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Learning probabilistic context-free grammars (PCFGs) from strings is a classic problem in computational linguistics since Horning ( 1969 ). Here we present an algorithm based on distributional learning that is a consistent estimator for a large class of PCFGs that satisfy certain natural conditions including being anchored (Stratos et al., 2016 ). We proceed via a reparameterization of (top–down) PCFGs that we call a bottom–up weighted context-free grammar. We show that if the grammar is anchored and satisfies additional restrictions on its ambiguity, then the parameters can be directly related to distributional properties of the anchoring strings; we show the asymptotic correctness of a naive estimator and present some simulations using synthetic data that show that algorithms based on this approach have good finite sample behavior.

Abstract Background People living with a long term condition may wish to be able to plan ahead, so that if in future they cannot make decisions, their wishes about their care will be known; this process is termed Advance Care Planning (ACP). In dementia, guidance stipulates that ACP discussions should take place whilst the person still has capacity to make decisions. However there is a lack of evidence on the effectiveness of ACP in influencing patient choice and resource use. The aims of this study are to determine the effectiveness of ACP in dementia care, identify the factors which facilitate the process in practice and provide a better understanding of the views and experiences of key stakeholders in order to inform clinical practice. Methods/Design The four phase project comprises a systematic review (Phase 1) and a series of qualitative studies (Phases 2 and 3), with data collection via focus groups and individual interviews with relevant stakeholders including people with dementia and their carers, health and social care professionals and representatives from voluntary organisations and the legal profession. The conduct of the systematic review will follow current best practice guidance. In phases 2 and 3, focus groups will be employed to seek the perspectives of the professionals; individual interviews will be carried out with people with dementia and their carers. Data from Phases 1, 2 and 3 will be synthesised in a series of team workshops to develop draft guidance and educational tools for implementing ACP in practice (Phase 4). Discussion In the UK, there is little published research on the effectiveness of ACP, despite its introduction into policy. This study was designed to explore in greater depth how ACP can best be carried out in routine practice. It affords the opportunity to develop both a theoretical and practical understanding of an area which both patients and professionals may find emotionally challenging. Importantly the study will also develop practical tools, which are grounded in practice, for all relevant stakeholders to enable the facilitation of timely and sensitive ACP discussions.