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+## Extending FunctionalCPD for Flexible Parameter Learning
+
+Contributors: @daehyun99
+
+### Introduction
+
+- In my view, pgmpy currently appears to place more emphasis on structure learning—especially `Causal Discovery`, which is closer to academic interests—than on practical `Parameter Learning`.
+<br>
+
+> Real-world Bayesian networks commonly have high-cardinality nodes. <br>
+> Issue #1776 shows another user hit 64 TiB allocation on an 82-node network. <br>
+> This is silently killing pgmpy adoption in production systems. [[1](https://github.com/pgmpy/pgmpy/issues/3203)]
+
+- The issue cited above suggests that, for certain domains, pgmpy may see lower adoption in practice.
+- This is because the current `DiscreteBayesianNetwork`(`DiscreteBN`) supports only `TabularCPD`, while `FunctionalBayesianNetwork`(`FunctionalBN`) supports only `FunctionalCPD`.
+- Given the practical need to handle diverse forms of data, adopting pgmpy in real-world settings can introduce significant constraints.
+<br>
+
+- In addition, pgmpy is considering support for multiple types of CPDs within a single Bayesian network. [[2](https://github.com/pgmpy/pgmpy/issues/2343)], [[3](https://github.com/pgmpy/pgmpy/issues/2344)]
+<br>
+
+- Therefore, this proposal suggests adding support so that `FunctionalBN` can handle multiple types of CPDs, such as `TabularCPD`, `LinearGaussianCPD`, and `skpro` models.
+
+![1](/2_Extending%20FunctionalCPD%20for%20Flexible%20Parameter%20Learning/99_Images/1.jpg)
+
+- The current inheritance structure of pgmpy is shown in the image above.
+- A previous consideration was to refactor `FunctionalBN` and `FunctionalCPD` to broaden their scope of support.
+- However, `_CoreGraph` has already been merged, and a refactoring of the `DAG` class is planned. [[11](https://github.com/pgmpy/pgmpy/issues/2376)]
+- Therefore, I believe that the refactoring of the `DAG` class (to inherit from `_CoreGraph`) should come first. [[4](https://github.com/pgmpy/pgmpy/issues/2385)]
+
+### Proposed Solution
+
+#### Create separate `Refactor_DAG` and `FunctionalBN` classes first, and then refactor them part to part later.
+
+![3](/2_Extending%20FunctionalCPD%20for%20Flexible%20Parameter%20Learning/99_Images/3.jpg)
+
+1. Create `Refactor_DAG` class.
+    - It will be inherting from `_CoreGraph` [[10](https://github.com/pgmpy/pgmpy/blob/5239dfe1f6ab4327e165209b3b9f36c9fa0b6b15/pgmpy/base/_base.py#L10)]
+    - It will be implementing minimal range of method.
+2. Create `FunctionalBN` class.
+3. Create `FunctionalCPD` class.
+4. Create `*Adapter` class. (`_BaseCPDAdapter`, `TabularCPDAdapter`, `LinearGaussianCPDAdapter`, `SkproAdapter`)
+5. Create `FunctionalEstimator` class.
+6. Create `FunctionalInference` class.
+7. (later) Refactoring `DAG` class part to part. (we can consider introducing `_GraphConverterMixin` in same time. [[7](https://github.com/pgmpy/pgmpy/issues/2933)])
+
+### Alternative Solutions
+
+#### A solution that directly refactors the `DAG`
+![2](/2_Extending%20FunctionalCPD%20for%20Flexible%20Parameter%20Learning/99_Images/2.jpg)
+
+- The `DAG` class carries a significant amount of responsibility, and many models inherit from it. [[12](https://github.com/pgmpy/pgmpy/blob/5239dfe1f6ab4327e165209b3b9f36c9fa0b6b15/pgmpy/base/DAG.py#L17)]
+
+> While refactoring the PDAG, we found that the GES algorithm also needs to be refactored. [[9](https://github.com/daehyun99/pgmpy/pull/70)]
+
+- Therefore, I believe that directly refactoring the `DAG` class, as shown above, would make it difficult to predict what kinds of issues might arise.
+- Also, It could block other issues that are currently in progress. [[5](https://github.com/pgmpy/pgmpy/issues/2835)], [[6](https://github.com/pgmpy/pgmpy/issues/3296)]
+- There is a possibility that existing classes such as `BIFReader`, `BIFWriter` may not function properly.
+- We need to reduce the scope of the project without blocking or interfering with other contributors’ work.
+
+<br>
+
+#### How to fit, run inference, and sample in the model [[8](https://github.com/pgmpy/pgmpy/pull/3260#issuecomment-4161481486)]
+```python
+
+model.fit()
+
+model.sample()
+
+model.query()
+
+```
+
+### Details of proposed solution
+✨: Optional(Nice to have)
+
+#### `_CoreGraph`
+| Method | Input | Return |
+| - | - | - |
+| `get_node()` | `node`,<br>`data`,<br>`include_models` | `node`'s info |
+| `get_nodes()` | `data`,<br>`include_models` | All `node`'s info |
+
+```python
+# DAG
+dag.get_node("B")
+{
+    node: "B",
+    parents: {"A", "C"},
+    children: {"F"},
+    roles: {"latents"},
+}
+
+FBN.get_node("B", data=True, include_models=True)
+{
+    node: "B",
+    parents: {"A", "C"},
+    children: {"F"},
+    roles: {"latents"},
+    distribution: {
+        # CPD instance or skpro's model
+    },
+    estimator: {
+        # estimator's config
+    },
+    d_info: {
+        # data info
+    }
+}
+
+```
+
+#### `Refactor_DAG`
+| Method | Input | Return |
+| - | - | - |
+| `__init__()` | `ebunch`,<br> `latents`,<br> `exposures`,<br> `outcomes`,<br> `roles` | - |
+| `add_edge()` | `u`,<br> `v`,<br> `edge_type` | - |
+| `add_edges_from()` | `ebunchs` | - |
+| `remove_edge()` | `u`,<br> `v`,<br> `edge_type` | - |
+| `remove_edges_from()` | `ebunchs` | - |
+| `is_valid_dag()` | - | `bool` |
+
+- The `is_valid_dag` verifies that it meets the conditions of the `DAG`, such as cycle confirmation. [[14]](https://github.com/pgmpy/pgmpy/pull/2579)
+
+#### `FunctionalBayesianNetwork`
+| Method | Input | Return |
+| - | - | - |
+| `__init__()` | `ebunch`,<br> `latents`,<br> `exposures`,<br> `outcomes`,<br> `roles` | - |
+| `add_cpd()` | `node`,<br> `distribution`,<br> `estimator` | - |
+| `add_cpds(*cpds)` | `FunctionalCPD` instance | - |
+| `remove_cpd()` | `node`| - |
+| `remove_cpds(*cpds)` | `FunctionalCPD` instance | - |
+| `get_cpd()` | `node` | `FunctionalCPD` |
+| `get_cpds()` | - | `list[FunctionalCPD]` |
+| `get_node()` | `node`,<br>`data`,<br>`include_models` | `node`'s info |
+| `get_nodes()` | `data`,<br>`include_models` | All `node`'s info |
+| `check_model()` | - | `bool` |
+| `get_cardinality()` | `node` | `cardinality: int` |
+| ✨`get_random()` | `n_nodes`,<br> `edge_prob`,<br> `n_states`,<br> `latents`,<br> `seed` | `FunctionalBayesianNetwork` |
+| ✨`get_random_cpds()` | `n_states`,<br> `latents`,<br> `seed` | `FunctionalCPD` |
+| ✨`save()` | `file_path`, `filetype` | - |
+| ✨`load()` | `file_path`, `filetype` | - |
+
+- `get_random` `get_random_cpds` [[6]((https://github.com/pgmpy/pgmpy/issues/3296))]
+- I am not yet aware of a format that supports multiple types of CPDs.
+- I am thinking of supporting only the Python pickle format for `save` and `load`. (Security issues related to the Python pickle format [[13](https://docs.python.org/3/library/pickle.html)])
+
+#### `FunctionalCPD`
+| Method | Input | Return |
+| - | - | - |
+| `__init__()` | `node(=variable)`,<br>`distribution`,<br>`estimator` | `FunctionalCPD` instance |
+| `fit()` | `data: pandas.DataFrame` | `FunctionalCPD` |
+| `predict_proba()` | `data: pandas.DataFrame` | `Distribution` instance |
+| `sample()` | `n_samples`,<br>`random_state` | `samples: pandas.DataFrame` |
+| ✨`plot()` | - | `matplotlib images` |
+| `state_dict()` | - | `dict[str, Any]` |
+
+- `plot() `:Users can visualize the form of the CPD and understand the data at a glance. [[15]](https://github.com/sktime/skpro/blob/main/examples/03_skpro_distributions.ipynb)
+- We can also visualize CPT data using matplotlib.
+
+![4](/2_Extending%20FunctionalCPD%20for%20Flexible%20Parameter%20Learning/99_Images/4.png)
+
+
+
+#### `_BaseCPDAdapter`, `TabularCPDAdapter`, `LinearGaussianCPDAdapter`, `SkproAdapter`
+| Method | Input | Return |
+| - | - | - |
+| `fit()` | `data: pandas.DataFrame` | `self` |
+| `predict_proba()` | `data: pandas.DataFrame` | `Distribution` instance |
+| `sample()` | `n_samples`,<br>`random_state` | `samples`: `pandas.DataFrame` or `skpro.Distribution` |
+| ✨`plot()` | - | `matplotlib images` |
+| `__repr__()` | - | - |
+
+#### `DiscreteBayesianNetwork`
+| Method | Input | Return |
+| - | - | - |
+| ✨`to_FunctionalBN()` | - | `FunctionalBayesianNetwork` |
+
+#### `FunctionalEstimator`
+| Method | Input | Return |
+| - | - | - |
+| `__init__()` | - | - |
+| `fit()` | `model`,<br>`data` | - |
+
+- Orchestration Parametor Learning with `FunctionalCPD`
+
+#### `FunctionalInference`
+| Method | Input | Return |
+| - | - | - |
+| `__init__()` | - | - |
+| `get_distribution()` | - | - |
+| `query()` | `variables` | - |
+
+- Orchestration Inference with `FunctionalCPD`
+
+### User journeys with the solution
+
+#### UseCase 1: Load the existing `DiscreteBN` and use it as a `FunctionalBN`.
+```python
+from pgmpy.example_models import load_model
+from pgmpy.parameter_estimator import MaximumLikelihoodEstimator, FunctionalEstimator
+from skpro.regression.bayesian import BayesianLinearRegressor
+
+alarm_model = load_model("bnlearn/alarm") # DiscreteBN
+alarm_samples = alarm_model.simulate(int(1e3))
+
+# Parameter Learning in Discrete Data
+DiscreteBN = DiscreteBayesianNetwork(ebunch=alarm_model.edges())
+
+est1 = MaximumLikelihoodEstimator()
+est1.fit(DiscreteBN, alarm_samples)
+
+alarm_model.nodes() # or alarm_model.get_nodes()
+# NodeView(('HISTORY', 'CVP', 'PCWP', 'HYPOVOLEMIA', 'LVEDVOLUME', 'LVFAILURE',  ...))
+
+# User Customizing setting
+FunctionalBN = DiscreteBN.to_FunctionBN()
+for node in alarm_model.nodes():
+    FunctionalBN.add_cpd(node=node, distribution="tabular", estimator=MaximumLikelihoodEstimator())
+FunctionalBN.add_cpd(node="HYPOVOLEMIA", distribution="Normal", estimator=BayesianLinearRegressor())
+
+# Parameter Learning
+est2 = FunctionalEstimator()
+FunctionalEstimator.fit(FunctionalBN, alarm_samples)
+
+# Inference
+infer = FunctionalInference(FunctionalBN)
+infer.query()
+
+```
+
+#### UseCase 2: SL -> PL
+
+![8](/2_Extending%20FunctionalCPD%20for%20Flexible%20Parameter%20Learning/99_Images/8.excalidraw.png)
+
+```python
+from pgmpy.example_models import load_model
+
+from pgmpy.causal_discovery import PC
+from pgmpy.models.functional import FunctionalBayesianNetwork as FunctionalBN
+from pgmpy.parameter_estimator import MaximumLikelihoodEstimator
+from pgmpy.factors.functional import FunctionalCPD
+
+from skpro.regression.bayesian import BayesianLinearRegressor
+
+alarm_model = load_model("bnlearn/alarm") # DiscreteBN
+alarm_samples = alarm_model.simulate(int(1e3))
+
+# Causal Discovery, Structural
+est1 = PC(ci_test='chi_square', variant="stable", max_cond_vars=4, return_type='dag')
+est1.fit(alarm_samples)
+
+# Analysis and Improvement of Structural Learning Results
+est.causal_graph_.to_graphviz()
+
+est.causal_graph_.get_nodes()
+
+
+# Parameter Learning
+FunctionalBN = FunctionalBN(est.causal_graph_)
+
+CVP = FunctionalCPD(node="CVP", distribution="tabular", estimator=MaximumLikelihoodEstimator())
+HYPOVOLEMIA = FunctionalCPD(node="HYPOVOLEMIA", distribution="normal", estimator=BayesianLinearRegressor())
+LVFAILURE = FunctionalCPD(node="LVFAILURE", distribution="linear", estimator=MaximumLikelihoodEstimator())
+
+FunctionalBN.add_cpds(CVP, HYPOVOLEMIA, LVFAILURE)
+
+# Parameter Learning
+est2 = FunctionalEstimator()
+est2.fit(FunctionalBN, alarm_samples)
+
+# Inference
+infer = FunctionalInference(FunctionalBN)
+infer.query()
+
+```
+
+### Reference
+- [[1]Issue #3203](https://github.com/pgmpy/pgmpy/issues/3203)
+- [[2]Issue #2343](https://github.com/pgmpy/pgmpy/issues/2343)
+- [[3]Issue #2344](https://github.com/pgmpy/pgmpy/issues/2344)
+- [[4]Issue #2385](https://github.com/pgmpy/pgmpy/issues/2385)
+- [[5]Issue #2835](https://github.com/pgmpy/pgmpy/issues/2835)
+- [[6]Issue #3296](https://github.com/pgmpy/pgmpy/issues/3296)
+- [[7]Issue #2933](https://github.com/pgmpy/pgmpy/issues/2933)
+- [[8]Pull Request #3260](https://github.com/pgmpy/pgmpy/pull/3260#issuecomment-4161481486)
+- [[9]Forked Repo - Pull Request #70](https://github.com/daehyun99/pgmpy/pull/70)
+- [[10]Issue #](https://github.com/pgmpy/pgmpy/blob/5239dfe1f6ab4327e165209b3b9f36c9fa0b6b15/pgmpy/base/_base.py#L10)
+- [[11]Issue #2376](https://github.com/pgmpy/pgmpy/issues/2376)
+- [[12]](https://github.com/pgmpy/pgmpy/blob/5239dfe1f6ab4327e165209b3b9f36c9fa0b6b15/pgmpy/base/DAG.py#L17)
+- [[13]Python 3.14.4 Documentation](https://docs.python.org/3/library/pickle.html)
+- [[14]Issue #2579](https://github.com/pgmpy/pgmpy/pull/2579)
+- [[15]skpro's examples file](https://github.com/sktime/skpro/blob/main/examples/03_skpro_distributions.ipynb)
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