---
title: "Reading Identification off the Diagram"
author: "Hsiu-Ting Yu"
date: "`r Sys.Date()`"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Reading Identification off the Diagram}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r setup, include = FALSE}
knitr::opts_chunk$set(collapse = TRUE, comment = "#>")
library(umg)
```

Because a well-formed UMG determines a likelihood, several necessary
conditions for identification can be read from the picture before any
estimation is attempted. The functions in this vignette operationalise
that idea. None of them replaces a formal identification analysis; each
catches a routine error at the drawing board, where errors are cheapest
to fix.

## Scaling marks for latent variables

Every latent continuous random vertex must have its scale fixed.
`umg_check_scaling()` reports the mechanism detected for each.

```{r scaling}
umg_check_scaling(umg_factor("F", paste0("y", 1:4)))   # marker loading
umg_check_scaling(umg_esem("F1", paste0("y", 1:4)))    # fixed variance
```

A latent vertex reported as `"none"` is the single most common
specification error in latent variable modelling, surfaced as a missing
mark rather than a nonconvergence.

## The counting rule

`umg_count_parameters()` applies the classical t-rule, comparing free
parameters against the distinct pieces of information the observed
variables supply. A non-negative `df` is necessary, not sufficient.

```{r counting}
umg_count_parameters(umg_factor("F", paste0("y", 1:6)))
```

## Label switching

Mixture and latent class models are identified only up to a permutation
of the classes.

```{r labelswitch}
umg_labelswitching(umg_mixture(umg_growth(4), c("I", "S")))
```

## Conditional independence and d-separation

`umg_dsep()` reads a conditional-independence claim off the diagram.
Covariance edges are treated as latent common causes (the projection of
an acyclic directed mixed graph onto a DAG).

```{r dsep}
med <- umg_mediation(direct = FALSE)
umg_dsep(med, "X", "Y")               # FALSE: connected through M
umg_dsep(med, "X", "Y", given = "M")  # TRUE: blocked by the mediator
```

The implied independencies of the model (its testable implications) are
enumerated under the local Markov property:

```{r impliedci}
umg_implied_ci(umg_mediation(direct = FALSE))
```

## A single summary

`umg_identify()` collects these readings into one printable object.

```{r identify}
umg_identify(umg_factor("F", paste0("y", 1:6)))
```