Conjugate Priors for Cluster-Specific Omics Profiles

Why Put Priors on Cluster Profiles?

In a Bayesian integrative clustering model, a sample’s cluster assignment determines which cluster-specific omics profile describes it. The cluster prevalence model concerns how common each cluster is; the profile model concerns the feature distributions within each cluster. Here the focus is on priors for the parameters of those feature distributions and on how observations update those priors.

A conjugate prior is a prior distribution whose posterior, after combining it with the specified likelihood, belongs to the same distribution family. Conjugacy gives transparent updates that can be computed from simple data summaries. It does not establish that the likelihood is scientifically adequate, nor does it remove the need to choose and assess prior parameters.

Let \(z_i=k\) mean that sample \(i\) is assigned to cluster \(k\). For a particular feature, the cluster data are the observations \(y_i\) for which \(z_i=k\); let \(n_k\) be their number. The updates below condition on these assignments. If assignments are uncertain, an update conditional on one allocation is not the same as a posterior that accounts for allocation uncertainty.

Gaussian Feature Means

For a continuous feature, suppose the observations in cluster \(k\) are conditionally independent and normally distributed with a common, known variance \(\sigma^2\) and unknown cluster mean \(\mu_k\). Use a Normal prior with mean \(m_0\) and variance \(s_0^2\). The prior variance \(s_0^2\) describes uncertainty about the cluster mean; it is distinct from the observation variance \(\sigma^2\).

The formula shows how precision, the reciprocal of variance, combines: posterior precision is prior precision plus the total data precision. Equivalently, the posterior mean is a precision-weighted average of the prior mean and the cluster sample mean. This result requires the observation variance to be known and shared across the observations in this update. It is not, by itself, an update for an unknown variance.

Poisson Count Rates

For a count feature, suppose counts from samples assigned to cluster \(k\) are conditionally independent Poisson observations with a common unknown rate \(\lambda_k\). Use a Gamma prior parameterized by shape \(a\) and rate \(b\), so its density is proportional to \(\lambda_k^{a-1}\exp(-b\lambda_k)\) for \(\lambda_k>0\). Under this convention, the prior mean is \(a/b\); \(b\) is a rate parameter, not a scale.

This form assumes equal exposure for each observation, or that the count has already been defined on a common exposure scale. If sample \(i\) has known exposure \(e_i\) and \(Y_i\sim\operatorname{Poisson}(e_i\lambda_k)\), the corresponding rate update instead has shape \(a+\sum_iY_i\) and rate \(b+\sum_i e_i\). Unequal sequencing depth or other exposure differences should not be ignored when the model requires them.

Bernoulli Probabilities

For a binary feature, let \(Y_i=1\) indicate the presence of a specified event and \(Y_i=0\) its absence. Suppose observations assigned to cluster \(k\) are independent Bernoulli draws with unknown probability \(p_k\) of a positive outcome. A Beta prior with positive shape parameters \(a\) and \(b\) is conjugate to this likelihood. Its mean is \(a/(a+b)\).

Why the Updates Work

The shared mechanism is multiplication of prior and likelihood. For the Gaussian model, keeping only terms involving \(\mu_k\), the log posterior density is $$ -\frac{1}{2s_0^2}(\mu_k-m_0)^2 -\frac{1}{2\sigma^2}\sum_{i:z_i=k}(y_i-\mu_k)^2 +\text{constant}. $$ The coefficient of \(-\mu_k^2/2\) is \(1/s_0^2+n_k/\sigma^2\), and the coefficient of \(\mu_k\) is \(m_0/s_0^2+\sum_i y_i/\sigma^2\). Completing the square therefore gives a Normal density with variance \(s_k^2=(1/s_0^2+n_k/\sigma^2)^{-1}\) and mean \(s_k^2(m_0/s_0^2+\sum_i y_i/\sigma^2)\), exactly the update above.

For the Poisson model, the likelihood terms depending on \(\lambda_k\) are proportional to \(\lambda_k^{\sum_iY_i}\exp(-n_k\lambda_k)\). Multiplying by the Gamma prior kernel gives $$ \lambda_k^{a+\sum_iY_i-1}\exp\{-(b+n_k)\lambda_k\}, $$ which is the kernel of a Gamma distribution with the stated shape and rate. For the Bernoulli model, the likelihood kernel is \(p_k^{r_k}(1-p_k)^{n_k-r_k}\). Multiplication by the Beta prior kernel gives \(p_k^{a+r_k-1}(1-p_k)^{b+n_k-r_k-1}\), the kernel of the stated Beta posterior. In both cases, the observed counts enter as sufficient summaries for these parameter updates.

Using Conjugacy in a Multi-Omics Profile

In a mixed-type cluster model, different features can have different likelihoods: a continuous feature may use a Gaussian model, a count feature a Poisson model, and a binary feature a Bernoulli model. Assigning a corresponding prior to each feature yields a set of feature-wise updates, conditional on the shared sample assignments and the model’s assumptions about dependence. For example, the conjugate update for one binary feature uses its own positive and negative counts; it does not require treating its numerical codes as Gaussian measurements.

Feature-wise conjugacy is simplest when features are modeled as conditionally independent given the cluster and their parameters, with independent priors on those parameters. If the model instead links feature parameters, or represents dependence between features, the marginal update for one feature may no longer have the simple form shown here. These formulas also assume that the observations assigned to the cluster are the observations appropriate for the likelihood. Missingness, normalization, and exposure must be handled consistently with the measurement model.

Diagnostics, Alternatives, and Limitations

Conjugacy makes updates easy to calculate, but it is not a diagnostic for model fit. Assess whether the chosen sampling distribution is plausible for each feature and whether its assumptions are credible in the cluster. For Gaussian features, inspect whether a shared known variance and a roughly Gaussian conditional distribution are defensible; strong outliers or markedly different spreads can make the mean update misleading. For counts, check whether variation is compatible with the Poisson assumption that conditional variance equals conditional mean, and whether exposure differences or excess zeros are important. For binary features, verify the outcome coding and consider whether samples plausibly share one probability within the cluster.

Prior choices also matter, especially when a cluster contains few observations. The Normal prior’s \(m_0\) and \(s_0^2\), Gamma prior’s \(a\) and \(b\), and Beta prior’s \(a\) and \(b\) affect the degree of shrinkage. Compare sensible prior settings and examine whether conclusions change materially. A proper prior with positive variance or positive shape parameters gives a well-defined update in these models, but a mathematically valid prior is not automatically scientifically appropriate.

Alternatives include nonconjugate priors or richer likelihoods that can represent extra variation, dependence, or uncertainty in additional parameters. Such choices can be more realistic but generally do not yield these simple closed-form updates. A key limitation here is that the Gaussian result treats \(\sigma^2\) as known; it does not account for uncertainty in that variance. Likewise, a single Poisson rate and a single Bernoulli probability may be too restrictive when samples differ systematically in exposure or other measured conditions.

• Mistake: interpreting a conjugate formula as evidence that its likelihood assumptions are true. Correction: conjugacy is an algebraic property of the prior-likelihood pair; assess the measurement model separately.
• Mistake: using the same update for unequal count exposures. Correction: include the known exposures in the Poisson likelihood; the posterior rate parameter then increases by their sum.
• Mistake: treating an update conditional on one cluster allocation as if it fully accounts for uncertain assignments. Correction: the formulas condition on the assigned samples; uncertainty in assignments requires accounting for alternative allocations in the broader model.