Model architecture

[dipanshugithub1]

Hi team, just wanted to clear some doubts regarding some meridian parameters:
1> When we are creating a GEO level hierarchical model, we are using the input data from a particular geo to regress against the target data from that particular GEO(with global priors) so that would mean that we are creating a different mu(t)(intercept) for every GEO and the only data which would be used to calculate the intercept would be from that particular GEO, right?
2> When creating knots for a time period with geo level hierarchical models (data with say 4 different geographies), are we considering all geographies together at a particular time point (4 records) or just that that particular geography for that time point (essentially 1 record)
( The article says: There are situations where you must set knots < n_times, for example, in a national-level model where you don't have multiple observations per time period and there are not enough degrees of freedom for each time period to get its own parameter. Note that some level of dimensionality reduction is necessary.)
3> The final intercept for a particular GEO would be the sum of the baseline intercept(tau(g)) for that GEO along with the time varying intercept?
4> Are we using some sort of global prior across the prior distributions for the latent variables of time varying intercept?

[dattatreyam23]

Hi @dipanshugithub1,

Thank you for contacting us.

The main difference between a hierarchical geo-level model and separate models for each geo is the sharing of information across geos. The geo-level hierarchical model assumes that the geo-level parameters (such as intercepts, media coefficients, control variables, etc.) are related to each other through a common underlying distribution. This information sharing leads to shrinkage effects and extreme geos are pulled toward the overall trend. Unless there are inherent differences across geos, this would lead to increased model stability and better estimation of causal effects.

Here’s a brief explanation of your questions:

1. Geo-level intercepts: No! μ₍t₎ describes the overall temporal effect and is not geo-specific. With that said, data in all geos will be used to estimate μ₍t₎.

2. Knots in multi-geo vs. single-geo: In multi-geo models (e.g. 4 geos), each time point has multiple observations (one per geo), so you can safely use knots = n_times. In contrast, single-geo models have just one data point per time, so fewer knots are needed to avoid overfitting. To answer the question directly, yes, a knot is set for all geos, not for a particular one.

   Using a geo-level model allows you to have more degrees of freedom in comparison to national-level models. If there are similarities between your geos in terms of the media impact mechanism and there is a requirement to accurately model the time-effects, then using a single geo-level hierarchical model should be preferred as it can allow you to increase the number of knots to model the $\mu_t$ parameter. Please check our documentation on choosing no. of knots for a detailed description.

3. Final intercept composition: Yes, the model’s total intercept for a geo and time is the sum of the geo baseline and the time-varying trend: τ₍g₎ + μ₍t₎. Refer to this related discussion for more information.

4. Priors on time knots: The latent time-knot parameters (bₖ) each have independent Normal priors. They share the same prior distribution but are not linked hierarchically. Check our intercept documentation for a more precise outlook.

Feel free to reach out if you have any questions or suggestions regarding Meridian.

Thank you,

Google Meridian Support Team

[dipanshugithub1]

Hi @dattatreyam23,
Thankyou for the response,
I just have some more doubts,
suppose i have 400 records, 100 records for each geo, now i want to make the time varying model for some media variable so i will have some two types of distributions for that media media variable the first is a distributions to capture time effect and the other to capture geo effect. Now how many y-likelihood equations would be there, if there is only one y-likelihood eq(i.e. 400 records are regressed at once) then should i multiply my transformed data for that media variable to both types of distributions or just the geo type distributions. And how will my data be arranged 4 records for a particular date together and a total of 100 knots which would mean 100 latent variables? so the first record of mu-likelihood will look like the data for that record multiplied to the geo level distribution and the latent variables along with their weights( the question is the latent variables and their weights will be same for the next 3 records as well right, because they are the same time, right?)
Thankyou guys for the patience and help.

[dattatreyam23]

Hello @dipanshugithub1,

Thank you for reaching out!

For starters you may check the sample geo-level dataset shared in GitHub for an example of how the data should be arranged for a geo-level model. You can see in this dataset that each (geo, time) point will be a separate record (row) in the dataset. The number of knots is customizable, if you have 100 time periods for a geo-level model the model defaults to setting 100 knots.

In GBHMMM, we assume the media impact mechanism is the same across all geos. So, the data from all geos for a particular time period is used together in estimating the model parameters. Consequently, as an example, a single posterior knots distribution at a particular knot location is obtained after bayesian updating using the data of all geos at that knot location. Only the geo intercept will vary for each geo, the rest of the model parameters will be shared across all geos in the model. Additionally, the time effects are only captured using knots, the media coefficients aren't assumed to be time-varying.

Coming to your queries regarding distributions and likelihood functions, I don't understand why you mentioned two types of distributions for a media variable. How are time and geos involved in the distribution? The likelihood function of y, by definition, is the product of individual density functions y_{g,t}. Therefore, there's only one likelihood function.

I look forward to your response so that I can better understand your query and help clarify the underlying workings of Meridian.

Google Meridian Support Team