AUTOZI

AUTOZI [#ref1]_ (Python class scvi.model.AUTOZI) is a model for assessing gene-specific levels of zero-inflation in scRNA-seq data.

Generative process

AUTOZI is very similar to scVI but employs a spike-and-slab prior for the zero-inflation mixture assignment for each gene. Whether the zero-inflation rate (\(\pi_{ng}\) in the original scVI model) is sampled from a set of non-negligible values (the “slab” component) or the set of negligible values (the “spike” component) is defined by \(m_g \sim Bernoulli(\delta_g)\) where \(\delta_g \sim Beta(\alpha, \beta)\). Thus, for each gene \(g\), the zero-inflation rate is defined, \(\pi_{ng} = (1-m_g)\pi_{ng}^{slab} + m_g \pi_{ng}^{spike}\).

The full generative model is as follows:

\begin{align} z_n &\sim N(0,I)\\ l_n &\sim LogNormal(l_u, l_\sigma^2)\\ \delta_g &\sim Beta(\alpha^g,\beta^g)\\ m_g &\sim Bernoulli(\delta_g)\\ \pi _{ng} &=( 1-m_{g}) \delta _{\{0\}} +m_{g} \delta _{\{h^{g}( z_{n})\}}\\ x_{ng}|z_n,l_n,m_g &\sim ZINB(l_nw_g(z_n), \theta_g, \pi_{ng})\\ \end{align}

Where \(w^g\) and \(h^g\) are neural networks taking in \(z_n\) and outputting the dropout rate and library size frequency respectively. The priors \(l_u\) and \(l_{\sigma^2}\) are the empircal mean and variance of the log library size per batch respectively. The priors for \(\delta_g\) are \(\alpha^g\) and \(\beta^g\) which by default are both set to 0.5 to enforce sparsity while maintaining symmetry. Finally, \(\delta_{\{x\}}\) denotes the Dirac distribution on \(x\).

Inference Procedure

To learn the parameters, we employ variational inference (see Variational Inference) with the following approximate posterior distribution:

\begin{align*} \bar{q} &= \prod ^{G}_{g=1} q( \delta _{g})\prod ^{N}_{n=1} q( z_{n} |x_{n}) q( l_{n} |x_{n}) \end{align*}

Tasks

To classify whether a gene \(g\) is or is not zero inflated, we call:

>>> outputs = model.get_alpha_betas()
>>> alpha_posterior = outputs['alpha_posterior']
>>> beta_posterior = outputs['beta_posterior']

Then Bayesian decision theory suggests the posterior probability of of zero-inflation is \(q(\delta_g < 0.5)\).

>>> from scipy.stats import beta
>>> threshold = 0.5
>>> zi_probs = beta.cdf(0.5, alpha_posterior, beta_posterior)