Can we pass to the limit in Poincaré-Jaynes-Bretthorst interpolation and deconvolution? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T18:28:39Z http://mathoverflow.net/feeds/question/47675 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/47675/can-we-pass-to-the-limit-in-poincare-jaynes-bretthorst-interpolation-and-deconvol Can we pass to the limit in Poincaré-Jaynes-Bretthorst interpolation and deconvolution? Pascal Orosco 2010-11-29T13:48:56Z 2010-11-30T15:49:31Z <p>In <em>Science and Hypothesis</em>, chapter XI, <em>The calculus of probabilities</em>, Henri Poincaré deals informally with the fundamental problem of interpolation. He concludes (see <a href="http://ia600308.us.archive.org/21/items/scienceandhypoth00poinuoft/scienceandhypoth00poinuoft.pdf" rel="nofollow">http://ia600308.us.archive.org/21/items/scienceandhypoth00poinuoft/scienceandhypoth00poinuoft.pdf</a>, page 206):</p> <p><em>Why, then, do I draw a curve without sinuosities? Because I consider a priori a law represented by a continuous function (or function the derivatives of which to a higher order are small), as more probable than a law not satisfying those conditions. But for this conviction the problem would have no meaning; interpolation would be impossible; no law could be deduced from a finite number of observations; science would cease to exist.</em></p> <p>To my knowledge, Poincaré interpolation was finally addressed in 1992 by Bretthorst using Jaynes' Principle of Maximum Entropy. Please refer to <em>Bayesian interpolation and deconvolution</em>, <a href="http://bayes.wustl.edu/glb/deconvolution.pdf" rel="nofollow">http://bayes.wustl.edu/glb/deconvolution.pdf</a>. The principle is quite simple:</p> <ul> <li>By Bayes' rule, we just need to assign the prior probability distribution for the smooth theoretical signal $u(t)$ to be estimated (eq. 10).</li> <li>Approximate for instance $\displaystyle \frac{\textrm{d}^{2}u}{\textrm{d}t^{2}}$ numerically by finite differences (eq. 16). We seek for our prior distribution among all continuous distributions having same Euclidean norm $\displaystyle\left\|\frac{\textrm{d}^{2}u}{\textrm{d}t^{2}}\right\|$ (eq. 17).</li> <li>Call the Principle of Maximum Entropy: finally, you get a multivariate Gaussian prior distribution (eq. 28) involving the fractional variance regularizer $\epsilon^{2}$.</li> <li>Assign a prior distribution for $\epsilon$ and compute its marginal posterior distribution (eq. 61). Then you get for instance its MAP estimate by maximization (fig. 3);</li> <li>Estimate $u_{j}$ by maximizing the marginal posterior $p(u_{j}|\epsilon,D,I)$ (eq. 52, fig. 4).</li> </ul> <p>Bretthorst considers that $u(t)$ is to be estimated at $\nu+2$ points where $\nu=\beta(N-1)+1$ (eq. 5), $N$ is the number of observations and $\beta$ is a positive integer. We can think for instance about a regular grid with step $\Delta t/\beta$. In order to make the finite differences (eq. 16) and consequently our estimates more accurate, we should certainly increase $\beta$ even if we want to estimate $u(t)$ at, say, only the $N$ measurement points. In practice, we actually observe that the interpolation converges quickly as $\beta$ increases. Therefore the following problems may arise:</p> <p><strong>Problem 1:</strong> Can we pass to the limit $\beta\to+\infty$ in the interpolation problem?</p> <p>$p(u_{j}|\epsilon,D,I)$ (eq. 52) involves the eigenvalues and the eigenvectors of the $j^{th}$ cofactor of the square matrix $g_{ik}=\epsilon^{2}R_{ik} + S_{ik}$ (eq. 43-52) of dimension $\nu+2$. Therefore, there are two possibilities:</p> <ul> <li>Either we can get those eigenvalues and eigenvectors analytically for any $\beta$ and $\epsilon$ or we can approximate them. Then, we would just need to take the limit in the series $h_{l}(u_{j})$ (eq. 43).</li> <li>Or we can get those eigenvalues and eigenvectors only for the infinite matrix $g_{ik}$. Then, we may need to justify that we can interchange the limits.</li> </ul> <p><strong>Problem 2:</strong> Can we pass to the limit $\beta\to+\infty$ in the deconvolution problem?</p> <p>Now the matrix $S_{ik}$ depends on the impulse response function $r(t)$ (eq. 112) so that we can't use the nice structure of the interpolation matrix (eq. 35).</p>