text:
\section{Introduction}

We propose a novel neural network architecture to ease the application of approximate Bayesian computation (ABC), a.k.a.~\textit{likelihood-free} inference. The architecture, called partially exchangeable network (PEN), uses partial exchangeability in Markovian data, allowing us to perform ABC inference for time series models with Markovian structure. Empirically, we also show that we can target non-Markovian time series data with PENs. Since the DeepSets architecture \cite{zaheer2017deep} turns out to be a special case of PEN, we can also perform ABC inference for static models. Our work is about automatically construct summary statistics of the data that are informative for model parameters. This is a main challenge in the practical application of ABC algorithms, since such summaries are often \textit{handpicked} (i.e.~ad-hoc summaries are constructed from model domain expertise), or these are automatically constructed using a number of approaches as detailed in \cref{sec:abc}. Neural networks have been previously used to automatically construct summary statistics for ABC. \citet{jiang2017learning} and \citet{creel2017neural} employ standard multilayer perceptron (MLP) networks for learning the summary statistics. \citet{chan2018likelihood} introduce a network that exploits the exchangeability property in exchangeable data. Our PEN architecture is a new addition to the tools for automatic construction of summary statistics, and PEN produces competitive inference results compared to \citet{jiang2017learning}, which in turn was shown outperforming the semi-automatic regression method by \citet{fearnhead2012constructing}. Moreover, our PEN architecture is more data efficient and when reducing the training data PEN outperforms \citet{jiang2017learning}, the factor of reduction being of order $10$ to $10^2$ depending on cases.

Our \textbf{main contributions} are:

\begin{itemize}
    \item Introducing the partially exchangeable networks (PENs) architecture;
    \item Using PENs to automatically learn summary statistics for ABC inference. We consider both static and dynamic models. In particular, our network architecture is specifically designed to learn summary statistics for dynamic models.
\end{itemize}

\section{Approximate Bayesian computation} \label{sec:abc}

Approximate Bayesian computation (ABC) is an increasingly popular inference method for model parameters $\theta$, in that it only requires the ability to produce artificial data from a stochastic model \textit{simulator} \cite{beaumont2002approximate,marin2012approximate}.  A simulator is essentially a computer program, which takes $\theta$, makes internal calls to a random number generator, and outputs a vector of artificial data. The implication is that ABC can be used to produce approximate inference when the likelihood function $p(y|\theta)$ underlying the simulator is intractable. As such ABC methods have been applied to a wide range of disciplines \cite{sisson2018handbook}. The fundamental idea in ABC is to generate parameter proposals $\theta^{\star}$ and accept a proposal if the simulated data $y^{\star}$ for that proposal is similar to observed data $y^{\text{obs}}$. Typically this approach is not suitable for high-dimensional data, and a set of summary statistics of the data is therefore commonly introduced to break the \textit{curse-of-dimensionality}. So, instead of comparing $y^{\star}$  to $y^{\text{obs}}$, we compare summary statistics of the simulated data $s^{\star}=S(y^{\star})$ to those of observed data $s^{\text{obs}}=S(y^{\text{obs}})$. Then we accept the proposed $\theta^{\star}$ if $s^{\star}$ is close to $s^{\text{obs}}$ in some metric. Using this scheme, ABC will simulate draws from the following approximate posterior of $\theta$
\begin{equation*}
    p^{\epsilon}_{\text{ABC}}(\theta | s^{\text{obs}}) \propto \int K_\epsilon(\Delta(s^{\star},s^{\text{obs}}))p(s^{\star}|\theta)p(\theta)ds^{\star},
\end{equation*}
where $p(\theta)$ is the prior of $\theta$, $\Delta$ is a distance function between observed and simulated summaries (we use a Mahalanobis distance, see the supplementary material in Appendix \ref{sec:suppl_mat}), $K_\epsilon(\cdot)$ is a kernel, which in all our applications is the uniform kernel returning 1 if $\Delta(s^{\star},s^{\text{obs}})<\epsilon$ and 0 otherwise, and $\epsilon>0$ is the so-called ABC-threshold. A smaller $\epsilon$ produces more accurate approximations to the true summaries posterior $p(\theta | s^{\text{obs}})$, though this implies a larger computational effort due to the increasing number of rejected proposals. An additional issue is that ideally we would like to target $p(\theta | y^{\text{obs}})$, not $p(\theta | s^{\text{obs}})$, but again unless sufficient statistics are available (impossible outside the exponential family), and since $\epsilon>0$, we have to be content with samples from $p^{\epsilon}_{\text{ABC}}$.

In this work we do not focus on \textit{how} to sample from $p^{\epsilon}_{\text{ABC}}(\theta | s^{\text{obs}})$ (see \citealp{sisson2018handbook} for possibilities). Therefore, we employ the simplest (and also most inefficient) ABC algorithm, the so called ``ABC rejection sampling'' \cite{pritchard1999population}. We will use the ``reference table'' version of ABC rejection sampling (e.g.~\citealp{cornuet2008inferring}), which is as follows:
\begin{itemize}
    \item Generate $\tilde{N}$ independent proposals $\theta^i\sim p(\theta)$,  and corresponding data  $y^i\sim p(y|\theta^i)$ from the simulator;
    \item Compute the summary statistics $s^i = S(y^i)$ for each $i=1,...,\tilde{N}$;
    \item Compute the distances $\Delta(s^{i}, s^{\text{obs}})$ for each $i=1,...,\tilde{N}$;
    \item Retain proposals $\theta^i$ corresponding to those $\Delta(s^{i}, s^{\text{obs}})$ that are smaller than the $x$-th percentile of all distances.
\end{itemize}
The retained $\theta^i$'s form a sample from $p^{\epsilon}_{\text{ABC}}$ with $\epsilon$ given by the selected $x$th percentile.
An advantage of this approach is that it allows to easily compare the quality of the ABC inference based on several methods for computing the summaries, under the same computational budget $\tilde{N}$. Moreover, once the ``reference table'' $(\theta^i,y^i)_{1\leq i \leq \tilde{N}}$ has been produced in the first step, we can recycle these simulations to produce new posterior samples using several methods for computing the summary statistics.

\subsection{Learning summary statistics}
Event though ABC rejection sampling is highly inefficient due to proposing parameters from the prior $p(\theta)$, this is not a concern for the purpose of our work.
In fact, our main focus is \textit{learning} the summary statistics $S(\cdot)$. This is perhaps the most serious difficulty affecting the application of ABC methodology to practical problems. In fact, we require summaries that are informative for $\theta$, as a replacement for the (unattainable) sufficient statistics. A considerable amount of research has been conducted on how to construct informative summary statistics (see \citealp{blum2013comparative} and \citealp{prangle2015summary} for an overview). However their selection is still challenging since no state-of-the-art methodology exists that can be applied to arbitrarily complex problems. \citet{fearnhead2012constructing} consider a regression-based approach where they also show that the best summary statistic, in terms of the minimal quadratic loss, is the posterior mean. The latter is however unknown since $p(\theta | y^{\text{obs}})$ itself is unknown. Therefore, they introduce a simulation approach based on a linear regression model
\begin{equation}
\theta_j^i = E(\theta_j | y^i) + \xi_j^i = b_{0_j} + b_j h(y^i) + \xi_j^i
\label{eq:linera_regerssion_model_fearnhead2012constructing}
\end{equation}
with $\xi_j^i$ some mean-zero noise. Here $j=1,...,\dim(\theta)$ and $h(y^i)$ is a vector of (non)-linear transformations of ``data'' $y^i$ (here $y^i$ can be simulated or observed data). Therefore \citet{fearnhead2012constructing} have $\dim(\theta)$ models to fit separately, one for each component of vector $\theta$. Of course, these fittings are to be performed \textit{before} ABC rejection is executed, so this is a step that anticipates ABC rejection, to provide the latter with suitable summary statistics. The parameters in each regression \eqref{eq:linera_regerssion_model_fearnhead2012constructing} are estimated by fitting the model by least squares to a new set of $N$ simulated data-parameter pairs ${(\theta^{i}, y^i)}_{1 \leq i \leq N}$ where, same as for ABC rejection, the $\theta^i$ are generated from $p(\theta)$ and the $y^i$ are generated from the model simulator conditionally on $\theta^i$. To clarify the notation: $N$ is the number of data-parameter pairs used to fit the linear regression model in \eqref{eq:linera_regerssion_model_fearnhead2012constructing}, while $\tilde{N}$ is the number of parameter-data pair proposals used in ABC rejection sampling. However the two sets of parameter-data pairs ${(\theta^{i}, y^i)}_{1 \leq i \leq N}$ and ${(\theta^{i}, y^i)}_{1 \leq i \leq \tilde{N}}$ are different since these serve two separate purposes. They are generated in the same way but independently of each other.  After fitting \eqref{eq:linera_regerssion_model_fearnhead2012constructing}, estimates $(\hat{b}_{0_j},\hat{b}_j)$ are returned and $\hat{b}_{0_j} + \hat{b}_j h(y)$ is taken as $j$th summary statistic, $j=1,...,\dim(\theta)$. We can then take $S_j(y^{\text{obs}}) =  \hat{b}_{0_j} + \hat{b}_j h(y^{\text{obs}})$ as $j$th component of $S(y^{\text{obs}})$, and similarly take $S_j(y^{\star}) = \hat{b}_{0_j} + \hat{b}_j h(y^{\star})$. The number of summaries is therefore equal to the size of $\theta$.

This approach is further developed in \citet{jiang2017learning} where a MLP  deep neural network regression model is employed, and replaces the linear regression model in \eqref{eq:linera_regerssion_model_fearnhead2012constructing}. Hence, \citet{jiang2017learning} has the following regression model
\begin{equation*}
\theta^i = E(\theta|y^i) + \xi^i = f_{\beta}(y^i) + \xi^i
\label{eq:linera_regerssion_model_dnn}
\end{equation*}
where $f_{\beta}$ is the MLP parametrized by the weights $\beta$. \citet{jiang2017learning} estimate $\beta$ from
\begin{equation} \label{eq:minimization_problem_dnn}
\min_{\beta}  \frac{1}{N} \sum_{i=1}^{N} \| f_{\beta}(y^i) - \theta^{i} \|^{2}_{2},
\end{equation}
where ${(\theta^{i}, y^i)}_{1\leq i \leq N}$ are the parameter-data pairs that the network $f_{\beta}$ is fitted to.
\ifx
After fitting the regression model we have that $S(y) = f_{\hat{\beta}}(y)$ is the summary statistic. The full scheme for running ABC in \citet{jiang2017learning} is as follows:

\begin{itemize}
    \item Generate a large number $N$ of parameter-data pairs ${(\theta^{i}, y^i)}_{1 \leq i \leq N}$, by sampling $\theta^{i}$ from the prior, and then simulate corresponding data set $y^{i}$ from the simulator;
    \item Train $f_{\beta}$ on the generated set of parameters-data;
    \item Run ABC where the network $f_{\hat{\beta}}$ is used to provide summary statistics.
\end{itemize}
\fi

The deep neuronal network with  multiple  hidden  layers considered in \citet{jiang2017learning}  offers  stronger  representational  power to approximate $E(\theta|y)$ (and hence learn an informative summary statistic),  compared  to using linear regression, if the posterior mean is a highly non-linear function of $y$. Moreover, experiments in \citet{jiang2017learning} show that indeed their MLP outperforms the linear regression approach in \citet{fearnhead2012constructing} (at least for their considered experiments), although at the price of a much larger computational effort. For this reason in our experiments we compare ABC coupled with PENs with the ABC MLP from \citet{jiang2017learning}.

In \citet{creel2017neural} a deep neural network regression model is used. He also introduces a pre-processing step such that instead of feeding the network with the data set $y^{\mathrm{obs}}$, the network is fed with a set of statistics of the data $s^{\mathrm{obs}}$. This means that, unlike in \citet{jiang2017learning}, in \citet{creel2017neural} the statistician must already know ``some kind'' of initial summary statistics, used as input, and then the network returns another set of summary statistics as output, and the latter are used for ABC inference. Our PENs do not require any initial specification of summary statistics.

\section{Partially exchangeable networks} \label{sec:pens}

Even though the likelihood function is intractable in the likelihood-free setting, we may still have insights into properties of the data generating process. To that end, given our data set $y \in \mathcal{Y}^M$ with $M$ units, we will exploit some of the invariance properties of its prior predictive distribution $p(y)=\int_\theta p(y|\theta)p(\theta)d\theta$. As discussed in \cref{sec:abc}, the regression approach to ABC \citep{fearnhead2012constructing} involves to learn the regression function $y \mapsto E(\theta | y)$, where $E(\theta | y)$ is the posterior mean. Our goal in this section is to leverage the invariances of the Bayesian model $p(y)$ to design deep neural architectures that are fit for this purpose.

\subsection{Exchangeability and partial exchangeability}

The simplest form of model invariance is \emph{exchangeability}. A model $p(y)$ is said to be exchangeable if, for all permutations $\sigma$ in the symmetric group $S_M$, $p(y)=p(y_{\sigma(1)},...,y_{\sigma(M)})$. For example, if the observations are independent and identically distributed (i.i.d.)~given the parameter, then $p(y)$ is exchangeable. A famous theorem of \citet{definetti1929}, which was subsequently generalized in various ways (see e.g.~the review of \citealp{diaconis1988}), remarkably shows that such conditionally i.i.d.~models are essentially the only exchangeable models.

If the model is exchangeable, it is clear that the function $y \mapsto E(\theta | y)$ is permutation invariant. It is therefore desirable that a neural network used to approximate this function should also be permutation invariant. The design of permutation invariant neural architectures has been the subject of numerous works, dating at least back to \citet[Chap. 2]{minsky1988perceptrons} and \citet{shawe1989}. A renewed interest in such architectures came about recently, notably through the works of \citet{ravanbakhsh2017deep}, \citet{zaheer2017deep}, and \citet{murphy2018janossy}---a detailed overview of this rich line of work can be found in \citet{bloem2019}. Most relevant to our work is the DeepSets architecture of \citet{zaheer2017deep} that we generalize to partial exchangeability, and the approach of \citet{chan2018likelihood}, who used permutation invariant networks for ABC.

However, the models considered in ABC are arising from intractable-likelihoods scenarios, which certainly are not limited to exchangeable data, quite the opposite, e.g.~stochastic differential equations \cite{picchini2014inference}, state-space models and beyond \cite{jasra2015approximate}. To tackle this limitation, we ask: \emph{could we use a weaker notion of invariance to propose deep architectures suitable for such models?} In this paper, we answer this question for a specific class of non-i.i.d.~models: Markov chains. To this end, we make use of the notion of \emph{partial exchangeability} studied by \citet{diaconis1980}. This property can be seen as a weakened version of exchangeability where $p(y)$ is only invariant to a subset of the symmetric group called \emph{block-switch transformations}. Informally, for $d \in \mathbb{N}$, a $d$-block-switch transformation interchanges two given disjoint blocks of $y \in \mathcal{Y}^M$ when these two blocks start with the same $d$ symbols and end with the same $d$ symbols.
\begin{definition}[\textbf{Block-switch transformation}]
For increasing indices $b=(i,j,k,l) \in \{0,\ldots,M \}^4$ such that $j-i\geq d$ and $l-k\geq d$, the \textbf{$d$-block-switch transformation} $T^{(d)}_b$ is defined as follows: if $y_{i:(i+d)}=y_{k:(k+d)}$ and $y_{(j-d):j}=y_{(l-d):l}$ then
\begin{align}
\everymath{\displaystyle}
y &= y_{1:i-1}
\tikz[baseline]{\node[fill=red!20,anchor=base] (b1ij) {$y_{i:j}$};}
y_{(j+1):(k-1)}
\tikz[baseline]{\node[fill=blue!20,anchor=base] (b1kl) {$y_{k:l}$};}
y_{(l+1):M} \\
T^{(d)}_b(y) &= y_{1:i-1}
\tikz[baseline]{\node[fill=blue!20,anchor=base] (b2kl) {$y_{k:l}$};}
y_{(j+1):(k-1)}
\tikz[baseline]{\node[fill=red!20,anchor=base] (b2ij) {$y_{i:j}$};}
y_{(l+1):M}.
\begin{tikzpicture}[overlay]
        \path[->,draw=red,shorten >=7.5pt,shorten <=7.5pt] (b1ij) edge  (b2ij);
        \path[->,draw=blue,shorten >=7.5pt,shorten <=7.5pt] (b1kl) edge  (b2kl);
\end{tikzpicture}%
\end{align}
If $y_{i:(i+d)} \neq y_{k:(k+d)}$ or $y_{(j-d):j} \neq y_{(l-d):l}$ then the block-switch transformation leaves $y$ unchanged: $T^{(d)}_b(y)=y$.
\end{definition}
\begin{definition}[\textbf{Partial exchangeability}] Let $A$ be a metric space.
A function $F: \mathcal{Y}^M \rightarrow A$ is said to be \textbf{$d$-block-switch invariant} if $F(y) = F(T_b^{(d)}(y))$ for all $y \in \mathcal{Y}$ and for all $d$-block-switch transformations $T_b^{(d)}$. Similarly, a model $p(y)$ is \textbf{$d$-partially exchangeable} if for all $d$-block-switch transformations $T_b^{(d)}$ we have $p(y)=p(T_b^{(d)}(y))$.
\end{definition}
Note that $0$-partial exchangeability reduces to exchangeability and that all permutations are $0$-block-switch transformations.

It is rather easy to see that, if $p(y|\theta)$ is a Markov chain of order $d$, then $p(y)$ is partially exchangeable (and therefore $y \mapsto E(\theta | y)$ is $d$-block-switch invariant). In the limit of infinite data sets, \citet{diaconis1980} showed that the converse was also true: any partially exchangeable distribution is conditionally Markovian. This result, which is an analogue of de Finetti's theorem for Markov chains, justifies that \emph{partial exchangeability is the right symmetry to invoke when dealing with Markov models}.

\subsection{From model invariance to network architecture}

When dealing with Markovian data, we therefore wish to model a regression function $y \mapsto E(\theta | y)$ that is $d$-block-switch invariant. Next theorem gives a general functional representation of such functions, in the case where $\mathcal{Y}$ is countable.

\begin{theorem}\label{th:decomposition}
Let $F: \mathcal{Y}^M \rightarrow A$ be $d$-block-switch invariant. If $\mathcal{Y}$ is countable, then there exist two functions  $\phi :\mathcal{Y}^{d+1}   \rightarrow  \mathbb{R}$ and $\rho :\mathcal{Y}^d \times \mathbb{R} \rightarrow  A$ such that
\begin{equation}
    \forall y \in \mathcal{Y}^M, \; F(y) = \rho \left(y_{1:d}, \sum_{i=1}^{M-d} \phi \left(y_{i:(i+d)} \right) \right).
    \label{eq:decomposition}
\end{equation}
\end{theorem}
\begin{proof}
Let $\sim$ be the equivalence relation over $\mathcal{Y}^M$ defined by $$x\sim y \iff \exists b_1,\ldots,b_k, \; \;   y = T_{b_1}^{(d)} \circ \cdots \circ T_{b_k}^{(d)} (x).$$
Let $\text{cl}:\mathcal{Y}^M \rightarrow \mathcal{Y}^M/{\sim}$ be the projection over the quotient set. According to the properties of the quotient set, since $F$ is $d$-block-switch invariant, there exists a unique function $g: \mathcal{Y}^M/{\sim} \rightarrow A$ such that $F = g \circ \text{cl}$.

Since $\mathcal{Y}$ is countable, $\mathcal{Y}^{d+1}$ is also countable and there exists an injective function $c: \mathcal{Y}^{d+1} \rightarrow \mathbb{N}$. Consider then the function
$$\nu: y \mapsto \left(y_{1:d},\sum_{i=1}^{M-d} 2^{-c(y_{i:(i+d)})} \right),$$
which is clearly $d$-block-switch invariant. There exists a unique function $h: \mathcal{Y}^M/{\sim} \rightarrow \nu(\mathcal{Y}^M)$ such that $\nu =  h \circ \text{cl}.$

We will now show that $h$ is a bijection. By construction, $h$ is clearly surjective. Let us now prove its injectivity. We thus have to show that, for all $x,y \in \mathcal{Y}^M$, $\nu(x)=\nu(y)$ implies $x \sim y$. Let $x,y \in \mathcal{Y}^M$ such that $\nu(x)=\nu(y)$. We have therefore $x_{1:d} = y_{1:d}$ and $$\sum_{i=1}^{M-d} 2^{-c(x_{i:(i+d)})}= \sum_{i=1}^{M-d} 2^{-c(y_{i:(i+d)})}.$$
The uniqueness of finite binary representations then implies that $\{x_{i:(i+d)}\}_{i\leq M-d}=\{y_{i:(i+d)}\}_{i\leq M-d}$. According to \citet[Proposition 27]{diaconis1980}, those two conditions imply that $x \sim y$, which shows that $h$ is indeed injective.

Since $h$ is a bijection, $\nu =  h \circ \text{cl}$ implies that $\text{cl} = h^{-1} \circ \nu$ which leads to $F = g \circ h^{-1} \circ \nu$. Finally, expanding this gives
$$ \forall y \in \mathcal{Y}^M, \; F(y) = g \circ h^{-1} \left(y_{1:d}, \sum_{i=1}^{M-d} 2^{-c(y_{i:(i+d)})}  \right),$$
which is the desired form with $\phi(y) = 2^{-c(y)}$ and $\rho = g \circ h^{-1}$.
\end{proof}

When $d=0$, the representation reduces to
\begin{equation}
     F(y) = \rho \left( \sum_{i=1}^{M} \phi \left(y_{i} \right) \right),
\end{equation}
and we exactly recover Theorem 2 from \citet{zaheer2017deep}---which also assumes countability of $\mathcal{Y}$---and the DeepSets representation. While an extension of our theorem to the uncountable case is not straightforward, we conjecture that a similar result holds even with uncountable $\mathcal{Y}$. A possible way to approach this conjecture is to study the very recent and fairly general result of \citet{bloem2019}. We note that the experiments on an autoregressive time series model in \cref{sec:AR2}, which is a Markovian process, support this conjecture.

\paragraph{Partially exchangeable networks}
The result in \cref{th:decomposition} suggests how to build $d$-block-switch invariant neural networks: we replace the functions $\rho$ and $\phi$ in \cref{eq:decomposition} by feed forward neural networks and denote this construction a $d$-partially exchangeable network (PEN-$d$ or PEN of order $d$). In this construction, we will call $\phi$ the \emph{inner network}, which maps a $d$-length subsequence $y_{i:i+d}$ into some representation $\phi(y_{i:i+d})$, and $\rho$ is the \emph{outer network} that maps the first $d$ symbols of the input, and the sum of the representations of all $d$-length subsequences of the input, to the output. We note that DeepSets networks are a special case of the PENs that corresponds to PEN-0.

\subsection{Using partially exchangeable networks for learning summary statistics for ABC}\label{sec:PENABC}

While PENs can by used for any exchangeable data, in this paper we use it for learning summary statistics in ABC. In particular, we propose the following regression model for learning the posterior mean
\begin{equation*}
       \theta^i = E(\theta|y^i) + \xi^i = \rho_{\beta_{\rho}}\biggl(y^i_{1:d}, \sum_{l = 1}^{M-d}  \phi_{\beta_{\phi}}(y^i_{l:l+d})\biggr) + \xi^i.
\end{equation*}
Here $\beta_{\phi}$ are the weights for the inner network, and $\beta_{\rho}$ are the weights for the outer network that maps its arguments into the posterior mean of the unknown parameters, which is the ABC summary we seek. When using PENs to learn the summary statistics we obtain the weights for the networks using the same criterion as in \cref{eq:minimization_problem_dnn}, except that instead of using the MLP network we use a PEN network for the underlying regression problem.

When targeting static models we employ a PEN-0, i.e.~a DeepSets network, since a static model can be viewed as a zero-order Markov model. For time series models we use a PEN-$d$, where $d>0$ is the order of the assumed data generating Markov process.

\section{Experiments} \label{sec:experiments}

We present four experiments: two static models (g-and-k and $\alpha$-stable distributions), and two time series models (autoregressive and moving average models). Full specification of the experimental settings is provided as supplementary material. The code was written in Julia 1.0.0 \cite{bezanson2017julia} and the framework Knet \cite{yuret2016knet} was used to build the deep learning models. The code can be found at \href{https://github.com/SamuelWiqvist/PENs-and-ABC}{https://github.com/SamuelWiqvist/PENs-and-ABC}. All experiments are simulation studies and the data used can be generated from the provided code. We compare approximate posteriors to the true posteriors using the Wasserstein distance, which we compute via the POT package \citep{flamary2017pot}. This distance can be sensitive to the number of posterior samples used, however, we observed that our results are fairly robust to variations in the number of samples. In all experiments we used 100 posterior samples to estimate the Wasserstein distance, except for the AR2 model where we used 500 samples.
We also employ two different MLP networks: ``MLP small'', where we use approximately the same number of weights as for the PEN-$d$ network; and ``MLP large'', which has a larger number of weights than PEN-$d$.

\subsection{g-and-k distribution} \label{sec:gandk}

The g-and-k distribution is defined by its quantile function via four parameters, and not by its probability density function since the latter is unavailable in closed form. This means that the likelihood function is ``intractable'' and as such exact inference is not possible. However, it is very simple to simulate draws from said distribution (see the supplementary material in Appendix \ref{sec:suppl_mat}), which means that g-and-k models are often used to test ABC algorithms \cite{prangle2017gk}.

The unknown parameters are $\theta = [A,B,g,k]$ (for full specification of the g-and-k distribution, see the supplementary material in Appendix \ref{sec:suppl_mat}). The prior distributions are set to $p(A) \sim \Gamma(2,1) $, $p(B) \sim \Gamma(2,1)$, $p(g) \sim \Gamma(2,0.5)$, and $p(k) \sim \Gamma(2,1)$ ($\Gamma(\alpha, \beta)$ is the Gamma distribution with shape parameter $\alpha$ and rate parameter $\beta$). We perform a simulation study with ground-truth parameters $A = 3$, $B = 1$, $g = 2$, $k = 0.5$ (same ground-truth parameter values as in \citealp{allingham2009bayesian}, \citealp{picchini2017approximate}, \citealp{fearnhead2012constructing}). Our data set comprises $M=1,000$ realizations from a g-and-k distribution.

We compare five different methods of constructing the summary statistics for ABC: (i) the handpicked summary statistics in \citet{picchini2017approximate}, i.e. $S(y) = [P_{20}, P_{40}, P_{60}, P_{80}, \text{skew}(y)]$ ($P_{i}$ is the $i$th percentile and $\text{skew}(y)$ is the skewness); (ii) ``MLP small''; (iii) ``MLP large''; (iv) a MLP network with a preprocessing step, denoted ``MLP pre'', where we feed the network with the empirical distribution function of the data instead of feeding it with the actual data; and (v) PEN-0 (DeepSets) since the data is i.i.d. the order of the Markov model is 0).

The probability density function for the g-and-k distribution can be approximated via finite differences, as implemented in the \texttt{gk} R package \cite{prangle2017gk}. This allow us to sample from an almost exact posterior distribution using standard Markov chain Monte Carlo (MCMC). We evaluate the inference produced using summaries constructed from the five methods (i--v) by comparing the resulting ABC posteriors to the ``almost exact'' posterior (computed using MCMC). ABC inferences are repeated over 100 independent data sets, and for a different number of training data observations for DNN models.
The results are presented in \cref{fig:res_gandk} and we can conclude that PEN-0 generates the best results. Furthermore, PEN-0 is also more data efficient since it performs considerably better than other methods with limited number of training observations. It seems in fact that PEN-0 requires 10 times less training data than ``MLP pre'' to achieve the same inference accuracy.
However all methods performed poorly when too few training observations are used.
The results also show that when MLP is fed with the observations it generates poor results, but if we instead use ``MLP pre'' and send in the empirical distribution function, in the spirit of \citet{creel2017neural}, we obtain considerably better results.

\begin{figure}[ht]
\vskip 0.2in
\begin{center}
\centerline{\includegraphics[width=1\columnwidth]{fig/gandk/res_gank_wasserstein}}
\caption{Results for g-and-k distribution: The estimated Wasserstein distances (mean over 100 repetitions) when comparing the MCMC posterior with ABC posteriors.
\label{fig:res_gandk}
\end{center}
\vskip -0.2in
\end{figure}

\subsection{$\alpha$-stable distribution} \label{sec:alphstable}

The $\alpha$-stable is a heavy-tailed distribution defined by its characteristic function (see supplementary material in Appendix \ref{sec:suppl_mat}). Its probability density function is intractable and inference is therefore challenging. Bayesian methods for the parameters can be found in e.g. \citet{peters2012likelihood} and \citet{ong2018variational}. Unknown parameters are $\theta = [\alpha, \beta, \gamma, \delta]$. We follow \citet{ong2018variational} and transform the parameters:
\begin{equation*}
    \tilde{\alpha} = \log \frac{\alpha-1.1}{2-\alpha}, \ \tilde{\beta} = \log \frac{\beta+1}{1-\beta}, \ \tilde{\gamma} = \log \gamma, \text{and} \ \tilde{\delta} = \delta.
\end{equation*}
This constraints the original parameters to $\alpha \in [1.1,2]$, $\beta \in [-1,1]$, and $\gamma > 0$. Independent Gaussian priors and ground-truth parameters  are as in \citet{ong2018variational}: $\tilde{\alpha}, \tilde{\beta}, \tilde{\gamma}, \tilde{\delta} \sim N(0,1)$; ground-truth values for the untransformed parameters are: $\alpha = 1.5$, $\beta = 0.5$, $\gamma = 1$, and $\delta = 0$. Observations consist of $M=1,000$ samples.

We compare methods for computing summary statistics as we did in \cref{sec:gandk} for the g-and-k distribution. However, since here the true posterior distribution is unavailable, we evaluate the different methods by comparing the root-mean square error (RMSE) between ground-truth parameter values and the ABC posterior means, see \cref{tab:alphastablermse}. From \cref{tab:alphastablermse} we conclude that PEN-0 performs best in terms of RMSE. Similarly to the g-and-k example we also see that ``MLP pre'' (see \cref{sec:gandk} for details) performs considerably better than MLP.
We now look at the resulting posteriors. In \cref{fig:alphastableposteriors} five posteriors from five independent experiments are presented (here we have used $5 \cdot 10^5$ training data observations). Inference results when using handpicked summary statistics are poor and for $\tilde{\gamma}$ the posterior resembles the prior. Posterior inference is worst for ``MLP large''. Results for ``MLP pre'' and PEN-0 are similar, at least in the case depicted in \cref{fig:alphastableposteriors} where we use $5 \cdot 10^5$ training data observations. However, in terms of RMSE, PEN-0 returns the best results when we reduce the number of training data observations.

\begin{table}[ht]
\caption{Results for $\alpha$-stable distribution. Root-mean square error (RMSE) when comparing posterior means to the ground-truth parameters (over 25 repetitions), for different methods of computing the summary statistics, and different number of training observations (between brackets).}
\label{tab:alphastablermse}
\vskip 0.15in
\begin{center}
\begin{small}
\begin{sc}
\resizebox{\columnwidth}{!}{
\begin{tabular}{lcccccc}
\toprule
{} &  Handpicked & MLP (small) & MLP (large) & MLP pre &  PEN-0  \\
\midrule
RMSE ($5\cdot10^5$) & 0.64   & 0.18 & 0.15 & 0.07 & 0.05 \\
RMSE ($10^5$)  &  0.64 & 0.19 & 0.17 & 0.07 & 0.06 \\
RMSE ($10^4$)  &  0.64 & 0.21 & 0.37 & 0.07 & 0.06\\
RMSE ($10^3$) & 0.64 & 0.72 & 0.62 & 0.40 & 0.07 \\
\bottomrule
\end{tabular}
}
\end{sc}
\end{small}
\end{center}
\vskip -0.1in
\end{table}

\begin{figure}[ht]
\vskip 0.2in
\centering

\subfigure[$\tilde{\alpha}$ (Handpicked) ]{\includegraphics[width=.18\columnwidth]{fig/alphastable/res_alpha_stable_alpha_handpicked}}\qquad
\subfigure[$\tilde{\alpha}$ (MLP large)]{\includegraphics[width=.18\columnwidth]{fig/alphastable/res_alpha_stable_alpha_mlp_1_large}}\qquad
\subfigure[$\tilde{\alpha}$ (MLP pre)]{\includegraphics[width=.18\columnwidth]{fig/alphastable/res_alpha_stable_alpha_mlp_pre1}}\qquad
\subfigure[$\tilde{\alpha}$ (PEN-0)]{\includegraphics[width=.18\columnwidth]{fig/alphastable/res_alpha_stable_alpha_deepsets1}}
\\
\subfigure[$\tilde{\beta}$ (Handpicked) ]{\includegraphics[width=.18\columnwidth]{fig/alphastable/res_alpha_stable_beta_handpicked}}\qquad
\subfigure[$\tilde{\beta}$ (MLP large)]{\includegraphics[width=.18\columnwidth]{fig/alphastable/res_alpha_stable_beta_mlp_1_large}}\qquad
\subfigure[$\tilde{\beta}$ (MLP pre)]{\includegraphics[width=.18\columnwidth]{fig/alphastable/res_alpha_stable_beta_mlp_pre1}}\qquad
\subfigure[$\tilde{\beta}$ (PEN-0)]{\includegraphics[width=.18\columnwidth]{fig/alphastable/res_alpha_stable_beta_deepsets1}}
\\
\subfigure[$\tilde{\gamma}$ (Handpicked) ]{\includegraphics[width=.18\columnwidth]{fig/alphastable/res_alpha_stable_gamma_handpicked}}\qquad
\subfigure[$\tilde{\gamma}$ (MLP large)]{\includegraphics[width=.18\columnwidth]{fig/alphastable/res_alpha_stable_gamma_mlp_1_large}}\qquad
\subfigure[$\tilde{\gamma}$ (MLP pre)]{\includegraphics[width=.18\columnwidth]{fig/alphastable/res_alpha_stable_gamma_mlp_pre1}}\qquad
\subfigure[$\tilde{\gamma}$ (PEN-0)]{\includegraphics[width=.18\columnwidth]{fig/alphastable/res_alpha_stable_gamma_deepsets1}}
\\
\subfigure[$\tilde{\delta}$ (Handpicked) ]{\includegraphics[width=.18\columnwidth]{fig/alphastable/res_alpha_stable_delta_handpicked}}\qquad
\subfigure[$\tilde{\delta}$ (MLP large)]{\includegraphics[width=.18\columnwidth]{fig/alphastable/res_alpha_stable_delta_mlp_1_large}}\qquad
\subfigure[$\tilde{\delta}$ (MLP pre)]{\includegraphics[width=.18\columnwidth]{fig/alphastable/res_alpha_stable_delta_mlp_pre1}}\qquad
\subfigure[$\tilde{\delta}$ (PEN-0)]{\includegraphics[width=.18\columnwidth]{fig/alphastable/res_alpha_stable_delta_deepsets1}}

\caption{Results for $\alpha$-stable distribution: Approximate marginal ABC posteriors. Results obtained using $5 \cdot 10^5$ training data observations. The green dashed line is the prior distribution. The colored lines show posteriors from 5 independent experiments. These posteriors are not cherry-picked.}
 \label{fig:alphastableposteriors}
\vskip -0.2in
\end{figure}

\subsection{Autoregressive time series model}\label{sec:AR2}

An autoregressive time series model of order two (AR(2)) follows:

$$y_l = \theta_1 y_{l-1} + \theta_2 y_{l-2} + \xi_l,\qquad \xi_l \sim N(0,1).$$

The AR(2) model is identifiable if the following are fulfilled: $\theta_2 < 1 + \theta_1, \theta_2 < 1 - \theta_1, \theta_2 > -1$ \cite{fuller1976introduction}. We let the resulting triangle define the uniform prior for the model. The ground-truth parameters for this simulation study are set to $\theta = [0.2, -0.13]$, and the data size is $M=100$.
AR(2) is a Markov model, hence and the requirement for PEN-$d$ with $d > 0$ is fulfilled.

We compare five methods for computing the summaries: (i) handpicked summary statistics, i.e. $S(y) = [\gamma(y,1), \gamma(y,2), \gamma(y,3), \gamma(y,4), \gamma(y,5)]$ ($\gamma(y,i)$ is autocovariance at lag $i$), which are reasonable summary statistics since autocovariances are normally employed in parameter estimation for autoregressive models, for instance when using the Yule–Walker equations; (ii) ``MLP small'' network; (iii) ``MLP large''; (iv) PEN-0 (DeepSets); and (v) PEN-2. Since AR(2) is a time series model it makes sense to use PEN-2, and PEN-0 results are reported only in the interest of comparison. Here we do not consider the ``MLP pre'' method used in Section \ref{sec:gandk} and \ref{sec:alphstable}, since the empirical distribution function does not have any reasonable meaning for time series data.
The likelihood function for AR(2) is known and we can therefore sample from the true posterior using MCMC.

Results are in \cref{fig:res_ar2}. PEN-2 outperforms MLP, for example we can see that the precision achieved when PEN-2 is trained on $10^3$ training observations can be achieved by MLP when trained on $10^5$ observations, implying an improvement of a $10^2$ factor. Approximate and exact posteriors are in \cref{fig:ar2_approx_posteriors} and we conclude that posteriors for both MLP and PEN-2 are similar to the true posterior when many training observations are used. However, the approximate posterior for MLP degrades significantly when the number of  training observations is reduced and is very uninformative with $10^3$ and even with $10^4$ observations, while for PEN-2 the quality of the approximate posterior distribution is only marginally reduced.
\begin{figure}[ht]
\vskip 0.2in
\begin{center}
\centerline{\includegraphics[width=1\columnwidth]{fig/ar2/res_ar2_wasserstein}}
\caption{Results for AR(2) model: Estimated Wasserstein distances (mean over 100 data sets) when comparing the true posterior with ABC posteriors, for varying sizes of training data when using DNN models.}
\label{fig:res_ar2}
\end{center}
\vskip -0.2in
\end{figure}
\begin{figure}[ht
\vskip 0.2in
\centering
\subfigure[Handpicked \textcolor{white}{some extra text ;)}]{\includegraphics[width=.27\columnwidth]{fig/ar2/res_ar2_posteriors_10_6_training_obs_handpicked}}\qquad
\subfigure[MLP large ($10^6$)]{\includegraphics[width=.27\columnwidth]{fig/ar2/res_ar2_posteriors_10_6_training_obs_mlp_large}}\qquad
\subfigure[PEN-2 ($10^6$) \textcolor{white}{some extra text ;)}]{\includegraphics[width=.27\columnwidth]{fig/ar2/res_ar2_posteriors_10_6_training_obs_bsn}}\qquad
\\
\hspace{2.9cm}
\subfigure[MLP large ($10^5$)]{\includegraphics[width=.27\columnwidth]{fig/ar2/res_ar2_posteriors_10_5_training_obs_mlp_large}}\qquad
\subfigure[PEN-2 ($10^5$) \textcolor{white}{some extra text ;)}]{\includegraphics[width=.27\columnwidth]{fig/ar2/res_ar2_posteriors_10_5_training_obs_bsn}}\qquad
\\
\hspace{2.9cm}
\subfigure[MLP large ($10^4$)]{\includegraphics[width=.27\columnwidth]{fig/ar2/res_ar2_posteriors_10_4_training_obs_mlp_large}}\qquad
\subfigure[PEN-2 ($10^4$) \textcolor{white}{some extra text ;)}]{\includegraphics[width=.27\columnwidth]{fig/ar2/res_ar2_posteriors_10_4_training_obs_bsn}}\qquad
\\
\hspace{2.9cm}
\subfigure[MLP large ($10^3$)]{\includegraphics[width=.27\columnwidth]{fig/ar2/res_ar2_posteriors_10_3_training_obs_mlp_large}}\qquad
\subfigure[PEN-2 ($10^3$) \textcolor{white}{some extra text ;)}]{\includegraphics[width=.27\columnwidth]{fig/ar2/res_ar2_posteriors_10_3_training_obs_bsn}}\qquad
\caption{Results for AR(2) model. The green line indicates the prior distribution, the contour plot is from the exact posterior and the blue dots are 100 samples from the several ABC posteriors. The number in parenthesis indicates number of observations in the training data set. These posteriors are not cherry-picked.}
\label{fig:ar2_approx_posteriors}
\vskip -0.2in
\end{figure}

\subsection{Moving average time series with observational noise model}
We consider a partially observed time series, with latent dynamics given by a moving average MA(2) model and observations perturbed with Gaussian noise:
\begin{align*}
\begin{cases}
y_l = x_l + \xi^y_l, \qquad \xi^y_l \sim N(0, \sigma_\epsilon = 0.3), \\
x_l = \xi_l + \theta_1 \xi^x_{l-1} + \theta_2 \xi^x_{l-2}, \qquad \xi^x_l \sim N(0, 1),
\end{cases}
\end{align*}
where the $\xi^x_l$ and $\xi^y_l$ are all independent.
An MA(2) process without observational noise is identifiable if $\theta_1\in[-2,2]$, $\theta_2\in[-1,1]$, and $\theta_2\pm\theta_1\ge -1$. Same as in \citet{jiang2017learning}, we define a uniform prior over this triangle. We use the same setting as in  \citet{jiang2017learning} and set the ground-truth parameters for the simulation study to $\theta = [0.6, 0.2]$. We only observe $\{y_l\}$ and the number of observations is $M=100$.

The latent dynamics are not Markovian, hence the Markov property required for PEN of order larger than 0 is not fulfilled, however, the quasi-Markov structure of the data might still allow us to successfully use PEN-$d$ with an order $d$ larger than 0. An additional complication is given by the observational noise $\xi_l^y$, further perturbing the dynamics.
Once more, we compare five methods for computing the summary statistics: (i) handpicked summaries $S(y) = [\gamma(y,1), \gamma(y,2)]$, i.e. we follow \citet{jiang2017learning}; (ii) ``MLP small''; (iii) ``MLP large''; (iv) PEN-0 (DeepSets); and (v) PEN-10. Same as for the AR(2) example, here PEN-0 results are reported only in the interest of a comparison with PEN-10, as for a time-series model it is expected from PEN-0 to be suboptimal.
Also in this case the likelihood function is available, and we can compute the true posterior distribution. Once more, we compare the approximate posteriors to the true posterior over 100 different data sets, see \cref{fig:res_ma2}. We conclude that PEN-10 performs slightly better than MLP when the training data set is large, and that PEN-10 outperforms MLP when we restrict the size of the training data. Once more, we notice that PEN-10 implies a factor $\geq 10$ in terms of savings on the size of the training data.

\begin{figure}[ht]
\vskip 0.2in
\begin{center}
\centerline{\includegraphics[width=1\columnwidth]{fig/ma2/res_ma2_wasserstein}}
\caption{Results for MA(2) model: Estimated Wasserstein distances (mean over 100 data sets) when comparing the true posterior with ABC posteriors.
\label{fig:res_ma2}
\end{center}
\vskip -0.2in
\end{figure}

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

Simulation experiments show that our partially exchangeable networks (PENs) achieve competitive results in learning summary statistics for use in ABC algorithms, outperforming the other deep learning methods that we have considered. Moreover, PENs require much smaller training data to achieve the same inference accuracy of competitors: in our experiments a reduction factor of order $10$ to $10^2$ was observed.

As mentioned in Section \ref{sec:abc}, in this work we were not focused on the specific ABC algorithm used for sampling, but only on learning summary statistics for ABC. However, in future work we plan to use  our  approach  for  constructing  summary  statistics  alongside  more  sophisticated variants of ABC methods, such as those which combine ABC with Markov chain  Monte  Carlo \cite{sisson2011likelihood}  or  sequential  techniques \cite{beaumont2009adaptive}.

\citet{murphy2018janossy} recently shed light on some limitations of the DeepSets architecture, and proposed to improve it by replacing the sum fed to the outer network by another pooling techinque called \emph{Janossy pooling}. Since the drawbacks they inspect are also likely to affect our architectures, extending Janossy pooling to the PEN framework might constitute a valuable improvement.

Our experiments show that the performance of the MLP networks using different choices for the number of weights is quite similar, and that PEN outperforms MLP even when MLP has access to a larger number of weights compared to PEN.
The main insight is that PENs by design incorporate the (partial) exchangeability property of the data, whereas the MLPs have to learn this property.
Exchangeability and partial exchangeability can in principle be expressed
in an MLP, but for small data sets these properties will be difficult to learn, and we expect that the model will overfit to the training data. One approach to alleviate this problem for MLPs is to perform data augmentation. However, it is not straightforward to perform data augmentation for continuous Markovian data, unless we have access to the underlying data generating process. In ABC the assumption is that we do have access to this process, but data generation may be computational expensive, and in a more general application we may not have access to the process.

Although we have applied the PEN architecture to the problem of learning summary statistics for ABC, notice that PEN is a general architecture and could be used for other applications. One example would be time series classification.

The main limitation for PEN is that it is designed for Markovian data or, when considering the special case of DeepSets (i.e. PEN-0), for exchangeable data. However, in the MA(2) example we achieve good inference results even though the MA(2) model is itself non-Markovian \textit{and} observations are perturbed with measurement noise.

\section*{Acknowledgements}

Research was partially supported by the Swedish Research Council (VR grant 2013-05167). We would also like to thank Joachim Hein and colleagues at LUNARC, Lund University, for helping out on setting up the GPU environment used for the simulations.

redpajama_set_name: RedPajamaArXiv


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Bass, A.J., P.C. Hemstra and L.J.V. Compagno (1986) Hexanchidae.: p. 45-47. In M.M. Smith and P.C. Heemstra (eds.) Smiths' sea fishes. Springer-Verlag, Berlin.
Compagno, L.J.V. and V.H. Niem (1998) Hexanchidae. Cowsharks, sixgill, and sevengill sharks.: p. 1208-1210. In K.E. Carpenter and V.H. Niem (eds.) FAO identification guide for fishery purposes. The Living Marine Resources of the Western Central Pacific. FAO, Rome.
Herman, J., M. Hovestadt-Euler, and D. C. Hovestadt / M. Stehman, Ed., 1994: Contributions to the study of the comparative morphology of teeth and other relevant ichthyodorulites in living supra-specific taxa of Chondrichthyan fishes. Addendum to part A, No. 1: Order Hexanciformes - Family: Hexanchidae. Odontological results ..... Bulletin de L'Institut Royal des Sciences Naturelles de Belgique 64. 147-163.


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Top Hedge Fund Books and e-Books – Part 4
Paul Milnes
Today, we look at seven more books and e-books that we consider to be essential reading for anybody that is looking to invest in – or play a part in a managing – a hedge fund, including another offering from Black Swan author Nassim Taleb, a straight-from-the-horse's-mouth account of George Soros' trading philosophies, and a book that manages to transcend a dreadful name to be one of the best books on the topic of the markets that has ever been written.
Antifragile: Things That Gain from Disorder
Nassim Nicholas Taleb (Penguin)
With his groundbreaking book The Black Swan, author Nassim Taleb demonstrated that unpredictable and highly improbable events underpin almost everything that happens in the world. Here, he urges us to look at uncertainty in a more positive light, deeming it a desirable – even necessary – factor in a number of important equations. In response to this new-found truth, Taleb presents the ideal of the "antifragile" – beyond the concepts of robustness and resilience.
While resilient or robust things resist shocks and stay the same, the antifragile just keeps improving. He draws an analogy with human bones, which get stronger when subjected to tension or stress, and other things in life that can benefit from volatility, turmoil, stress, and disorder. Furthermore, it is immune to errors in prediction and has additional protection from adverse events.
For example, how did the sinking of the Titanic save lives, and why do city-states perform better than nation-states? With this book, Taleb provides us with a blueprint for surviving in a world where uncertainty is the only certainty.
Clifford Asness and Antti Ilmanen (Institutional Investor)
This extended article, originally published in Institutional Investor and now available to view for free online, has gone on to become a foundation text for many within the investment industry. It gives one of the most concise and convincing arguments for the use of techniques such as leverage and derivatives trading to boost annual returns above the psychologically-important 5% mark.
Diversification has long been a watchword of the investment industry, but the concepts associated with it are changing all the time. In an earlier age, diversification might have meant owning stock in more than one sector, or investing in both stocks and debt securities. These days, it is all about investing in sometimes complex financial instruments alongside more traditional investments in order to give stronger – and crucially, more reliable – returns.
Diversification and Beyond: The Comparative Benefits of Multi-Strategy Investing by DE Shaw & Co
Back in the early days of hedge funds, there was only one strategy that counted – the long/short 'hedging' model that gave a name to this new investment class. However, over time hedge fund managers have begun to see the benefits in employing other strategies alongside this, or even instead of this, in order to bring greater returns and market neutrality, which was ultimately the goal of the original hedge fund strategy.
This study by the D.E. Shaw Group aims to show how a multi-strategy approach can reap greater rewards, and bring more stability, than a dedicated single-strategy approach. While there is something to be said for sticking to what you are good at, many of these specialist approaches are highly dependent on market conditions for their success. And that, as this study makes clear, is why it can pay to diversify with regards to investment strategies.
Soros on Soros
George Soros (Wiley)
As hedge fund managers go, they don't get much more famous than George Soros, AKA The Man Who Broke The Bank of England. His record-breaking success with the Quantum Fund brought him some profile in the 1980s, but it wasn't until he made over $1 bilion by shorting the pound – and forcing the UK to exit the European Exchange Rate Mechanism in the process – that he became a household name.
There have been many books about George Soros, and he has written a lot of them himself. However, unless you have a keen analytical mind (and a stomach for what most would deem to be overly-dense prose) then you're better off sticking with the third-party accounts of his life and work. However, this book, which overcomes Soros' weighty writing style by engaging him instead in two lengthy interviews, manages to bridge the gap between readability and from-the-horses-mouth detail. As an introduction to the investment methods of one of the greatest market thinkers of all time, this book is unparalleled.
Michael Lewis (W.W. Norton & Co)
If you wanted to profit from the financial crisis, then the best thing you could have done is to bet against – or 'short' – the collateralized debt obligation bubble that brought the global financial system to its knees. This book tells the stories of those who did, highlighting the eccentric nature of these contrarian investors who believed the bubble was going to burst.
Those interviewed for the book include Meredith Whitney, who predicted the demise of Citigroup and Bear Stearns; Steve Eisman, an outspoken hedge fund manager; Greg Lippmann, a Deutsche Bank trader; Dr. Michael Burry, an ex-neurologist who created Scion Capital despite suffering from blindness in one eye and Asperger's syndrome; Eugene Xu, a quantitative analyst who created the first CDO market by matching buyers and sellers; and the founders of Cornwall Capital, who started a hedge fund in their garage with $110,000 and built it into $120 million when the market crashed..
As well as highlighting those who benefited from the crash, it shines a spotlight on some of those who suffered the most, such as Merrill's $300 million CDO manager Wing Chau, Howie Hubler, who lost $9 billion in one trade – the biggest ever – and AIG, which suffered over $99 billion in losses. This acclaimed book was shortlisted for the 2010 Financial Times Business Book of the Year Award, and spent 28 weeks on the New York Times non-fiction bestseller list.
Benjamin Graham (Harper Business Essentials)
Many of the books we have covered in this series were written after the financial crisis, but market principles go much deeper than the latest event-inspired revisions. In fact, some of the best books about investing were written decades ago, and in most cases history has been kinder to them than you might expect.
The Intelligent Investor is one such book, and has been considered essential reading for market investors for over 35 years now, with Warren Buffett describing it as the most important book he has ever read. It's basically the foundation text for the principle of "value investing", a strategy of buying into fundamentally good businesses at times when their stock is underpriced. Although prescriptive, this philosophy is in fact applicable to a wide variety of investment strategies, and the success of investors such as Warren Buffett are a testimony to the soundness of these principles.
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Joel Greenblatt (Prentice Hall)
Sometimes, great books have terrible-sounding titles, and this book is an exemplary case. While the approach of mixing humour with finance can often backfire, in the masterful hands of Joel Greenblatt, it only serves to illustrate his insights with greater clarity. As a fund manager who had achieved 50% compound annual growth for ten years at the time of publication, his arguments have added conviction.
It lays out a series of investment principles, such as the need to concentrate on looking for bargains, limiting downside risk, and loading up on your best ideas. The author has a particular preference for special and distressed situations which have a high uncertainty but a low risk, and looking in places where other investors are not. For those that have an interest in the distressed investment strategy, this is a highly informative – and readable – book that maps out the key advantages of this approach.
I am a writer based in London, specialising in finance, trading, investment, and forex. Aside from the articles and content I write for IntelligentHQ, I also write for euroinvestor.com, and I have also written educational trading and investment guides for various websites including tradingquarter.com. Before specialising in finance, I worked as a writer for various digital marketing firms, specialising in online SEO-friendly content. I grew up in Aberdeen, Scotland, and I have an MA in English Literature from the University of Glasgow and I am a lead musician in a band. You can find me on twitter @pmilne100.
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We always have fun discussions on our Facebook page and love to get to know all of our readers better over there. We recently asked how people use their FoodSaver or other vacuum sealers and got a TON of great responses. We compiled them all here for a quick, easy-to-read list. If you have a FoodSaver, break it out and start using it for some of these things! If you don't have one yet, you can check them out at the FoodSaver website.
20. Seal shredded zucchini to use in stews/breads/etc.
40. Meals in a jar!


text:
© 2017 ScoutOut, LLC All Rights Reserved
GET SMART ABOUT YOUR GAME. GET EVALUATED
MIKE MOTTAU
NHL SCOUT - CHICAGO
GARRETT PETERS
NHL SCOUT - ST. LOUIS
IAN MORAN
FORMER NHL PLAYER
After spending four years at Boston College, Mike went on to play thirteen seasons of professional hockey. His career was split between the AHL and NHL, finishing with 321 total games in the NHL. Mike suited up for the US on the international stage on four occasions, once at the World Junior Championship andthree times at the World Championships. Since his retirement in 2014, Mike has stayed involved in hockey and is currently a Pro Scout for the Chicago Blackhawks
NHL players are often quoted saying,"I wish I knew what I know now, when I was younger."
Growth and development as a player comes through hard work, receiving feedback from coaches and becoming a student of the game.
ScoutOut enables athletes to easily request an individual evaluation during an event or anytime throughout the regular season. Our network of high level hockey representatives and scouts from NHL, OHL, USHL, NAHL and Prep/Private schools are thrilled about the opportunity to offer their many years of experience and expertise in development to motivated athletes.
High quality video is captured by ScoutOut videographers at an event. The video is then tagged for each player and shared with our scouts. Our scouts will then analyze your game and provide you with a professional report outlining your strengths and areas of suggested development. COST OF A REPORT IS $85
Garrett finished his junior hockey career with three seasons in the QMJHL, spending two years with the Halifax Mooseheads and his final year with the Drummondville Voltigeurs. Following juniors, he played 3 years of pro hockey, from there he moved behind the bench. In 2015-2016, Garrett was Head Coach and GM of the Providence Capitals in USPHL Elite along with a being a scout for the Halifax Mooseheads. Currently Garrett is an Amateur Scout with the St. Louis Blues.
Ian was drafted in the 6th round of the 1990 NHL Draft out of High School before heading off to Boston College. After this second season at BC he turned pro following his second season. Ian enjoyed a long pro career, playing 14 seasons professional hockey in North America and European leagues, amassing 489 total games in the NHL. Ian is now helping the next generation hone their player abilities as a skills coach in the New England Area.
CREATE YOUR PROFILE & GET SCOUTED
YOUR VIDEO SCOUTS


text:
backbonebeerexample
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Backbone example using a beer list

redpajama_set_name: RedPajamaGithub


I grew up in churches that had all the answers. Heaven was up, hell was down, and we knew who was going in which direction. God created the world in six days--one hundred and forty-four hours. The Red Sea parted just like in the movie. The whale swallowed Jonah. If the Bible said that Jonah swallowed the whale, we would have made that a test of faith. We reduced the mystery of the Unknowable God to Four Spiritual Laws that would fit on a post card and still leave room for "The Bible says it. I believe it. That settles it." We didn't have room for questions, because we were certain of everything.


Sport. One of the best things ever. Gets you going in the morning or helps you get your thoughts away after a long day. Sport is the perfect drug in my opinion: besides being pleasurable, it's also good for your health. I love sport.
I decided to play football and go to the gym here. Football is a bigger subject so I will cover that in my next post and only talk about the gym this time. The annual fee for Fitness as they've named it here is 65€. If you look at how much you have to pay in one month than this is a really good price. With that price comes good quality. First, to start going to the gym you have to do a 'Fitness Intake' where you have to show your competence with gym equipment. It sounds frightening but actually you just do your own training for an hour and they will look if you do anything that threatens the others or yourself. There were around 10 people doing the intake at the same time with me and none of them failed. The fact that the gym does something like this really shows their concern for the health of the clients not just for the money they make.
Another thing is that the gym is well equipped. You have your areas for dumbbells, bars, cardio, stretching and a variety of different strength-exercise equipment. The only problem is probably the size. There are many students in Groningen so the gym gets extremely crowded in the evening. The earlier you get there the better, or just go in the weekend. The final signs of quality are my favorite rules in the gym here: flip-flops are forbidden and a towel is mandatory. Besides eliminating the chance of having to use sweaty equipment, people wearing proper sports clothing and cleaning up after themselves creates a good atmosphere.
I believe the gym-enthusiasts have been addressed now. But that's only one opportunity out of the 86 courses offered by ACLO. If you're looking for something more than lifting weights on your own(you can do it in a group as well) then you've got plenty to choose from. I was tempted to take part of a sport with an attention-grabbing name like SurvivalRun, African dance, BOM or Krav Maga but I chose to play football. I've played football for a big part of my life and I just enjoy it a lot. However, more information about that next time!
more blogs from "Blog talent"


text:
Il santuario della Madonna dell'Oriente è un edificio religioso situato in comune di Sanfront (CN).

Storia

Il santuario è situato a monte della frazione Bollano, nel vallone del torrente Croesio, nei pressi del confine con il comune di Paesana. La sua origine è senza dubbio antica e la chiesa viene citata già in un documento del 1245 secondo il quale, con altri beni, era stata donata dal vescovo Gandolfo di Torino alla Prevostura di Oulx.. Secondo la tradizione sarebbe stata costruita in memoria del miracoloso salvataggio di un bambino dall'annegamento nel corso dello straripamento del vicino torrente Croesio.

Nel 1847 Luigi De Bartolomeis scriveva che presso Sanfront "sui monti a destra del Po sorge in sito elevato il santuario della Madonna d'Oriente tenuto in grande venerazione dai fedeli, che vi si recano processionando nelle festività d'ogni anno".

Il santuario è tuttora oggetto di devozione da parte dei fedeli della zona; tutti gli anni a inizio settembre si svolge una festa alla quale partecipano principalmente gli abitanti di Paesana e di Sanfront.

Descrizione

L'edificio sorge al centro di un'area prativa circondata da grossi esemplari di castagno. Si presenta a pianta rettangolare, con l'abside orientata verso nord-est. La chiesa, con copertura in lose, è affiancata su due lati da un ampio porticato, mentre l'abside e la facciata sud-orientale sono libere. Il campanile sorge isolato, e ai suoi piedi si trova un piccolo edificio che la gente del posto identifica come casa dell'eremita. L'interno della chiesa è riccamente decorato, mentre all'esterno, oltre a decorazioni pittoriche nel portico, sono ancora presenti alcune sbiadite tracce di affreschi sul lato sud-orientale e sulla cosiddetta casa dell'eremita. L'attuale santuario è caratterizzato nel suo complesso da uno stile tardo-barocco.

Note

Altri progetti

Collegamenti esterni

Sanfront
Madonna dell'Oriente
Sanfront
Chiese della diocesi di Saluzzo

redpajama_set_name: RedPajamaWikipedia


Please read carefully as the menu options have been updated!
Did you recently notice that the menu options have changed on your Nest Thermostat? Unlike your smart phone, where you must acknowledge and kick off any software updates, the Nest updates automatically. At some point this month your device would have been upgraded to software version 5.9.3-6. This update was the 5th made to the software this year (a frequency that is likely to continue). Whether you interact with your thermostat via your phone or by standing in front of the device it's a good idea to understand the new features.
The settings menu on your Nest thermostat is where you go to change the device's behavior. You can change what you see when the device detects your presence, set a schedule, etc. The type of system the device controls for you and the wires that are connected to it will dictate what options you have available to you. Here is a summary of the more important features currently available to optimize your Nest. If you don't see a feature mentioned here available on your device's menu that means your system does not support that feature.
Home/Away Assist – the Nest has two operating modes; Home and Away. When operating in Home mode the program/learned schedule for heating and cooling is used. When operating in Away mode the Eco Temperatures will be used to save energy when the house is empty. Enable this feature to automatically switch to Away mode when the thermostat senses that nobody's home, or you canmanually toggle the home and away setting from your phone.
Eco – the heating and cooling temperatures used when in Away mode. This feature helps you save energy.
Safety Temp – the heating and cooling temperatures that help protect your home from things like frozen pipes. These temperatures are maintained regardless of what mode your thermostat is in. The default settings are 40℉ for the heat with cooling set to off.
Sensor Schedule – adjust the Nest Temperature Sensor schedule for ideal comfort in the different areas of that zone.
Nest Sense – for if you are interested in having the thermostat learn about you and your home. Your system type will dictate which combinate of sensors and algorithms are used.
Reminders – turn this on if you want a reminder to change your air filter.
Nest Protect – if you have a Nest Protect installed and on the same account, use this option to set the system to shut off in the event of a smoke and/or CO emergency.
Display – here is where you can have the device wake up based on motion sensed or when you press it. Also specify what data appears on the screen first.
Brightness – for when the device is in a room where low light is best.
Energy Programs – local utility companies have established some energy saving programs that customers can opt into. Applicable programs will be listed here.
Click Sound – press here to toggle the device between having an audible click when turned or not.
Nest App – for verifying which Nest account the device is connected to.
Network – for verifying which Wi-Fi network the device is connected to.
Lock – allows only authorized users to adjust settings.
Equipment – shows what was selected during the initial setup.
Nest Pro – displays the contact information for Smart Home Solutions (provided we did your installation).
Software – check to see if you are on the latest version.
Technical Info – menu of data that is used when troubleshooting issues with the thermostat.


text:
29-й Венецианский международный кинофестиваль проходил в Венеции, Италия, с 25 августа по 7 сентября, 1968 года.

Из-за студенческих демонстраций, прокатившихся по всей Европе, фестиваль потерял своё значение на целое десятилетие — молодёжь требовала закрыть «фашистский» и «буржуазный» кинофестиваль.

Фестиваль распался на несколько программ, стал внеконкурсным, «Золотые львы» больше не вручались. С 1973 по 1978 год фестиваль вообще не проводился..

Жюри
 Гуидо Пиовене (Италия) (председатель жюри)
 Жак Дониоль-Валькроз (Франция)
 Акира Ивасаки (Япония)
 Роджер Мэнвелл (Великобритания)
 Иштван Немешкюрти (Венгрия)
 Винсенте Антонио Пинеда (Испания)
 Эдгар Райц (Западная Германия)

Разделы

Фильмы в конкурсе
 Ballade pour un chien, фильм Жерар Верже (Франция)
 Silenzio e grido, фильм Миклош Янчо (Венгрия)
 Das Schloß, фильм Рудольф Нольте (Западная Германия)
 Después del diluvio, фильм Хасинто Эстева (Испания)
 Diario di una schizofrenica, фильм Нело Ризи (Италия)
 Артисты под куполом цирка: беспомощны, фильм Александр Клюге (Западная Германия)
 Лица, фильм Джон Кассаветис (США)
 I muri, фильм Андраш Ковач (Венгрия)
 Fuoco!, фильм Джан Витторио Бальди (Италия)
 Галилео, фильм Лилиана Кавани (Италия/Болгария)
 Kierion, фильм Димостенис Теос (Греция)
 La schiuma dei giorni, фильм Шарль Бельмонт (Франция)
 Обнажённое детство, фильм Морис Пиала (Франция) (Фильм-открытие)
 Il Socrate, фильм Робер Лапужед (Франция/Западная Германия)
 Me and My Brother, фильм Роберт Фрэнк (США)
 Монтерей-Поп, фильм Д.А. Пеннебейкер (США)
 Богоматерь Турции, фильм Кармело Бене (Италия)
 Партнёр, фильм Бернардо Бертолуччи (Италия)
 Podne, фильм Младомир «Пуриша» Джорджевич (Югославия)
 Sept jours ailleurs, фильм Марин Кармиц (Франция)
 Стресс втроем, фильм Карлос Саура (Испания)
 Саммит, фильм Джорджо Бонтемпи (Италия)
 Tell Me Lies, фильм Питер Брук (Великобритания)
 Теорема, фильм Пьер Паоло Пазолини (Италия)
 Колесо праха, фильм Питер Эммануель Голдман (США)
 Quattordici o guerra, фильм Барри Ширз (США)
 Дезертиры и странники, фильм Юрай Якубиско (Чехословакия/Италия)

Награды
 Золотой лев: Артисты под куполом цирка: Беспомощны фильм Александр Клюге
 Специальный приз жюри Венецианского кинофестиваля: Богоматерь Турции, фильм Кармело Бене и Сократ, фильм Роберт Лапужед
 Кубок Вольпи за лучшую мужскую роль: Джон Марли — Лица
 Кубок Вольпи за лучшую женскую роль: Лаура Бетти — Теорема
 Почётное упоминание: Кьерион, фильм Димостенис Теос

Примечания

Ссылки
 29-й Венецианский кинофестиваль
 Официальный сайт кинофестиваля в Венеции
 29-й Венецианский международный кинофестиваль в архиве Биеннале

Август 1968 года
Сентябрь 1968 года
1968 год в Италии
Кинофестивали 1968 года


text:
Churchill's Legacy
by Alan Watson (Author)
Non-Fiction History History by country Europe Great Britain World War II
Churchill's Legacy describes how Churchill wielded his influence in post-war politics to enable the restoration of Europe through two key speeches in 1946. Having first helped bring victory to the Allies in 1945, Churchill went on to preserve the freedom of the world by gaining the support of the United States in the restoration of Europe.
In Fulton Missouri, Churchill alerted America to the reality of 'Uncle Joe' - a tyrant determined to dominate Europe at any cost. Churchill called for an Anglo-American alliance based on their shared values and the deterrent of America's possession of the atomic bomb. Churchill also urged the Americans to recognise the debt they owed Britain for opposing Hitler in 1940. In doing so, he contributed to the US thinking behind the need for the Marshall Plan.
In Zurich, Churchill boldly proposed a partnership between France and Germany: a United States of Europe. The hatred stirred up by the war had to be replaced by partnership for Europe to recover its economic vitality and regain its moral stature. Together, the Anglo-American Alliance and a United States of Europe led by France and Germany would have the power to 'smite the crocodile' of Soviet ambition.
To understand what Churchill intended with these two speeches requires perspective. The daring of his imagination and the scale of his architecture for a new Western Alliance was extraordinary. At the time, not many recognized the symmetry of what was proposed. At Churchill's funeral in 1965, commentators bemoaned the end of an era. In truth, Churchill was the catalyst of a new era-one built upon effective defence, economic revival, and European unity. His speeches have been added to UNESCO'S International Memory of the World Register.


text:
The Big Gundown ist ein Avantgardealbum von John Zorn. Es wurde von September 1984 bis September 1985 von Don Hünerburg für Nonesuch Records aufgenommen, zusammengestellt und 1986 veröffentlicht wurde. Im Jahr 2000 wurde das Album in einer erweiterten Form auf Zorns Tzadik-Label wiederveröffentlicht. (15th Anniversary Edition).

Die Musik

Das Album des US-amerikanischen Komponisten, Saxophonisten und Multi-Instrumentalisten
John Zorn ist schon im Titel eine Hommage an den italienischen Filmkomponisten
Ennio Morricone. Es enthält auf eine radikale Weise verarbeitete Coverversionen von Filmmusiken des Komponisten zu Filmen von Claude Lelouch, Gillo Pontecorvo, Bernardo Bertolucci, Elio Petri und besonders Sergio Leone. Ursprünglich vorgesehener Titel war daher auch Once Upon a Time in the Lower East Side, als Anspielung auf Leones Klassiker Es war einmal in Amerika. Der letztendlich gewählte Titel des Albums bezieht sich auf einen Spaghetti-Western aus dem Jahr 1966, The Big Gundown von Sergio Sollima mit Lee Van Cleef in der Hauptrolle, zu dem Morricone damals die Musik schrieb. Der Film lief in Deutschland unter dem Titel Der Gehetzte der Sierra Madre.

Zorn wählte bewusst neben Musikern seines regulären Umfeldes aus der New Yorker Jazzfusion- und Artrockavantgarde wie Bill Frisell, Arto Lindsay, Fred Frith und Wayne Horvitz zusätzlich eher dem Jazz-Mainstream oder anderen Genres zuzuordnende Musiker aus, darunter den Mundharmonikaspieler Toots Thielemans für den Titel Poverty aus One Upon a Time in America, den Organisten John Patton für seine Interpretation von Erotico aus dem Film The Burglars sowie die Sängerin Diamanda Galás und den Gitarristen Vernon Reid für Metamorfosi, ein Thema aus Elio Petris Film Der Weg der Arbeiterklasse ins Paradies.

The Big Gundown legte die Grundlage für nachfolgende Konzeptalben Zorns wie Spillane 1987 und Spy Vs Spy 1988.

Die Titel
 "The Big Gundown" – 7:26
 "Peur Sur La Ville" – 4:16
 "Poverty" (Once Upon A Time In America) – 3:49
 "Milano Odea" – 3:02
 "Erotico (The Burglars)" – 4:27
 "Battle Of Algiers" – 3:50
 "Giu La Testa" (Duck You Sucker!) – 6:06
 "Metamorfosi" (La Classe Operaia Va In Paradiso) – 4:37
 "Tre Nel 5000" – 4:37
 "Once Upon a Time in the West" – 8:44

 Weitere Titel der Ausgabe von 2000:
11. "The Sicilian Clan" – 3:20
12. "Macchie Solari" 3:29
13. "The Ballad Of Hank McCain" – 5:27
14. "Svegliati & Uccidi" 3:03
15. "Chi Mai" – 3:06
16. "The Ballad Of Hank McCain (instrumental) – 5:28"

Alle Kompositionen stammen von Ennio Morricone, außer "Tre Nel 5000" von John Zorn

Wirkung und Bewertung
Richard Cook und Brian Morton sehen John Zorn in der Rolle eines künstlerischen Puppenspielers und vergleichen ihn mit der
Rolle George Martins bei den Aufnahmen der Beatles sowie des frühen Frank Zappa. Sie halten das Album "für in jeder Hinsicht bemerkenswert" und zählen es zu den wichtigsten Alben der 1980er Jahre. Sie bewerteten das Werk mit der höchsten Punktzahl, zusätzlich aufgewertet durch eine Krone.

Guy Peters wertet das Album als genre-sprengend und eines der risikoreichsten der 1980er Jahre. Das Album könne als einer der erfolgreichsten Befürworter des vorherrschenden dekonstruktivistischen Ansatzes in der Literatur und der Musik gewertet werden: "John Zorn geht beständig auf der schmalen Grenze zwischen Bewunderung und Nachahmung, was auch eine Konsequenz aus der Ausdrucksstärke des Werkes von Morricone selbst ist."

Literatur
 Richard Cook, Brian Morton: The Penguin Guide to Jazz on CD. 6. Auflage. Penguin, London 2002, ISBN 0-14-051521-6.

Weblinks
 Tzadik Katalog
 Scott Yanow im All Music Guide

Einzelnachweise

Album (Jazz)
Konzeptalbum
Album 1986
John Zorn
Ennio Morricone