Solving the equation for the maximum a posteriori estimate

1969 ◽  
Vol 57 (5) ◽  
pp. 814-815
Author(s):  
A. Guida
Keyword(s):  
Maximum A Posteriori ◽  
A Posteriori ◽  
Posteriori Estimate ◽  
A Posteriori Estimate ◽  
Maximum A Posteriori Estimate

scholarly journals Bayesian ODE solvers: the maximum a posteriori estimate

2021 ◽  
Vol 31 (3) ◽  
Author(s):  
Filip Tronarp ◽  
Simo Särkkä ◽  
Philipp Hennig
Keyword(s):  
Reproducing Kernel ◽  
Convergence Rates ◽  
Numerical Solutions ◽  
Linear Time ◽  
Maximum A Posteriori ◽  
A Posteriori ◽  
Scattered Data Approximation ◽  
Posteriori Estimate ◽  
A Posteriori Estimate ◽  
Maximum A Posteriori Estimate

AbstractThere is a growing interest in probabilistic numerical solutions to ordinary differential equations. In this paper, the maximum a posteriori estimate is studied under the class of $$\nu $$ ν times differentiable linear time-invariant Gauss–Markov priors, which can be computed with an iterated extended Kalman smoother. The maximum a posteriori estimate corresponds to an optimal interpolant in the reproducing kernel Hilbert space associated with the prior, which in the present case is equivalent to a Sobolev space of smoothness $$\nu +1$$ ν + 1 . Subject to mild conditions on the vector field, convergence rates of the maximum a posteriori estimate are then obtained via methods from nonlinear analysis and scattered data approximation. These results closely resemble classical convergence results in the sense that a $$\nu $$ ν times differentiable prior process obtains a global order of $$\nu $$ ν , which is demonstrated in numerical examples.

scholarly journals Incorporating word embeddings in unsupervised morphological segmentation

2020 ◽  
pp. 1-21
Author(s):  
Ahmet Üstün ◽  
Burcu Can
Keyword(s):  
Semantic Information ◽  
Maximum A Posteriori ◽  
Word Embeddings ◽  
A Posteriori ◽  
Low Resource ◽  
A Posteriori Estimate ◽  
Morphological Segmentation ◽  
Vector Representations ◽  
Turkish Language ◽  
Maximum A Posteriori Estimate

Abstract We investigate the usage of semantic information for morphological segmentation since words that are derived from each other will remain semantically related. We use mathematical models such as maximum likelihood estimate (MLE) and maximum a posteriori estimate (MAP) by incorporating semantic information obtained from dense word vector representations. Our approach does not require any annotated data which make it fully unsupervised and require only a small amount of raw data together with pretrained word embeddings for training purposes. The results show that using dense vector representations helps in morphological segmentation especially for low-resource languages. We present results for Turkish, English, and German. Our semantic MLE model outperforms other unsupervised models for Turkish language. Our proposed models could be also used for any other low-resource language with concatenative morphology.

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