The field of probabilistic numerics (PN), loosely speaking, attempts to provide a statistical treatment of the errors and/or approximations that are made en route to the output of a deterministic numerical method, e.g. the approximation of an integral by quadrature, or the discretised solution of an ordinary or partial differential equation. This decade has seen a surge of activity in this field. In comparison with historical developments that can be traced back over more than a hundred years, the most recent developments are particularly interesting because they have been characterised by simultaneous input from multiple scientific disciplines: mathematics, statistics, machine learning, and computer science. The field has, therefore, advanced on a broad front, with contributions ranging from the building of overarching generaltheory to practical implementations in specific problems of interest. Over the same period of time, and because of increased interaction among researchers coming from different communities, the extent to which these developments were -- or were not -- presaged by twentieth-century researchers has also come to be better appreciated. Thus, the time appears to be ripe for an update of the 2014 Tübingen Manifesto on probabilistic numerics[Hennig, 2014, Osborne, 2014d,c,b,a] and the position paper[Hennig et al., 2015] to take account of the developments between 2014 and 2019, an improved awareness of the history of this field, and a clearer sense of its future directions. In this article, we aim to summarise some of the history of probabilistic perspectives on numerics (Section 2), to place more recent developments into context (Section 3), and to articulate a vision for future research in, and use of, probabilistic numerics (Section 4).
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The UK government was given details last December of allegedly extensive contacts between the Trump campaign and Moscow, according to court papers. Reports by Christopher Steele, a former MI6 officer, on possible collusion between the the Trump camp and the Kremlin are at the centre of a political storm in the US over Moscow's role in getting Donald Trump elected. It was not previously known that the UK intelligence services had also received the dossier but Steele confirmed in a court filing earlier this month that he handed a memorandum compiled in December to a "senior UK government national security official acting in his official capacity, on a confidential basis in hard copy form". The court papers say Steele decided to pass on the information he had collected because it was "of considerable importance in relation to alleged Russian interference in the US presidential election", that it "had implications for the national security of the US and the UK" and "needed to [be] analysed and further investigated/verified". The December memo alleged that four Trump representatives travelled to Prague in August or September in 2016 for "secret discussions with Kremlin representatives and associated operators/hackers", about how to pay hackers secretly for penetrating Democratic party computer systems and "contingency plans for covering up operations".