Estimating Product-Stage Value Elasticities Utilizing Hierarchical Bayesian

$gamma_{c(i),t}$ is a set of category-by-time dummy variables to seize the typical 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 customary and customary in lots of regression-based fashions to regulate for unobserved confounders. This (pooled) regression mannequin permits us to get better the typical elasticity $beta$ throughout all $N$ items. This might imply that the shop may goal a mean value degree 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 items have a pure grouping (product classes), we would be capable of establish the typical elasticity of every class by working separate regressions (or interacting the value elasticity with the product class) for every class utilizing solely items from that class. This might 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 knowledge, we may even run separate regressions for every particular person product to acquire extra granular elasticities.

Nonetheless, real-world knowledge usually presents challenges: some merchandise have minimal value variation, brief gross sales histories, or class imbalance throughout classes. Underneath these restrictions, working separate regressions to establish product elasticity would seemingly result in giant customary 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 completely different groupings whereas preserving heterogeneity. With the HB formulation, it’s attainable to run one single regression (just like the pooled case) whereas nonetheless recovering elasticities on the product degree, permitting for granular optimizations.

Understanding Hierarchical Bayesian Fashions

At its core, HB is about recognizing the pure construction in our knowledge. Relatively than treating all observations as utterly unbiased (many separate regressions) or forcing them to observe similar patterns (one pooled regression), we acknowledge that observations can cluster into teams, with merchandise inside every group sharing related patterns. The “hierarchical” side refers to how we set up our parameters in several ranges. In its most simple format, we may have:

• A International parameter that applies to all knowledge.
• 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 degree of pooling. For instance, if we predict there are not any similarities throughout classes, we may take away the worldwide parameter. If we predict 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 throughout the teams, we will add one other hierarchical layer. The chances are countless!

The “Bayesian” side refers to how we replace our beliefs about these parameters primarily based on noticed knowledge. We first begin with a proposed prior distribution that symbolize our preliminary perception of those parameters, then replace them iteratively to get better a posterior distributions that includes the data from the info. In apply, because of this 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 items 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) need to get better 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 degree 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 primary priors for them to keep up 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 degree. To check whether or not these priors are appropriately specified, we’d do a prior predictive checks (not lined on this article) to see whether or not our prior beliefs can get better the info 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 knowledge, its elasticity will probably be pulled towards the class elasticity $beta_{c(i)}$. Equally, classes with fewer merchandise will probably be influenced extra by the worldwide elasticity, which derives its imply from all class elasticities. The fantastic thing about this strategy is that the diploma of “pooling” occurs robotically primarily based on the info. Merchandise with a lot of value variation will keep estimates nearer to their particular person knowledge patterns, whereas these with sparse knowledge 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 knowledge, defining the mannequin, and becoming the mannequin to the info. We shut out with some visualizations of the outcomes.

Information Producing Course of

We simulate gross sales knowledge 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 developments, 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 items bought from a Poisson distribution. We then filter to maintain solely items with greater than 100 items bought (helps accuracy of estimates, not a needed step), and are left with $N=11,798$ merchandise over $T = 156$ durations (weekly knowledge for 3 years). From this dataset, the true international elasticity is $beta_g = -1.6$, with category-level elasticities starting from $beta_{c(i)} in [-1.68, -1.48]$.

Remember the fact that this DGP ignores a variety of real-world intricacies. We don’t mannequin any components that would collectively have an effect on each costs and demand (akin to promotions), and we don’t mannequin any confounders. This instance is solely meant to indicate that we will get better the product-specific elasticity beneath a wells-specified mannequin, and doesn’t goal to cowl find out how to appropriately establish that value is exogenous. Nonetheless, I counsel 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 knowledge for value elasticity of demand evaluation.
    Information is generated by
    """
    if seed is just not None:
        np.random.seed(seed)
    
    # Class demand and developments
    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]
    developments = category_time_trends[:, np.newaxis] * np.ones((C, T-1))
    cumulative_effects = np.cumsum(developments + 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, dimension=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 components (-20% to +20%)
    change_factors = 1 + np.random.uniform(-0.2, 0.2, dimension=(N, T-1))
    
    # Create a matrix to carry multipliers
    multipliers = np.ones((N, T-1))
    
    # Apply change components solely the place modifications ought to happen
    multipliers[change_mask] = change_factors[change_mask]
    
    # Apply the modifications 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, dimension=N)
    # Get time results for every product's class (form: N x T)
    time_effects = category_demand_paths
[:, np.newaxis, :].squeeze(1)
    
    # Guarantee time results do not go unfavourable
    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, dimension=(N, T))
    
    # Get class base demand for every product
    category_base = category_base_demand

    
    # 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 house to get charge parameters
    lambda_vals = np.exp(log_lambda)

    # Generate items bought
    units_sold = np.random.poisson(lambda_vals)  # Form: (N, T)
    
    # Create index arrays for all mixtures 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 items bought
    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

# Hold solely items 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"- Authentic variety of merchandise: {original_product_count}")
    print(f"- Merchandise with > {min_units} items: {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:

• Authentic variety of merchandise: 20000
• Merchandise with > 100 items: 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 mixtures utilizing pd.factorize(). This permits us to seelct the proper parameter for every commentary. We then convert the value (logged) and items collection into JAX arrays, then create plates that corresponds to every of our parameter teams. These plates retailer the parameters values for every degree of the hierarchy, together with storing the parameters representing the mounted 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 customary deviation of 1.
• product_a: $N=11,798$ product-specific elasticities with priors centered on their respective class parameters and customary 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 charge parameter with clipping for numerical stability. Lastly, we name on the info plate to pattern from a Poisson distribution with the calculated charge parameter. The optimization algorithm will then discover the values of the parameters that finest match the info utilizing stochastic gradient descent. Under is a graphical illustration of the mannequin to indicate 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 items collection to jax arrays
    log_price = jnp.log(df.value.values)
    final result = jnp.array(final result) if final result is just not 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("knowledge", dimension=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 summary, SVI is a gradient-based optimization methodology to attenuate the KL-divergence between a proposed posterior distribution to the true posterior distribution by minimizing the ELBO. It is a completely different estimation method from Markov-Chain Monte Carlo (MCMC), which samples straight from the true posterior distribution. In real-world purposes, SVI is extra environment friendly and simply scales to giant 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 charge 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 working 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 charge 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 get better the posterior distribution of the parameters by feeding within the ensuing parameters and the preliminary dataset. We can’t name the parameters svi_result.params straight 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 customary 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 degree, 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 okay, v in samples.gadgets():
    if okay 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 will see that the estimated elasticities follows the trail of the true elasticities, with a Imply Absolute Error of round 0.0724. Factors in pink represents merchandise whose 95% CI doesn’t include the true elasticity, whereas factors in blue symbolize 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 degree. We will see that the SVI estimates carefully observe the sample of the true elasticities however with some noise, significantly because the elasticities change into an increasing number of unfavourable. On the highest proper panel, we plot the connection between the error of the estimated elasticities and the true elasticity values. As true elasticities change into an increasing number of unfavourable, our mannequin turns into much less correct.

For the category-level and global-level elasticities, we will create the posteriors utilizing two strategies. We will both boostrap all product-level elasticities throughout the class, or we will get the category-level estimates straight from the posterior parameters. After we take 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. Nonetheless, 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 straight use the group-level parameters as an alternative of bootstrapping the extra granular estimates. When trying on the international degree, 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 customary errors.

Issues

1. HB underestimates posterior variance: One downside of utilizing SVI for the estimation is that it underestimates the posterior variance. Whereas we are going to cowl this subject intimately in a later article, the target operate 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 total 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 throughout the similar hierarchy. Because of this if are any spillover results (akin to cannibalization or cross-price elasticity), it might not be accounted for within the confidence bounds. Because of this mean-field assumption, the uncertainty estimates are usually overly assured, leading to confidence intervals which can be too slim and fail to correctly seize the true parameter values on the anticipated charge. We will see within the high left determine that solely 9.7% of the product-level elasticities cowl their true elasticity. In a later publish, we are going to embrace some options to this downside.
2. Significance of priors: When utilizing HB, priors matter considerably extra in comparison with customary Bayesian approaches. Whereas giant datasets usually enable the chance to dominate priors when estimating international parameters, hierarchical buildings modifications this dynamic and scale back the efficient pattern sizes at every degree. In our mannequin, the worldwide parameter solely sees 10 category-level observations (not the total dataset), classes solely draw from their contained merchandise, and merchandise rely solely on their very own observations. This lowered efficient pattern dimension causes shrinkage, the place outlier estimates (like very unfavourable elasticities) get pulled towards their class means. This highlights the significance of prior predictive checks, since misspecified priors could 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])
    
    # Information 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('pink')   # Exterior CI bounds
    
    # Proportion of factors inside bounds
    within_bounds_pct = colours.depend('blue') / len(colours) * 100
    
    # Plot knowledge
    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 knowledge
    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}", 
                remodel=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 info
    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 knowledge
    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: Apart from estimating value elasticity of demand, HB fashions even have quite a lot of different makes use of in Information Science. In retail, HB fashions can forecast demand for current shops and clear up the cold-start downside for brand new shops by borrowing info from shops/networks which have already been established and are clustered throughout the similar hierarchy. For suggestion programs, HB can estimate user-level preferences from a mix of person and item-level traits. This construction allows related suggestions to new customers primarily based 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 related items primarily based 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 assessments to cut back the required pattern dimension for experiments by incorporating prior information. This strategy 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.

Ultimate 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 akin 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 knowledge buildings which can be usually 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 circumstances for this class of mannequin, so please attain out with questions, insights, or examples by way of 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 creator.