Estimating Heterogeneous Birth Probabilities using the EM Algorithm: A Probabilistic Approach to Population Dynamics Modeling

Ecology, epidemiology, demography, and evolutionary biology are only a few of the many scientific disciplines that face the fundamental problem of modelling population dynamics. Birth is a crucial part of these models since it determines how populations develop and change over time. All people are considered to have the same reproductive capacity in traditional population models, which leads them to presume that birth probability are uniform. Nevertheless, it is important to note that this assumption is greatly simplified because there is a great deal of genetic, environmental, behavioural, and social variation in actual populations (Böhning, D. 2000). Birth rates can vary among members of the same community for a variety of reasons, including but not limited to age, health, social standing, and the availability of resources. Neglecting to account for this diversity might cause the models to draw biassed or oversimplified conclusions, which in turn reduces their predictive potential and practicality. Due to the complexity of real-world populations, statistical frameworks that can explicitly describe and predict diverse birth probabilities are in high demand (Caswell, H. 2001).

Using probabilistic models in conjunction with the Expectation-Maximization (EM) method is one potential way to tackle this difficulty. For maximum likelihood estimate using latent (unobserved) variables or partial data, the well-established EM algorithm iterative technique can be used. It is possible that researchers may not have access to all of the necessary data, such as complete birth records or aggregate counts, to completely disclose the underlying heterogeneity when studying population dynamics, and that individual birth probabilities are latent variables affected by unobserved factors (Rubin, D. B. 1977). Up until convergence, the EM method repeatedly refines parameter estimates by alternating between the E-step, which involves estimating the expected values of the latent variables, and the M-step, which involves maximising the probability of the observed data given these estimates. For models where concealed or incomplete data makes direct likelihood maximisation problematic, EM's repeated refinement is a great fit (Dunson, D. B., Vehtari, A., 2013).

Let us pretend that we are dealing with a population of size N, where the birth probability pi is unknown and may differ from one person to the next. Assuming that the birth events of each individual follow a Bernoulli distribution with parameter pi allows one to describe the heterogeneity of birth probability (Samanta, T. 2006). Here, Xi is the outcome of the delivery, where 1 is the result if the person gives birth within a specific time range and 0 is the result otherwise. Keeping an eye on the numbers  considering the likelihood of birthis expressed as:

When individual probabilities are unseen and diverse, it is tough to directly estimate the vector p. A more reasonable assumption would be that these probabilities come from a latent subpopulation with its own unique birth probability parameter, or that they are chosen from a mixed distribution (Brooks, S. P. 2009). As an example, one may propose a finite mixture model that assumes the population is divided into K subgroups, each with its own birth probability.  and mixing proportions ​ such that  Then, each person's birth is simulated as belonging to one of these subgroups, with the memberships of these subgroups being hidden (unknown). Although the EM framework finds this latent structure accessible, it hinders likelihood maximisation.

Assuming that each individual iii is a member of subgroup k, the complete-data probability may be expressed as:

while maximising the anticipated log-likelihood, and adjusting the parameters (M-step):

The EM approach eventually finds parameter estimates that represent the subpopulation-specific heterogeneity in birth probability by repeatedly applying these stages. A wide range of ecological and demographic datasets, including those with missing or noisy individual data, can benefit from this probabilistic mixture modelling approach because it permits a flexible representation of population heterogeneity without requiring explicit individual-level covariates (Peel, D. 2000).

For a better grasp of population dynamics, the capacity to represent diverse birth probabilities is crucial. Because it takes into account different ways of reproducing, it improves our capacity to foretell how populations will expand, remain stable, or react to environmental changes Doak, (D. F. 2002). Conservation biology, public health planning, and resource management rely heavily on these improved models because they help to explain the range of birth rates, which is essential for making policy choices and targeted actions. Research on population dynamics may be taken to new heights with the help of the EM algorithm since its probabilistic framework can be expanded to include more complicated issues, such as birth rates that change over time, covariate effects, or the possibility of combined modelling of mortality and migration (Newman, M. E. J. 2010).

Dorazio, R. M. (2008) Typical deterministic methods used in population dynamics models presume that people act consistently, especially with regard to reproduction and birth rates. The importance of individual-level variability, which is crucial for correct population process modelling, has been highlighted in recent advances, nevertheless. There are a lot of environmental and biological variables that contribute to differences in reproductive success, such as age, genetic diversity, social structure, and resource availability. Management plans and projections made without taking this variability into account are likely to be unsuccessful. For accurate predictions, modelling frameworks need to take variety into consideration, as population patterns in biological and human systems are getting more and more complicated. By include unpredictability in birth probability and capturing the stochastic character of individual behaviour, probabilistic models have become excellent instruments to tackle this difficulty. To better comprehend population dynamics, these models permit more adaptable and data-driven methods. To better understand the ecological limits and evolutionary factors that impact reproduction, scientists have begun to incorporate individual variability into their models. In conservation planning, these kinds of frameworks are extremely helpful for determining how to distribute resources across subpopulations based on their fertility rates.

Otto, S. P., & Day, T. (2007) As computing resources and longitudinal data have become more accessible, the application of probabilistic approaches in ecological and demographic studies has grown swiftly. Incorporating the intrinsic variability and uncertainty seen in populations, these models provide a significant departure from deterministic forecasts. The ability of probabilistic modelling to capture uncertainty in characteristics such as migration rates, birth rates, and mortality rates is one of its main advantages. The use of probability distributions in birth modelling allows for a more comprehensive description of the range of reproductive behaviours by individuals or groups in terms of the likelihood of reproductive occurrences. These methods shine when dealing with populations or species that exhibit highly varied life histories, as average rates miss important dynamics. Probabilistic models also make it easier to test hypotheses and make predictions when dealing with incomplete or censored data, which are typical issues in demography research conducted in the field. These models are getting better at representing the complexity of biology by adding features like hierarchical structures, latent variables, and time-varying parameters. The advancement of Bayesian and machine learning approaches, which enhance demographic inference even more, is in line with this probabilistic trend.

K. H., & Norris, J. L. (2003) In statistical analysis, the Expectation-Maximization (EM) method has emerged as a crucial tool for dealing with incomplete, unobserved, or latent data. When direct computing is not an option, its iterative nature offers a realistic alternative to maximum likelihood estimation. Partial survey answers, unregistered births, or undetected subgroup memberships are frequent sources of incomplete data in population modelling. To fill these gaps, the EM method optimises the likelihood function (M-step) based on estimated anticipated values of hidden variables (E-step). Because of this, it is highly useful for predicting hidden variation in birth probability. The technique improves parameter estimates with each iteration by giving probabilistic weights to unseen categories, including behavioural characteristics or subgroup identities. Stable and interpretable results, especially when dealing with sparse data, are guaranteed by its ability to converge to a local maximum of the likelihood function. In addition, EM-based models may be modified to account for mixture distributions, in which each individual is believed to be a member of many subpopulations with different birth probability. Because of these features, the EM algorithm is highly appealing to demographic researchers who are tackling complicated real-world data.

Turchin, P. (2003) A statistical framework is provided by mixture models and latent class techniques to handle unobserved variability within populations. Parameters in these models, including birth probabilities, are thought to be derived from a combination of several underlying subpopulations. Since real-world populations frequently include several invisible demographic or behavioural strata, this way of thinking is in line with reality. Mixture models break down complicated distributions into smaller parts by estimating parameters for each class and assigning probability to each individual's probable subgroup membership. These models, when applied to birth data, can reveal subgroups with high or low fertility, as well as other hidden reproductive patterns that conventional approaches can miss. By going a step further and presenting a probabilistic categorisation of people, the latent class method sheds light on the structure and variety of reproductive behaviour. When it is not feasible or practical to directly determine subgroup identification, these strategies are especially helpful in demographic studies that employ surveys or follow participants over time. In addition, by taking the uncertainty in subgroup membership into consideration, parameter estimation is improved when iterative estimate methods like the EM algorithm are used in conjunction with mixed models.

Pollock, K. H. (2002) A paradigm shift has occurred in the analysis and interpretation of demographic processes due to the incorporation of statistical learning techniques into population biology. Datasets pertaining to fertility and population increase are among the many huge and complicated ones that have seen an uptick in the use of machine learning, Bayesian inference, and probabilistic modelling for the purpose of extracting meaningful patterns. When the fundamental processes are driven by latent variables or when typical parametric assumptions do not apply, these approaches become extremely effective. Statistical learning enables the discovery of hidden correlations between personal traits and gestational behaviour in birth modelling. Unsupervised learning and clustering algorithms, for instance, may identify subgroups in a population with drastically different birth probability, all without making any assumptions about what constitutes a subgroup. Furthermore, these methods are capable of dealing with ecological and human population research' typical problems, such as high-dimensional data, missing values, and nonlinear interactions. Combining statistical learning approaches with probabilistic algorithms, such as EM, improves model interpretability, prediction accuracy, and parameter estimation. As a result, scientists are better able to construct data-driven models that can adjust to the intricate nature of real-world systems.

The theoretical and computational foundation for estimating the odds of heterogeneous births in a population with various subgroups is described in this section. The key concept is to represent the population as a combination of subpopulations, with different birth probabilities for each. We use the Expectation-Maximization (EM) technique to estimate the maximum likelihood even when partial data is available, because subpopulation membership is latent.

Probabilistic Model Formulation

We think of a population that is split into K hidden subpopulations, and each of these subpopulations has its unique birth probability θk.  Denoted as Xi, the observed birth outcome (e.g., birth incidence within a defined time frame) for the i-th individual, with N being the total number of people

We presuppose the subsequent creative procedure:

• For each individual i, a latent variable indicates membership in one of the K subpopulations, with prior probabilities satisfying
•   Given  the birth event  Bernoulli(θk​).

We want to estimate the parameters of this mixture model, which captures birth process heterogeneity ​ from observed birth data N​, where the subpopulation labels are unobserved.

Likelihood Function

If the Zi's were observed, the complete-data probability would be:

With Zi as a margin, the incomplete-data probability is:

Since the latent structure makes direct maximisation impossible, we use the EM method instead.

Expectation-Maximization (EM) Algorithm

Until convergence, the EM algorithm repeats two stages recursively:

E-Step (Expectation Step):

Find the likelihood (or responsibility) that person iii is a part of subpopulation k, using the present parameter estimations Ϙ(t):

M-Step (Maximization Step):

Revise the settings by referring to the duties

In relation to the parameters, these revisions improve the predicted complete-data log-likelihood to the greatest extent possible.

Convergence Criteria:

The process is repeated until:

A predetermined threshold ϵ is not exceeded when the increase in the log-likelihood between rounds falls below it.

Minimal changes have been made to the parameters

Simulation and Estimation Procedure

We estimate with the suggested EM-based method and do simulations of synthetic datasets with known parameters to verify the model.

Step-by-step process:

1. Data Generation:

·        Choose

·        Sample  Categorical for each individual.

·        Generate Bernoulli

2. Initialization:

·         Start with values determined by chance or by using a heuristic (like k-means clustering on X)

3. EM Execution:

• E-step and M-step should be repeated.
• To keep track of convergence, record the log-likelihood at each stage.

4. Model Selection:

• Utilise model selection techniques like the Bayesian Information Criterion (BIC) or the Akaike Information Criterion (AIC) to ascertain the ideal number of components, K.

5. Performance Metrics:

• Compare the estimated parameters with the real values (in the simulated data) to determine the correctness of the estimation.
• Using maximal a posteriori assignment, determine how well subpopulation labels classify:

Even in cases when subgroup identities are unknown and only binary birth outcomes are known, our EM-based probabilistic model may effectively estimate the populations' heterogeneous birth probability.

Here we show the outcomes of the EM-based estimate approach applied to both simulated and real-world demographic data. The accuracy of parameter estimate, resilience to sample size and noise, competence of model selection, and adaptability to actual population data are the main considerations in the evaluation. Furthermore, we investigate the algorithm's ability to reveal latent heterogeneity in birth probability, a crucial component of accurate population modelling and strategic policymaking.

Synthetic Data Experiments

In order to thoroughly assess the suggested approach, we initially built synthetic population datasets using known ground truths. Subpopulations with different birth probability were represented in each dataset, which was created using a specified blend of Bernoulli distributions. At first, we zeroed down on a two-part mixture model with parameters:

·         (mixing proportions),

·         (birth probabilities),

·         individuals.

We conducted the EM technique using a random starting point in fifty separate simulations and recorded parameter estimations when they converged. The mean absolute error (MAE) between the estimated and true values, as well as the standard deviation and mean of the estimations, are summarised in Table 4.1.

Table 1. Parameter Estimation Accuracy for Simulated Data (K = 2, N = 1000)

The outcomes demonstrate that the EM algorithm reliably approached values close to the actual truth. Minimal and consistent estimating errors were seen throughout all runs. These results confirm that the method is statistically sound when tested in controlled settings. Figure 4.1 shows that the log-likelihood values rose monotonically with each repetition.

Figure 1: Convergence of Log-Likelihood Over Iterations

Unsupervised clustering of people based on their observed binary birth outcomes was also effective, with a classification accuracy of subpopulation membership of 92.8% on average (using maximum a posteriori estimate).

Real-World Case Study: Rural Fertility Patterns

We used actual demographic data from a rural India socioeconomic survey that included 1,200 married women between the ages of 15 and 49 to evaluate the model's practical applicability. A binary variable was used to record each woman's birth result during the last two years. Given the lack of subpopulation markers, unsupervised learning is the most appropriate approach.

The results are presented in Table 2, which shows the use of the EM method with K=2.

Table 2. Estimated Parameters from Real-World Fertility Data

The findings showed that there were two hidden categories with very differing rates of birth. These results are consistent with the known age, education, and income-based demographic fertility stratifications. Approximately one-third of the participants were in the group with a lower birth rate, which is in line with the current tendency for some rural communities to delay having children and utilise contraception.

When stratified data are not available, the model's utility in demographic inference is demonstrated by its ability to detect heterogeneity without previous labelling or grouping.

Sensitivity Analysis

We ran sensitivity studies by changing the sample size and the number of components K to see how resilient the method was. Table 3 displays the impact of sample size on the accuracy of estimations for a given K=2.

Table 3: Mean Absolute Error (MAE) vs. Sample Size

Errors decreased by about 65% from N=500 to N=2000, which is consistent with expectations that bigger samples will enhance estimation accuracy. The EM method grows well and gains advantages from bigger population data, as seen here.

We also checked if the model could accurately determine how many subpopulations there actually are. In order to compare fitted models with K=2,3, and 4, we first created synthetic data with K=3 components and then utilised the Bayesian Information Criterion (BIC). Table 4.4 displays the average outcomes.

Table 4: BIC Values for Model Selection (True K=3)

We can validate that our technique can detect the number of unique reproductive subgroups with little overfitting, since the BIC properly favoured the real model in over 92% of trials.

Discussion

Convergence speed was mainly unaffected by initialisation, and the EM algorithm converged within 20-25 iterations for the majority of datasets. As anticipated in mixture models with overlapping components, convergence slowed and parameter identifiability declined in situations when birth probabilities were quite near, such as θ1 = 0.45 and θ2 = 0.55. In situations where subpopulations were found, the posterior probability γik, as determined by entropy, showed distinct divisions. An important indicator of model dependability, entropy rose for ambiguous data, suggesting uncertainty in subpopulation assignment.

A strong probabilistic framework for modelling population dynamics with individual-level variability is presented in the paper on calculating heterogeneous birth probability using the EM method. Using the Expectation-Maximization method, the model overcomes the shortcomings of conventional homogenous assumptions by accurately capturing the latent variation in birth rates among subpopulations. As is typical in demographic research, the EM method iteratively refines parameter estimates, allowing for reliable inference even when data is insufficient or noisy. More realistic and nuanced insights into population dynamics are provided by this strategy, which improves the capacity to anticipate patterns of population increase by directly incorporating uncertainty and variability into the model structure. In addition, the method can handle enormous datasets seen in sociological and ecological research since it is computationally efficient and scalable. Customised studies that account for biological, environmental, or social aspects are also made possible by the probabilistic framework, which allows for the insertion of variables impacting birth probability. For a more complete picture of population changes and their predictions, this study recommends using a mix of statistical learning methods and demographic modelling. This technique might be further developed in future studies by including migration and mortality dynamics or by using hierarchical models to account for variation at several levels.