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Then, I will show how we can represent this tabular dataset in an Undirected Graphical Model. I will use the [`adult`](https://archive.ics.uci.edu/dataset/2/adult) dataset from uci repo as an example. + + + +```python +X = pd.read_csv("adult.csv").fillna(value="NA") +n, p = X.shape +``` + + +```python +print_markdown(X.head().to_markdown()) +``` + + +| | age | workclass | fnlwgt | education-num | marital-status | occupation | relationship | race | sex | capital-gain | capital-loss | hours-per-week | native-country | income>50K | +|---:|------:|------------:|---------:|----------------:|-----------------:|-------------:|---------------:|-------:|------:|---------------:|---------------:|-----------------:|-----------------:|-------------:| +| 0 | 23 | 5 | 4 | 12 | 2 | 8 | 3 | 0 | 1 | 2 | 0 | 39 | 0 | 0 | +| 1 | 34 | 1 | 4 | 12 | 0 | 4 | 2 | 0 | 1 | 0 | 0 | 12 | 0 | 0 | +| 2 | 22 | 0 | 13 | 8 | 1 | 6 | 3 | 0 | 1 | 0 | 0 | 39 | 0 | 0 | +| 3 | 37 | 0 | 15 | 6 | 0 | 6 | 2 | 4 | 1 | 0 | 0 | 39 | 0 | 0 | +| 4 | 12 | 0 | 22 | 12 | 0 | 5 | 0 | 4 | 0 | 0 | 0 | 39 | 12 | 0 | + + +### Histograms + +One compact representation of tabular data is by a [histogram](https://en.wikipedia.org/wiki/Histogram). The univariate histogram is the empirical distribution of a variable: + +$$\tilde P(\text{Occupation} = j) = \frac{1}{N}\sum_N I\{x_{i, \text{occupation}} = j\}$$ + + + +```python +render_plotly_html(px.histogram(X, x="occupation")) +``` + + + + + + + +
+
+ + + + +Histograms can also express marginal distributions between multiple variables: + +$$\tilde P(\text{Sex} = k, \text{Occupation} = j) = \frac{1}{N}\sum_N I\{x_{i, \text{sex}} = k, x_{i, \text{occupation}} = j\}$$ + +This will be a 2-dimensional structure, however, we can always "unroll" the counts into one long vector: + + + +```python +render_plotly_html( + px.histogram( + x=X["sex"].map(str) + " / " + X["occupation"].map(str), + labels={"x": "Sex / Occupation"}, + ) +) +``` + + + + + + + +
+
+ + + + +In this way, we can compute joint histograms of as many variables as we wish (i.e. 3, 4, ...). We can also compute marginal distributions conditioned upon another variable (also known as conditional distributions): + +$$\tilde P(\text{Occupation}=j \mid \text{Sex}=k)=\frac{\tilde P(\text{Occupation}=j,\text{Sex}=k))}{\tilde P(\text{Sex}=k)}$$ + + + +```python +render_plotly_html(px.histogram(X, x="occupation", facet_row="sex")) +``` + + + + + + + +
+
+ + + + +### Undirected Graphical Models + +For discrete data like the `adult` dataset, histograms directly correspond to [marginal distributions](https://en.wikipedia.org/wiki/Marginal_distribution) or marginals. The name marginal refers to the fact that these counts were often computed "in the margins" of a table. Marginal distributions are naturally modeled by [Undirected Graphical Models (UGM)](https://ermongroup.github.io/cs228-notes/representation/undirected/). You can think of a UGM as a graph data structure specifically designed for storing probability distributions. UGMs factor a joint distribution into several _cliques_, each clique representing an unnormalized probability distribution called a _factor_: + +$$ +\begin{gather*} +P(x_1, ..., x_n) = \frac{1}{Z}\prod_{c \in C} \phi_c(x_c), \\ +Z = \sum_{x_1, ..., x_n} \prod_{c \in C} \phi_c(x_c) +\end{gather*} +$$ + +$Z$ is a normalization constant to ensure that the joint distribution sums to 1. Internally, a factor is just a function that maps configurations of that factor's scope to nonnegative values: + +$$ +\begin{equation*} +\phi(x_1, x_2) = \begin{cases} + 10 & x_1 = x_2\\ + 5 & x_1 \neq x_2\\ + 1 & \text{otherwise} + \end{cases} +\end{equation*} +$$ + + +To model a tabular dataset with a UGM, we can add a factor for each marginal we care to measure from our dataset. For example, we might be interested in modeling the joint distribution: + +$$ +\begin{gather*} +P(\text{age}, \text{race}, \text{marital-status}, \text{education}, \text{sex}) = \frac{\phi(\text{age}, \text{race}, \text{marital-status}) \cdot \phi(\text{marital-status}, \text{education}, \text{sex})}{Z} +\end{gather*} +$$ + + +We can visualize this UGM by adding a node for each clique. Edges are drawn between cliques with shared variables (i.e. `marital-status`) which is known as a *separation set* or _sepset_: + +```mermaid +flowchart TD + A(age, race, marital-status) ---|marital-status| B(marital-status, education, sex) +``` + + +### Factored Representations + +Why is it important to factor the distribution like this? Why not just express all the variables (age, race, marital-status, education, sex), in one large clique instead of two parts: + +```mermaid +flowchart TD +A(age, race, marital-status, education, sex) +``` + +An important consideration in modeling a tabular dataset with a UGM is the size of each clique's _state space_, the space of all possible states. First, compute the # of distinct values +of each column: + + +```python +# Print out the number of unique values associated with this dataset. +print_markdown( + X.nunique().rename("Unique Values").sort_values(ascending=False).to_markdown() +) +``` + + +| | Unique Values | +|:---------------|----------------:| +| hours-per-week | 96 | +| fnlwgt | 77 | +| age | 74 | +| capital-loss | 48 | +| native-country | 42 | +| capital-gain | 23 | +| education-num | 16 | +| occupation | 15 | +| workclass | 9 | +| marital-status | 7 | +| relationship | 6 | +| race | 5 | +| sex | 2 | +| income>50K | 2 | + + +Taking our example from above, we would need to allocate an array of size **82,880** for $\phi(\text{age}, \text{race}, \text{marital-status}, \text{education}, \text{sex})$: + + + +```python +74 * 5 * 7 * 16 * 2 +``` + + + + + 82880 + + + +Certainly, if we tried to include every variable it becomes exponentially large: + + + +```python +np.prod(X.nunique()) +``` + + + + + 46019837303193600 + + + +It turns out, the way we _factored_ the joint distribution $P(\text{age}, \text{race}, \text{marital-status}, \text{education}, \text{sex})$ into two factors (i.e. $\phi(\text{age}, \text{race}, \text{marital-status})$ and $\phi(\text{marital-status}, \text{education}, \text{sex})$) is very important for [making operations on a UGM tractable](https://ermongroup.github.io/cs228-notes/inference/jt/). + +> When using UGMs to model tabular datasets it is important to limit the product of the cardinality of the individual factors. If you wish to model several columns, consider "factoring" the joint distribution into smaller factors, only including variables that are highly dependent on each other in a specific factor. +> {: .prompt-warning } + + +[`pgmpy`'s `FactorDict`](https://github.com/pgmpy/pgmpy/blob/dev/pgmpy/factors/FactorDict.py) has a utility function to convert a tabular dataset into a dictionary of factors. We can use this to calculate the size of hypothetical graphical models for different factorizations. First, lets start with no factorization at all (i.e. one large clique with all the variables): + + + +```python +from pgmpy.factors import FactorDict + +# Model all of the variables in one large clique. +empirical_factor = FactorDict.from_dataframe( + df=X, + marginals=[ + ("age", "race", "marital-status", "education-num", "sex"), + ], +) +pprint(empirical_factor) +``` + + {('age', 'race', 'marital-status', 'education-num', 'sex'): } + + + +```python +# Compute total size (measured in # of states) of the empirical marginals +np.prod( + empirical_factor[ + ("age", "race", "marital-status", "education-num", "sex") + ].cardinality +) +``` + + + + + 82880 + + + +Which matches the calculation we made above. We can also investigate a particular subset of the variables in a [`DiscreteFactor`](https://github.com/pgmpy/pgmpy/blob/dev/pgmpy/factors/discrete/DiscreteFactor.py) (through the `marginalize` operation). For example, if we wanted to know the empirical distribution of `sex` and `race` then: + + + +```python +sex_race_empirical = empirical_factor[ + ("age", "race", "marital-status", "education-num", "sex") +].marginalize( + [ + "age", + "marital-status", + "education-num", + ], + inplace=False, +) +print(sex_race_empirical) +``` + + +---------+--------+-----------------+ + | race | sex | phi(race,sex) | + +=========+========+=================+ + | race(0) | sex(0) | 13027.0000 | + +---------+--------+-----------------+ + | race(0) | sex(1) | 28735.0000 | + +---------+--------+-----------------+ + | race(1) | sex(0) | 517.0000 | + +---------+--------+-----------------+ + | race(1) | sex(1) | 1002.0000 | + +---------+--------+-----------------+ + | race(2) | sex(0) | 185.0000 | + +---------+--------+-----------------+ + | race(2) | sex(1) | 285.0000 | + +---------+--------+-----------------+ + | race(3) | sex(0) | 155.0000 | + +---------+--------+-----------------+ + | race(3) | sex(1) | 251.0000 | + +---------+--------+-----------------+ + | race(4) | sex(0) | 2308.0000 | + +---------+--------+-----------------+ + | race(4) | sex(1) | 2377.0000 | + +---------+--------+-----------------+ + + +These are "empirical marginals" in the sense that they are equal to the histogram we would have computed for $\tilde P(\text{race}, \text{sex})$. Indeed, sampling from these histograms would be the same as sampling with replacement from the original dataset which is itself equivalent with the [_Non-Parametric Bootstrap_](). + + +But lets assume that 82880 is too large to fit into memory on our machine. Now, lets compute a factored representation of the same variables: + + + +```python +factors = FactorDict.from_dataframe( + df=X, + marginals=[ + ("age", "race", "marital-status"), + ("marital-status", "education-num", "sex"), + ], +) +pprint(factors) +``` + + {('age', 'race', 'marital-status'): , + ('marital-status', 'education-num', 'sex'): } + + +Which models the interaction of `(age, race)` & `(education-num, sex)` through the common variable `marital-status`. We can compute the sum of the clique's sizes: + + + +```python +# Compute total size (measured in # of states) of the clique tree. +sum(np.prod(v.cardinality) for k, v in factors.items()) +``` + + + + + 2814 + + + +We can see a savings of `1 - 2814 / 82880 ~ 96%` when making this conditional independence assumption! + + + +```python +1 - 2814 / 82880 +``` + + + + + 0.9660472972972973 + + + +References + +- [Stanford CS228 Probabilistic Graphical Models](https://ermongroup.github.io/cs228-notes/representation/undirected/) +- [Computer Age Statistical Inference, Chapter 10](https://hastie.su.domains/CASI/) + + +## Maximum Likelihood Estimates + +We have talked about how to represent a probability distribution with a UGM and the tradeoffs between graph structure and memory consumption. Now, we will discuss fitting a factored UGM to a dataset and how "close" we are to the unfactored representation. + + + +```python +from pgmpy.inference.ExactInference import BeliefPropagation +from pgmpy.estimators import MaximumLikelihoodEstimator +from pgmpy.models import JunctionTree +``` + +First, lets create a [`JunctionTree`](https://github.com/pgmpy/pgmpy/blob/dev/pgmpy/models/JunctionTree.py) with the same nodes from our `factors` from before (Section 20.4 of [Machine Learning - A Probabilistic Perspective](https://probml.github.io/pml-book/)). Instead of using the empirical counts as the potentials, lets compute the MLE of the factors. + + + +```python +junction_tree = JunctionTree() +# Same nodes from factor. +junction_tree.add_nodes_from(factors.keys()) +# Add factors, except set all the factors to the identity factor. +junction_tree.add_factors(*[i.identity_factor() for i in factors.values()]) +# Add edges between the two nodes. +junction_tree.add_edge(*list(factors.keys())) +# Use `check_model` to ensure required properties of junction tree. +assert junction_tree.check_model() +print_markdown(nx.to_pandas_edgelist(junction_tree).to_markdown()) +``` + + +| | source | target | weight | +|---:|:----------------------------------|:-------------------------------------------|:---------| +| 0 | ('age', 'race', 'marital-status') | ('marital-status', 'education-num', 'sex') | | + + +To fit the parameters of a `JunctionTree` with Maximum Likelihood, we can use an approach called [Iterative Proportional Fitting](https://www.cs.cmu.edu/~epxing/Class/10708-14/scribe_notes/scribe_note_lecture8.pdf). For the special case of a Junction Tree, this algorithm converges in a single iteration (Section 19.5.7 of [Machine Learning - A Probabilistic Perspective](https://probml.github.io/pml-book/)). + + +```python +# Create estimator. +mle = MaximumLikelihoodEstimator(model=junction_tree, data=X) +mle +``` + + + + + + + + + +```python +junction_tree.clique_beliefs = mle.get_parameters() +pprint(junction_tree.clique_beliefs) +``` + + {('age', 'race', 'marital-status'): , + ('marital-status', 'education-num', 'sex'): } + + +With our MLE factors, we can now use the model to perform marginal inference on the `sex` and `race` variables. + + + +```python +sex_race_query = BeliefPropagation(model=junction_tree).query(variables=["race", "sex"]) +modeled_marginals = sex_race_query.values.flatten() +labels = list(str(i) for i in itertools.product(*sex_race_query.state_names.values())) +``` + + +```python +plot_df = pd.DataFrame( + { + "Type": ["Modeled"] * len(modeled_marginals) + + ["Observed"] * len(modeled_marginals) + + ["Rel Diff"] * len(modeled_marginals), + "Labels": labels * 3, + "Marginals": list(modeled_marginals * n) + + list(sex_race_empirical.values.flatten()) + + list(modeled_marginals * n / sex_race_empirical.values.flatten() - 1.0), + } +) +fig = px.bar( + plot_df, + x="Labels", + y="Marginals", + facet_row="Type", + text_auto=True, + color=np.where(plot_df["Marginals"] < 0, "red", "green"), + color_discrete_map={"red": "red", "green": "green"}, +) +# fig.update_traces(marker_color=) +fig.update_yaxes(matches=None) +render_plotly_html(fig) +``` + + + + + + + +
+
+ + + + + +```python +1 - np.average(np.abs(modeled_marginals * n / sex_race_empirical.values.flatten() - 1)) +``` + + + + + 0.929148505709527 + + + +On average we "preserved" >92% of the empirical counts of the original dataset, while only using ~4% of the memory consumption. We can also compute the [Jensen Shannon Divergence](https://docs.scipy.org/doc/scipy/reference/generated/scipy.spatial.distance.jensenshannon.html) between the modeled & empirical counts: + + + +```python +scipy.spatial.distance.jensenshannon( + modeled_marginals, sex_race_empirical.values.flatten() / n +) +``` + + + + + 0.02030564509025804 + + + +Where [Jensen Shannon Divergence is between 0 and 1](https://en.wikipedia.org/wiki/Jensen%E2%80%93Shannon_divergence#Bounds), with 0 indicating highest similarity and 1 indicating lowest similarity. + +References +- [Machine Learning - A Probabilistic Perspective](https://probml.github.io/pml-book/) + +## Optimization +`pgmpy` also has a [`MirrorDescentEstimator`](https://github.com/pgmpy/pgmpy/blob/dev/pgmpy/estimators/MirrorDescentEstimator.py) that uses a [proximal method](https://en.wikipedia.org/wiki/Proximal_gradient_method) and [convex optimization](https://en.wikipedia.org/wiki/Convex_optimization) to estimate potential values. For completeness, I include this as well even though it did not perform as well as `MaximumLikelihoodEstimator`. + + +```python +from pgmpy.estimators import MirrorDescentEstimator +``` + + +```python +# Reset the same junction tree to identity factors. +junction_tree.clique_beliefs = FactorDict( + {k: v.identity_factor() for k, v in junction_tree.clique_beliefs.items()} +) +# Setup the MirrorDescentEstimator +mirror_descent = MirrorDescentEstimator(model=junction_tree, data=X) +# Estimate potential values based off the L1 distance between the modeled & empirical marginals. +junction_tree = mirror_descent.estimate(marginals=factors, metric="L1", iterations=200) +``` + + + 0%| | 0/200 [00:00 + + +
+
+ + + + +Repeat the quality metrics. It seems the [`MirrorDescentEstimator`](https://github.com/pgmpy/pgmpy/blob/dev/pgmpy/estimators/MirrorDescentEstimator.py) does not perform as well as the `MaximumLikelihoodEstimator`, but is still able to compress most of the statistical relationships with 4% of the parameters. + + +```python +1 - np.average(np.abs(modeled_marginals * n / sex_race_empirical.values.flatten() - 1)) +``` + + + + + 0.810944530455735 + + + + +```python +scipy.spatial.distance.jensenshannon( + modeled_marginals, sex_race_empirical.values.flatten() / n +) +``` + + + + + 0.05387315667585743 + + + +References +- [Probabilistic Machine Learning: Advanced Topics (Supplementary Material, Section 6.1.5)](https://probml.github.io/pml-book/) +- [Graphical-model based estimation and inference for differential privacy (Appendix A.)](https://arxiv.org/pdf/1901.09136.pdf) + +## Summary +In this post I talked about tabular datasets and how we can model their marginals with UGMs. UGMs have the ability to factor, allowing them to express a joint distribution with less parameters. We can learn potential values for a UGM using `MaximumLikelihoodEstimator` and [`MirrorDescentEstimator`](https://github.com/pgmpy/pgmpy/blob/dev/pgmpy/estimators/MirrorDescentEstimator.py), all with the `pgmpy` library. We saw that an estimated UGM becomes a compact representation of the table's joint distribution. + + +```python +import subprocess + +_ = subprocess.run( + [ + "jupyter", + "nbconvert", + "--to", + "markdown", + "--output", + "~/Documents/jakee417.github.io/_includes/tabular.md", + "tabular.ipynb", + ], +) +``` diff --git a/_javascript/.DS_Store b/_javascript/.DS_Store index 2253d7e2128450f919fe4c7677e5114fae1a7a41..505804fd779d090ec41f98837448d755c76d26e3 100644 GIT binary patch delta 57 zcmZoMXffE}!=!V=pyr_7gQJcN3=HfHxeWOXDGa3yISi=`#ku(@E=f80NkAD6h^oyM HOg*9i3>XqP delta 57 zcmZoMXffE}!=y8ZQ#!$O_fba%1_pM9T!wsx6oyiU9EMbe;@tccm!zEhB%lljMAhaB HrXEoM$+r*M diff --git a/_posts/2024-04-15-ugm-tabular.md b/_posts/2024-04-15-ugm-tabular.md new file mode 100644 index 00000000000..be47aea8719 --- /dev/null +++ b/_posts/2024-04-15-ugm-tabular.md @@ -0,0 +1,15 @@ +--- +title: Undirected Graphical Models for Tabular Data +author: jake +date: 2024-04-15 12:00:00 +0800 +categories: [Math] +tags: [] +math: true +mermaid: true +image: + path: /assets/img/ugm.png + alt: Converting a tabular dataset into a graphical model +--- +A post about modeling tabular datasets using Undirected Graphical Models with a python package called [pgmpy](https://github.com/pgmpy/pgmpy). + +{% include tabular.md %} \ No newline at end of file diff --git a/_sass/.DS_Store b/_sass/.DS_Store index fe56580c6c119e9030fc118bc3c19f60b86a8828..acb804750e8e0b3f3c24b98ed8de11d165e66e89 100644 GIT binary patch delta 197 zcmZoMXffEJ!NkPjN}g$xi}`a6n`y z8;VOZ?wRZ@u4mEu3n<`WME)mX2@j7Wk}D>PjN}g$xi}`a6n`y z8;VOZ?waf^u4hqg4HRZ#$YLnY&3Az-n{-+-!E-m*5U3o&2#DO|jp7oF7bc$)H)Km& a0#ZDgQ9=UB+$=Bgj*tEh;e0~KF3??lkTan%19Lr=GM94ac(kV+)7~51t_GK8$ z7=y~vU@FTngCT1(#y-aE-*@+Y&iS0jeRu!<`~7$RDNNq4*ZX>1*YkQ_&+B^i`&BE` zeS2i~?AWnmpV{S$Ham9g;_lcXeEz%Lz)!S>_F;j)c81!Rp5IY5Aopd*j>9|5E*f9I z?Y1-_nyqA)ud@ng_MN(5VzztlbC1rY9O2*2|8V;Go##R?JmY24fcL&aB6TmdXij#FxPsa!nj4`E^myB7IP-cLgNq?6cYLFz~OWM@r8wd{LXkmy#hEz;Es(SY~HvDzrbD|<5eEFX)LjKpb 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