PMC

2D illustration of Laplace smoothing regularization and soft-priors encoding of a tumor (magenta) and prostate (cyan).

Probabilities generated by Laplace and SGT smoothing for1, 4, and 6 years of observations.

(a) Raw image; (b) Bilateral smoothing results; (c) Anisotropic smoothing results; (d) Gray transform of b; (e) Gray transform of c; (f) ResNet_SegNet result of a; (g) VGG_Unet result of a; (h) Intersection of f with c; (i) kernel of h. (j) kernel of h. (k) Otsu result of h; (l) Canny result of h; (m) ED result of h; (n) ISEF result of h; (o) Binary Laplace of h; (p) Zero-Crosing of o; (q) Shen-Castan of h; (r) Expanded Canny from L1; (s) Expanded ED from L1; (t) Expanded Shen-Castan from L1.

GUI for surface modeling. Graphical user interface for surface modeling: Disparity data (left) and segmentation data (right) are combined in module 3 to approximate surfaces to 3-d point clouds according to different modeling options. Surfaces are either fitted according to planar, quadratic or cubic functions or smoothed using curvature flow or Laplace smoothing.

Analysis of paGLM-2 MAP estimators on RGC data binned at 1.66 ms with adaptive interval selection and evidence optimization in a Poisson GLM with exponential nonlinearity. a. The approximate log marginal likelihood with a ridge prior computed via the Laplace approximation or via the quadratic approximation () (left) and the exact and paGLM-2 MAP estimates with the optimal ridge prior value (right). b. The exact and paGLM-2 MAP estimates with a smoothing prior.

Example fits for models using a) normal, b) Laplace, and c) horseshoe priors where observations are drawn from normal distributions with SD = 4.5. Plots show true functions (dashed gray lines), posterior medians (solid dark gray lines), and associated 95% Bayesian credible intervals (BCI; gray bands) for each θ. Values between observed locations are interpolated for plotting.

Association heatmap of major Mtb lineages and spoligotype families with patient’s region of origin. Shown are the no. of samples in each category, with row- and column-wise sample sizes indicated above and on the right of the heatmap. Colors indicate strength and direction (from blue, strongly negative, to red, strongly positive) of the association between lineage/spoligotype family and region of origin, expressed as fold-change of the observed count in each category relative to the expected count under the hypothesis of independence. Laplace smoothing was applied to proportions to avoid zero fold-changes for zero counts. Spoligotype families belonging to the Euro-American lineage are prefixed with EAL. Other lineages are designated by lineage name.

Simulated examples for nonparametric regression: (a) m(x) = 3 sin(2.5x) + 2 exp(−5x²), n = 50, σ = 0.5; (b) m(x) = x^2/5 − cos(πx), n = 100, σ = 0.8. The estimates (solid lines) with credible bands (dashed lines) using INLA are compared to the estimates (dash-dotted lines) using cubic smoothing spline regression. The true functions are denoted by the dotted lines.

A table representing the model space of experiments is instantiated with a pseudocount of one (a form of Laplace smoothing). The symbols along the left indicate the classes of experiments involving an Agent, A: Positive Intervention (A ↑), Positive Non-intervention (A⌀^↑), Negative Non-intervention (A⌀^↓), and Negative Intervention (A ↓). The symbols along the top indicate the results recorded in a Target, B: increase (B+), no change (B0), and decrease (B−). This particular instantiation of the scoring table encodes four (5 − 1) Positive Interventions that caused the Target to increase, one (2 − 1) Positive Non-intervention that caused the Target to decrease, and one (2 − 1) Negative Non-intervention that caused the Target to decrease. There are thus five experiments suggesting an Excitatory relation (green regions), and one experiment suggesting an Inhibitory relation (red region).

Scatter plot of population for the linear regression simulation

Top: Analysis of emotional gait data from Roether et al. () with PCA, ICA and Anechoic demixing for different numbers of sources. The bars represent model evidences computed with Laplace approximation, relative to the lowest observed model evidence (PCA, 1 source). The anechoic analyses were carried out either without smoothing (black, f₀ → ∞) or with the optimal f₀ = 7 Hz (blue) for the wave kernel (see Equation 26). Error bars are standard errors, computed across trials. The best model (highest evidence) is the anechoic mixture with three sources and f₀ = 7 Hz, followed by the SIM model with f₀ = 7 Hz. PCA and ICA are significantly worse for any number of sources. Bottom: Detailed cutoff frequency analysis of the AMM model (left) and the SIM model (right), at their respective best number of sources I. The LAP score (relative to the SIM model with I = 1 set to 50) for both models peaks at f₀ = 7Hz. However, the best AMM model's approximate posterior probability is larger than the best SIM posterior by a factor of ≈ 10¹⁹. For details, see text.

A. Peripheral basophil activation testing to Ara h 2 to identify degranulation as percentage of CD63^hi basophils by flow cytometry. B. Basophil sensitivity (ED50) and AUC (AUC, filled green) in representative SU (left, blue) and TD (right, red) subjects. C, D. Longitudinal basophil sensitivity and AUC using a generalized additive mixed model (with smoothing by Laplace approximate) to Ara h 2 (C) and peanut (D) in SU (blue) or TD (dashed, red). The y-axis is logarithmically scaled for basophil AUC, the peanut avoidance period is shaded in grey, and the black arrow marks 3 months of OIT. Labeled visits on the x-axis include pre-OIT (Bl), build-up visits (B), maintenance (M), post-OIT oral food challenge (O), post-avoidance double blinded food challenge (D), and the follow-up visits (F). Lines indicate means and shaded regions indicate standard deviation. X-axis lettered labels refer to visits, PreOIT (P), Build-up (B), Maintainance (M), Post-OIT OFC (O), Post-avoidance DBFC (D), Follow-up (F). E. The proportion of the variance in Ara h 2 ED50 change from baseline to post-avoidance explained by quantitative measurements of serum Ara h 2 specific immunoglobulins.