Posterior over ontology nodes
The toy theorem panel shows how NuClass keeps probability mass on both internal and leaf nodes using a stop-at-node formulation.
Research project website
NuClass v2
Adaptive multi-scale integration for robust cell annotation across histopathology and spatial imaging data.
This page combines the theorem intuition and the ontology explorer into one interface. The top section explains the training and Bayes-risk decision rule on a toy tree, and the second section lets you explore the larger NuClass cell hierarchy used for annotation.
Model view
Coarse labels, subtree likelihood, Bayes-risk inference
Interactive view
Search, filter, expand, and inspect the cell ontology
Proof access
Direct PDF preview and download from this page
Use case
Histopathology and spatial imaging cell annotation
Training with coarse supervision
Coarse labels become subtree events, so the likelihood aggregates all descendants that remain consistent with the observed label.
Decision rule on the tree
Prediction is based on Bayes risk under tree distance, which makes ontology-aware decisions different from plain posterior argmax.
NuClass theorem toy
The toy tree uses the ontology A -> {B, C}, C -> {D, E}. Adjust the logits to see how posterior mass changes, how a coarse label becomes a subtree likelihood during training, and how Bayes-risk decoding can choose a different prediction than posterior argmax.
Posterior from logits
p(S=v|x)=exp(zv)/sum_u exp(zu)
sum p = 1.0
argmax p = A
Bayes a-hat = A
Training with subtree likelihood
Fine: L=p(S=v|x). Coarse: L=p(S in subtree(u)|x)=sum_w A[u,w]p(S=w|x).
Loss l=-log L.
Observed label
Type
Likelihood L
0.0
Loss l = -log L
0.0
-
Inference by Bayes risk
ER(a)=sum_v p(S=v|x)R[a,v], a-hat=argmin ER(a)
ER(a-hat)
0.0
ER(argmax)
0.0
Show ER(a) for all nodes A,B,C,D,E
Toy ontology view
Each node shows posterior p(S=node|x) and risk ER(a=node).
subtree(y-tilde)
y-tilde
argmax
Bayes
A
root
p
0.0
ER
0.0
B
leaf
p
0.0
ER
0.0
C
internal
p
0.0
ER
0.0
D
leaf
p
0.0
ER
0.0
E
leaf
p
0.0
ER
0.0
Show matrices from the proof ancestor matrix A and risk matrix R
Ancestor matrix A (A[u,v]=1 if u is ancestor of v)
Risk matrix R (tree edge distance)
NuClass ontology explorer
This larger tree lets you inspect the cell hierarchy directly. You can change layouts, control visible depth, filter lineages, restrict to dataset nodes, and search for specific cell types while keeping the visual language aligned with the theorem panel above.
Cell type tree explorer
Horizontal labels, overlap control, lineage filters, depth control,
and search-to-node highlighting.
stats
Display
3
Lineage filter
Search
Locate expands the path and highlights the first match.
ready
Proof preview
The proof PDF is embedded here for quick reference. If your browser blocks inline PDF rendering, use the preview or download buttons instead.
NuClass tree proof PDF
The same proof document is available for direct browser preview and local download from the buttons on the right.