Google Merdian Budget Optimization when KPI is non revenue

[Romi-del]

Hi Google,
I tried multiple iterations when kpi is non revenue and seeing in flexible budget it does not allocate like when kpi is revenue and it does not show cpik. Is fixed budget is used for Google Merdian to see how budget is reallocated across channel. Since I have seen in fixed budget they are allocating rather than flexible budget

[cpulavarthi]

Hello @Romi-del,

Thank you for contacting us!

Fixed budget optimization in Meridian allows you to maximize the ROI or incremental KPI in a fixed budget scenario. The optimal allocation is found by maximizing the return on investment whenever the kpi_type is either revenue or when the kpi_type is non-revenue but revenue_per_kpi is known. When the kpi_type is set to non-revenue and revenue_per_kpi is unknown, the fixed budget optimization finds the optimal allocation that maximizes the incremental KPI.

Flexible budget optimization isn’t the default option in Meridian, you need to set fixed_budget = False while running the optimize() method to run a flexible budget optimization. Flexible budget optimization allows you to input either a specific minimum ROI target or a marginal ROI target to optimize the budget allocation.

You can refer to our documentation on budget optimization scenarios for more information and sample code snippets to run each type of optimization.

Since the flexible budget optimization requires either a target ROI or mROI, and the workings are straightforward when the KPI is revenue. When the KPI is non-revenue, then the ROI is defined as Incremental KPI per Cost (IKPC). In this scenario, ROI is expressed as KPI units per dollar spent. The metric CPIK is the reciprocal of this ROI. So if you run flexible budget optimization using a standard ROI constraint like target_roi=1.0, then you are essentially instructing the optimizer to find a budget where you get 1.0 conversion per $1 spent (i.e., a $1.00 CPA). If your actual CPA (CPIK) is $50.00, your actual ROI (IKPC) is (1 / 50 = 0.02). Because a target of (1.0) is mathematically impossible for your channels to achieve, the flexible budget optimizer fails to allocate budget or returns zero. To run a flexible budget optimization for a non-revenue KPI, you must set your target_roi as the reciprocal of your target CPIK (target Cost Per Acquisition).

You can extract the calculated CPIK values from optimization results which contain the cpik metric as follows:

non_optimized_df = optimization_results.nonoptimized_data.to_dataframe()
optimized_df = optimization_results.optimized_data.to_dataframe()

Is fixed budget used to see reallocation?

Yes. In a Fixed Budget Scenario, the total budget is constrained to a static limit (typically your historical spend), and the optimizer reallocates and shifts budget across channels to maximize your total incremental KPI. This is why you see reallocation occurring immediately in your fixed budget runs. It is the correct scenario to use if your total budget is locked and your goal is to find the most efficient channel mix.

Hope this answers all of your questions. Feel free to reach out if you have any further questions or suggestions for us.

Google Meridian Support Team

[Romi-del]

Hi just one another question when KPI is non revenue where KPI is some tickets_sold like that, I have tried using this code: [import numpy as np
import tensorflow_probability as tfp
import tensorflow as tf
from meridian.model import prior_distribution

p_mean = 0.30 # prior mean proportion of KPI incremental due to all media
p_sd = 0.25 # prior std dev proportion of KPI incremental to all media
kpi_total = df['consideration_score'].sum()

Calculate total media spend

total_media_spend = df[media_spends].sum().sum()

Calculate roi_mean

roi_mean = p_mean * kpi_total / total_media_spend

Calculate the sum of squares of total spend for each media channel

sum_of_squared_total_spends = np.sum(np.power(df[media_spends].sum(), 2))

Calculate roi_sd

roi_sd = p_sd * kpi_total / np.sqrt(sum_of_squared_total_spends)

lognormal_sigma = np.sqrt(np.log(roi_sd2 / roi_mean2 + 1))
lognormal_mu = np.log(roi_mean * np.exp(-lognormal_sigma**2 / 2))

Correctly wrap the tfp.distributions.LogNormal within meridian.model.prior_distribution.PriorDistribution

roi_prior = prior_distribution.PriorDistribution(
roi_m=tfp.distributions.LogNormal(
loc=np.float32(lognormal_mu),
scale=np.float32(0.08),
),

slope_m=tfp.distributions.LogNormal(
    loc=tf.math.log(2.5),   # stronger saturation
    scale=0.1
),

ec_m=tfp.distributions.LogNormal(
    loc=tf.math.log(0.15),
    scale=0.1
),

# Added beta_m prior for the Hill function shape parameter
beta_m=tfp.distributions.LogNormal(
    loc=tf.math.log(1.0), # Common starting point for Hill function shape
    scale=0.1
)

)] to set p_mean and p_sd and using that I am getting positive baseline and incremental, but the real problem is with CPIK since kpi is non revenue I am using CPIK as a parameter but I am getting an infinite CPIK using this log normal distribution in media. Could you please what distribution and whether should we take prior or contribution in model spec and what distribution non media should follow here, such that I can avoid the infinite CPIK.

[cpulavarthi]

Hello @Romi-del,

Thank you for sharing the details of your model setup.

The "infinite CPIK" you are encountering happens when the Incremental KPI (the denominator of the CPIK calculation) for a channel evaluates to exactly zero. Because CPIK is Cost / Incremental KPI, a zero in the denominator results in infinity. This typically occurs when a prior is overly restrictive or improperly scaled, causing the MCMC sampler to collapse a channel's contribution to zero.

Looking at your code, there are a few structural issues and deviations from Meridian's best practices that are likely contributing to this mathematical impossibility. Here is how you can adjust your setup to resolve this:

1. Hardcoded scale in the prior definition: You calculated lognormal_sigma based on your total media spend and KPI variance, but you hardcoded scale=np.float32(0.08) when defining the roi_m prior. An arbitrary scale of 0.08 is extremely narrow and is likely forcing the distribution into a space that prevents the model from properly estimating incremental KPI, causing the zero-denominator issue.
2. Prior Selection: It seems to me like you are trying to set a custom total paid media contribution prior for your model. However, you defined both roi_m and beta_m priors in your PriorDistribution block. Meridian will only use one of these depending on how you set the media_prior_type parameter in your ModelSpec ('roi', 'contribution', or 'coefficient'), and it will ignore the others. Please check our documentation on custom total paid media contribution priors to adjust your code accordingly. The same page also has information on custom IKPC priors and channel-level contribution priors. You may choose to use one of them if you wish to incorporate your intuition on channel-wise performance.
3. You set the slope_m prior to be a LogNormal distribution. We strongly recommend against customizing slope_m priors, as the default Deterministic(1.0) distribution is used to ensure concave Hill curves which in turn allows the budget optimization algorithm to produce a global optimum.
4. For the non-media variables, Meridian uses the TruncatedNormal(0.0, 0.1, -1.0, 1.0) prior as a default. Non-media variables are allowed to have negative contributions by default and we recommend using the same. If you have prior knowledge that the contribution of a particular variable is strictly positive or negative, this should be incorporated into the prior.

By removing the hardcoded scale and relying on Meridian's default shape parameters (slope_m), your model should stabilize, preventing the Incremental KPI from collapsing to zero and resolving the infinite CPIK issue.

Please let us know if the issue persists after making these adjustments!

Google Meridian Support Team