{
"cells": [
{
"cell_type": "markdown",
"id": "51e3591e",
"metadata": {},
"source": [
"# MBG/NBD Model\n",
"\n",
"In this notebook we show how to fit a MBG/NBD model in PyMC-Marketing. We compare the results with the [`lifetimes`](https://github.com/CamDavidsonPilon/lifetimes) package (no longer maintained). The model is presented in the paper: Batislam E. P., Denizel M., Filiztekin A. (2007) [Empirical validation and comparison of models for customer base analysis](https://www.sciencedirect.com/science/article/abs/pii/S0167811607000171)"
]
},
{
"cell_type": "markdown",
"id": "68f7ba7e",
"metadata": {},
"source": [
"## Prepare Notebook"
]
},
{
"cell_type": "code",
"execution_count": 1,
"id": "81c950fb",
"metadata": {},
"outputs": [],
"source": [
"import arviz as az\n",
"import matplotlib.pyplot as plt\n",
"import pandas as pd\n",
"import xarray as xr\n",
"from fastprogress.fastprogress import progress_bar\n",
"from lifetimes import ModifiedBetaGeoFitter\n",
"\n",
"from pymc_marketing import clv\n",
"\n",
"# Plotting configuration\n",
"az.style.use(\"arviz-darkgrid\")\n",
"plt.rcParams[\"figure.figsize\"] = [12, 7]\n",
"plt.rcParams[\"figure.dpi\"] = 100\n",
"plt.rcParams[\"figure.facecolor\"] = \"white\"\n",
"\n",
"%load_ext autoreload\n",
"%autoreload 2\n",
"%config InlineBackend.figure_format = \"retina\""
]
},
{
"cell_type": "markdown",
"id": "6e4b3b25",
"metadata": {},
"source": [
"## Read Data\n",
"\n",
"We use the `CDNOW` dataset (see lifetimes [quick-start](https://lifetimes.readthedocs.io/en/latest/Quickstart.html))."
]
},
{
"cell_type": "code",
"execution_count": 2,
"id": "a99638b5",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
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frequency
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recency
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T
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monetary_value
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0
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2
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30.43
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38.86
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22.35
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1
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1.71
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11.77
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"text/plain": [
" frequency recency T monetary_value\n",
"0 2 30.43 38.86 22.35\n",
"1 1 1.71 38.86 11.77\n",
"2 0 0.00 38.86 0.00\n",
"3 0 0.00 38.86 0.00\n",
"4 0 0.00 38.86 0.00"
]
},
"execution_count": 2,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"data_path = \"https://raw.githubusercontent.com/pymc-labs/pymc-marketing/main/data/clv_quickstart.csv\"\n",
"\n",
"df = pd.read_csv(data_path)\n",
"\n",
"df.head()"
]
},
{
"cell_type": "markdown",
"id": "35cc830b",
"metadata": {},
"source": [
"Recall from the `lifetimes` documentation the following definitions:\n",
"\n",
"> - `frequency` represents the number of repeat purchases the customer has made. This means that it’s one less than the total number of purchases. This is actually slightly wrong. It’s the count of time periods the customer had a purchase in. So if using days as units, then it’s the count of days the customer had a purchase on.\n",
"> - `T` represents the age of the customer in whatever time units chosen (weekly, in the above dataset). This is equal to the duration between a customer’s first purchase and the end of the period under study.\n",
"> - `recency` represents the age of the customer when they made their most recent purchases. This is equal to the duration between a customer’s first purchase and their latest purchase. (Thus if they have made only 1 purchase, the recency is 0.)"
]
},
{
"cell_type": "markdown",
"id": "192738eb",
"metadata": {},
"source": [
"```{tip}\n",
"We rename the index column to `customer_id` as this is required by the model\n",
"```"
]
},
{
"cell_type": "code",
"execution_count": 3,
"id": "6eb810b4",
"metadata": {},
"outputs": [],
"source": [
"data = (\n",
" df.reset_index()\n",
" .rename(columns={\"index\": \"customer_id\"})\n",
" .drop(columns=\"monetary_value\")\n",
")"
]
},
{
"cell_type": "markdown",
"id": "91cb4b6f",
"metadata": {},
"source": [
"## Model Specification\n",
"\n",
"The MBG/NBD model is a probabilistic model that describes the buying behavior of a customer in the non-contractual setting. It is based on the following assumptions for each customer:\n",
"\n",
"### Dropout after first purchase\n",
"\n",
"Contrasting with the BG/NBD model, in the MBG/NBD a customer may drop out at time zero with probability p. \n",
"This leads to the following individual level likelihood function: \n",
"\n",
"$$ L(\\lambda, p | X=x, T) = (1 - p)^{x+1} \\lambda^x \\exp(\\lambda T) + p(1-p)^x \\lambda^x \\exp(-\\lambda t_x)$$\n",
"\n",
"Compare the previous expresion with the regular BG/NBD likelihood:\n",
"\n",
"$$ L(\\lambda, p | X=x, T) = (1 - p)^{x} \\lambda^x \\exp(\\lambda T) + \\delta_{x>0} p(1-p)^{x-1} \\lambda^x \\exp(-\\lambda t_x)$$\n"
]
},
{
"cell_type": "markdown",
"id": "6183c040",
"metadata": {},
"source": [
"## Model Fitting\n",
"\n",
"Estimating such parameters is very easy in PyMC-Marketing. We instantiate the model in a similar way:"
]
},
{
"cell_type": "code",
"execution_count": 4,
"id": "13835b98",
"metadata": {},
"outputs": [],
"source": [
"model = clv.ModifiedBetaGeoModel(data=data)"
]
},
{
"cell_type": "markdown",
"id": "36c28cce-5988-4db8-b21d-07bee206e508",
"metadata": {},
"source": [
"And *build* the model to see the model configuration:"
]
},
{
"cell_type": "code",
"execution_count": 5,
"id": "8c828d6b-d65b-47f4-aa59-48984d7ae4fc",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"MBG/NBD\n",
" alpha ~ Weibull(2, 10)\n",
" r ~ Weibull(2, 1)\n",
" phi_dropout ~ Uniform(0, 1)\n",
" kappa_dropout ~ Pareto(1, 1)\n",
" a ~ Deterministic(f(kappa_dropout, phi_dropout))\n",
" b ~ Deterministic(f(kappa_dropout, phi_dropout))\n",
"recency_frequency ~ ModifiedBetaGeoNBD(a, b, r, alpha, )"
]
},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"model.build_model()\n",
"model"
]
},
{
"cell_type": "markdown",
"id": "0a5fc96b-eedd-43ae-bd70-5ae388f1f636",
"metadata": {},
"source": [
"Notice the additional `phi_dropout` and `kappa_dropout` priors. These were added to the default configuration to improve performance, but can be omitted when specifying a custom `model_config` with `a` and `b`.\n",
"\n",
"The specified model structure can also be visualized:"
]
},
{
"cell_type": "code",
"execution_count": 6,
"id": "2ffc4d86-01ae-455f-af14-40c617948b33",
"metadata": {},
"outputs": [
{
"data": {
"image/svg+xml": [
"\n",
"\n",
"\n",
"\n",
"\n"
],
"text/plain": [
""
]
},
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"model.graphviz()"
]
},
{
"cell_type": "markdown",
"id": "07ab143f",
"metadata": {},
"source": [
"We can now fit the model. The default sampler in PyMC-Marketing is the No-U-Turn Sampler (NUTS). We use the default $4$ chains and $1000$ draws per chain."
]
},
{
"cell_type": "markdown",
"id": "c2130e31",
"metadata": {},
"source": [
"```{note}\n",
"It is not necessary to build the model before fitting it. We can fit the model directly.\n",
"```"
]
},
{
"cell_type": "code",
"execution_count": 7,
"id": "d23e4726",
"metadata": {},
"outputs": [
{
"name": "stderr",
"output_type": "stream",
"text": [
"Initializing NUTS using jitter+adapt_diag...\n",
"Multiprocess sampling (4 chains in 4 jobs)\n",
"NUTS: [alpha, r, phi_dropout, kappa_dropout]\n"
]
},
{
"data": {
"application/vnd.jupyter.widget-view+json": {
"model_id": "174b3ee3e6bb49eda2dcedf6fcd19eb5",
"version_major": 2,
"version_minor": 0
},
"text/plain": [
"Output()"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/html": [
"\n"
],
"text/plain": []
},
"metadata": {},
"output_type": "display_data"
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"Sampling 4 chains for 1_000 tune and 2_000 draw iterations (4_000 + 8_000 draws total) took 10 seconds.\n"
]
}
],
"source": [
"sample_kwargs = {\n",
" \"draws\": 2_000,\n",
" \"chains\": 4,\n",
" \"target_accept\": 0.9,\n",
" \"random_seed\": 42,\n",
"}\n",
"\n",
"idata_mcmc = model.fit(**sample_kwargs)"
]
},
{
"cell_type": "code",
"execution_count": 8,
"id": "f677d6c3",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"\n",
"
"
]
},
"metadata": {
"image/png": {
"height": 711,
"width": 1211
}
},
"output_type": "display_data"
}
],
"source": [
"# PPC histogram plot\n",
"clv.plot_expected_purchases_ppc(model, ppc=\"posterior\");"
]
},
{
"cell_type": "markdown",
"id": "393969db",
"metadata": {},
"source": [
"## Some Applications\n",
"\n",
"Now that you have fitted the model, we can use it to make predictions. For example, we can predict the expected probability of a customer being alive as a function of time (steps). Here is a snippet of code to do that:"
]
},
{
"cell_type": "markdown",
"id": "a1c538d0",
"metadata": {},
"source": [
"### Expected Number of Purchases"
]
},
{
"cell_type": "markdown",
"id": "0cfd1f3e",
"metadata": {},
"source": [
"Let us take a sample of users:"
]
},
{
"cell_type": "code",
"execution_count": 46,
"id": "74a66344",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"