Maximum A Posteriori Estimation Pdf Download
Sandrine Willert
In Bayesian statistics, a maximum a posteriori probability (MAP) estimate is an estimate of an unknown quantity, that equals the mode of the posterior distribution. The MAP can be used to obtain a point estimate of an unobserved quantity on the basis of empirical data. It is closely related to the method of maximum likelihood (ML) estimation, but employs an augmented optimization objective which incorporates a prior distribution (that quantifies the additional information available through prior knowledge of a related event) over the quantity one wants to estimate. MAP estimation can therefore be seen as a regularization of maximum likelihood estimation.
Now assume that a prior distribution g \displaystyle g over θ \displaystyle \theta exists. This allows us to treat θ \displaystyle \theta as a random variable as in Bayesian statistics. We can calculate the posterior distribution of θ \displaystyle \theta using Bayes' theorem:
The denominator of the posterior distribution (so-called marginal likelihood) is always positive and does not depend on θ \displaystyle \theta and therefore plays no role in the optimization. Observe that the MAP estimate of θ \displaystyle \theta coincides with the ML estimate when the prior g \displaystyle g is uniform (i.e., g \displaystyle g is a constant function).
as c \displaystyle c goes to 0, the Bayes estimator approaches the MAP estimator, provided that the distribution of θ \displaystyle \theta is quasi-concave.[1] But generally a MAP estimator is not a Bayes estimator unless θ \displaystyle \theta is discrete.
In many types of models, such as mixture models, the posterior may be multi-modal. In such a case, the usual recommendation is that one should choose the highest mode: this is not always feasible (global optimization is a difficult problem), nor in some cases even possible (such as when identifiability issues arise). Furthermore, the highest mode may be uncharacteristic of the majority of the posterior.
Finally, unlike ML estimators, the MAP estimate is not invariant under reparameterization. Switching from one parameterization to another involves introducing a Jacobian that impacts on the location of the maximum.[2]
The GloVe word embedding model relies on solving a global optimization problem, which can be reformulated as a maximum likelihood estimation problem. In this paper, we propose to generalize this approach to word embedding by considering parametrized variants of the GloVe model and incorporating priors on these parameters. To demonstrate the usefulness of this approach, we consider a word embedding model in which each context word is associated with a corresponding variance, intuitively encoding how informative it is. Using our framework, we can then learn these variances together with the resulting word vectors in a unified way. We experimentally show that the resulting word embedding models outperform GloVe, as well as many popular alternatives.
The site is secure.
The https:// ensures that you are connecting to the official website and that any information you provide is encrypted and transmitted securely.
We propose a new framework to jointly improve spatial resolution and remove fixed structural patterns for coherent fiber bundle imaging systems, based on inverting a principled forward model. The forward model maps a high-resolution representation to multiple images modeling random probe motions. We then apply a point spread function to simulate low-resolution figure bundle image capture. Our forward model also uses a smoothing prior. We compute a maximum a posteriori (MAP) estimate of the high-resolution image from one or more low-resolution images using conjugate gradient descent. Unique aspects of our approach include (1) supporting a variety of possible applicable transformations; (2) applying principled forward modeling and MAP estimation to this domain. We test our method on data synthesized from the USAF target, data captured from a transmissive USAF target, and data from lens tissue. In the case of the USAF target and 16 low-resolution captures, spatial resolution is enhanced by a factor of 2.8.
Let \(\mathbb T = [0,T], \ T < \infty \), \(f :\mathbb T \times \mathbb R^d \rightarrow \mathbb R^d\), \(y_0 \in \mathbb R^d\) and consider the following ordinary differential equation (ODE):
Classically, the error of a numerical solver is quantified in terms of the worst-case error. However, in applications where a numerical solution is sought as a component of a larger statistical inference problem (see, for example, Matsuda and Miyatake 2019; Kersting et al. 2020), it is desirable that the error can be quantified with the same semantic, that is to say, probabilistically (Hennig et al. 2015; Oates and Sullivan 2019). Hence, the recent endeavour to develop probabilistic ODE solvers.
The rest of the paper is organised as follows. In Sect. 2, the solution of the ODE (1) is formulated as a Bayesian inference problem. In Sect. 3, the associated MAP problem is stated and the iterated extended Kalman smoother for computing it is presented (Bell 1994). In Sect. 4, the connection between MAP estimation and optimisation in a certain reproducing kernel Hilbert space is reviewed. In Sect. 5, the error of the MAP estimate is analysed, for which polynomial convergence rates in the fill distance are obtained. These rates are demonstrated in Sect. 7, and the paper is finally concluded by a discussion in Sect. 8.
Let \(\varOmega \subset \mathbb R\), then for a (weakly) differentiable function \(u :\varOmega \rightarrow \mathbb R^d\), its (weak) derivative is denoted by Du, or sometimes \(\dotu\). The space of m times continuously differentiable functions from \(\varOmega \) to \(\mathbb R^d\) is denoted by \(C^m(\varOmega ,\mathbb R^d)\). The space of absolutely continuous functions is denoted by \(\mathrm AC(\varOmega ,\mathbb R^d)\). The vector-valued Lesbegue spaces are denoted by \(\mathcal L_p(\varOmega ,\mathbb R^d)\) and the related Sobolev spaces of m times weakly differentiable functions are denoted by \(H_p^m(\varOmega ,\mathbb R^d)\), that is, if \(u \in H^m_p(\varOmega ,\mathbb R^d)\) then \(D^m u \in \mathcal L_p(\varOmega ,\mathbb R^d)\). The norm of \(y \in \mathcal L_p(\varOmega ,\mathbb R^d)\) is given by
For a positive definite matrix \(\varSigma \), its symmetric square root is denoted by \(\varSigma ^1/2\), and the associated Mahalanobis norm of a vector a is denoted by \(\Vert a\Vert _\varSigma = a^\mathsf T \varSigma ^-1 a\).
The present approach involves defining a probabilistic state-space model, from which the approximate solution to (1) is inferred. This is essentially the same approach as that of Tronarp et al. (2019b). The class of priors considered is defined in Sect. 2.1, and the data model is introduced in Sect. 2.2.
solves a certain stochastic differential equation. Furthermore, let \(\\mathrm e_m\_m=0^\nu \) be the canonical basis on \(\mathbb R^\nu +1\) and \(\mathrm I_d\) is the identity matrix in \(\mathbb R^d\times d\), it is then convenient to define the matrices \(\mathrm E_m = \mathrm e_m \otimes \mathrm I_d, \ 0 \le m \le \nu \). That is, the mth sub-vector of X is given by
where X takes values in \(\mathbb R^d(\nu +1)\) and the mth sub-vector of X is given by \(X^m = D^m Y\) and takes values in \(\mathbb R^d\) for \(0 \le m \le \nu \). The transition densities for X are given by (Srkk and Solin 2019)
Once the degree of smoothness \(\nu \) has been selected, the parameters \(\varSigma (t_0^-)\), \(\F_m\_m=0^\nu \), and \(\varGamma \) need to be selected. Some common sub-classes of (2) are listed below.
for some \(\lambda ,\sigma ^2 > 0\), and \(\varSigma (t_0^-)\) is set to the stationary covariance matrix of the resulting X process. If \(d > 1\), then each coordinate of the solution can be modelled by an individual Mateŕn process.
Many popular choices of Gaussian processes not mentioned here also have state-space representations or can be approximated by a state-space model (Karvonen and Sarkk 2016; Tronarp et al. 2018; Hartikainen and Srkk 2010; Solin and Srkk 2014). A notable example is Gaussian processes with squared exponential kernel (Hartikainen and Srkk 2010). See Chapter 12 of Srkk and Solin (2019), for a thorough exposition.
For the Bayesian formulation of probabilistic numerical methods, the data model is defined in terms of an information operator (Cockayne et al. 2019). In this paper, the information operator is given by
then \(\mathcal Z[Y](t) = \mathcal S_z[X](t) = z(t,X(t))\). Furthermore, it is necessary to account for the initial condition, \(X^0(0) = y_0\), and with small additional cost the initial condition of the derivative can also be enforced \(X^1(0) = f(0,y_0)\).
The properties of the Nemytsky operator are entirely determined by the vector field f. For instance, if \(f \in C^\alpha (\mathbb T\times \mathbb R^d, \mathbb R^d)\), \(\alpha \ge 0\), then \(\mathcal S_f\) maps \(C^\nu (\mathbb T,\mathbb R^d)\) to \(C^\min (\nu ,\alpha )(\mathbb T,\mathbb R)\), which is fine for present purposes. However, in the subsequent convergence analysis it is more appropriate to view \(\mathcal S_f\) (and \(\mathcal Z\)) as a mapping between different Sobolev spaces, which is possible if \(\alpha \) is sufficiently large (Valent 2013).
where \(\text Proj(A) = A (A^\mathsf T A)^-1 A^\mathsf T\) is the projection matrix onto the column space of A. By (13a) and \(\varSigma _F(t_n^-) \succ 0\), the dimension of the column space of \(\varSigma _F^1/2(t_n^-) C^\mathsf T(t_n)\) is readily seen to be d. That is, the null space of \(\varSigma _F(t_n)\) is of dimension d. By (14a) and (14c), it is also seen that \(\varSigma _F(t_n)\) and \(\varSigma _S(t_n)\) share null space. This rank deficiency is not a problem in principle since the addition of \(Q(h_n)\) in (12b) ensures \(\varSigma _F(t_n^-)\) is of full rank. However, in practice \(Q(h_n)\) may become numerically singular for very small step sizes.
7fc3f7cf58