the state of any \emph{healthy} plant there to
\emph{infected}, and otherwise has no effect.  All lifetimes and
infection processes are considered independent.  The initial state
of the system is typically defined by one or more nodes being
occupied by infected plants, the remaining nodes being occupied by
healthy plants. The epidemic may \emph{die out} at some finite time
at which the set of infected nodes first becomes empty, or, on an
infinite graph only, it is possible that it may continue forever.

Thus, for any infected node $i$, the event~$E_{ij}$ that any
neighbouring node $j$ receives at least one infection from node~$i$
has probability $p_{ij}=1-\mathbb{E}[\exp(-\lambda_{ij}\tau_i)]$
(where $\mathbb{E}$ denotes expectation).  Note that, for any given
node $i$, even though the infection processes are independent, the
events~$E_{ij}$ are themselves independent if and only if the random
lifetime $\tau_i$ is a constant.  We now suppose that this is the
case and that furthermore, for all ordered pairs $(i,j)$ of
neighbours, we have $p_{ij}=p$ for some probability $p$. Suppose
further that it is possible to observe neither the time evolution of
the epidemic nor the edges of the graph by which infections travel,
but only the initially infected set of nodes and the set of nodes
which are at some time infected and thus ultimately in the empty
state~\textit{R}.  It is then not difficult to see, and is indeed
well known~\cite{Grassberger1983,Kuulasmaa&Zachary}, that the
epidemic may be probabilistically realised as an unoriented
\emph{bond percolation} process on the graph in which each edge is
independently \emph{open} with probability $p$ (or \textit{closed}
with probability $1-p$), and in which the set of nodes which are
ever infected consists of those nodes reachable along open
\emph{paths} (chains of open edges) from those initially infected.
(Note that the ability to use an \emph{unoriented} bond percolation
process requires both the assumptions that the above events $E_{ij}$
are independent and that $p_{ij}=p_{ji}$ for all $i,j$; in the
absence of \emph{either} of these assumptions one would in general
need to consider an oriented process with the appropriate dependence
structure~\footnote{Non-constant lifetime distributions $\tau_i$ can
  similarly lead to other interesting percolation processes.  For,
  example, site percolation may be approximated arbitrarily closely by
  a lifetime distribution which with some sufficiently small
  probability takes some sufficiently large value, and which otherwise
  takes the value zero.}.)

In the present paper we consider the epidemic to take place on some
subset $P$ of $\mathbb{Z}^d$, where we allow $P=\mathbb{Z}^d$ as a
possibility. Two sites (nodes) are considered \emph{neighbours} if
and only if they are distance $1$ apart and thus $\mathbb{Z}^d$
induces a graph, called the \textit{$d$-dimensional cubic lattice}
$\mathbb{L}^d=(\mathbb{Z}^d,\mathbb{E}^d)$, with the set
$\mathbb{E}^d$ of edges connecting neighbours. Thus in the case
$P=\mathbb{Z}^2$, for example, each node has 4 neighbours. Notice
that the set $P$ also induces a graph with respect to the lattice
$\mathbb{L}^d$---we shall denote this graph $\Pi$. This may be
considered as a model for \emph{nearest-neighbour} interaction.

We assume furthermore that initially there is a single infected
site, and that all other sites in $\Pi$ are occupied by healthy
individuals.

\subsection{Incomplete observations}
The probability $p$ introduced above is considered to be unknown,
possibly depending, through the experimental design, on other
parameters (for example, on the spacing of the grid). We consider
inference under each of the following two scenarios:
\renewcommand{\labelenumi}{(\roman{enumi})}
\begin{enumerate}
\item the observations consist of the set of sites which are ever
infected, so that the routes by which infections travel are not
observed; note that, in terms of the bond percolation process, this
corresponds to one's knowledge of the connected component containing
the initially infected site---the location of this site within the
component not being relevant to inference for $p$ (see below);
\item all that is
observed is the \emph{size} of the set of sites which are ever
infected.
\end{enumerate}
We refer further to the former of these two scenarios as
$\mathcal{S}1$ and to the latter scenario as $\mathcal{S}2$.

Since for any given set of initially infected sites, the
distribution of the set of ever-infected sites in an \textit{SIR}
epidemic with constant infectious times is the same as for the
corresponding unoriented bond percolation process (with the same
initial set of the nodes involved), a final snapshot of such
epidemic can be seen as an open cluster of the corresponding
percolation process.

\begin{figure}[ptb]
\centering
\includegraphics[width=1.8in]{Figures/outbreak.eps}
\caption{An open cluster (black solid dots) containing the origin (a
black dot in a circle) as a result of percolation simulation on
$\mathbb{L}^2$. Here the bond percolation probability $p$ was taken
to be $0.478$; the solid segments represent open bonds. The open
cluster can be seen as a finite outbreak of an epidemic with
constant infectious periods and neighbour-to-neighbour infection
intensity spread rate $\lambda_{ij}\approx 0.65$ (since
$0.478=1-e^{-0.65}$). The dotted lines depict directions along which
infection did not spread (from black to grey dots); thus, grey dots
depict individuals which remain healthy and the dotted lines
represent those bonds that must be closed given the knowledge of the
cluster set.}
\label{fig:outbreak
\end{figure}

Figure~\ref{fig:outbreak} depicts an open cluster obtained by
simulation of percolation process on the integer lattice
$\mathbb{Z}^2$ for $p=0.478$. This connected component containing
the origin can be seen as a final (and finite) outbreak of an
\textit{SIR} epidemic process of the kind discussed above. The
origin (or, indeed, any other vertex of the open cluster) may be
considered to be the site where the initially inoculated individual
has been placed. Clearly, the realised bond structure is not the
only possible way which results in the site configuration from
Figure~\ref{fig:outbreak}. However, the distribution of this site
configuration as an extinct \textit{SIR} epidemic coincides with
that of the corresponding unoriented bond percolation process.

\section{Parameter estimation}

\subsection{Scenario $\mathcal{S}1$}
Let $\Pi$ be a (proper or improper) subgraph of $\mathbb{L}^d$ which
contains the origin and let $\mathcal{C}$ be an open cluster of a
percolation process on $\Pi$ containing the origin. The set of nodes
$\mathcal{C}$ represents a snapshot of an extinct outbreak of our
spatial \textit{SIR} epidemic evolved on $\Pi\subseteq\mathbb{L}^2$.

Let us introduce some additional notions. Let $G=(V,E)$ be a locally
finite graph and let $G'=(V',E')$ be a subgraph of $G$ (that is
$V'\subseteq V$ and $E'\subseteq E$). By the \textit{saturation} of
the graph $G'$ with respect to $G$ we understand the graph
$\tilde{G}=(\tilde{V},\tilde{E})$ such that
$$
\tilde{V}=V'\text{ and }\tilde{E}=\{(x,y)\,|\,x,y\in
V'\,\&\,(x,y)\in E\}.
$$
We denote the saturation of $G'$ with respect to $G$ by $\satur_GG'$
or, in cases when it is clear from the context with respect to what
graph the saturation takes place, by $\satur G'$. A graph $G'$ whose
saturation coincides with itself is called a \textit{fully saturated
graph}. For example, the fully saturated graph (with respect to
$\mathbb{L}^2$) is obtained from the graph depicted in
Figure~\ref{fig:outbreak} by connecting all pairs of neighbouring
black sites (according to the 4-neighbourhood relationship). Note
that the operation of saturation may also be applied solely to a
subset of vertices of the original graph, since it does not make use
of the edges of the subgraph-operand.

To distinguish between the boundary points of a graph and their
neighbours, which are not in the graph, we introduce the notions of
the \textit{surface} and the \textit{frontier} of the graph (again,
with respect to another graph). Let us denote by $\partial G$ the
surface of $G$ in $\Pi$, $G\subseteq\Pi$, that is to say the set
$$\partial G:=\{x\in G: \, \exists y\in\Pi\backslash G \text{ s.t.\ }x\text{ and }y\text{ are neighbours in }
\Pi\},$$ and by $\Gamma_G$ the frontier of $G$ in $\Pi$, i.e.\ the
set $\partial (\Pi\backslash G)$.

In order to identify the likelihood function we introduce the set
$\mathcal{G}(\mathcal{C})$ of all \textit{connected} subgraphs of
$\Pi$ with $\mathcal{C}$ as a vertex set. Note that the set
$\mathcal{G}(\mathcal{C})$ is necessarily nonempty.  For each
$G\in\mathcal{G}(\mathcal{C})$ the number of edges between the
vertices of the graph $G$ and the elements of its frontier
$\Gamma_{G}$ is constant---we denote it by $w_\mathcal{C}$. Finally,
we denote the total number of edges in $G$ by $e(G)$.

The probability that $\mathcal{C}$ represents the set of
ever-infected sites and that the edges of $G$ correspond to those
routes along which the infection travelled is
$$
\mathbb{P}_p(G)=p^{e(G)}(1-p)^{e(\satur{\mathcal{C}})-e(G)+w_\mathcal{C}},
$$
and the likelihood function associated with the observed set
$\mathcal{C}$ of ever-infected sites is given by

$$
\mathcal{L}(p)=\mathbb{P}_p(\mathcal{C})=\sum\limits_{G\in\mathcal{G}(\mathcal{C})}\mathbb{P}_p(G).
$$

Hence, under assumption of a uniform prior for $p$, its posterior
distribution $\pi(p\,|\,\mathcal{C})$ is a mixture of beta
distributions:
$$
\pi(p\,|\,\mathcal{C})\propto
\sum\limits_{k}r(k)\thinspace\text{Beta}\left(k+1,e(\satur{\mathcal{C}})-k+w_\mathcal{C}+1\right),
$$
where
$$r(k):=\#\{G\in\mathcal{G}(\mathcal{C})\,|\, e(G)=k\}.$$

It is not feasible to calculate $\pi(p\,|\,\mathcal{C})$ in the
above form, since it is hard to enumerate all corresponding graphs
and, thus, to calculate efficiently the coefficients $r(k)$. We
describe therefore an McMC algorithm that allows one to sample from
the distribution $\pi(p\,|\,\mathcal{C})$ under the uniform prior on
$p$, that is, effectively, to estimate the likelihood function of
$p$.

Our Markov chain explores the joint space
$(0,1)\times\mathcal{G}(\mathcal{C})$ of possible values for $p$ and
graphs from $\mathcal{G}(\mathcal{C})$. The stationary distribution
of the chain is the joint posterior distribution of $p$ and
$G\in\mathcal{G}(\mathcal{C})$. The description of the chain is
given in Algorithm~\ref{alg:MCMC_S1} (Appendix~\ref{appendix:mcmc}).
This Markov chain explores the set of all connected graphs
$\mathcal{G}(\mathcal{C})$ by simply deleting or adding an edge from
the current graph preserving the connectivity of the given site
configuration~$\mathcal{C}$.

\begin{figure}[ptb]
\centering
\includegraphics[width=3.2in]{Figures/post_n_likelihood_corrected.eps}
\caption{The solid line corresponds to the likelihood function
evaluated for the complete information (both the sites and edges are
known) on the cluster $\mathcal{C}$ from Figure~\ref{fig:outbreak}.
The histogram is based on a large sample drawn from the McMC applied
to the site configuration $\mathcal{C}$ (nodes only).}
\label{fig:post_n_likelihood
\end{figure}

We apply Algorithm~\ref{alg:MCMC_S1} to the site configuration
$\mathcal{C}$ from Figure~\ref{fig:outbreak} (black dots only).
Figure~\ref{fig:post_n_likelihood} depicts the likelihood function
of the model parameter for the complete observations (i.e.\ nodes
and edges of the cluster) and a histogram of the sample obtained
from the posterior distribution $\pi(p\,|\,\mathcal{C})$ when the
prior distribution is uniform on the interval $(0,1)$.

Mixing properties of Markov chains described by
Algorithm~\ref{alg:MCMC_S1} are briefly discussed in
Appendix~\ref{appendix:Alg12_and_S12}.

\subsection{Scenario $\mathcal{S}2$}
\label{sec:S2}

Under this scenario only the size $n$ of the outbreak of our
\textit{SIR} epidemic evolving on $\Pi=\mathbb{L}^d$ is given.

Let $\mathcal{G}_n$ be the set of all possible connected graphs on
$n$ vertices including the origin. These graphs represent the
outbreaks of the size $n$ and we distinguish all isomorphic graphs
which have different locations or orientations.

We denote the number of edges of $\Pi$ between the vertices of the
graph $G\in\mathcal{G}_n$ and the vertices of its frontier
$\Gamma_{G}$ by $w(G)$.

Given the epidemic size $n$, the inference on $p$ involves
evaluation of the likelihood function
$\mathcal{L}_n(p):=\mathbb{P}_p(|\mathcal{C}|=n)$ which can be
represented as follows:
$$
\mathcal{L}_n(p)=\sum\limits_{G\in\mathcal{G}_n}\mathbb{P}_p(G).
$$
As previously, under assumption of a uniform prior for $p$ its
posterior distribution $\pi(p\,|\, |\mathcal{C}|=n)$ is a mixture of
beta distributions:
\begin{equation}
\pi(p\,|\,
|\mathcal{C}|=n)\propto\sum\limits_{s,k,l}q(s,k,l)\thinspace
\text{Beta}\left(k+1,s-k+l+1\right), \label{eq:posteriorS2}
\end{equation}
where
$$q(s,k,l):=\#\{G\in\mathcal{G}_n\,|\, e(\satur{G})=s,e(G)=k,w(G)=l\}.$$

\begin{figure}[ptb]
\begin{center}
\includegraphics[width=3.2in]{Figures/connected_component_inference.eps}
\end{center}
\caption{Likelihood functions $\mathcal{L}_n(p)$ ($n=25,50,70$)
obtained using the McMC from Algorithm~\ref{alg:MCMC_S2}
and McMC sample histogram of $\mathcal{L}_n(p)$ for $n=70$.
\label{fig:likelihoods_for_connected_components
\end{figure}

This, again, represents a hard enumeration problem. However,
inference on $p$ can be made using the McMC technique.
Algorithm~\ref{alg:MCMC_S2} (Appendix~\ref{appendix:mcmc}) contains
a description of a Markov chain which serves the purpose of sampling
from the posterior distribution $\pi(p\,|\,|\mathcal{C}|=n)$, given
the prior distribution of $p$ is uniform on [0,1]. The chain
explores the joint space $(0,1)\times \mathcal{G}_n$ of possible
values for the percolation parameter $p$ and all possible connected
graphs on $n$ nodes, by deleting a vertex from a graph and adding a
vertex from its frontier.

Figure~\ref{fig:likelihoods_for_connected_components} depicts plots
of the likelihood function $\mathcal{L}_n$ when $n=25,\,50,\,70$ and
a histogram of a sample drawn from $\pi(p\,|\,|\mathcal{C}|=70)$.
The graphs of the likelihoods were obtained by smoothing the
histograms of corresponding samples generated by the McMC presented
in Algorithm~\ref{alg:MCMC_S2}. Note that, as might reasonably be
expected, the values of $p$ maximising the likelihood increase with
$n$.

Mixing properties of Markov chains described by
Algorithm~\ref{alg:MCMC_S2} are briefly discussed in
Appendix~\ref{appendix:Alg12_and_S12}. Two video examples of dynamic
graph updates illustrating Algorithms~\ref{alg:MCMC_S1} and
\ref{alg:MCMC_S2} can be found at the WEB address~\cite{BejanWEB}.

\subsection{Convergence analytical results for $\mathcal{S}2$}
Percolation exhibits a phenomenon of \textit{criticality}, this
being central the percolation theory: as $p$ increases, the sizes of
clusters (connected components) also increase, and there is a
critical value of $p_c$ at which there appears a cluster which
dominates the rest of the pattern. If $p<p_c$, then with probability
one all open clusters are finite, but when $p>p_c$ there is a single
infinite open cluster almost surely~\cite{Grimmett99}. Note, that
the critical percolation parameter $p_c$ depends on the dimension
$d$ of the lattice. Bond percolation on the square lattice seems to
be most studied to date of all percolation processes. The critical
probability $p_c$ in the case of a square lattice is $\frac{1}{2}$.
What follows, however, holds for any lattice $\mathbb{Z}^d$, $d\geq
3$.

\subsubsection{Asymptotic behaviour of the maximum likelihood estimates of~$p$}
Denote by $\mathcal{C}(x)$ the open cluster (connected component)
which contains the vertex $x$. Let us write
$\chi(p)=\mathbb{E}_p|\mathcal{C}|$ for the mean number of vertices
in the open cluster at the origin. Using the translation invariance
of the process on $\mathbb{Z}^d$, we have
$\chi(p)=\mathbb{E}_p|\mathcal{C}(x)|$ for all vertices $x$.
Percolation theory tells us that if $p<p_c$, then $\chi(p)<\infty$.
When $p>p_c$, then $\chi(p)=\infty$ and the function $\chi$ is not
of a much interest in this case. Instead, one studies the function
$\chi^{f}(p)=\mathbb{E}_p[|\mathcal{C}|:|\mathcal{C}|<\infty]$. The
function $\chi(p)$ ($\chi^{f}(p)$) monotonically increases to
infinity as $p\uparrow p_c$ ($p\downarrow p_c$). It is
known~\cite{Grimmett99} that there is no infinite open cluster when
$p=p_c$ for percolation on the square lattice $\mathbb{Z}^2$ and on
the lattice $\mathbb{Z}^d$, where $d\geq 19$; this is also believed
to be true for all other values of $d$. How likely it is to observe
an open cluster of size $n$ when $n$ is very large? What value of
$p$ should one suggest if one happened to observe a large epidemic
of size $n$?

Intuitively, it is unlikely that very large but finite epidemics may
be obtained for values of $p$ significantly different from $p_c$ (if
$p\ll p_c$, then it is unlikely that the epidemic will attain a very
large size; if $p\gg p_c$, then it is equally unlikely that, having
attained a sufficiently large size $n$, the epidemic will have
burned out). Intuition suggests therefore that the likelihood
function for $p$, given that the size $n$ of the connected component
containing the origin is growing, should be increasingly
concentrated around $p_c$.

Let $\mathcal{L}_n(p)=\mathbb{P}_p(|\mathcal{C}|=n)$ be the
probability that an open cluster $\mathcal{C}$ from a percolation
process with the edge density $p$ is of size $n$. (Equivalently,
assume that we observe an \textit{SIR} spread of infection via
nearest-neighbour interactions on $\mathbb{Z}^d$  and that the final
size of its outbreak is $n$; when $n$ is fixed the function
$\mathcal{L}_n(p)$ can be regarded as the likelihood function for
the percolation probability parameter $p$.)

Let $\hat{p}_n$ be the maximum likelihood estimate for $p$ derived
from $\mathcal{L}_n(p)$. For any $p$ define also
$L_n(p)=\mathcal{L}_n(p)/\mathcal{L}_n(p_c)$.

\begin{theorem}
The sequence of maximum likelihood estimates $\hat{p}_n$ for $p$
converges to the critical probability $p_c$. \label{thm:mle}
\end{theorem}

The proof of this theorem is based on the following lemma, together
with a generally accepted regularity condition.

\begin{lemma}
For any $p\in(0,1)$ different from $p_c$ the following holds:
$$
\liminf\limits_{n\to\infty}L_n(p)=0.
$$
Moreover, this convergence is uniform for any closed interval which
does not contain $p_c$.
\label{lem:p_c}
\end{lemma}

The proof of the lemma and the theorem can be found in
Appendix~\ref{appendix:proofs}.

As mentioned above, it is reasonable to expect the sequence of the
maximum likelihood estimates $\hat{p}_n$ to be monotonic (see
Figure~\ref{fig:likelihoods_for_connected_components}). This
observation gives rise to the following conjecture.
\begin{conjecture}
The sequence of $\{\hat{p}_n\}$ converges to $p_c$ monotonically
from the left. \label{conj:posteriors}
\end{conjecture}

Furthermore, we formulate the corresponding companion conjecture for
the sequence of posterior distribution, using the notion of a delta
sequence~\cite{Arfken1985}.
\begin{conjecture}
Provided $p\in\supp \pi(\cdot)$ the functional sequence
$\{\pi(p\,|\,n)\}_{n=1}^{\infty}$ is a delta sequence which
generates the delta function $\delta(p-p_c)$.
\label{conj:posterior_asympt}
\end{conjecture}
Thus, we believe that the limiting posterior distribution of the
percolation parameter is a one-point mass distribution at $p=p_c$,
or the Dirac delta function $\delta(p-p_c)$.

\subsubsection{Combinatorial characterisation of large percolation clusters on $\mathbb{L}^d$}
The theoretical results obtained and conjectured previously for
inference under scenario $\mathcal{S}2$ can be used to derive their
combinatorial analogues regarding the relative number of
realisations of the process with the cluster size $n$. Under
scenario $\mathcal{S}2$ the posterior distribution $\pi(p\,|\,
|\mathcal{C}|=n)$ can be seen as a mixture of beta distributions, as
in (\ref{eq:posteriorS2}). As in Section~\ref{sec:S2}, let
$q(s,k,l)$ be the number of all possible graphs $G$ which may
correspond to an open cluster $\mathcal{C}$ (the vertex set of $G$
and $\mathcal{C}$ coincide), and such that $k$ is the number of
edges in $G$, $s$ is the number of edges in the saturation of $G$,
and $l$ is the number of edges of $\Pi$ between the surface and the
frontier of $G$. Conjecture~\ref{conj:posterior_asympt} would
suggest that the number $q(s,k,l)$ of graphs $G$ corresponding to
open clusters $\mathcal{C}$ which could emerge as a result of the
percolation process with parameter $p_c$ and for which it holds that
\begin{equation}
\frac{k+1}{s+l+2}\approx p_c,
\end{equation}
is far greater than the number of all other graphs. This is so,
since the sequence of beta distributions
$\text{Beta}(\alpha_n,\beta_n)$ is a delta sequence generating the
delta function at $p_c$ if and only if
$\alpha_n/(\alpha_n+\beta_n)\to p_c$.

Thus, in percolation processes on $\mathbb{L}^d$ the number of
finite graphs corresponding to open clusters of size $n$ (where $n$
is large) that satisfy the condition
\begin{equation}
\frac{e(G)+1}{e(\satur{G})+w(G)+2}\approx p_c
\label{eq:approx_combinatorics_Ld}
\end{equation}
largely exceeds the number of all other connected components on $n$
nodes. In other words, a typical graph corresponding to an open
cluster of a large size in percolation process on $\mathbb{L}^d$ is
a connected subgraph of $\mathbb{L}^d$ characterised
by~(\ref{eq:approx_combinatorics_Ld}). In particular, when $d=2$:
\begin{equation}
e(G)-w(G)\approx e(\satur{G})-e(G);
\label{eq:approx_combinatorics_L2}
\end{equation}
that is the number of open edges in $G$ is approximately equal to
the total number of closed edges and edges between $G$ and its
frontier.

\subsubsection{Large percolation clusters as rare events}
When $n$ is large, the appearance of finite open clusters of size
$n$ is highly unlikely: the distribution of the cluster size
(hypothetically) decays as $n^{-1-1/\delta}$, $\delta>0$, when
$p=p_c$, and the decay is exponential (sub-exponential) when $p<p_c$
($p>p_c$). Large finite percolation clusters can therefore be viewed
as rare events. Since the state space of the McMC proposed for
inference on the percolation parameter $p$ under scenario
$\mathcal{S}2$ and described in Algorithm~\ref{alg:MCMC_S2} involves
the set of all open clusters on $n$ nodes, this algorithm can be
readily used in order to obtain realisations of these rare events.

\section{Bayesian optimal designs}
\subsection{Utility based optimal designs}
\label{subsection:utility_based_optimal_designs}
\subsubsection{General description}
The experimental design problem can be conveniently approached
within the Bayesian framework~\cite{Chaloner_Verdinelli1995}.
Suppose we study a stochastic process for which we formulate a model
$M$, characterised by a model parameter $\theta$. The model $M$ specifies a sampling density
 $f_d(y\,|\,\theta)$ for the
experimental outcome $y$ obtained from observation of the process under experimental
set-up $d$
given the value of the model parameter $\theta$. Our knowledge about
$\theta$ is described by a prior distribution $\pi(\theta)$.
Whenever the choice of set-up $d$ is under our control there
arises the question of selecting the optimal $d$ under which one
should observe the stochastic process. Such prescribed conditions
are referred to as a {\em design}, and the optimal design is found
under optimality criteria specifically tailored to
the context and the purpose of the experiment.

By employing a utility function $u(d,y,\theta)$ one can specify the
purpose of the experiment and measure the value of its outcome $y$
accordingly. The methodology of posing and solving utility-based
optimal design problems within the Bayesian paradigm became somewhat
standard~\cite{Muller1999,Cook&Gibson&Gilligan}. The design has to
be chosen before performing the experiment and one may choose to
maximise the expectation of the utility function $u(d,y,\theta)$
with respect to $\theta$ and $y$ \cite{Muller1999}:
\begin{equation}
d_{\text{max}}=\arg\max\limits_{d\in \mathcal{D}}U(d),
\label{eq:optimal_d}
\end{equation}
where
\begin{equation}
U(d)=\int\limits_{\Theta}\int\limits_{Y}
u(d,y,\theta)f_d(y\,|\,\theta)\pi(\theta)\, \text{d}\theta
\text{d}y. \label{eq:utility}
\end{equation}
Here $\mathcal{D}$ is the set of possible designs. The set of
possible outcomes $y$ of the experiment is denoted by $Y$. The
experiment is defined by a model $f_d(y\,|\,\theta)$, i.e.\ by the sampling density
 of $y$ conditional on $\theta$ for a given design~$d$.

When the purpose of the experiment is to infer the model parameter,
a sensible choice to measure the utility of the experiment under
design $d$ might be one that represents the information gained on
$\theta$ from the experiment. One of the standard choices is the
Kullback--Leibler (KL) divergence $D_{\text{KL}}$ between the prior
distribution of $\theta$, $\pi(\theta)$, and its posterior
distribution $\pi(\theta\,|\,y,d)$ obtained following the experiment
yielding data~$y$:
\begin{equation}
D_{\text{KL}}\{\pi(\theta\,|\,y,d)\parallel
\pi(\theta)\}=\int\limits_{\Theta}\log
\frac{\pi(\theta\,|\,y,d)}{\pi(\theta)}\pi(\theta\,|\,y,d)\,\text{d}\theta,
\label{eq:defining_KL_divergence}
\end{equation}
where $\pi(\theta\,|\,y,d)\propto f_d(y\,|\,\theta)\pi(\theta)$.

Introduced by Kullback and Leibler~\cite{Kulback&Leibler1951} this
information measure was justified by Lindley~\cite{Lindley56} and
Bernardo~\cite{Bernardo1979} and is related to the Shannon
information measure.

The KL divergence is a random variable, being a function of
the random outcome $y$. In order to find the
the design $d_{\text{max}}$ that is optimal with respect to this choice of utility
measure one should maximise the expected value of the KL divergence
over the set of observables $Y$ as a function of the design variate
$d$:
\begin{equation}
U_{\text{KL}}(d)=\mathbb{E}_Y[D_{\text{KL}}\{\pi(\theta\,|\,y,d)\parallel
\pi(\theta)\}], \label{eq:KL_expected_divergence}
\end{equation}
this being invariant under a reparametrisation of the model in terms
of $\theta$.

When the experimental motivation consists in increasing one's
knowledge on the model parameter the expected KL divergence
$U_{\text{KL}}(d)$ coincides with (\ref{eq:utility}) taking
$u(d,y,\theta)=\log \frac{\pi(\theta\,|\,y,d)}{\pi(\theta)}$. If,
however, the purpose of the experiment is to inform someone
who holds a different prior to our own and we wish to advise them on which design to use,
using our own `superior' knowledge
of the system under study, then the expected KL divergence should be calculated
in the form prescribed by (\ref{eq:KL_expected_divergence}) when the
integration over the space of observables $Y$ is to be carried out
using one's superior knowledge. We refer to these contrasting
experimental scenarios as \textit{progressive design} and
\textit{instructive design}~\footnote{In \cite{Cook&Gibson&Gilligan}
these experimental scenarios are called \textit{progressive} and
\textit{pedagogic} designs.}, respectively, and discuss them next
in more detail.

\subsubsection{Progressive design}
Under this scenario there is an experimenter $\mathscr{E}$ who holds
prior beliefs regarding the parameter, $\theta$, in the form of a prior
distribution $\pi(\theta)$ (perhaps obtained from earlier observation
of  the process) and whose goal is to design an optimal experiment in order to
increase this knowledge. Using KL divergence as the measure of information gain,
this experimenter should maximise the expected KL
divergence (\ref{eq:KL_expected_divergence}):

\begin{align}
U_{\text{KL}}(d)&=\mathbb{E}_Y\mathbb{E}_\Theta\left. \left[\log\frac{\pi(\theta\,|\,y,d)}{\pi(\theta)}\,\right|\,y\right] \nonumber\\
&=\int\limits_{\Theta}\int\limits_{Y}
\log\frac{\pi(\theta\,|\,y,d)}{\pi(\theta)}f_d(y,\theta)\,
\text{d}\theta \text{d}y,\nonumber
\end{align}
so that $U_{\text{KL}}(d)$ coincides with (\ref{eq:utility}) where
$u(d,y,\theta)=\log{\frac{\pi(\theta\,|\,y)}{\pi(\theta)}}$. Notice,
however, that the expected information gain $U_{\text{KL}}(d)$ can
be written as follows:

\begin{align}
U_{\text{KL}}(d)=&\int\limits_{Y}\int\limits_{\Theta}\log
\pi(\theta\,|\,y,d)f_d(y\,|\,\theta)\pi(\theta)\,\text{d}\theta\text{d}y\nonumber\\
&-\int\limits_{\Theta}\log \pi(\theta)
\pi(\theta)\,\text{d}\theta,\nonumber
\end{align}
where the second term is the Shannon entropy of $\pi(\theta)$, which
does not depend on $d$. Thus, in order to obtain the solution
$d_{\text{max}}$ in this case, it suffices to maximise the first
term only.

\subsubsection{Instructive design}
In contrast to the progressive design scenario, in the instructive
case there is an experimenter $\mathscr{E}$, holding a prior
$\pi(\theta)$, and a better (or differently) informed trainer (or
instructor) $\mathscr{T}$ whose knowledge about the model parameter
is summarised in a distribution $\pi^*(\theta)$. (Of particular
interest is a special case where the instructor knows the true value
of $\theta$.) The aim here is to maximise the change in
experimenter's belief from $\pi(\theta)$ to $\pi(\theta\,|\,y)$ by
designing an experiment using the superior knowledge
$\pi^*(\theta)$.

This latter optimisation problem can be formulated as follows:
\begin{align}
d^*_\text{max}&=\arg\max\limits_{d\in \mathcal{D}}U^*_{\text{KL}}(d),\label{eq:optimal_d_instructive}\\
U^*_{\text{KL}}(d)&=\int\limits_{Y}
D_{\text{KL}}\{\pi(\theta\,|\,y,d)\parallel \pi(\theta)\}f^*(y)\,
\text{d}y,\label{eq:utility_instructive}
\end{align}
where, as before, $\pi(\theta\,|\,y)$ is derived as in
(\ref{eq:defining_KL_divergence}), and $f^*(y)$ is the instructor's
prior predictive distribution for $y$:
$$
f^*(y)=\int\limits_{\Theta}f_d(y\,|\,\theta)\pi^*(\theta)\,\text{d}\theta.
$$
In particular, if the instructor $\mathscr{T}$ knows the exact value
of $\theta$, $\theta^*$, and hence $\pi^*(\theta)$ is the Dirac
function $\delta(\theta-\theta^*)$, then
$f^*(y)=f_d(y\,|\,\theta^*)$, so that
\begin{equation}
U^*_{\text{KL}}(d)=\int\limits_{\mathcal{Y}}
D_{\text{KL}}\{\pi(\theta\,|\,y,d)\parallel
\pi(\theta)\}f_d(y\,|\,\theta^*)\,\text{d}y.
\label{eq:instructive_delta}
\end{equation}

Note that the `instructive' expected utility $U^*_{\text{KL}}$ is
generally not symmetric with respect to the exchange of
$\pi(\theta)$ and $\pi^*(\theta)$: if the experimenter $\mathscr{E}$
and the instructor $\mathscr{T}$ change their roles, then the
corresponding optimal instructive designs are different, unless
their knowledge about $\theta$ is identical, that is
$\pi(\theta)=\pi^*(\theta)$ almost everywhere.

\subsection{Simulation-based evaluation of the expected utility}
\label{section:simulation based evaluation of expected utility}
Generally, the solution to the optimal design problems
(\ref{eq:optimal_d}-\ref{eq:utility}) and
(\ref{eq:optimal_d_instructive}-\ref{eq:utility_instructive}) cannot
be obtained analytically. This is mainly due to the following three
problems. First, the design space $\mathcal{D}$ can be complicated,
with many design variables, some of them having a continuous range of
values. Second, even if the design space has a simple structure, the
utility function may not be simple to evaluate, so the expected
utility $U(d)$ cannot be obtained explicitly. Finally, for
incomplete observations of highly non-linear stochastic processes
such as epidemic models, the likelihood is not usually available in
a closed form, and this results in computationally intensive
evaluations or sampling methods.

A review of analytical and approximate numerical solutions to
Bayesian optimal design problems for traditional experimental
design involving linear and non-linear models can be found in
\cite{Verdinelli1992,Chaloner_Verdinelli1995}.

M\"{u}ller~\cite{Muller1999} reviews simulation-based methods for
optimal design problems where the expected utility $U(d)$ is
evaluated by Monte-Carlo simulation. In its simplest form an
estimate $\widehat{U}$ of $U$ for any given design $d$ in the
progressive case is as follows:
\begin{equation}
\widehat{U}(d)=\frac{1}{M}\sum\limits_{i=1}^{M}u(d,\theta_i,y_i),
\label{eq:MCevaluation}
\end{equation}
where $\{(\theta_i,y_i),i=1,\ldots,M\}$ is a Monte-Carlo sample
generated values:
\begin{equation}
\theta_i\sim \pi(\theta), \thickspace y_i\sim p_d(y\,|\,\theta).
\end{equation}
The expected utility $U$ may, in particular, be based on the KL
divergence.

Analogously, the expected utility $U^*_{\text{KL}}$ under
the instructive scenario, when the instructor knows the true value of
the model parameter, $\theta^*$, can be evaluated using the
following scheme:

\begin{equation}
\widehat{U}^*_{\text{KL}}(d)=\frac{1}{M}\sum\limits_{i=1}^{M}\widehat{KL}(y^*_i,d),
\label{eq:*MCevaluation}
\end{equation}
where $y^*_i\sim f(y\,|\,\theta^*,d)$, $i=1,\ldots,M$, and
$\widehat{KL}(y_i,d)$ is an estimate of the KL divergence
$D_{\text{KL}}\{\pi(\theta\,|\,y^*,d)\parallel \pi(\theta)\}$ that
can be obtained via numerical integration of
$\log\frac{\pi(\theta\,|\,y,d)}{\pi(\theta)}$ with respect to the
posterior $\pi(\theta\,|\,y,d)$. The former function, in turn, may
need to be evaluated through simulation methods---perhaps using an
McMC scheme.

Evaluation of a continuous expected utility surface $U(d)$ by
(\ref{eq:MCevaluation}) or (\ref{eq:*MCevaluation}) can be achieved by
computing its values on a discretised grid of points and further
smoothing of the obtained set of values in order to approximate the
expected utility landscape. This, however, may be problematic when
$d\in\mathcal{D}$ is a multidimensional design parameter. An
alternative method is provided by the augmented probability
simulation approach which is studied in
\cite{Clyde&Mueller&Parmigiani1995,Bielza&Mueller&Rios-Insu}, and
reviewed in \cite{Muller1999}.

The augmented probability simulation approach assumes that
$u(d,\theta,y)$ is a non-negative bounded function. (This condition,
although not always automatically satisfied, can be easily achieved
by correspondingly modifying the utility function.) An artificial
density, proportional to $u(d,\theta,y)f_d(y\,|\,\theta)\pi(\theta)$
can then be defined for $(d,\theta,y)$ jointly~\cite{Muller1999}:
\begin{equation}
h(d,\theta,y)\propto u(d,\theta,y)f_d(y\,|\,\theta)\pi(\theta),
\end{equation}
so that its marginal in $d$ is proportional to the expected utility
$U(d)$. Sampling from $h(\cdot,\cdot,\cdot)$ can be used to obtain a
sample from its marginal using the Metropolis--Hastings McMC scheme
described in \cite{Muller1999}.

When using the augmented probability modelling approach one
approximates the optimal design by an empirical mode of the marginal
density of $d$ under the artificial distribution $h$, that is by the
mode of histogram formed from the first components of the set of
samples obtained. Notice that the evaluation of $u(d,\theta,y)$ at
each sampling step may in general itself involve an McMC sampling
from the posterior $\pi(\theta\,|\,y,d)$, so that the approach is
potentially very expensive computationally.

\subsection{Inner-outer plots}
We now apply the utility-based Bayesian approach introduced above in
order to design optimal experiments for our \textit{SIR} epidemic
model with incomplete observations of the kind described by
scenario~$\mathcal{S}1$. We do so by identifying a (finite) class of
designs (node configurations) which we call \textit{inner-outer
plots}. These are plots of bounded size which are obtained by
removing some nodes from the underlying grid. We refer to this
process as \textit{sparsification of the grid}.

A typical example of an inner-outer plot in $\mathbb{Z}^2$ is given
in Figure~\ref{fig:inner_outer_design}. The inner-outer plot
depicted consists of two parts---\textit{inner} and \textit{outer}
regions (hence the name of the design). Some sites are removed from
the outer region of the plot in a regular manner, while all sites of
the inner region are preserved. Our motivation in considering
sparsified grids is to show that intuitively obvious designs, such
as completely dense grids, can be improved. The justification of the
grid sparsification is that it impedes the ability of the epidemic
to spread, which makes possible better inference when $p$ is large,
while the more dense inner part of the design works better when $p$
is small. A formal description of inner-outer plots follows next.

\begin{figure}[ptb]
\begin{center}
\includegraphics[width=2.8in]{Figures/ioplot.eps}
\end{center}
\caption{An example of an inner-outer $(m,r)$-plot (nodes only) in
$\mathbb{Z}^2$. In this inner-outer plot $m=9$ and $r=3$. The plot
is bounded by an $N\times N$ square, where $N=21$ can be calculated using~(\ref{eq:m_on_N&r}).
Depicted is the saturation of an $(9,3)$-plot with respect to $\mathbb{L}^2$.
\label{fig:inner_outer_design
\end{figure}

Let us assume that $m$ is an odd positive integer and
$r\in\mathbb{N}\cup\{0\}$. An \textit{inner-outer $(m,r)$-plot
$\Pi^{(d)}_0(m,r)$ in $\mathbb{Z}^d$ with centre at the origin} is a
$d$-dimensional box $B_N^{(d)}$ with side-length $N=m+4r$ and some
vertices removed as follows:

\begin{equation}
\Pi_0(m,r):=
\begin{cases}
B_N^{(d)}, & r=0\\
B_N^{(d)}\setminus C_{N}^{(d)}(m,r), & r>0,
\end{cases}
\label{eq:inner-outer_definition}
\end{equation}
where $C_{N}^{(d)}(m,r)$ is the set
\begin{align}
\{&x\in B_N^{(d)}:\, \parallel x
\parallel_\infty=m+2j+1,j=0,\ldots,r-1\,\&\,\nonumber\\
&\&\parallel x\parallel_1\equiv0\bmod 2\}.\nonumber
\end{align}
Here $\parallel x \parallel_1=\sum\limits_{i=1}^d|x_i|$ and
$\parallel x
\parallel_\infty=\max (|x_1|,\ldots,|x_d|)$ for any
$x=(x_1,\ldots,x_d)\in\mathbb{Z}^2$ and the box $B_N^{(d)}$ is
defined as follows~\footnote{Note that $N$ is a positive integer
number since $m$ is odd.}:
\begin{align}
B_N^{(d)}:&=[-(N-1)/2,(N-1)/2]^d\nonumber\\
          &=\{x\in\mathbb{Z}^d:\,
\parallel x
\parallel_\infty\leq (N-1)/2\}.\nonumber
\end{align}
We call any plot that can be obtained by translating the plot
$\Pi^{(d)}_0(m,r)$ in $\mathbb{Z}^d$ an \textit{inner-outer
$(m,r)$-plot}, or simply \textit{an inner-outer plot}, and denote it
by $\Pi^{(d)}(m,r)$.

The total number of nodes contained in an $(m,r)$-plot can be
calculated by subtracting the total number of the nodes removed from
the outer plot (see (\ref{eq:inner-outer_definition})) as follows:
\begin{align}
T&=N(m,r)^2-4\sum\limits_{i=1}^{r}\left(\frac{m-1}{2}+2i-1\right)\nonumber\\
 &=(m+3r)^2+r(3r+2), \label{eq:Total_no_nodes_sparsified_plot}
\end{align}
where, as before,
\begin{equation}
N(m,r)=m+4r. \label{eq:m_on_N&r}
\end{equation}

The inner-outer $(m,r)$-plot depicted in
Figure~\ref{fig:inner_outer_design} is from $\mathbb{Z}^2$. The size
of the inner plot is $m\times m$, where $m=9$, and there are $r=3$
`circles' (with respect to the metric $\parallel \cdot
\parallel_\infty$) in the outer plot from which every second node is
removed. The size of the bounding box is $N\times N$, where
$N=m+4r=21$. The total number of nodes $T$ that this node
configuration contains can be calculated using
(\ref{eq:Total_no_nodes_sparsified_plot}) and is equal to $357$.

\subsection{Inference for \textit{SIR} epidemics on inner-outer plots}
We consider an \textit{SIR} epidemic with constant infectious times
which arises at the central site of an inner-outer plot
$\Pi=\Pi^{(d)}(m,r)$ from $\mathbb{Z}^d$ and evolves on that plot
according to the nearest-neighbour interaction rule, that is using
edges from $\mathbb{L}^d$ with endpoints from $\Pi$,  subject to
being restricted by the plot boundary. We consider this model in the
context of scenario $\mathcal{S}1$, that is when the only
information available about the outcome of the epidemic is its site
configuration $\mathcal{C}$. As before, this model is equivalent to
that of percolation on $\Pi^{(d)}(m,r)$.

\begin{figure*}[ptb]
\centering
\includegraphics[width=5.0in]{Figures/9025.eps}
\caption{Left: site configuration obtained on the inner-outer
$(13,2)$-plot with $p=0.9$.
Right: McMC histogram of an McMC sample for $p$ assuming a uniform prior for this parameter.
\label{fig:9025
\end{figure*}

\begin{figure*}[ptb]
\centering
\includegraphics[width=5.0in]{Figures/08616.eps}
\caption{Left: site configuration obtained on the inner-outer
$(23,4)$-plot with  $p=0.86$.
Right: histogram of an McMC sample for $p$ assuming a uniform prior for this parameter.
\label{fig:08616
\end{figure*}

We note that Algorithm~\ref{alg:MCMC_S1} and
Algorithm~\ref{alg:MCMC_S2} can be readily used for making inference
on the percolation probability parameter $p$ given the sites of an
open cluster $\mathcal{C}$, or merely its size in the case of
Algorithm~\ref{alg:MCMC_S2}, generated from a percolation process on
any locally finite graph~\footnote{By a locally finite graph we mean
a graph with no vertices of infinite degree.}.

Figure~\ref{fig:08616} shows the plot of a simulated open
percolation cluster $\mathcal{C}$ obtained on the inner-outer plot
$\Pi^{(2)}(13,2)$ with $p=0.9$ (left plot) and an estimate of the likelihood
formed from a histogram estimate of the
posterior density of $p$ obtained using the Markov chain described in
Algorithm~\ref{alg:MCMC_S1} with $\mathcal{C}$ as the input data, with a
uniform prior density for $p$.
Figure~\ref{fig:9025} shows similar plots for a realisation of the
percolation process on $\Pi^{(2)}(23,4)$ with $p=0.86$.

\subsection{Optimal design problem and design space}
We consider now an optimal design problem for the percolation
process on node sets of limited size. We adopt the utility-based
Bayesian approach formulated in
Section~\ref{subsection:utility_based_optimal_designs}. The choice
of the design space $\mathcal{D}$ can be made in a number of ways,
possibly reflecting such restrictions as limitations on the number
of experimental units or the dimensions of the experimental plot. In
the context of inner-outer plots the former restriction would mean
that $T$ from (\ref{eq:Total_no_nodes_sparsified_plot}) is bounded,
whereas the latter condition is equivalent to bounding the quantity
$N(m,r)$. A combination of these conditions or some other
information can be also taken into account when identifying the
design space. One advantage of grid-based plot designs is that they
can greatly reduce the dimensionality of the design space compared
with designs that allow units to be placed at arbitrary locations.

We assume now that $N$ is odd and fixed and that the design space has
the form
$$
\mathcal{D}=\{d\,|\, d=\Pi^{(2)}(m,r)\thickspace \&\thickspace
m+4r=N\}
$$
For example, if $N=19$ then, as can be derived easily from
(\ref{eq:m_on_N&r}), the design space consists of the following
designs:
\begin{align}
\mathcal{D}=\{&\Pi^{(2)}(19,0),\,\Pi^{(2)}(15,1),\,\Pi^{(2)}(11,2),\,\nonumber\\
              &\Pi^{(2)}(7,3),\,\Pi^{(2)}(3,4)\}.\nonumber
\end{align}
The set of observables $Y$ is the set of all connected components on
$d\in\mathcal{D}$ containing the central node.

\section{Practical implementation}
\label{subsection:practical_implementation} Given a finite design
space $\mathcal{D}$ it is fairly straightforward to solve the
optimal design problem for the percolation model on inner-outer plots
using the tools developed above. We now make some comments about
solving the problem under each of the two design scenarios.

\subsection{Progressive design: expected utility evaluation through augmented modelling}
Since $\mathcal{D}$ is finite we choose to identify the design that
maximises the expected utility function in the progressive case
using augmented modelling as described in
Section~\ref{section:simulation based evaluation of expected
utility}. Recall that this is based on an artificial distribution
$h(d,p,y)\propto u(d,p,y)f_d(y\,|\,p)\pi(p)$ from which samples are
generated using a Metropolis--Hastings sampler. The optimal design $d^*$
is identified then as a value of $d$ at which the marginal of $h$ is
maximised.

\subsection{Instructive design: Monte Carlo evaluation of the expected utility}
We treat the `instructive' case differently from that of the
`progressive' one because of the form of the expected utility under
this scenario. Recall that in the `instructive' case whenever the
instructor knows the true value $p^*$ of the model parameter $p$ one
can write the expected utility based on the Kullback--Leibler
divergence as follows (compare with (\ref{eq:instructive_delta})):
$$
U^*_{\text{KL}}(d)=\sum\limits_{y\in\mathcal{Y}}
f_d(y\,|\,p^*)\int\limits_0^1\log\frac{\pi(p\,|\,y,d)}{\pi(p)}\pi(p\,|\,y,d)\,\text{d}p,
$$
where $f_d(y\,|\,p^*)$ is the likelihood function evaluated at the
true value of the model parameter $p^*$ given the open cluster $y$
at the centre of the inner-outer plot $d$.

It is clear that to evaluate the expected utility
$U^*_{\text{KL}}(d)$ via standard Monte Carlo simulation, then a
Markov chain has to be run to evaluate the posterior
$\pi(p\,|\,y,d)$ each time we sample for a new observation (an open
cluster) $y$ and also the potentially time-consuming integration has
to be done with respect to the model parameter~$p$. This integration
can be implemented in the following way. Since we can sample from
the posterior $\pi(p\,|\,y,d)$ via McMC
(Algorithm~\ref{alg:MCMC_S1}) for any given open cluster $y$, we do
so and then fit the beta distribution (or some other distribution)
to the McMC sample obtained in order to perform integration
numerically in a more efficient, albeit, approximate manner.

Thus, the expected utility evaluation scheme for inner-outer plots
in the `instructive' case and scenario $\mathcal{S}1$ can be
described as follows.

For each inner-outer plot $d\in\mathcal{D}$ we do the following:
\begin{enumerate}
\item generate a random sample of $M$ independent connected clusters $\{y_i\}_{i=1}^{M}$
on $\Pi(m,r)$: $y_i\sim f(y\,|\,p^*,d)$;
\item perform $M$ McMC's in order to obtain $M$ independent samples for the posterior distribution
$\pi(p\,|\,y,d)$;
\item fit a Beta distribution to each of the samples obtained; refer to the fitted distributions as
$\pi(p\,|\,y_i,d)$;
\item evaluate numerically the integrals
$$I_i:=\int_{0}^{1}\log\frac{\pi(p\,|\,y_i,d)}{\pi(p)}\pi(p\,|\,y_i,d)\,\text{d}p;$$
\item estimate the expected utility: $\bar{U}_M=\frac{1}{M}\sum\limits_{i=1}^{M}I_i.$
\end{enumerate}

It is natural to ask how well the true posteriors can be fitted by a
Beta distribution (step~(iii) of the above scheme). Experience
suggests that such a fit never affects the outcome of the analysis
on the qualitative level unless the prior distribution has more than
one local mode or its support is smaller than the entire interval
$(0,1)$. Recall that the purpose of this step is to make evaluation
of the integrals $I_i$ easier and faster, and hence the family of
Beta distributions is only one possible choice. For instance, if the
prior distribution $\pi(p)$ is uniform on $(0,1)$, then the
integrals $I_i$ represent the entropies of the fitted distributions.
In the case when the fitting is done by beta distributions, each of
these integrals can be quickly calculated using the following
analytical formula for the entropy of the beta distribution
$\text{Beta}(\alpha,\beta)$:
\begin{align}
\ent{\text{Beta}(\alpha,\beta)}=&\log
B(\alpha,\beta)+(\alpha+\beta-2)\psi(\alpha+\beta)\nonumber\\
&-(\alpha-1)\psi(\alpha)-(\beta-1)\psi(\beta),\nonumber
\end{align}
where $\psi$ is the \textit{digamma function}, $\psi(z)=\Gamma
'(x)/\Gamma(z)$. Other choices of prior distribution may
suggest alternative and more appropriate fitting distributions to facilitate
the calculation of the integrals $I_i$,
$i=1,\ldots,M$.

\subsection{Example}
In our example we consider all inner-outer plots in $\mathbb{L}^2$
whose sizes do not exceed $N=11$. There are only three such plots:
$\Pi^{(2)}(3,2)$, $\Pi^{(2)}(7,1)$, and $\Pi^{(2)}(11,0)$. For ease
of reference we mark them $\text{A}$, $\text{B}$, and $\text{C}$
respectively (as depicted in Figure~\ref{fig:inner_outer_sample}).
Thus, the design space $\mathcal{D}=\{\text{A},\text{B},\text{C}\}$
consists of three designs, among which $\text{A}$ is the most
sparsified plot whereas no nodes are removed from $\text{C}$ at all.

\begin{figure}[ptb]
\begin{center}
\includegraphics[width=3.5in]{Figures/spars_plots.eps}
\end{center}
\caption{\textit{Inner-outer} design plots $A$, $B$, and $C$ form the design space $\mathcal{D}=\{A,B,C\}$.
\label{fig:inner_outer_sample
\end{figure}

Figure~\ref{fig:inner_outer_hist} represents graphically the results
of the comparison of designs from $\mathcal{D}$ under both
`progressive' and `instructive' scenarios when the prior distribution
$\pi(p)$ is uniform on the interval $(0,1)$. The left panel of the
figure corresponds to the former scenario and depicts a histogram of
a sample corresponding to the marginal of the artificial augmenting
distribution $h(d,p,y)\propto u(d,p,y)f(y\,|\,p,d)\pi(p)$ in
$d\in\mathcal{D}$. The right panel corresponds to the latter
scenario and shows the Monte Carlo estimated values of the expected
utilities and 95\% credibility intervals for each of the three
considered designs ($M=1500$, see (\ref{eq:MCevaluation}) in
Section~\ref{section:simulation based evaluation of expected
utility}) assuming that the instructor knows $p$ precisely
so that $\pi^*(p)=\delta(p-0.9)$.

\begin{figure*}[ptb]
\begin{center}
\includegraphics[width=5.4in]{Figures/histo.eps}
\end{center}
\caption{Left: sample histogram for the marginal of $h(d,p,y)$ in
$d$, $d\in\{\text{A},\text{B},\text{C}\}$, under progressive design
and $\pi(p)\sim\text{U}(0,1)$. Right: evaluated expected utility
under instructive design with $\pi^*(p)\equiv\delta(p-0.9)$ and 95\%
credibility intervals ($M=1500$) for the plots $\text{A}$,
$\text{B}$, and $\text{C}$,
under instructive design.
\label{fig:inner_outer_hist
\end{figure*}

The plots from Figure~\ref{fig:inner_outer_hist} indicate that the
solutions to the optimal design problem under the two scenarios are
different from each other. The `moderately sparsified'
plot~$\text{B}$ maximises the expected utility in the progressive
case, that is in the case when the experiment is designed by a
single experimenter. If, however, it is the instructor who knows the
true value of the model parameter ($p=0.9$) and wishes to choose the
best inner-outer plot from the set $\mathcal{D}$ for a more
uninformed experimenter to use (instructive scenario), then the
optimal plot is the `mostly sparsified' inner-outer plot $\text{A}$.
Notably, the `mostly dense' plot $\text{C}$ would be the worst
choice in the instructive case, whereas it outperforms the `mostly
sparsified' plot $\text{A}$ but is worse than the `moderately
sparsified' plot $\text{B}$ in the progressive case.  These results
demonstrate the moderately counter-intuitive finding that the
optimal design is not necessarily that with the maximal number of
experimental units and that a 'less-is-more' principle may be in
operation.  On another level, this finding may not be entirely
unpredictable in the instructive case.  The optimal design's
combination of a dense centre and sparsified outer region may help
to increase the probability that the experimental epidemic succeeds
in establishing itself but does not lead to saturation of the
lattice before it dies out.  This in turn may represent an outcome
that yields much information on the unknown~$p$.

\section{Concluding comments}

In this paper we have considered a number of question relating to
the observation of percolation processes on finite lattices.  We
have demonstrated the value of McMC techniques for parameter
estimation under different sampling regimes and shown how Bayesian
experimental design techniques can be used to identify the optimal
lattices on which to carry out experiments.

We introduced inner-outer plot designs by sparsifying the underlying
grid. Allowing one to control excessive connectivity in outer
regions of experimental plots, these designs help to separate
different values of the model parameter $p$ when $p\gg p_c$, while
giving the epidemic a chance to establish in inner regions when
$p\ll p_c$. Inner-outer plots can be further generalised by
considering more than two levels of grid sparsification, according
to the principle used in our work: the farther the region is from
the centre of the plot the more intensive sparsification it
receives. To an extent, this process can be viewed as switching to
similarly shaped lattices with percolation parameter taking its
values from the set $\{p,p^2,p^3,\ldots\}$ in decreasing fashion.

Although we have worked largely within framework of percolation
theory, the connection between percolation and spatio-temporal
epidemic models with nearest-neighbour interactions mean that the
results may have significant implications for and application to the
study of epidemic systems.

\begin{acknowledgments}
We thank Dr~Alex Cook for useful discussions at early stages of this
work. A.~Iu.~B. acknowledges James Watt scholarship, Overseas
Research Student Awards Scheme support and funding provided through
the TIME project (UK EPSRC grant EP/C547632/1).
\end{acknowledgments}

redpajama_set_name: RedPajamaArXiv


Our dog, Gilligan, just had surgery this past weekend. Nothing major he just had a cyst removed from his leg. However, he has to wear this cone so he won't pull out his stitches. As you can see he doesn't like it.
» Filed Under: Big Dave's Log,Dogs.


text:
WeThotUWasAToad
asked on 9/6/2015
Custom pronunciation in MS Office text-to-speech (TTS)
Is there a way to define custom pronunciation or change existing pronunciation for specific words in MS Office (2010 & 2013) text-to-speech (TTS)?
With the exception of Excel, MS Office (2010 & 2013) apps come with a built-in TTS function. It can be activated by first adding the "Speak selected text" shortcut icon to the Quick Access Toolbar (QAT) and then, after selecting the text to be spoken, clicking that icon.
I have found that it works pretty well — certainly well enough for a listener to understand the basic gist or content contained in whatever selected text is being read. However, when confronted with certain specific words or phrases — particularly those which are probably not in its original vocabulary — TTS offers its best guess at the correct pronunciation but, not infrequently, gets it wrong.
Is there any way to access the MS Office TTS vocabulary (either through a user-interface settings box somewhere or by the use of VBA) in order to define correct pronunciation for words which it currently mispronounces?
Microsoft OfficeMicrosoft WordWord ProcessorsVoice RecognitionVisual Basic Classic
Martin Liss
regmigrant
This question has been classified as abandoned and is closed as part of the Cleanup Program. See the recommendation for more details.
I started with Experts Exchange in 2004 and it's been a mainstay of my professional computing life since. It helped me launch a career as a programmer / Oracle data analyst

redpajama_set_name: RedPajamaCommonCrawl


text:
Edward Fabián Díaz Cárdenas (born 19 August 1994) is a Colombian professional racing cyclist, who is currently suspended from the sport after testing positive for continuous erythropoietin receptor activator (CERA) at the 2017 Vuelta a Colombia.

References

External links

1994 births
Living people
Colombian male cyclists
People from Nobsa
Sportspeople from Boyacá Department
21st-century Colombian people

redpajama_set_name: RedPajamaWikipedia


David is a Psychologist working at the intersection of people, science and technology. His work as a coach aims to optimise individuals through an evidence-based approach which results in improvements to metrics that matter to individuals and the business. He is passionate about connecting with people and empowering them to their full potential. This passion has led him to found the UX Psychology Community which connects and empowers over 2,000 leaders in the digital space around in Melbourne, London and Berlin. He likes to keep his mind and body healthy through exercise, yoga, meditation and cooking.


text:
Software Market Forecast and Vendor Rankings: Security Software, 2016-21
May 2017 •
Ovum's security software market forecast provides revenue opportunity estimates across key segments - web content and email protection, endpoint and mobile device protection, identity and access management (IAM), information/data protection, security management, and network security-for the period 2016-21, spanning four regions, 20 countries, and 16 vertical industries.
The forecast includes revenue opportunity estimates across key subsegments, including web filtering, data loss prevention, DDoS prevention, and security information and event management. The forecast also now includes revenues for the outsourced and managed security services market.
Gain insight into the spending on security software across different regional markets for the period 2016-21.
Understand how the spending on security software across different verticals will change through 2021.
How will the share of different subsegments under security software change during 2016-21?
How will the size of the security software market in the Americas, EMEA, and Asia-Pacific change through 2021?
$ 7500 - 1 User License - PDF $ 22500 - Corporate License - PDF
Topics related to Cybersecurity
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The global cybersecurity market exhibited strong growth during 2015-2020. Looking forward, the analyst expects the market to grow at a CAGR of around 11% during 2021-2026. Keeping in mind the uncertainties of COVID-19, we are continuously tracking and evaluating the direct as well as the indirect influence of the pandemic...
Global Healthcare Cyber Security Market By Threat Type, By End Use, By Solution, By Regional Outlook, Industry Analysis Report and Forecast, 2021 - 2027
298 pages • By KBV Research • Oct 2021
The Global Healthcare Cyber Security Market size is expected to reach $28.4 billion by 2027, rising at a market growth of 16.3% CAGR during the forecast period. Cyber security technologies and solutions are constantly getting advanced because of the ever-changing threat landscape, where threats are becoming more sophisticated and damaging....
IDC FutureScape: Worldwide Future of Digital Infrastructure 2022 Predictions
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This IDC study provides IDC's 2022 top 10 predictions for the future of digital infrastructure. "Digital infrastructure spans compute, storage, network, infrastructure software including virtualization and containers and the automation, AI/ML analytics, and security software and cloud services needed to maintain and optimize both legacy and...
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The global DDoS Protection and Mitigation market size is expected to grow from USD 3.3 billion in 2021 to USD 6.7 billion by 2026, at a Compound Annual Growth Rate (CAGR) of 15.1% from 2021 to 2026. The major factors fueling the DDoS Protection and Mitigation market include Growing demand for sophisticated security solutions, Evolving regulatory...
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Abstract: Global Quantum Cryptography Market to Reach US$291.9 Million by the Year 2026 Quantum cryptography harnesses the quantum-mechanical properties of light for creating a secure encryption and enabling secure data transfer between any two parties. Quantum cryptography presents a...
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Global Privileged Access Management Solutions Market 2022-2026 The analyst has been monitoring the privileged access management solutions market and it is poised to grow by $ 4.28 bn during 2022-2026, progressing at a CAGR of 22.81% during the forecast period. Our report on the privileged access management solutions market...
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text:
Is Essex Property Trust Inc (ESS) A Good Stock To Buy?
Is Essex Property Trust Inc (NYSE:ESS) a good place to invest some of your money right now? We can gain invaluable insight to help us answer that question by studying the investment trends of top investors, who employ world-class Ivy League graduates, who are given immense resources and industry contacts to put their financial expertise to work. The top picks of these firms have historically outperformed the market when we account for known risk factors, making them very valuable investment ideas.
Essex Property Trust Inc (NYSE:ESS) shareholders have witnessed a decrease in enthusiasm from smart money recently. ESS was in 16 hedge funds' portfolios at the end of March. There were 19 hedge funds in our database with ESS positions at the end of the previous quarter. Our calculations also showed that ESS isn't among the 30 most popular stocks among hedge funds.
Hedge funds' reputation as shrewd investors has been tarnished in the last decade as their hedged returns couldn't keep up with the unhedged returns of the market indices. Our research has shown that hedge funds' large-cap stock picks indeed failed to beat the market between 1999 and 2016. However, we were able to identify in advance a select group of hedge fund holdings that outperformed the market by 40 percentage points since May 2014 through May 30, 2019 (see the details here). We were also able to identify in advance a select group of hedge fund holdings that'll significantly underperform the market. We have been tracking and sharing the list of these stocks since February 2017 and they lost 30.9% through May 30, 2019. That's why we believe hedge fund sentiment is an extremely useful indicator that investors should pay attention to.
[caption id="attachment_30621" align="aligncenter" width="487"] Cliff Asness of AQR Capital Management[/caption]
AQR CAPITAL MANAGEMENT
Let's go over the recent hedge fund action regarding Essex Property Trust Inc (NYSE:ESS).
How have hedgies been trading Essex Property Trust Inc (NYSE:ESS)?
At the end of the first quarter, a total of 16 of the hedge funds tracked by Insider Monkey held long positions in this stock, a change of -16% from the previous quarter. The graph below displays the number of hedge funds with bullish position in ESS over the last 15 quarters. With hedge funds' positions undergoing their usual ebb and flow, there exists an "upper tier" of noteworthy hedge fund managers who were boosting their stakes meaningfully (or already accumulated large positions).
ESS_june2019
Among these funds, Renaissance Technologies held the most valuable stake in Essex Property Trust Inc (NYSE:ESS), which was worth $229 million at the end of the first quarter. On the second spot was AEW Capital Management which amassed $160.5 million worth of shares. Moreover, Adage Capital Management, AQR Capital Management, and Citadel Investment Group were also bullish on Essex Property Trust Inc (NYSE:ESS), allocating a large percentage of their portfolios to this stock.
Since Essex Property Trust Inc (NYSE:ESS) has experienced declining sentiment from the entirety of the hedge funds we track, we can see that there lies a certain "tier" of funds who sold off their positions entirely by the end of the third quarter. Intriguingly, Richard Driehaus's Driehaus Capital dumped the largest stake of the "upper crust" of funds tracked by Insider Monkey, valued at about $5.6 million in stock, and Ray Dalio's Bridgewater Associates was right behind this move, as the fund dumped about $1.7 million worth. These moves are important to note, as aggregate hedge fund interest dropped by 3 funds by the end of the third quarter.
Let's now take a look at hedge fund activity in other stocks similar to Essex Property Trust Inc (NYSE:ESS). These stocks are AMETEK, Inc. (NYSE:AME), Harris Corporation (NYSE:HRS), American Water Works Company, Inc. (NYSE:AWK), and Pembina Pipeline Corp (NYSE:PBA). This group of stocks' market valuations match ESS's market valuation.
[table] Ticker, No of HFs with positions, Total Value of HF Positions (x1000), Change in HF Position AME,29,1038604,-3 HRS,25,783732,2 AWK,28,609712,0 PBA,15,102666,1 Average,24.25,633679,0 [/table]
As you can see these stocks had an average of 24.25 hedge funds with bullish positions and the average amount invested in these stocks was $634 million. That figure was $447 million in ESS's case. AMETEK, Inc. (NYSE:AME) is the most popular stock in this table. On the other hand Pembina Pipeline Corp (NYSE:PBA) is the least popular one with only 15 bullish hedge fund positions. Essex Property Trust Inc (NYSE:ESS) is not the least popular stock in this group but hedge fund interest is still below average. Our calculations showed that top 20 most popular stocks among hedge funds returned 6.2% in Q2 through June 19th and outperformed the S&P 500 ETF (SPY) by nearly 3 percentage points. A small number of hedge funds were also right about betting on ESS, though not to the same extent, as the stock returned 4.3% during the same time frame and outperformed the market as well.


text:
Q: selecting a single checkbox selects all the other checkboxes - flutter I am new to flutter. When I try to select a single card that in a ListView.builder, all the other cards that are in my list are selected.
I know that this is an issue of indexing. But, i am using the indexing to got through a list of data that i'm getting from the DB. So, the index in the itembuilder is int and not bool.
My Code
                shrinkWrap: true,
                itemCount: snapshot.data.length,
                itemBuilder: (context, index) {
                  ServiceProvider serviceProvider = snapshot.data[index];
                  return new Card(
                    shape: selected
                        ? new RoundedRectangleBorder(
                        side: new BorderSide(color: Colors.blue, width: 2.0),
                        borderRadius: BorderRadius.circular(4.0))
                        : new RoundedRectangleBorder(
                        side: new BorderSide(color: Colors.white, width: 2.0),
                        borderRadius: BorderRadius.circular(4.0)),
                    child: new Padding(
                      padding: const EdgeInsets.all(5.0),
                      child: new Column(
                        crossAxisAlignment: CrossAxisAlignment.start,
                        children: <Widget>[
                          new Text(serviceProvider.name, style: TextStyle(fontSize: 20, height: 1.5)),
                          new Text(serviceProvider.serviceType, style: TextStyle(fontSize: 15, height: 1.5),),
                          new Text(serviceProvider.phoneNumber, style: TextStyle(fontSize: 15, height: 1.5),),
                          new Text(serviceProvider.address, style: TextStyle(fontSize: 15, height: 1.5),),
                          new Text(serviceProvider.location, style: TextStyle(fontSize: 15, height: 1.5),),
                          new Text(serviceProvider.emailAddress, style: TextStyle(fontSize: 15, height: 1.5),),
                          Container(
                            child: ButtonBar(
                              children: <Widget>[
                                new Checkbox(
                                    value: selected,
                                    onChanged: (value) {
                                      setState(() {
                                        selected = value;
                                        print(selected);
                                      });
                                      selectedName.add(serviceProvider.name);
                                      print(selectedName);
                                    }),
                                new RaisedButton(
                                  child: Text('Discard'),
                                  onPressed: () {
                                    setState(() {
                                      deleteMethod(serviceProvider.name);
                                    });
                                  },
                                ),
                                new RaisedButton(
                                  child: Text('Verified'),
                                  onPressed: () {
                                    setState(() {
                                      updateMethod(serviceProvider.name);
                                    });
                                  },
                                ),
                              ],
                            ),
                          )
                        ],
                      ),
                    ),
                  );
                },
              );```

How do i go about this?

A: Keep a Set of the selected objects (in your case, ServiceProviders, assuming their hashCode and equals are correctly overridden, which you can do in IntelliJ/Android Studio by alt-inserting them). You can then see if the set contains an object (to figure out whether or not it's selected), remove an object (to remove selection), or add an object (to select).
To do this, keep the following at the top of your State class:
final selectedServiceProviders = Set<ServiceProvider>(); // creates an empty set (nothing selected by default)

Then, where you use setState to change the value of the selected boolean, simply use setState to modify the selectedServiceProviders with the corresponding method described in the first paragraph (add() or remove()). To check whether an object is selected, you can call selectedServiceProviders.contains(object);

redpajama_set_name: RedPajamaStackExchange


text:
The Last Shift (2020)
Watch The Last Shift (2020) Online
Stanley, an aging fast food worker, prepares to work his final graveyard shift after 38 years. When he's asked to train his replacement, Jevon, Stanley's weekend takes an unexpected turn.
Director: Andrew Cohn
Actors: Allison Tolman, Birgundi Baker, Da'Vine Joy Randolph, Ed O'Neill, Richard Jenkins, Shane Paul McGhie
Tags:Added Today FMovies Movies 2020
Impractical Jokers: The Movie (2020)
The story of a humiliating high school mishap from 1992 that sends the Impractical Jokers on the road competing in hidden-camera challenges for the chance to turn back the clock…
Genre: Comedy, Trending
Our Brand Is Crisis (2015)
A feature film based on the documentary "Our Brand Is Crisis", which focuses on the use of American political campaign strategies in South America.
Running Scared (1986)
Two street-wise Chicago cops have to shake off some rust after returning from a Key West vacation to pursue a drug dealer that nearly killed them in the past.
Genre: Action, Comedy, Crime, Thriller
Sam Montgomery is a tomboyish, unpopular girl at school. She has been text messaging a somebody named Nomad for a few months and he asks her to meet him at…
Genre: Comedy, Family, Romance
Highpoint (1982)
James Hatcher embezzles ten million dollars from a joint mafia/CIA operation, leaving them squabbling with each other. Unemployed Lewis Kinney gets caught up in the intrigue, and must try to…
The Fantasticks (2000)
Two rural teens sing and dance their way through a forbidden romance and a dangerous travelling carnival.
Genre: Comedy, Music, Romance
School Daze (1988)
Fraternity and sorority members clash with other students at a historically black college during homecoming weekend.
Genre: Comedy, Drama, Musical
Lesbian Vampire Killers
With their women having been enslaved by a pack of lesbian vampires, the remaining menfolk of a rural town send two hapless young lads out onto the moors as a…
Genre: Action, Comedy, Fantasy, Horror
Cheech & Chong's The Corsican Brothers (1984)
Two brothers who can feel each others' pain and pleasure mess up the French revolution.
Mr. Morgan's Last Love
The film centres on a retired, widowed professor living in Paris who develops a special relationship with a younger French woman.
Country: Belgium, France, Germany, USA
The Private Eyes (1980)
The lord and lady of a capacious manor are killed, and the lord's ghost seems to have returned to knock off the staff one by one, causing Inspector Winship and…
Genre: Comedy, Mystery
Top Secret! (1984)
Popular and dashing American singer Nick Rivers travels to East Germany to perform in a music festival. When he loses his heart to the gorgeous Hillary Flammond, he finds himself…
Genre: Comedy, Music
Trailer: The Last Shift (2020)


text:
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Culture of Virtuosity
Impellam Group
A win-win situation: the benefits of employee engagement for a business and its people
Written by Impellam Group
06 January 2020 | 5 mins
The success of any business is predicated on its people. For people to perform at their best, they need to be engaged — and robust employee engagement programmes can help facilitate this.
As the Fourth Industrial Revolution unfolds, opportunities for innovation seem limitless. Businesses have better access to disruptive new technologies than ever, and the automation of mundane tasks can help save time and money.
Despite fears of a robot takeover of service jobs, employees are still the biggest competitive advantage that any company, regardless of industry, has at its fingertips. As the leadership expert and keynote speaker, Brigette Hyacinth asserts,
"Your employees are your most valuable asset. Don't take them for granted or treat them poorly. They are your best brand ambassadors. Employees who have been pushed to the point where they no longer care will not go the extra mile. They will not take the initiative to solve problems. They will end up treating customers the same way you treat them."
People drive bottom-line growth — but only if their intrinsic needs are consistently being met.
Indeed, according to Deloitte research conducted across over 3,300 businesses around the world, 'culture and engagement' was deemed the most important issue for companies. Of all the challenges facing HR — including leadership, learning and development (L&D), and workforce capability — engagement is seen as the most critical and least understood.
As this blog will explore, employee engagement is vital for both business success and individual happiness. Engaged people are more likely to feel fulfilled at work and have a sense of purpose, while employers can trust them to deliver on their outputs.
Benefits to a business
1) Productivity and performance
The UK has a massive productivity problem. According to The Economist, one-third of British businesses have seen no increase in productivity since the millennium.
Australian business has had a similar setback. In June 2018, a Productivity Commission report revealed that labour productivity in the country was essentially flatlining. After a generation in which productivity had grown at almost 2% annually, the figure dropped to a "troubling" 0.2%.
Enter engagement. A 2006 study by Gallup looked at over 23,000 business units around the world and compared the productivity of those with the highest engagement scores versus those with the lowest engagement scores. The results? The most engaged people were 18% more productive than the least engaged.
Further analysis from call centres run by UK-based insurance company, More Than, also found that branches with higher levels of employee engagement had 35% less downtime between calls. Engagement enables productivity.
But what comes first? Engagement or performance? Understanding the relationship between the two can feel like something of a chicken-and-egg quandary for businesses. Should organisations engage with their people and hope they fulfil their duties, or should they demand proof of productivity before engaging?
As it turns out, overwhelming evidence suggests that the process always starts with engagement. In their 1999 book, First Break All the Rules: What the Greatest Managers Do Differently, Marcus Buckingham and Curt Coffman stated that the relationship from engagement to performance is four times stronger than the reverse. Meanwhile, Aston University identified a robust link from engagement to performance but not the other way round. The process only works left to right: engagement facilitates performance.
2) Customer satisfaction
"The customer comes first."
We've all heard this oft-repeated adage at some point in our lives. The problem? It's simply not true.
Right Management found that 70% of their more engaged people had a good understanding of customer needs. For disengaged people, the figure was a lowly 17%.
In NHS trusts, patient satisfaction is significantly higher where there are higher levels of employee engagement. Professor Michael West of the Centre for Performance-led HR at Lancaster University concluded that "employee engagement emerged as the best predictor of NHS trust outcomes." Put simply, hospitals with higher engagement scores have lower mortality rates.
3) Innovation
Extensive research from the Chartered Institute of Personnel and Development (CIPD) suggests that higher levels of engagement lead to more innovative work behaviour. Engaged people are far more likely to search out new methods and techniques to transform innovative ideas into useful applications. Further analysis published by Gallup found that 59% of engaged people say that work brings out their creativity, whereas only 3% of those considered disengaged felt creative at work.
4) Profit
Over a five-year period, Kenexa found that organisations with engagement scores in the upper quartile had twice the annual net income and seven times the shareholder return than businesses in the bottom 25%. Companies with engaged people have 8.6% higher profit margins than companies with bored people.
Marks and Spencer, for example, found that stores with improving engagement delivered an average of £62 million more in sales every year than stores with declining engagement. Meanwhile, Dorothy Perkins discovered that environments characterised by high engagement delivered 12% higher growth in sales, a 10% improvement in operating savings, and 35% lower stock loss.
Benefits for an organisation's people
1) Wellbeing and health
Research across the UK demonstrates a strong correlation between employee engagement and psychological wellbeing. According to the British Psychological Society's psychometrics, for a test to be predictive, it needs a correlation of 20%. CIPD data shows that the link between engagement and wellbeing is 35% — almost double the prediction threshold.
The CIPD also found that those who are absorbed in their work are three times as likely to have positive emotions (enthusiasm, cheerfulness, contentedness) than negative ones (stress, depression, anxiety). Gallup, meanwhile, found that 54% of disengaged people say that work affects their health negatively; for engaged people, the number is only 12%. The key takeaway? Engaged people take significantly less sick days per year.
The Corporate Leadership Council (CLC) has reported that highly engaged businesses have the potential to reduce staff turnover by 87%. The study also found that disengaged people are four times as likely to leave a business than engaged ones. Similarly, Royal Bank of Scotland (RBS) data suggested that branches with the lowest engagement scores had almost double the voluntary turnover as highly engaged branches.
3) Health and safety
Other research has found that engaged organisations are actually safer. In some cases, the least engaged businesses were on average experiencing 62% more accidents than those high engagement scores in the upper quartile. The Olympic Delivery Authority (ODA), for example, had an accident frequency rate of half the construction industry average and put it all down to employee engagement.
If one thing is clear from these benefits, it's that everyone gains. The only losers are a business' competitors. When engagement is high, individuals, managers, shareholders, and customers associated with a business all stand to win. By driving engagement across a business, the conditions are set for trust and virtuosity to flourish. As such, the business case for widespread employee engagement needs direct CEO and CFO oversight and needs to be a fundamental part of business strategy.
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Are you losing out on extraordinary talent? Consider a skills-based approach to recruitment and redeployment.
One of the most unexpected developments of the pandemic is the widening skills gap organizations are grappling with. Even before the arrival of COVID-19, a considerable chasm existed between labor supply and demand, but with the acceleration of digital transformation around the world, the need for in-demand and new skills has simply skyrocketed. For many employers, competing in limited pools of talent is increasingly untenable. So how can they overcome this challenge in a talent-scarce environment?
A meaningful International Women's Day, by Penny Queller, Chief Customer Officer at Impellam Group
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Just days after confirming the status of their relationship, Future stepped out with his boo Ciara for a double date with fellow Atlanta superstar couple Clifford "T.I." Harris and his wife Tiny.
According to SandraRose, the couples were spotted leaving Crustacean's restaurant in Beverly Hills yeserday (January 28, 2013).
T.I. and Tiny are already filming season 3 of T.I. and Tiny: The family Hustle. VH1′s hit family oriented show returns in March. Ciara is reportedly dating Future, whose anemic song with Rihanna, titled "Love", is in heavy rotation on the radio.
Ciara and one of Future's baby mamas have been been at war with each other for at least a week on social networking websites.
On a related note, KatTalesTV was on the scene as the couples exited the restaurant and tried unsuccessfully to get them to speak on the drama between Future's ex-girl and his next-girl.
What do you think these two "power couples" are up to?


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Citroen 'Ami EV' – An adorable off-road buggy
in Automobile
The My Ami Buggy includes canvas doors that can be removed, an LED light bar, and knobby tyres.
by Hammad Khalid December 19, 2021, 10:03 pm 29 Views
Citroen My Ami Buggy Concept is a free and distinct portrayal of an object created to spend leisure time in a new way – a pioneer with an extroverted, colourful, and muscular style that seeks to stay both basic and utilitarian.
The Citroen teams created this entirely liberated Ami, targeted to anyone looking for an electrified, easy-to-use car with a strong personality and fantastic charm.
Citroen says in a news statement that the buggy is intended to be whimsical, something "that every youngster from 14 to 77 would like to add to their Christmas list," and that it was inspired by "non-automotive domains such as building games, industrial design, and fashion." The spare tyre on the roof is suggestive of a casually placed beanie.
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The khaki, black, and yellow colour scheme is the first thing that catches the eye, giving it an earthy camouflage sense while being very visible.
Another noticeable feature is the absence of doors. The standard ones have been replaced with detachable canvas doors with zippers, which provide much-needed air circulation and add to the buggy's attractiveness. The knobbier and wider tyres will allow you to drive on rougher roads, but keep in mind that ground clearance isn't great. It also features headlamp guards and a spare knobby tyre installed on the roof, which provide a "sense of adventure" and increase the adorability.
"My Ami Buggy is a proposition in line with the philosophy of Ami, which is not a car. We have therefore looked for inspiration in the world of construction games for the fun and functional side, industrial design for ergonomics and aesthetics, encompassing everyday objects (furniture, lighting, etc.) and fashion accessories (sneakers, sports equipment, glasses, etc.). The simple, extruded shapes of the camera and smartphone holders, for example, were inspired by the work of designers who created essential and timeless consumer objects in the second half of the last century. My Ami Buggy Concept needed to be functional and simple, in the purest spirit of iconic and contemporary industrial objects."
Samuel Pericles – Designer of My Ami Buggy
As reported by Engadget, other additions include an LED light bar, removable "nomadic speaker," a roof cap to protect against the sun and a graphics package that includes the words "Pilot" and "01" on the driver's seat and "Copilot" and "02" on the passenger seat. It also offers thicker memory foam in the seats, new storage bins on the dashboard and even a range of luggage designed to "fit perfectly into the vehicle's interior spaces," according to Citroën. Other features like smartphone clamps, bottle holders and more were all built using 3D printing so that they can be reproduced on demand.
The Ami EV has a top speed of 28 MPH and a range of 44 miles on its 5.5 kWh battery. Teenagers aged 14 and above are also permitted to drive in France and other parts of Europe. The Ami Buggy idea must have been a lot of fun for the company's designers, but there's no news on whether Citroen will ever manufacture and sell it – though the kid in all of us definitely hopes so!
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The Vault Gym's personal training service is second to none. The PT team is over twenty-five strong and all of them are very well respected within the industry.
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Alma Verdell Howard, 85, a lifelong resident of Thatcher passed away Tuesday July 21, 2015, at his home. Verdell was a lifelong farmer in the Gila Valley and was well respected for his farming skills.
Verdell was born July 19, 1930, in Hubbard to Enos and Amy Bryce Howard. He attended school in Thatcher, graduating in the class of '49.
Later in life he served on the Thatcher Town Council for 25 years, was a member of the Board of Directors of the former Eastern Arizona Bank and also Stockman's Bank, was President of the Graham County Canal Board, and a partner in the Quail Ridge Development. Verdell was active in his church, serving as the Sunday School President and later as the First Councilor in the Thatcher Third Ward Bishopric.
Verdell was a devoted husband, father and grandfather. He is preceded in death by his grandson, Jeremy Curtis, two sisters, Wanda Moody and Mavis and a brother, Zenos Howard.
He leaves behind his sweet wife, Nelda Howard, three daughters, Diane (Jay) Layton, Marla (James) Wright, and Pam Curtis, as well as 15 grandchildren and 36 great-grandchildren.
A public viewing and visitation will be held Monday July 27 from 3 to 5 PM at the Howard Residence, 3177 W. 1st Street, Thatcher AZ. The family will hold a private service at a different time.