Notation¶
This page defines the notation. Every guide, docstring and notebook in this repository follows it.
Symbols¶
Random variables are capital Latin letters, for example \(Y\). Their realizations, observed or sampled values, are lowercase Latin letters, for example \(y\). The cumulative distribution function of \(Y\) is \(F_Y\) and the density is \(f_Y\). A hat marks an estimate: \(\hat{F}\) is the fitted distribution, \(F\) the true and often unknown one.
Sets and vectors are bold, for example \(\mathbf{X} = [X_1, \dots, X_J]\).
A single optimized parameter is a lowercase Greek letter, for example \(\vartheta\) or \(\beta\). A vector of parameters is bold lowercase, for example \(\boldsymbol{\vartheta} = (\vartheta_1, \ldots, \vartheta_M)^\top\). A tuple that collects several parameter groups, for example all weight matrices of a network, is bold capital, for example \(\boldsymbol{\Theta}\).
A function that a parameter vector defines carries it as a subscript: \(h_{\boldsymbol{\vartheta}}\) is the monotone transformation with coefficients \(\boldsymbol{\vartheta}\).
The model's symbols¶
| Symbol | Meaning | In code |
|---|---|---|
| \(X_j\), \(Y\) | the variables of the DAG | node names in the spec |
| \(\mathrm{pa}(X_j)\) | the causal parents of \(X_j\) | node_parents |
| \(U_j\) | the standard-logistic latent of node \(j\) | u = flow.abduct(df) |
| \(h_{\boldsymbol{\vartheta}}\) | the monotone transformation of a node | the I term; theta in docstrings |
| \(\vartheta_k\) | the ordinal cutpoints, elements of \(\boldsymbol{\vartheta}\) | ordinal_cutpoints |
| \(\beta\) | a linear-shift coefficient; \(e^\beta\) is an odds ratio | LS; flow.ls_coefficients() |
| \(g_j\) | a complex shift, an NN of one term's parents | CS |
| \(\beta_0\) | the constant treatment effect of a VC term |
beta0 |
| \(b_{\boldsymbol{\Theta}}\) | the penalized effect-modification network | the VC net; units= sets its layers |
| \(\lambda\) | the L2 weight on \(b_{\boldsymbol{\Theta}}\) | penalty= |
| \(\sigma\) | the logistic function | torch.sigmoid |
Every node's model is one additive formula on the latent scale. model.md writes it out. For a continuous node the shifts add. For an ordinal node the shift subtracts inside the sigmoid: \(P(Y \le k \mid \mathrm{pa}) = \sigma(\vartheta_k - \text{shift})\).