Introduction to the Bass Diffusion Model

What is the Bass Model?

The Bass diffusion model, developed by Frank Bass in 1969, is a mathematical model that describes how new products get adopted in a population over time. It’s widely used in marketing to forecast sales of new products, especially when historical data is limited or non-existent.

The model captures the entire lifecycle of product adoption, from introduction to saturation, making it a powerful tool for product planning and marketing strategy development.

The Motivation Behind the Bass Model

Before the Bass model, companies struggled to predict the adoption patterns of new products. Traditional forecasting methods often failed because they couldn’t account for the social dynamics that drive product adoption. Frank Bass recognized that product adoption follows a distinct pattern:

• Initial slow growth: When a product first launches, adoption starts slowly

• Rapid acceleration: As more people adopt, word-of-mouth spreads and adoption accelerates

• Eventual saturation: Eventually, the market becomes saturated and adoption slows down

The Bass model provides a mathematical framework to capture these patterns, enabling businesses to make more informed decisions about production planning, inventory management, and marketing resource allocation.

Mathematical Formulation

The Bass model is based on a differential equation that describes the rate of adoption:

\[\frac{f(t)}{1-F(t)} = p + q F(t)\]

Where:

• \(F(t)\) is the installed base fraction (cumulative proportion of adopters)

• \(f(t)\) is the rate of change of the installed base fraction (\(f(t) = F'(t)\))

• \(p\) is the coefficient of innovation or external influence

• \(q\) is the coefficient of imitation or internal influence

The solution to this equation gives the adoption curve:

\[F(t) = \frac{1 - e^{-(p+q)t}}{1 + (\frac{q}{p})e^{-(p+q)t}}\]

The adoption rate at time \(t\) is given by:

\[f(t) = (p + q F(t))(1 - F(t))\]

Alternatively, this can be written as:

\[f(t) = \frac{(p+q)^2 \cdot e^{-(p+q)t}}{p \cdot (1+\frac{q}{p}e^{-(p+q)t})^2}\]

Key Components of the Bass Model Implementation

The Bass model implementation in PyMC Marketing consists of several key components:

1. Adopters - The number of new adoptions at time \(t\):

\[\text{adopters}(t) = m \cdot f(p, q, t)\]

2. Innovators - Adoptions driven by external influence (advertising, etc.):

\[\text{innovators}(t) = m \cdot p \cdot (1 - F(p, q, t))\]

3. Imitators - Adoptions driven by internal influence (word-of-mouth):

\[\text{imitators}(t) = m \cdot q \cdot F(p, q, t) \cdot (1 - F(p, q, t))\]

4. Peak Adoption Time - When the adoption rate reaches its maximum:

\[\text{peak} = \frac{\ln(q) - \ln(p)}{p + q}\]

The total number of adopters over time is the sum of innovators and imitators, which equals \(\text{adopters}(t)\). All of these components are directly implemented in the PyMC model, allowing us to analyze each aspect of the diffusion process separately.

Understanding the Relationship Between Components

A key insight of the Bass model is how it decomposes adoption into two sources:

\[\text{adopters}(t) = \text{innovators}(t) + \text{imitators}(t)\]

At each time point:

• Innovators (\(m \cdot p \cdot (1 - F(t))\)) represents new adoptions coming from people who are influenced by external factors like advertising

• Imitators (\(m \cdot q \cdot F(t) \cdot (1 - F(t))\)) represents new adoptions coming from people who are influenced by previous adopters

As time progresses:

• Initially, innovators dominate the adoption process when few people have adopted (\(F(t)\) is small)

• Later, imitators become the primary source of new adoptions as the word-of-mouth effect grows

• Eventually, both decrease as the market approaches saturation (\(F(t)\) approaches 1)

The cumulative adoption at any time point is:

\[\text{Cumulative Adoption}(t) = m \cdot F(t)\]

This means that as \(t \to \infty\), the cumulative adoption approaches the total market potential \(m\):

\[\lim_{t \to \infty} \text{Cumulative Adoption}(t) = m\]

Therefore, the Bass model provides a complete accounting of the market:

• At each time point, new adopters are either innovators or imitators

• Over the entire product lifecycle, all potential adopters (m) eventually adopt the product

• The model tracks both the adoption rate (new adopters per time period) and the cumulative adoption (total adopters to date)

This structure enables marketers to understand not just how many people will adopt over time, but also the driving forces behind adoption at different stages of the product lifecycle.

Understanding the Key Parameters

The model has three main parameters:

• Market potential (m): Total number of eventual adopters (the ultimate market size)

• Innovation coefficient (p): Measures external influence like advertising and media - typically \(0.01-0.03\)

• Imitation coefficient (q): Measures internal influence like word-of-mouth - typically \(0.3-0.5\)

Parameter Interpretation

• A higher p value indicates stronger external influence (advertising, marketing)

• A higher q value indicates stronger internal influence (word-of-mouth, social interactions)

• The ratio q/p indicates the relative strength of internal vs. external influences

• The peak of adoption occurs at time

\[t^* = \frac{\ln(q / p)}{p + q}\]

Innovators vs. Imitators

The Bass model distinguishes between two types of adopters:

1. Innovators: People who adopt independently of others’ decisions, influenced mainly by mass media and external communications

   • Mathematically represented as: \(\text{innovators}(t) = m \cdot p \cdot (1 - F(p, q, t))\)

2. Imitators: People who adopt because of social influence and word-of-mouth from previous adopters

   • Mathematically represented as: \(\text{imitators}(t) = m \cdot q \cdot F(p, q, t) \cdot (1 - F(p, q, t))\)

Real-World Applications

The Bass model has been successfully applied to forecast the adoption of various products and technologies:

• Consumer durables: TVs, refrigerators, washing machines

• Technology products: Smartphones, computers, software

• Pharmaceutical products: New drugs and treatments

• Entertainment products: Movies, games, streaming services

• Services and subscriptions: Banking services, subscription plans

Business Value: Why the Bass Model Matters to Executives and Marketers

From a business perspective, the Bass diffusion model provides substantial competitive advantages and ROI benefits:

1. Resource Optimization and Cash Flow Management

• Production Planning: Avoid costly overproduction or stockouts by accurately forecasting demand curves

• Marketing Budget Allocation: Optimize spending across the product lifecycle, investing more during key inflection points

• Supply Chain Efficiency: Coordinate with suppliers and distributors based on predicted adoption rates

• Cash Flow Optimization: Better predict revenue streams, improving financial planning and investor relations

2. Strategic Decision Making

• Launch Timing: Determine the optimal time to enter a market based on diffusion patterns

• Pricing Strategy: Implement dynamic pricing strategies aligned with the adoption curve

• Competitive Analysis: Compare your product’s adoption parameters with competitors to identify strengths and weaknesses

• Product Portfolio Management: Make informed decisions about when to phase out older products and introduce new ones

3. Risk Mitigation

• Scenario Planning: Test different market assumptions and external factors through parameter variations

• Early Warning System: Identify deviations from expected adoption curves early, enabling faster intervention

• Investment Justification: Provide data-driven forecasts to justify R&D and marketing investments to stakeholders

4. Performance Measurement

• Marketing Effectiveness: Measure the impact of marketing campaigns on the innovation coefficient (p)

• Word-of-Mouth Strength: Quantify the power of your brand’s social influence through the imitation coefficient (q)

• Total Market Potential: Validate or adjust your total addressable market estimates (m)

In today’s data-driven business environment, companies that effectively utilize models like Bass diffusion gain a significant competitive edge through more precise forecasting, better resource allocation, and strategic market timing.

Bayesian Extensions

In this notebook, we show how to generate simulated data from the Bass model and fit a Bayesian model to it. The Bayesian formulation offers several advantages:

• Uncertainty quantification through prior distributions on parameters

• Hierarchical modeling for multiple products or markets

• Incorporation of expert knowledge through informative priors

• Full probability distributions for future adoption forecasts

What we’ll do in this notebook

In this notebook, we’ll:

1. Set up parameters for a Bass model simulation

2. Generate simulated adoption data for multiple products

3. Fit the Bass model to our simulated data using PyMC

4. Visualize the adoption curves

Prepare Notebook

from typing import Any

import arviz as az
import matplotlib.pyplot as plt
import numpy as np
import numpy.typing as npt
import pandas as pd
import pymc as pm
import xarray as xr

from pymc_marketing.bass.model import create_bass_model
from pymc_marketing.plot import plot_curve
from pymc_marketing.prior import Prior, Scaled

az.style.use("arviz-darkgrid")
plt.rcParams["figure.figsize"] = [12, 7]
plt.rcParams["figure.dpi"] = 100

%load_ext autoreload
%autoreload 2
%config InlineBackend.figure_format = "retina"

seed: int = sum(map(ord, "bass"))
rng: np.random.Generator = np.random.default_rng(seed=seed)

Setting Up Simulation Parameters

First, we’ll set up the parameters for our simulation. This includes:

• The time period for our simulation (in weeks)

• The number of products to simulate

• Start dates for the simulation period

def setup_simulation_parameters(
    n_weeks: int = 52,
    n_products: int = 9,
    start_date: str = "2023-01-01",
    cutoff_start_date: str = "2023-12-01",
) -> tuple[
    npt.NDArray[np.int_],
    pd.DatetimeIndex,
    pd.DatetimeIndex,
    list[str],
    pd.Series,
    dict[str, Any],
]:
    """Set up initial parameters for the Bass diffusion model simulation.

    Parameters
    ----------
    n_weeks : int
        Number of weeks to simulate
    n_products : int
        Number of products to include in the simulation
    start_date : str
        Starting date for the simulation period
    cutoff_start_date : str
        Latest possible start date for products

    Returns
    -------
    T : numpy.ndarray
        Time array (weeks)
    possible_dates : pandas.DatetimeIndex
        All dates in the simulation period
    possible_start_dates : pandas.DatetimeIndex
        Possible start dates for products
    products : list
        List of product names
    product_start : pandas.Series
        Start date for each product
    coords : dict
        Coordinates for PyMC model
    """
    # Set a seed for reproducibility
    seed = sum(map(ord, "bass"))
    rng = np.random.default_rng(seed)

    # Create time array and date range
    T = np.arange(n_weeks)
    possible_dates = pd.date_range(start_date, freq="W-MON", periods=n_weeks)
    cutoff_start_date = pd.to_datetime(cutoff_start_date)
    cutoff_start_date = cutoff_start_date + pd.DateOffset(weeks=1)
    possible_start_dates = possible_dates[possible_dates < cutoff_start_date]

    # Generate product names and random start dates
    products = [f"P{i}" for i in range(n_products)]
    product_start = pd.Series(
        rng.choice(possible_start_dates, size=len(products)),
        index=pd.Index(products, name="product"),
    )

    coords = {"T": T, "product": products}
    return T, possible_dates, possible_start_dates, products, product_start, coords

Creating Prior Distributions

For our Bayesian Bass model, we need to specify prior distributions for the key parameters:

• m (market potential): How many units can potentially be sold in total

• p (innovation coefficient): Rate of adoption from external influences

• q (imitation coefficient): Rate of adoption from internal/social influences

• likelihood: The probability distribution that models the observed adoption data

For the market potential m we use a scaling trick to specify a scale-free prior and then add a global factor:

def create_bass_priors(factor: float) -> dict[str, Prior | Scaled]:
    """Define prior distributions for the Bass model parameters.

    Returns
    -------
    dict
        Dictionary of prior distributions for m, p, q, and likelihood

    Notes
    -----
    - m: Market potential (scaled Gamma distribution)
    - p: Innovation coefficient (Beta distribution)
    - q: Imitation coefficient (Beta distribution)
    - likelihood: Observation model (Negative Binomial)
    """
    return {
        # We use a scaled Gamma distribution for the market potential.
        "m": Scaled(Prior("Gamma", mu=1, sigma=0.001, dims="product"), factor=factor),
        "p": Prior("Beta", mu=0.03, sigma=0.01 / 2, dims="product"),
        "q": Prior("Beta", mu=0.38, sigma=0.1 / 2, dims="product"),
        "likelihood": Prior("NegativeBinomial", n=1.5, dims="product"),
    }

Let’s generate and visualize the priors.

FACTOR = 50_000
priors = create_bass_priors(factor=FACTOR)

fig, ax = plt.subplots(nrows=2, ncols=1, figsize=(15, 12))

priors["p"].preliz.plot_pdf(ax=ax[0])
ax[0].set(title="Innovation Coefficient (p)")
priors["q"].preliz.plot_pdf(ax=ax[1])
ax[1].set(title="Imitation Coefficient (q)")
fig.suptitle(
    "Prior Distributions for Bass Model Parameters",
    fontsize=18,
    fontweight="bold",
    y=0.95,
);

Observe we have chosen the priors within the usual ranges of empirical studies:

• Innovation coefficient (p): Measures external influence like advertising and media - typically \(0.01-0.03\)

• Imitation coefficient (q): Measures internal influence like word-of-mouth - typically \(0.3-0.5\)

Generate Synthetic Data

With the generative Bass model, we can generate a synthetic dataset by sampling from the prior and choosing one particular sample to use as observed data. For this purpose we define two auxiliary functions.

def sample_prior_bass_data(model: pm.Model) -> xr.DataArray:
    """Generate a sample from the prior predictive distribution of the Bass model.

    Parameters
    ----------
    model : pymc.Model
        The PyMC model to sample from

    Returns
    -------
    xarray.DataArray
        Simulated adoption data
    """
    with model:
        idata = pm.sample_prior_predictive(random_seed=rng)
    return idata["prior"]["y"].sel(chain=0, draw=0)


def transform_to_actual_dates(bass_data, product_start, possible_dates) -> pd.DataFrame:
    """Transform simulation data from time index to calendar dates.

    Parameters
    ----------
    bass_data : xarray.DataArray
        Simulated bass model data
    product_start : pandas.Series
        Start date for each product
    possible_dates : pandas.DatetimeIndex
        All dates in the simulation period

    Returns
    -------
    pandas.DataFrame
        Adoption data with actual calendar dates
    """
    bass_data = bass_data.to_dataset()
    bass_data["product_start"] = product_start.to_xarray()

    df_bass_data = (
        bass_data.to_dataframe().drop(columns=["chain", "draw"]).reset_index()
    )
    df_bass_data["actual_date"] = df_bass_data["product_start"] + pd.to_timedelta(
        7 * df_bass_data["T"], unit="days"
    )

    return (
        df_bass_data.set_index(["actual_date", "product"])
        .y.unstack(fill_value=0)
        .reindex(possible_dates, fill_value=0)
    )

Now we can generate the observed data:

# Setup simulation parameters
T, possible_dates, _, products, product_start, coords = setup_simulation_parameters()

# Create and configure the Bass model
generative_model = create_bass_model(t=T, coords=coords, observed=None, priors=priors)

# Sample and select one "observed" dataset.
bass_data = sample_prior_bass_data(generative_model)
actual_data = transform_to_actual_dates(bass_data, product_start, possible_dates)

Sampling: [m_unscaled, p, q, y]

The actual_data data frame has the typical format of a real dataset.

actual_data

product	P0	P1	P2	P3	P4	P5	P6	P7	P8
2023-01-02	0	0	36	0	0	0	0	0	0
2023-01-09	0	0	194	0	0	0	0	0	0
2023-01-16	0	0	324	0	0	0	0	0	0
2023-01-23	0	0	79	0	0	0	0	0	0
2023-01-30	0	25	236	0	0	11	0	0	0
2023-02-06	0	125	578	0	0	93	0	0	0
2023-02-13	0	406	502	0	0	59	0	0	0
2023-02-20	0	113	24	0	0	169	0	0	0
2023-02-27	0	654	256	87	0	30	0	0	0
2023-03-06	0	417	160	6	0	10	0	0	0
2023-03-13	99	228	190	52	0	125	0	0	0
2023-03-20	55	358	81	1443	0	1088	0	0	0
2023-03-27	208	924	57	239	0	171	0	0	0
2023-04-03	188	191	46	366	0	68	0	0	0
2023-04-10	51	478	176	244	0	216	0	0	0
2023-04-17	725	40	36	118	0	50	0	0	0
2023-04-24	284	18	34	506	0	65	0	0	0
2023-05-01	237	151	37	68	0	63	0	0	0
2023-05-08	400	66	54	43	0	51	0	0	0
2023-05-15	64	20	19	52	0	58	0	0	0
2023-05-22	72	7	2	24	0	84	0	0	0
2023-05-29	202	8	5	36	0	14	0	0	0
2023-06-05	362	16	4	3	0	23	0	0	0
2023-06-12	98	11	0	18	0	8	0	0	0
2023-06-19	28	4	0	15	0	5	0	0	0
2023-06-26	22	4	0	8	0	20	0	0	0
2023-07-03	31	1	0	2	0	4	0	0	0
2023-07-10	4	1	0	3	0	6	0	0	0
2023-07-17	19	1	0	0	0	3	0	0	0
2023-07-24	2	0	0	0	0	0	0	0	66
2023-07-31	7	1	0	2	0	0	0	0	72
2023-08-07	3	0	0	0	0	0	0	0	82
2023-08-14	0	0	0	0	0	1	0	0	14
2023-08-21	3	0	0	1	0	1	0	0	236
2023-08-28	0	0	0	0	0	1	0	0	337
2023-09-04	1	0	0	0	0	0	0	243	401
2023-09-11	2	0	0	0	0	0	111	337	196
2023-09-18	0	1	0	0	0	0	184	234	110
2023-09-25	0	0	0	0	0	0	135	1011	336
2023-10-02	0	0	0	0	0	0	92	362	255
2023-10-09	0	0	0	0	0	0	206	246	89
2023-10-16	0	0	0	0	0	0	1108	246	263
2023-10-23	0	0	0	0	0	0	96	479	20
2023-10-30	0	0	0	0	91	0	612	361	429
2023-11-06	0	0	0	0	50	0	70	63	17
2023-11-13	0	0	0	0	196	0	59	157	12
2023-11-20	0	0	0	0	95	0	454	207	47
2023-11-27	0	0	0	0	289	0	134	253	36
2023-12-04	0	0	0	0	710	0	101	26	18
2023-12-11	0	0	0	0	28	0	108	89	18
2023-12-18	0	0	0	0	108	0	68	100	15
2023-12-25	0	0	0	0	82	0	74	6	4

On the other hand, the bass_data has the same data as arrays indexed by time (relative) and product.

Let’s visualize both.

fig, ax = plt.subplots(
    nrows=2, ncols=1, figsize=(15, 12), sharex=False, sharey=True, layout="constrained"
)

# Plot raw simulated data (by time step)
bass_data.to_series().unstack().plot(ax=ax[0])
ax[0].legend(
    title="Product", title_fontsize=14, loc="center left", bbox_to_anchor=(1, 0.5)
)
ax[0].set(
    title="Simulated Weekly Adoption by Product (Time Steps)",
    xlabel="Time Step (Weeks)",
    ylabel="Number of Adoptions",
)

# Plot data with actual calendar dates
actual_data.plot(ax=ax[1])
ax[1].legend(
    title="Product", title_fontsize=14, loc="center left", bbox_to_anchor=(1, 0.5)
)
ax[1].set(
    title="Simulated Weekly Adoption by Product (Calendar Dates)",
    xlabel="Date",
    ylabel="Number of Adoptions",
)

fig.suptitle(
    "Bass Diffusion Model - Simulated Product Adoption", fontsize=18, fontweight="bold"
);

Fit the Model

We are now ready to fit the model and generate the posterior predictive distributions.

# We condition the model on observed data.
with pm.observe(generative_model, {"y": bass_data.values}) as model:
    idata = pm.sample(
        tune=1_500,
        draws=2_000,
        chains=4,
        nuts_sampler="nutpie",
        compile_kwargs={"mode": "NUMBA"},
        random_seed=rng,
    )

    idata.extend(
        pm.sample_posterior_predictive(
            idata, model=model, extend_inferencedata=True, random_seed=rng
        )
    )

Sampler Progress

Total Chains: 4

Active Chains: 0

Finished Chains: 4

Sampling for now

Estimated Time to Completion: now

Progress	Draws	Divergences	Step Size	Gradients/Draw
	3500	0	0.69	7
	3500	0	0.69	7
	3500	0	0.68	7
	3500	0	0.69	7

Sampling: [y]

We do not have any divergences. Let’s look at the summary of the parameters.

az.summary(data=idata, var_names=["p", "q", "m"])

	mean	sd	hdi_3%	hdi_97%	mcse_mean	mcse_sd	ess_bulk	ess_tail	r_hat
p[P0]	0.028	0.004	0.022	0.035	0.000	0.000	19625.0	5784.0	1.0
p[P1]	0.031	0.004	0.024	0.038	0.000	0.000	22744.0	6087.0	1.0
p[P2]	0.029	0.004	0.022	0.035	0.000	0.000	21865.0	6482.0	1.0
p[P3]	0.032	0.004	0.025	0.039	0.000	0.000	18618.0	5779.0	1.0
p[P4]	0.027	0.003	0.021	0.033	0.000	0.000	19278.0	6548.0	1.0
p[P5]	0.024	0.003	0.017	0.030	0.000	0.000	19607.0	5645.0	1.0
p[P6]	0.030	0.004	0.023	0.037	0.000	0.000	19315.0	6526.0	1.0
p[P7]	0.034	0.004	0.027	0.041	0.000	0.000	18583.0	6109.0	1.0
p[P8]	0.025	0.003	0.019	0.032	0.000	0.000	21261.0	6134.0	1.0
q[P0]	0.387	0.014	0.360	0.413	0.000	0.000	19869.0	5773.0	1.0
q[P1]	0.387	0.015	0.360	0.414	0.000	0.000	20627.0	5563.0	1.0
q[P2]	0.372	0.013	0.346	0.396	0.000	0.000	18371.0	6149.0	1.0
q[P3]	0.455	0.019	0.417	0.488	0.000	0.000	21950.0	5866.0	1.0
q[P4]	0.319	0.012	0.297	0.340	0.000	0.000	21452.0	5974.0	1.0
q[P5]	0.353	0.013	0.329	0.377	0.000	0.000	25087.0	6500.0	1.0
q[P6]	0.375	0.014	0.348	0.400	0.000	0.000	22732.0	6189.0	1.0
q[P7]	0.330	0.012	0.308	0.353	0.000	0.000	19555.0	6141.0	1.0
q[P8]	0.336	0.012	0.315	0.358	0.000	0.000	20631.0	5849.0	1.0
m[P0]	50000.162	49.220	49911.114	50092.819	0.336	0.622	21522.0	6217.0	1.0
m[P1]	49999.776	50.442	49911.002	50098.496	0.362	0.702	19469.0	5585.0	1.0
m[P2]	50000.023	49.510	49908.858	50094.166	0.321	0.639	23623.0	6301.0	1.0
m[P3]	50000.153	49.263	49904.000	50089.367	0.336	0.623	21453.0	5811.0	1.0
m[P4]	50000.109	48.947	49912.149	50097.660	0.340	0.687	20831.0	5712.0	1.0
m[P5]	49999.750	48.974	49902.908	50086.002	0.338	0.610	21047.0	6166.0	1.0
m[P6]	50000.284	49.741	49909.945	50094.064	0.333	0.675	22285.0	5602.0	1.0
m[P7]	49999.871	49.816	49906.288	50094.007	0.331	0.643	22643.0	6009.0	1.0
m[P8]	50000.335	49.921	49902.734	50091.605	0.344	0.664	21006.0	5100.0	1.0

_ = az.plot_trace(
    data=idata,
    var_names=["p", "q", "m"],
    compact=True,
    backend_kwargs={"figsize": (12, 7), "layout": "constrained"},
)
plt.gcf().suptitle("Model Trace", fontsize=16);

Overall, the diagnostics and trace look good.

Next, we look into the posterior distributions of the parameters.

ax, *_ = az.plot_forest(idata["posterior"]["p"], combined=True)
ax.axvline(x=priors["p"].parameters["mu"], color="gray", linestyle="--")
ax.get_figure().suptitle("Innovation Coefficient (p)", fontsize=18, fontweight="bold")

Text(0.5, 0.98, 'Innovation Coefficient (p)')

ax, *_ = az.plot_forest(idata["posterior"]["q"], combined=True)
ax.axvline(x=priors["q"].parameters["mu"], color="gray", linestyle="--")
ax.get_figure().suptitle("Imitation Coefficient (q)", fontsize=18, fontweight="bold")

Text(0.5, 0.98, 'Imitation Coefficient (q)')

We do see some heterogeneity in the parameters, but overall they are centered around the true values (from the generative model).

Examining Posterior Predictions for Specific Products

Let’s look at the posterior predictive distributions to see how well our model captures the simulated data.

fig, axes = plt.subplots(
    nrows=3, ncols=3, figsize=(15, 12), sharex=True, sharey=True, layout="constrained"
)

idata["posterior_predictive"]["y"].pipe(plot_curve, {"T"}, axes=axes)

for i, ax in enumerate(axes.flatten()):
    ax.plot(T, bass_data[:, i], color="black")

fig.suptitle("Posterior Predictive vs Observed Data", fontsize=18, fontweight="bold");

Overall, the model does a good job of capturing the data.

Next, we look into the adopters, which represent the expected value of the likelihood.

fig, axes = plt.subplots(
    nrows=3, ncols=3, figsize=(15, 12), sharex=True, sharey=True, layout="constrained"
)

idata["posterior"]["adopters"].pipe(plot_curve, {"T"}, axes=axes)

for i, ax in enumerate(axes.flatten()):
    ax.plot(T, bass_data[:, i], color="black")

fig.suptitle("Adopters vs Observed Data", fontsize=18, fontweight="bold");

This show the fit is indeed quite reasonable.

We can also evaluate the model goodness by looking into the cumulative data:

fig, axes = plt.subplots(
    nrows=3, ncols=3, figsize=(15, 12), sharex=True, sharey=True, layout="constrained"
)

idata["posterior"]["adopters"].cumsum(dim="T").pipe(plot_curve, {"T"}, axes=axes)

for i, ax in enumerate(axes.flatten()):
    ax.plot(T, bass_data[:, i].cumsum(), color="black")

fig.suptitle("Adopters  Cumulative vs Observed Data", fontsize=18, fontweight="bold");

We can enhance this view by looking into the components of the model: innovators and imitators (in orange and green, respectively).

fig, axes = plt.subplots(
    nrows=3, ncols=3, figsize=(15, 12), sharex=True, sharey=True, layout="constrained"
)

idata["posterior"]["adopters"].cumsum(dim="T").pipe(
    plot_curve, {"T"}, colors=3 * 3 * ["C0"], axes=axes
)

idata["posterior"]["innovators"].pipe(
    plot_curve, {"T"}, colors=3 * 3 * ["C1"], axes=axes
)
idata["posterior"]["imitators"].pipe(
    plot_curve, {"T"}, colors=3 * 3 * ["C2"], axes=axes
)

for i, ax in enumerate(axes.flatten()):
    ax.plot(T, bass_data[:, i].cumsum(), color="black")

fig.suptitle("Innovators vs Imitators", fontsize=18, fontweight="bold");

Finally, we can inspect the peak of the adoption curve.

ax, *_ = az.plot_forest(idata["posterior"]["peak"], combined=True)
ax.get_figure().suptitle("Peak", fontsize=18, fontweight="bold");

This fits the observed data quite well. Let’s see for example the product P4.

fig, ax = plt.subplots()

product_id = 4

bass_data[:, product_id].plot(ax=ax, color="black")

idata["posterior"]["adopters"].sel(product=f"P{product_id}").pipe(
    plot_curve, {"T"}, axes=ax
)

peak_hdi = az.hdi(idata["posterior"]["peak"].sel(product=f"P{product_id}"))["peak"]
ax.axvspan(
    peak_hdi.sel(hdi="lower").item(),
    peak_hdi.sel(hdi="higher").item(),
    color="C1",
    alpha=0.4,
)

ax.set_title(f"Peak Product {products[product_id]}", fontsize=18, fontweight="bold");

%load_ext watermark
%watermark -n -u -v -iv -w -p nutpie,pymc_marketing,pytensor

Last updated: Fri Apr 25 2025

Python implementation: CPython
Python version       : 3.12.9
IPython version      : 9.0.2

nutpie        : 0.14.3
pymc_marketing: 0.13.1
pytensor      : 2.30.3

pandas        : 2.2.3
numpy         : 2.1.3
pymc          : 5.22.0
matplotlib    : 3.10.1
arviz         : 0.21.0
pymc_marketing: 0.13.1
xarray        : 2025.3.1

Watermark: 2.5.0