Module NonparBayes¶

Introduction¶

The procedure is as follows. Consider the diffusion process \((x_t\colon 0 \le t \le T)\) given by

\(dx_t = b(x_t) dt + dw_t\)

where the drift b is expressed as linear combination

\(f(x) = \sum_{i =1}^n c_i \phi_i(x)\)

(see Module Schauder) and prior distribution on the coefficients

\(c_i \sim N(0,\xi_i)\)

Then the posterior distribution of \(b\) given observations \(x_t\) is given by

\(c_i | x_s \sim N(W^{-1}\mu, W^{-1})\) \(W = \Sigma + (\operatorname{diag}(\xi))^{-1}\),

with the nxn-matrix

\(\Sigma_{ij} = \int_0^T \phi_i(x_t)\phi_j(x_t) dt\)

and the n-vector

\(\mu_i = \int_0^T \phi_i(x_t) d x_t\).

Using the recursion detailed in Module Schauder, one rather computes

\(\Sigma^\prime_{ij} = \int_0^T \psi_i(x_t)\psi_j(x_t) dt\)

and the n-vector

\(\mu^\prime_i = \int_0^T \psi_i(x_t) d x_t\)

and uses pickup_mu!(mu) and pickup_Sigma!(Sigma) to obtain \(\mu\) and \(\Sigma\).

Optional additional basis functions¶

One can extend the basis by additional functions, implemented are variants. B1 includes a constant, B2 two linear functions

B1: \(\phi_1 \dots \phi_n, c\)
B2: \(\phi_1 \dots \phi_n, \max(1-x, 0), \max(x, 0)\)

To compute mu, use

mu = pickup_mu!(fe_mu(y,L, 0))
mu = fe_muB1(mu, y);

or

mu = pickup_mu!(fe_mu(y,L, 0))
mu = fe_muB2(mu, y);

Reference¶

Functions taking y` without parameter [a,b] expect ``y to be shifted into the intervall [0,1].

NonparBayes.pickup_mu!(mu)¶: computes mu from mu’

NonparBayes.drop_mu!(mu)¶: Computes mu’ from mu.

NonparBayes.pickup_Sigma!(Sigma)¶: Transforms Sigma’ into Sigma.

NonparBayes.drop_Sigma!(Sigma)¶: Transforms Sigma into Sigma’.

NonparBayes.fe_mu(y, L, K)¶: Computes mu’ from the observations y using 2^L-1 basis elements and returns a vector with K trailing zeros (in case one ones to customize the basis.

NonparBayes.fe_muB1(mu, y)¶: Append \(\mu_{n+1} = \int_0^T \phi_{n+1} d x_t\) with \(\phi_{n+1} = 1\).

NonparBayes.fe_muB2(mu, y)¶: Append \(\mu_{n+1} = \int_0^T \phi_{n+1} d x_t\) with \(\phi_{n+1} = \max(1-x, 0)\) and \(\mu_{n+2} = \int_0^T \phi_{n+2} d x_t\) with \(\phi_{n+2} = \max(x, 0)\)

NonparBayes.fe_Sigma(y, dt, L)¶: Computes the matrix Sigma’ from the observations y uniformly spaced at distance dt using 2^L-1 basis elements.

NonparBayes.bayes_drift(x, dt, a, b, L, xirem, beta, B)¶

Performs estimation of drift on observations x in [a,b] spaced at distance dt using the Schauder basis of level L and level wise coefficients decaying at rate beta. A Brownian motion like prior is obtained for beta= 0.5. The K remaining optional basiselements have variance xirem.

The result is returned as [designp coeff se] where coeff are coefficients of finite elements with maximum at the designpoints designp and standard error se.

Observations outside [a,b] may influence the result through phi_{n+1}, ..., phi_{n+K}

NonparBayes.visualize_posterior(post[, truedrift])¶: Plot 2r*se wide marginal credibility bands, where post is the result of bayes_drift and truedrift the true drift (if known :-) ).