Estimating Product-Degree Value Elasticities Utilizing Hierarchical Bayesian

$gamma_{c(i),t}$ is a set of category-by-time dummy variables to seize the common demand in every category-time interval and $delta_i$ is a set of product dummies to seize the time-invariant demand shifter for every product. This “fixed-effect” formulation is commonplace and customary in lots of regression-based fashions to manage for unobserved confounders. This (pooled) regression mannequin permits us to recuperate the common elasticity $beta$ throughout all $N$ models. This may imply that the shop may goal a mean value stage throughout all merchandise of their retailer to maximise the income:

$$underset{textrm{Value}_t}{max} ;;; textrm{Value}_{t}cdotmathbb{E}(textrm{Amount}_{t} | textrm{Value}_{t}, beta)$$

If these models have a pure grouping (product classes), we’d be capable of determine the common elasticity of every class by operating separate regressions (or interacting the value elasticity with the product class) for every class utilizing solely models from that class. This may imply that the shop may goal common costs in every class to maximise category-specific income, such that:

$$underset{textrm{Value}_{c(i),t}}{max} ;;; textrm{Value}_{c(i),t}cdotmathbb{E}(textrm{Amount}_{c(i),t} | textrm{Value}_{c(i),t}, beta_{c(i)})$$

With adequate information, we may even run separate regressions for every particular person product to acquire extra granular elasticities.

Nevertheless, real-world information typically presents challenges: some merchandise have minimal value variation, brief gross sales histories, or class imbalance throughout classes. Underneath these restrictions, operating separate regressions to determine product elasticity would doubtless result in massive commonplace errors and weak identification of $beta$. HB fashions addresses these points by permitting us to acquire granular estimates of the coefficient of curiosity by sharing statistical power each throughout totally different groupings whereas preserving heterogeneity. With the HB formulation, it’s doable to run one single regression (just like the pooled case) whereas nonetheless recovering elasticities on the product stage, permitting for granular optimizations.

Understanding Hierarchical Bayesian Fashions

At its core, HB is about recognizing the pure construction in our information. Reasonably than treating all observations as utterly impartial (many separate regressions) or forcing them to observe equivalent patterns (one pooled regression), we acknowledge that observations can cluster into teams, with merchandise inside every group sharing comparable patterns. The “hierarchical” side refers to how we manage our parameters in numerous ranges. In its most elementary format, we may have:

• A International parameter that applies to all information.
• Group-level parameters that apply to observations inside that group.
• Particular person-level parameters that apply to every particular particular person.

This technique is versatile sufficient so as to add or take away hierarchies as wanted, relying on the specified stage of pooling. For instance, if we expect there are not any similarities throughout classes, we may take away the worldwide parameter. If we expect that these merchandise haven’t any pure groupings, we may take away the group-level parameters. If we solely care in regards to the group-level impact, we will take away the individual-level parameter and have the group-level coefficients as our most granular parameter. If there exists the presence of subgroups nested inside the teams, we will add one other hierarchical layer. The probabilities are limitless!

The “Bayesian” side refers to how we replace our beliefs about these parameters based mostly on noticed information. We first begin with a proposed prior distribution that characterize our preliminary perception of those parameters, then replace them iteratively to recuperate a posterior distributions that comes with the data from the information. In observe, which means that we use the global-level estimate to tell our group-level estimates, and the group-level parameters to tell the unit-level parameters. Models with a bigger variety of observations are allowed to deviate extra from the group-level means, whereas models with a restricted variety of observations are pulled nearer to the means.

Let’s formalize this with our value elasticity instance, the place we (ideally) wish to recuperate the unit-level value elasticity. We estimate:

$$log(textrm{Models}_{it})= beta_i log(textrm{Value})_{it} +gamma_{c(i),t} + delta_i + epsilon_{it}$$

The place:

• $beta_i sim textrm{Regular}(beta_{cleft(iright)},sigma_i)$
• $beta_{c(i)}sim textrm{Regular}(beta_g,sigma_{c(i)})$
• $beta_gsim textrm{Regular}(mu,sigma)$

The one distinction from the primary equation is that we change the worldwide $beta$ time period with product-level betas $beta_i$. We specify that the unit stage elasticity $beta_i$ is drawn from a traditional distribution centered across the category-level elasticity common $beta_{c(i)}$, which is drawn from a shared international elasticity $beta_g$ for all teams. For the unfold of the distribution $sigma$, we will assume a hierarchical construction for that too, however on this instance, we simply set fundamental priors for them to take care of simplicity. For this utility, we assume a previous perception of: ${ mu= -2, sigma= 1, sigma_{c(i)}=1, sigma_i=1}$. This formulation of the prior assumes that the worldwide elasticity is elastic, 95% of the elasticities fall between -4 and 0, with a typical deviation of 1 at every hierarchical stage. To check whether or not these priors are accurately specified, we might do a prior predictive checks (not coated on this article) to see whether or not our prior beliefs can recuperate the information that we observe.

This hierarchical construction permits info to stream between merchandise in the identical class and throughout classes. If a specific product has restricted value variation information, its elasticity shall be pulled towards the class elasticity $beta_{c(i)}$. Equally, classes with fewer merchandise shall be influenced extra by the worldwide elasticity, which derives its imply from all class elasticities. The great thing about this method is that the diploma of “pooling” occurs robotically based mostly on the information. Merchandise with plenty of value variation will keep estimates nearer to their particular person information patterns, whereas these with sparse information will borrow extra power from their group.

Implementation

On this part, we implement the above mannequin utilizing the Numpyro bundle in Python, a light-weight probabilistic programming language powered by JAX for autograd and JIT compilation to GPU/TPU/CPU. We begin off by producing our artificial information, defining the mannequin, and becoming the mannequin to the information. We shut out with some visualizations of the outcomes.

Knowledge Producing Course of

We simulate gross sales information the place demand follows a log-linear relationship with value and the product-level elasticity is generated from a Gaussian distribution $beta_i sim textrm{Regular}(-2, 0.7)$. We add in a random value change each time interval with a $50%$ likelihood, category-specific time traits, and random noise. This provides in multiplicatively to generate our log anticipated demand. From the log anticipated demand, we exponentiate to get the precise demand, and draw realized models offered from a Poisson distribution. We then filter to maintain solely models with greater than 100 models offered (helps accuracy of estimates, not a crucial step), and are left with $N=11,798$ merchandise over $T = 156$ durations (weekly information for 3 years). From this dataset, the true international elasticity is $beta_g = -1.6$, with category-level elasticities starting from $beta_{c(i)} in [-1.68, -1.48]$.

Take into account that this DGP ignores lots of real-world intricacies. We don’t mannequin any elements that would collectively have an effect on each costs and demand (similar to promotions), and we don’t mannequin any confounders. This instance is solely meant to point out that we will recuperate the product-specific elasticity below a wells-specified mannequin, and doesn’t intention to cowl methods to accurately determine that value is exogenous. Nevertheless, I recommend that readers check with Causal Inference for the Courageous and True for an introduction to causal inference.

import numpy as np
import pandas as pd

def generate_price_elasticity_data(N: int = 1000,
                                   C: int = 10,
                                   T: int = 50,
                                   price_change_prob: float = 0.2,
                                   seed = 42) -> pd.DataFrame:
    """
    Generate artificial information for value elasticity of demand evaluation.
    Knowledge is generated by
    """
    if seed will not be None:
        np.random.seed(seed)
    
    # Class demand and traits
    category_base_demand = np.random.uniform(1000, 10000, C)
    category_time_trends = np.random.uniform(0, 0.01, C)
    category_volatility = np.random.uniform(0.01, 0.05, C)  # Random volatility for every class
    category_demand_paths = np.zeros((C, T))
    category_demand_paths[:, 0] = 1.0
    shocks = np.random.regular(0, 1, (C, T-1)) * category_volatility[:, np.newaxis]
    traits = category_time_trends[:, np.newaxis] * np.ones((C, T-1))
    cumulative_effects = np.cumsum(traits + shocks, axis=1)
    category_demand_paths[:, 1:] = category_demand_paths[:, 0:1] + cumulative_effects

    # product results
    product_categories = np.random.randint(0, C, N)
    product_a = np.random.regular(-2, .7, measurement=N)
    product_a = np.clip(product_a, -5, -.1)
    
    # Preliminary costs for every product
    initial_prices = np.random.uniform(100, 1000, N)
    costs = np.zeros((N, T))
    costs[:, 0] = initial_prices
    
    # Generate random values and whether or not costs modified
    random_values = np.random.rand(N, T-1)
    change_mask = random_values < price_change_prob
    
    # Generate change elements (-20% to +20%)
    change_factors = 1 + np.random.uniform(-0.2, 0.2, measurement=(N, T-1))
    
    # Create a matrix to carry multipliers
    multipliers = np.ones((N, T-1))
    
    # Apply change elements solely the place adjustments ought to happen
    multipliers[change_mask] = change_factors[change_mask]
    
    # Apply the adjustments cumulatively to propagate costs
    for t in vary(1, T):
        costs[:, t] = costs[:, t-1] * multipliers[:, t-1]
    
    # Generate product-specific multipliers
    product_multipliers = np.random.lognormal(3, 0.5, measurement=N)
    # Get time results for every product's class (form: N x T)
    time_effects = category_demand_paths[product_categories][:, np.newaxis, :].squeeze(1)
    
    # Guarantee time results do not go damaging
    time_effects = np.most(0.1, time_effects)
    
    # Generate interval noise for all merchandise and time durations
    period_noise = 1 + np.random.uniform(-0.05, 0.05, measurement=(N, T))
    
    # Get class base demand for every product
    category_base = category_base_demand[product_categories]
    
    # Calculate base demand
    base_demand = (category_base[:, np.newaxis] *
                   product_multipliers[:, np.newaxis] *
                   time_effects *
                   period_noise)

    # log demand
    alpha_ijt = np.log(base_demand)

    # log value
    log_prices = np.log(costs)

    # log anticipated demand
    log_lambda = alpha_ijt + product_a[:, np.newaxis] * log_prices

    # Convert again from log area to get fee parameters
    lambda_vals = np.exp(log_lambda)

    # Generate models offered
    units_sold = np.random.poisson(lambda_vals)  # Form: (N, T)
    
    # Create index arrays for all combos of merchandise and time durations
    product_indices, time_indices = np.meshgrid(np.arange(N), np.arange(T), indexing='ij')
    product_indices = product_indices.flatten()
    time_indices = time_indices.flatten()
    
    # Get classes for all merchandise
    classes = product_categories[product_indices]
    
    # Get all costs and models offered
    all_prices = np.spherical(costs[product_indices, time_indices], 2)
    all_units_sold = units_sold[product_indices, time_indices]
    
    # Calculate elasticities
    product_elasticity = product_a[product_indices]

    df = pd.DataFrame({
        'product': product_indices,
        'class': classes,
        'time_period': time_indices,
        'value': all_prices,
        'units_sold': all_units_sold,
        'product_elasticity': product_elasticity
    })
    
    return df

# Maintain solely models with >X gross sales
def filter_dataframe(df, min_units = 100):
    temp = df[['product','units_sold']].groupby('product').sum().reset_index()
    unit_filter = temp[temp.units_sold>min_units]['product'].distinctive()
    filtered_df = df[df['product'].isin(unit_filter)].copy()

    # Abstract
    original_product_count = df['product'].nunique()
    remaining_product_count = filtered_df['product'].nunique()
    filtered_out = original_product_count - remaining_product_count
    
    print(f"Filtering abstract:")
    print(f"- Unique variety of merchandise: {original_product_count}")
    print(f"- Merchandise with > {min_units} models: {remaining_product_count}")
    print(f"- Merchandise filtered out: {filtered_out} ({filtered_out/original_product_count:.1%})")

    # International and class elasticity
    global_elasticity = filtered_df['product_elasticity'].imply()
    filtered_df['global_elasticity'] = global_elasticity

    # Class elasticity
    category_elasticities = filtered_df.groupby('class')['product_elasticity'].imply().reset_index()
    category_elasticities.columns = ['category', 'category_elasticity']
    filtered_df = filtered_df.merge(category_elasticities, on='class', how='left')

    # Abstract
    print(f"nElasticity Data:")
    print(f"- International elasticity: {global_elasticity:.3f}")
    print(f"- Class elasticities vary: {category_elasticities['category_elasticity'].min():.3f} to {category_elasticities['category_elasticity'].max():.3f}")
    
    return filtered_df

df = generate_price_elasticity_data(N = 20000, T = 156, price_change_prob=.5, seed=42)
df = filter_dataframe(df)
df.loc[:,'cat_by_time'] = df['category'].astype(str) + '-' + df['time_period'].astype(str)
df.head()

Filtering abstract:

• Unique variety of merchandise: 20000
• Merchandise with > 100 models: 11798
• Merchandise filtered out: 8202 (41.0%)

Elasticity Data:

• International elasticity: -1.598
• Class elasticities vary: -1.681 to -1.482

Mannequin

We start by creating indices for merchandise, classes, and category-time combos utilizing pd.factorize(). This enables us to seelct the right parameter for every commentary. We then convert the value (logged) and models collection into JAX arrays, then create plates that corresponds to every of our parameter teams. These plates retailer the parameters values for every stage of the hierarchy, together with storing the parameters representing the fastened results.

The mannequin makes use of NumPyro’s plate to outline the parameter teams:

• global_a: 1 international value elasticity parameter with a $textrm{Regular}(-2, 1)$ prior.
• category_a: $C=10$ category-level elasticities with priors centered on the worldwide parameter and commonplace deviation of 1.
• product_a: $N=11,798$ product-specific elasticities with priors centered on their respective class parameters and commonplace deviation of 1.
• product_effect: $N=11,798$ product-specific baseline demand results with a typical deviation of three.
• time_cat_effects: $(T=156)cdot(C=10)$ time-varying results particular to every category-time mixture with a typical deviation of three.

We then reparameterize the parameters utilizing The LocScaleReparam() argument to enhance sampling effectivity and keep away from funneling. After creating the parameters, we calculate log anticipated demand, then convert it again to a fee parameter with clipping for numerical stability. Lastly, we name on the information plate to pattern from a Poisson distribution with the calculated fee parameter. The optimization algorithm will then discover the values of the parameters that greatest match the information utilizing stochastic gradient descent. Under is a graphical illustration of the mannequin to point out the connection between the parameters.

import jax
import jax.numpy as jnp
import numpyro
import numpyro.distributions as dist
from numpyro.infer.reparam import LocScaleReparam

def mannequin(df: pd.DataFrame, final result: None):
    # Outline indexes
    product_idx, unique_product = pd.factorize(df['product'])
    cat_idx, unique_category = pd.factorize(df['category'])
    time_cat_idx, unique_time_cat = pd.factorize(df['cat_by_time'])

    # Convert the value and models collection to jax arrays
    log_price = jnp.log(df.value.values)
    final result = jnp.array(final result) if final result will not be None else None

    # Generate mapping
    product_to_category = jnp.array(pd.DataFrame({'product': product_idx, 'class': cat_idx}).drop_duplicates().class.values, dtype=np.int16)

    # Create the plates to retailer parameters
    category_plate = numpyro.plate("class", unique_category.form[0])
    time_cat_plate = numpyro.plate("time_cat", unique_time_cat.form[0])
    product_plate = numpyro.plate("product", unique_product.form[0])
    data_plate = numpyro.plate("information", measurement=final result.form[0])

    # DEFINING MODEL PARAMETERS
    global_a = numpyro.pattern("global_a", dist.Regular(-2, 1), infer={"reparam": LocScaleReparam()})

    with category_plate:
        category_a = numpyro.pattern("category_a", dist.Regular(global_a, 1), infer={"reparam": LocScaleReparam()})

    with product_plate:
        product_a = numpyro.pattern("product_a", dist.Regular(category_a[product_to_category], 2), infer={"reparam": LocScaleReparam()})
        product_effect = numpyro.pattern("product_effect", dist.Regular(0, 3))

    with time_cat_plate:
        time_cat_effects = numpyro.pattern("time_cat_effects", dist.Regular(0, 3))

    # Calculating anticipated demand
    def calculate_demand():
        log_demand = product_a[product_idx]*log_price + time_cat_effects[time_cat_idx] + product_effect[product_idx]
        expected_demand = jnp.exp(jnp.clip(log_demand, -4, 20)) # clip for stability 
        return expected_demand

    demand = calculate_demand()

    with data_plate:
        numpyro.pattern(
            "obs",
            dist.Poisson(demand),
            obs=final result
        )
    
numpyro.render_model(
    mannequin=mannequin,
    model_kwargs={"df": df,"final result": df['units_sold']},
    render_distributions=True,
    render_params=True,
)

Estimation

Whereas there are a number of methods to estimate this equation, we use Stochastic Variational Inference (SVI) for this explicit utility. As an outline, SVI is a gradient-based optimization technique to reduce the KL-divergence between a proposed posterior distribution to the true posterior distribution by minimizing the ELBO. This can be a totally different estimation method from Markov-Chain Monte Carlo (MCMC), which samples instantly from the true posterior distribution. In real-world purposes, SVI is extra environment friendly and simply scales to massive datasets. For this utility, we set a random seed, initialize the information (household of posterior distribution, assumed to be a Diagonal Regular), outline the educational fee schedule and optimizer in Optax, and run the optimization for 1,000,000 (takes ~1 hour) iterations. Whereas the mannequin might need converged beforehand, the loss nonetheless improves by a minor quantity even after operating the optimization for 1,000,000 iterations. Lastly, we plot the (log) losses.

from numpyro.infer import SVI, Trace_ELBO, autoguide, init_to_sample
import optax
import matplotlib.pyplot as plt

rng_key = jax.random.PRNGKey(42)
information = autoguide.AutoNormal(mannequin, init_loc_fn=init_to_sample)
# Outline a studying fee schedule
learning_rate_schedule = optax.exponential_decay(
    init_value=0.01,
    transition_steps=1000,
    decay_rate=0.99,
    staircase = False,
    end_value = 1e-5,
)

# Outline the optimizer
optimizer = optax.adamw(learning_rate=learning_rate_schedule)
svi = SVI(mannequin, information, optimizer, loss=Trace_ELBO(num_particles=8, vectorize_particles = True))

# Run SVI
svi_result = svi.run(rng_key, 1_000_000, df, df['units_sold'])
plt.semilogy(svi_result.losses);

Recovering Posterior Samples

As soon as the mannequin has been educated, we will can recuperate the posterior distribution of the parameters by feeding within the ensuing parameters and the preliminary dataset. We can not name the parameters svi_result.params instantly since Numpyro makes use of an affline transformation on the back-end for non-Regular distributions. Due to this fact, we pattern 1000 occasions from the posterior distribution and calculate the imply and commonplace deviation of every parameter in our mannequin. The ultimate a part of the next code creates a dataframe with the estimated elasticity for every product at every hierarchical stage, which we then be part of again to our unique dataframe to check whether or not the algorithm recovers the true elasticity.

predictive = numpyro.infer.Predictive(
    autoguide.AutoNormal(mannequin, init_loc_fn=init_to_sample),
    params=svi_result.params,
    num_samples=1000
)

samples = predictive(rng_key, df, df['units_sold'])

# Extract means and std dev
outcomes = {}
excluded_keys = ['product_effect', 'time_cat_effects']
for ok, v in samples.gadgets():
    if ok not in excluded_keys:
        outcomes[f"{k}"] = np.imply(v, axis=0)
        outcomes[f"{k}_std"] = np.std(v, axis=0)

# product elasticity estimates
prod_elasticity_df = pd.DataFrame({
    'product': df['product'].distinctive(),
    'product_elasticity_svi': outcomes['product_a'],
    'product_elasticity_svi_std': outcomes['product_a_std'],
})
result_df = df.merge(prod_elasticity_df, on='product', how='left')

# Class elasticity estimates
prod_elasticity_df = pd.DataFrame({
    'class': df['category'].distinctive(),
    'category_elasticity_svi': outcomes['category_a'],
    'category_elasticity_svi_std': outcomes['category_a_std'],
})
result_df = result_df.merge(prod_elasticity_df, on='class', how='left')

# International elasticity estimates
result_df['global_a_svi'] = outcomes['global_a']
result_df['global_a_svi_std'] = outcomes['global_a_std']
result_df.head()

Outcomes

The next code plots the true and estimated elasticities for every product. Every level is ranked by their true elasticity worth (black), and the estimated elasticity from the mannequin can also be proven. We are able to see that the estimated elasticities follows the trail of the true elasticities, with a Imply Absolute Error of round 0.0724. Factors in purple represents merchandise whose 95% CI doesn’t include the true elasticity, whereas factors in blue characterize merchandise whose 95% CI incorporates the true elasticity. Provided that the worldwide imply is -1.598, this represents a mean error of 4.5% on the product stage. We are able to see that the SVI estimates intently observe the sample of the true elasticities however with some noise, notably because the elasticities turn out to be an increasing number of damaging. On the highest proper panel, we plot the connection between the error of the estimated elasticities and the true elasticity values. As true elasticities turn out to be an increasing number of damaging, our mannequin turns into much less correct.

For the category-level and global-level elasticities, we will create the posteriors utilizing two strategies. We are able to both boostrap all product-level elasticities inside the class, or we will get the category-level estimates instantly from the posterior parameters. After we have a look at the category-level elasticity estimates on the underside left, we will see that the each the category-level estimates recovered from the mannequin and the bootstrapped samples from the product-level elasticities are additionally barely biased in the direction of zero, with an MAE of ~.033. Nevertheless, the boldness interval given by the category-level parameter covers the true parameter, in contrast to the bootstrapped product-level estimates. This implies that when figuring out group-level elasticities, we must always instantly use the group-level parameters as an alternative of bootstrapping the extra granular estimates. When wanting on the international stage, each strategies incorporates the true parameter estimate within the 95% confidence bounds, with the worldwide parameter out-performing the product-level bootstrapping, at the price of having bigger commonplace errors.

Issues

1. HB underestimates posterior variance: One downside of utilizing SVI for the estimation is that it underestimates the posterior variance. Whereas we’ll cowl this subject intimately in a later article, the target perform for SVI solely takes under consideration the distinction in expectation of our posited distribution and the true distribution. Because of this it doesn’t think about the complete correlation construction between parameters within the posterior. The mean-field approximation generally utilized in SVI assumes conditional (on the earlier hierarchy’s draw) independence between parameters, which ignores any covariances between merchandise inside the identical hierarchy. Because of this if are any spillover results (similar to cannibalization or cross-price elasticity), it could not be accounted for within the confidence bounds. On account of this mean-field assumption, the uncertainty estimates are usually overly assured, leading to confidence intervals which might be too slim and fail to correctly seize the true parameter values on the anticipated fee. We are able to see within the high left determine that solely 9.7% of the product-level elasticities cowl their true elasticity. In a later publish, we’ll embody some options to this drawback.
2. Significance of priors: When utilizing HB, priors matter considerably extra in comparison with commonplace Bayesian approaches. Whereas massive datasets usually permit the probability to dominate priors when estimating international parameters, hierarchical buildings adjustments this dynamic and cut back the efficient pattern sizes at every stage. In our mannequin, the worldwide parameter solely sees 10 category-level observations (not the complete dataset), classes solely draw from their contained merchandise, and merchandise rely solely on their very own observations. This lowered efficient pattern measurement causes shrinkage, the place outlier estimates (like very damaging elasticities) get pulled towards their class means. This highlights the significance of prior predictive checks, since misspecified priors can have outsized affect on the outcomes.

def elasticity_plots(result_df, outcomes=None):
    # Create the determine with 2x2 grid
    fig = plt.determine(figsize=(12, 10))
    gs = fig.add_gridspec(2, 2)
    
    # product elasticity
    ax1 = fig.add_subplot(gs[0, 0])
    
    # Knowledge prep
    df_product = result_df[['product','product_elasticity','product_elasticity_svi','product_elasticity_svi_std']].drop_duplicates()
    df_product['product_elasticity_svi_lb'] = df_product['product_elasticity_svi'] - 1.96*df_product['product_elasticity_svi_std']
    df_product['product_elasticity_svi_ub'] = df_product['product_elasticity_svi'] + 1.96*df_product['product_elasticity_svi_std']
    df_product = df_product.sort_values('product_elasticity')
    mae_product = np.imply(np.abs(df_product.product_elasticity-df_product.product_elasticity_svi))
    colours = []
    for i, row in df_product.iterrows():
        if (row['product_elasticity'] >= row['product_elasticity_svi_lb'] and 
            row['product_elasticity'] <= row['product_elasticity_svi_ub']):
            colours.append('blue')  # Inside CI bounds
        else:
            colours.append('purple')   # Outdoors CI bounds
    
    # Share of factors inside bounds
    within_bounds_pct = colours.depend('blue') / len(colours) * 100
    
    # Plot information
    ax1.scatter(vary(len(df_product)), df_product['product_elasticity'], 
                colour='black', label='True Elasticity', s=20, zorder=3)
    
    ax1.scatter(vary(len(df_product)), df_product['product_elasticity_svi'], 
                colour=colours, label=f'SVI Estimate (MAE: {mae_product:.4f}, Protection: {within_bounds_pct:.1f}%)', 
                s=3, zorder=2)
    ax1.set_xlabel('Product Index (sorted by true elasticity)')
    ax1.set_ylabel('Elasticity Worth')
    ax1.set_title('SVI Estimates of True Product Elasticities')
    ax1.legend()
    ax1.grid(alpha=0.3)
    
    # Relationship between MAE and true elasticity
    ax2 = fig.add_subplot(gs[0, 1])
    
    # Calculate MAE for every product
    temp = result_df[['product','product_elasticity', 'product_elasticity_svi']].drop_duplicates().copy()
    temp['product_error'] = temp['product_elasticity'] - temp['product_elasticity_svi']
    temp['product_mae'] = np.abs(temp['product_error'])
    correlation = temp[['product_mae', 'product_elasticity']].corr()
    
    # Plot information
    ax2.scatter(temp['product_elasticity'], temp['product_error'], alpha=0.5, s=5, colour = colours)
    ax2.set_xlabel('True Elasticity')
    ax2.set_ylabel('Error (True - Estimated)')
    ax2.set_title('Relationship Between True Elasticity and Estimation Accuracy')
    ax2.grid(alpha=0.3)
    ax2.textual content(0.5, 0.95, f"Correlation: {correlation.iloc[0,1]:.3f}", 
                rework=ax2.transAxes, ha='heart', va='high',
                bbox=dict(boxstyle='spherical', facecolor='white', alpha=0.7))
    
    # Class Elasticity
    ax3 = fig.add_subplot(gs[1, 0])
    
    # Distinctive classes and elasticities
    category_data = result_df[['category', 'category_elasticity', 'category_elasticity_svi', 'category_elasticity_svi_std']].drop_duplicates()
    category_data = category_data.sort_values('category_elasticity')
    
    # Bootstrapped means from product elasticities inside every class
    bootstrap_means = []
    bootstrap_ci_lower = []
    bootstrap_ci_upper = []
    
    for cat in category_data['category']:
        # Get product elasticities for this class
        prod_elasticities = result_df[result_df['category'] == cat]['product_elasticity_svi'].distinctive()
        
        # Bootstrap means
        boot_means = [np.mean(np.random.choice(prod_elasticities, size=len(prod_elasticities), replace=True)) 
                     for _ in range(1000)]
        
        bootstrap_means.append(np.imply(boot_means))
        bootstrap_ci_lower.append(np.percentile(boot_means, 2.5))
        bootstrap_ci_upper.append(np.percentile(boot_means, 97.5))
    
    category_data['bootstrap_mean'] = bootstrap_means
    category_data['bootstrap_ci_lower'] = bootstrap_ci_lower
    category_data['bootstrap_ci_upper'] = bootstrap_ci_upper
    
    # Calculate MAE
    mae_category_svi = np.imply(np.abs(category_data['category_elasticity_svi'] - category_data['category_elasticity']))
    mae_bootstrap = np.imply(np.abs(category_data['bootstrap_mean'] - category_data['category_elasticity']))
        
    # Plot the information
    left_offset = -0.2
    right_offset = 0.2
    x_range = vary(len(category_data))
    ax3.scatter(x_range, category_data['category_elasticity'], 
                colour='black', label='True Elasticity', s=50, zorder=3)
    
    # Bootstrapped product elasticity
    ax3.scatter([x + left_offset for x in x_range], category_data['bootstrap_mean'], 
                colour='inexperienced', label=f'Bootstrapped Product Estimate (MAE: {mae_bootstrap:.4f})', s=30, zorder=2)
    for i in x_range:
        ax3.plot([i + left_offset, i + left_offset], 
                [category_data['bootstrap_ci_lower'].iloc[i], category_data['bootstrap_ci_upper'].iloc[i]], 
                colour='inexperienced', alpha=0.3, zorder=1)
    
    # category-level SVI estimates
    ax3.scatter([x + right_offset for x in x_range], category_data['category_elasticity_svi'], 
                colour='blue', label=f'Class SVI Estimate (MAE: {mae_category_svi:.4f})', s=30, zorder=2)
    for i in x_range:
        ci_lower = category_data['category_elasticity_svi'].iloc[i] - 1.96 * category_data['category_elasticity_svi_std'].iloc[i]
        ci_upper = category_data['category_elasticity_svi'].iloc[i] + 1.96 * category_data['category_elasticity_svi_std'].iloc[i]
        ax3.plot([i + right_offset, i + right_offset], [ci_lower, ci_upper], colour='blue', alpha=0.3, zorder=1)

    ax3.set_xlabel('Class Index (sorted by true elasticity)')
    ax3.set_ylabel('Elasticity')
    ax3.set_title('Comparability with True Class Elasticity')
    ax3.legend()
    ax3.grid(alpha=0.3)

    # international elasticity
    ax4 = fig.add_subplot(gs[1, 1])
    temp = result_df[['product','product_elasticity_svi','global_elasticity']].drop_duplicates()
    bootstrap_means = [np.mean(np.random.choice(np.array(temp['product_elasticity_svi']), 100)) for i in vary(10000)]
    global_means = np.random.regular(result_df['global_a_svi'].iloc[0], result_df['global_a_svi_std'].iloc[0], 10000)
    true_global = np.distinctive(temp.global_elasticity)[0]
    p_mae = np.abs(np.imply(bootstrap_means) - true_global)
    g_mae = np.abs(np.imply(global_means) - true_global)
    
    # Plot information
    ax4.hist(bootstrap_means, bins=100, alpha=0.3, density=True, 
             label=f'Product Elasticity Bootstrap (MAE: {p_mae:.4f})')
    ax4.hist(global_means, bins=100, alpha=0.3, density=True, 
             label=f'International Elasticity Distribution (MAE: {g_mae:.4f})')
    ax4.axvline(x=true_global, colour='black', linestyle='--', linewidth=2, 
                label=f'International Elasticity: {true_global:.3f}', zorder=0)
    ax4.set_xlabel('Elasticity')
    ax4.set_ylabel('Frequency')
    ax4.set_title('International Elasticity Comparability')
    ax4.legend()
    ax4.grid(True, linestyle='--', alpha=0.7)

    # Present
    plt.tight_layout()
    plt.present()

elasticity_plots(result_df)

Conclusion

Alternate Makes use of: Other than estimating value elasticity of demand, HB fashions even have a wide range of different makes use of in Knowledge Science. In retail, HB fashions can forecast demand for present shops and resolve the cold-start drawback for brand new shops by borrowing info from shops/networks which have already been established and are clustered inside the identical hierarchy. For suggestion techniques, HB can estimate user-level preferences from a mix of person and item-level traits. This construction permits related suggestions to new customers based mostly on cohort behaviors, step by step transitioning to individualized suggestions as person historical past accumulates. If no cohort groupings are simply obtainable, Okay-means can be utilized to group comparable models based mostly on their traits.

Lastly, these fashions may also be used to mix outcomes from experimental and observational research. Scientists can use historic observational uplift estimates (advertisements uplift) and complement it with newly developed A/B checks to cut back the required pattern measurement for experiments by incorporating prior information. This method creates a steady studying framework the place every new experiment builds upon earlier findings slightly than ranging from scratch. For groups dealing with useful resource constraints, this implies sooner time-to-insight (particularly when mixed with surrogate fashions) and extra environment friendly experimentation pipelines.

Closing Remarks: Whereas this introduction has highlighted a number of purposes of hierarchical Bayesian fashions, we’ve solely scratched the floor. We haven’t deep dived into granular implementation elements similar to prior and posterior predictive checks, formal goodness-of-fit assessments, computational scaling, distributed coaching, efficiency of estimation methods (MCMC vs. SVI), and non-nested hierarchical buildings, every of which deserves their very own publish.

However, this overview ought to present a sensible start line for incorporating hierarchical Bayesian into your toolkit. These fashions provide a framework for dealing with (normally) messy, multi-level information buildings which might be typically seen in real-world enterprise issues. As you start implementing these approaches, I’d love to listen to about your experiences, challenges, successes, and new use instances for this class of mannequin, so please attain out with questions, insights, or examples by my e-mail or LinkedIn. If in case you have any suggestions on this text, or wish to request one other subject in causal inference/machine studying, please additionally be happy to achieve out. Thanks for studying!

Note: All photos used on this article is generated by the writer.