Introduction to fido::Pibble
Justin Silverman
Source:vignettes/introduction-to-fido.Rmd
introduction-to-fido.RmdAn introduction to fido
fido (Justin D. Silverman et al. 2019) is a loose acronym for “(Bayesian) Multinomial Logistic-Normal Models”. In particular the development of fido stems from the need for fast inference for time-invariant MALLARD models(Justin D. Silverman et al. 2018). fido is very fast! It uses closed form solutions for model gradients and Hessians written in C++ to preform MAP estimation in combination with parameter uncertainty estimation using a Laplace Approximation. One of the main models in fido is the function pibble which fits a Multinomial Logistic-Normal Linear Regression model.
So what is a fido model exactly? First let me give the broad description from 10,000ft up: Basically its a model for multinomial count data (e.g., each sample contains the counts of \(D\) “types of things”). Importantly, unlike the more common Poisson count models, the multinomial models a “competition to be counted” (i.e., cases in which counting more of one type of thing means that I have less resources available to count other types of things).
This may seem vague so let me give an example. Pretend there is a ball pit with red, green, and blue balls. Pretend that the ball pit is very large and I don’t know the total number of balls in the ball pit, yet I want to say something about the relative number of red, blue, and green balls in the pit. One way I may choose to measure the ball pit is by grabbing an armful of balls and counting the number of balls of each color (e.g., in one armful I may collect 5 red, 3 blue, and 6 green). My arms can only contain so many balls (in this example about 14) and so if I were to have (randomly) gotten another green ball in my armful (making 7 total) I would likely not have been able to measure one of the red or blue balls; hence the “competition to be counted”. It turns out that this type of sampling occurs all the time in many situations (Wikipedia has an example with political polling). Perhaps one of the most notable examples of this type of count data occurs with modern high-throughput sequencing studies such as 16S rRNA studies to profile microbial communities or bulk/single-cell RNA-seq studies to study expression profiles of cells. In all cases, transcripts are sequenced and the number of different types of transcripts are counted. The important part is that sequencing only samples a small portion of the total genetic material available and leads to similar competition to be counted.
The pibble model
Pibble is one type of fido model. In particular its a fido model for multivariate linear regression.
Let \(Y\) denote an \(D\times N\) matrix of counts. Let us denote the \(j\)-th column of \(Y\) as \(Y_j\). Thus each “sample” in the dataset is a measurement of the relative amount of \(D\) “types of things”. Suppose we also have have covariate information in the form of a \(Q\times N\) matrix \(X\).
The following is the pibble model including likelihood and priors: \[ \begin{align} Y_j & \sim \text{Multinomial}\left(\pi_j \right) \\ \pi_j & = \phi^{-1}(\eta_j) \\ \eta_j &\sim N(\Lambda X_j, \Sigma) \\ \Lambda &\sim MN_{(D-1) \times Q}(\Theta, \Sigma, \Gamma) \\ \Sigma &\sim W^{-1}(\Xi, \upsilon) \end{align} \] Here \(MN_{(D-1) \times Q}\) denotes a Matrix Normal distribution for a matrix \(\Lambda\) of regression coefficients of dimension \((D-1)\times Q\). Essentially you can think of the Matrix normal as having two covariance matrices one describing the covariation between the rows of \(\Lambda\) (\(\Sigma\)) and another describing the covariation of the columns of \(\Lambda\) (\(\Gamma\)). and \(W^{-1}\) refers to the Inverse Wishart distribution (which is a common distribution over covariance matrices). The line \(\pi_j = \phi^{-1}(\eta_j)\) represents a transformation between the parameters \(\pi_j\) which exist on a simplex (e.g., \(\pi_j\) must sum to 1) and the transformed parameters \(\eta_j\) that exist in real space. In particular we define \(\phi^{-1}\) to be the inverse additive log ratio transform (which conversely implies that \(\eta_j = ALR(\pi_j)\)) also known as the identified softmax transform (as it is more commonly known in the Machine Learning community). While I will say more on this later in this tutorial, one thing to know is that I have the model implemented using the ALR transform as it is computationally simple and fast; the results of the model can be viewed as if any number of transforms had been used (instead of the ALR) including the isometric log-ratio transform, or the centered log-ratio transform.
Before moving on, I would like to give a more intuitive description of pibble. Essentially the main modeling component of pibble is the third equation above (\(\eta_j \sim N(\Lambda X_j, \Sigma)\)) which is just a multivariate linear model. That is, \(X\) are your covariates (which can be continuous, discrete, binary, etc…), and \(\Sigma\) is the covariance matrix for the regression residuals.
Example analysis of microbiome data
This analysis is the same as that presented in the fido manuscript (Justin D. Silverman et al. 2019). I will reanalyze a previously published study comparing microbial composition in the terminal ileum of subjects with Crohn’s Disease (CD) to healthy controls (Gevers et al. 2014). To do this I will fit a pibble model using CD status, inflammation status and age as covariates (plus a constant intercept term).
library(MicrobeDS)
library(phyloseq)
library(dplyr)
library(fido)
set.seed(899)
data("RISK_CCFA")
# drop low abundant taxa and samples
dat <- RISK_CCFA %>%
subset_samples(disease_stat!="missing",
immunosup!="missing") %>%
subset_samples(diagnosis %in% c("no", "CD")) %>%
subset_samples(steroids=="false") %>%
subset_samples(antibiotics=="false") %>%
subset_samples(biologics=="false") %>%
subset_samples(biopsy_location=="Terminal ileum") %>%
tax_glom("Family") %>%
prune_samples(sample_sums(.) >= 5000,.) %>%
filter_taxa(function(x) sum(x > 3) > 0.10*length(x), TRUE)Create Design Matrix and OTU Table
sample_dat <- as.data.frame(as(sample_data(dat),"matrix")) %>%
mutate(age = as.numeric(as.character(age)),
diagnosis = relevel(factor(diagnosis, ordered = FALSE), ref="no"),
disease_stat = relevel(factor(disease_stat, ordered = FALSE), ref="non-inflamed"))
X <- t(model.matrix(~diagnosis + disease_stat+age, data=sample_dat))
Y <- otu_table(dat)
# Investigate X and Y look like
X[,1:5]
#> 1939.SKBTI.0175 1939.SKBTI047 1939.SKBTI051 1939.SKBTI063
#> (Intercept) 1.00000 1.00000 1.00 1.00000
#> diagnosisCD 1.00000 1.00000 1.00 1.00000
#> disease_statinflamed 0.00000 1.00000 1.00 1.00000
#> age 15.16667 14.33333 15.75 13.58333
#> 1939.SKBTI072
#> (Intercept) 1.00
#> diagnosisCD 1.00
#> disease_statinflamed 1.00
#> age 15.75
Y[1:5,1:5]
#> OTU Table: [5 taxa and 5 samples]
#> taxa are rows
#> 1939.SKBTI.0175 1939.SKBTI047 1939.SKBTI051 1939.SKBTI063 1939.SKBTI072
#> 4442127 0 9 0 14 2
#> 74305 1 2 35 1 0
#> 663573 36 1 0 2 1
#> 2685602 10 264 211 276 83
#> 4339819 0 37 42 70 22Next specify priors. We are going to start by specifying a prior on the covariance between log-ratios \(\Sigma\). I like to do this by thinking about a prior on the covariance between taxa on the log-scale (i.e., between the log of their absolute abundances not the log-ratios). I will refer to this covariance on log-absolute abundances \(\Omega\). For example, here I will build a prior that states that the mean of \(\Omega\) is the identity matrix \(I_D\). From From Aitchison (1986), we know that if we assume that the taxa have a covariance \(\Omega\) in terms of log-absolute abundance then their correlation in the \(\text{ALR}_D\) is given by \[ \Sigma = G \Omega G^T \] where \(G\) is a \(D-1 \times D\) matrix given by \(G = [I_{D-1}; -1_{D-1}]\) (i.e., \(G\) is the \(\text{ALR}_D\) contrast matrix). Additionally, we know that the Inverse Wishart mode is given by \(\frac{\Xi}{\upsilon + D}\). Finally, note that \(\upsilon\) essentially controls our uncertainty in \(\Sigma\) about this prior mean. Here I will take \(\upsilon = D+3\). This then gives us \(\Xi = (\upsilon - D) GIG^T\). We scale \(\Xi\) by a factor of 1/2 to make \(Tr(\Xi)=D-1\).
upsilon <- ntaxa(dat)+3
Omega <- diag(ntaxa(dat))
G <- cbind(diag(ntaxa(dat)-1), -1)
Xi <- (upsilon-ntaxa(dat))*G%*%Omega%*%t(G)Finally I specify my priors for \(\Theta\) (mean of \(\Lambda\)) and \(\Gamma\) (covariance between columns of \(\Lambda\); i.e., covariance between the covariates). I will center my prior for \(\Lambda\) about zero, and assume that the covariates are independent.
I strongly recommend users perform prior predictive checks to make
sure their priors make sense to them. fido makes this easy, all
the main fitting functions (e.g., pibble) will
automatically sample from the prior predictive distribution if
Y is left as NULL (e.g., without data your
posterior is just your prior).
priors <- pibble(NULL, X, upsilon, Theta, Gamma, Xi)
print(priors)
#> pibblefit Object (Priors Only):
#> Number of Samples: 250
#> Number of Categories: 49
#> Number of Covariates: 4
#> Number of Posterior Samples: 2000
#> Contains Samples of Parameters:Eta Lambda Sigma
#> Coordinate System: alr, reference category: 49The main fitting functions in the fido package output
special fit objects (e.g., pibble outputs an object of
class pibblefit). These fit objects are just lists with
some extra metadata that allows special method dispatch. For example, if
you call print on a pibblefit object you will get a nice
summary of what is in the object.
Note: Currently, the function pibble takes
expects inputs and outputs in the “default” coordinate system; this is
simply the ALR coordinate system where the last category (49 above) is
taken as reference (this will be generalized in future versions). More
specifically for a vector \(x\)
representing the proportions of categories \(\{1, \dots, D\}\) we can write \[x^* = \left( \log \frac{x_1}{x_D}, \dots, \log
\frac{x_{D-1}}{x_D}\right).\] As mentioned above however, I have
designed fido to work with many different coordinate systems
including the ALR (with respect to any category), CLR, ILR, or
proportions. To help transform things between these coordinate systems I
have written a series of transformation functions that transform any
pibblefit object into a desired coordinate system.
Importantly, pibblefit objects keep track of what
coordinate system they are currently in so as a user you only need to
specify the coordinate system that you want to change into. Keep in mind
that covariance matrices cannot be represented in proportions and so
visualizations or summaries based on covariance matrices will be
suppressed when pibblefit objects are in the proportions
coordinate system. As an example, lets look at viewing a summary of the
prior for \(\Lambda\) with respect to
the CLR coordinate system1.
priors <- to_clr(priors)
summary(priors, pars="Lambda", gather_prob=TRUE, as_factor=TRUE, use_names=TRUE)
#> $Lambda
#> # A tibble: 784 x 9
#> Parameter coord covariate val .lower .upper .width .point .interval
#> <chr> <int> <int> <dbl> <dbl> <dbl> <dbl> <chr> <chr>
#> 1 Lambda 1 1 0.0509 -0.528 0.596 0.5 mean qi
#> 2 Lambda 1 2 0.00199 -0.567 0.575 0.5 mean qi
#> 3 Lambda 1 3 -0.0373 -0.620 0.532 0.5 mean qi
#> 4 Lambda 1 4 -0.00205 -0.554 0.549 0.5 mean qi
#> 5 Lambda 2 1 0.00991 -0.564 0.535 0.5 mean qi
#> 6 Lambda 2 2 -0.00000782 -0.572 0.580 0.5 mean qi
#> 7 Lambda 2 3 -0.0247 -0.570 0.518 0.5 mean qi
#> 8 Lambda 2 4 0.0165 -0.536 0.589 0.5 mean qi
#> 9 Lambda 3 1 0.0138 -0.584 0.620 0.5 mean qi
#> 10 Lambda 3 2 -0.00502 -0.542 0.581 0.5 mean qi
#> # ... with 774 more rowsBy default the summary function returns a list (with
possible elements Lambda, Sigma, and
Eta) summarizing each posterior parameter based on
quantiles and mean (e.g., p2.5 is the 0.025 percentile of the posterior
distribution). As this type of table may be hard to take in due to how
large it is, pibblefit objects also come with a default
plotting option for each of the parameters. Also the returned plot
objects are ggplot objects so normal ggplot2
commands work on them. Before doing that though we are going to use one
of the names functions for pibblefit objects
to provide some more specific names for the covariates (helpful when we
then plot).
names_covariates(priors) <- rownames(X)
p <- plot(priors, par="Lambda")
#> Scale for 'colour' is already present. Adding another scale for 'colour', which will
#> replace the existing scale.
p + ggplot2::xlim(c(-10, 10)) 
This looks fairly reasonable to me. So I am going to go ahead and fit
the model with data. fido provides a helper method called
refit that we will use to avoid passing prior parameters
again.
priors$Y <- Y # remember pibblefit objects are just lists
posterior <- refit(priors, optim_method="lbfgs")Unlike the main pibble function, the refit
method can be called on objects in any coordinate system and all
transformations to and from the default coordinate system are handled
internally2. This is one nice thing about using the
refit method. That said, new objects added to the
pibblefit object need to be added in the proper coordinates
For example, if we wanted to replace our prior for \(\Xi\) for an object in CLR coordinates, we
would had to transform our prior for Xi to CLR coordinates
before adding it to the priors object.
Now I are also going to add in the taxa names to make it easier to interpret the results.
tax <- tax_table(dat)[,c("Class", "Family")]
tax <- apply(tax, 1, paste, collapse="_")
names_categories(posterior) <- taxBefore doing anything else lets look at the posterior predictive
distribution to assess model fit. This can be accessed through the
method ppc3.
ppc(posterior) + ggplot2::coord_cartesian(ylim=c(0, 30000))
There are a few things to note about this plot. First, when zoomed out like this it looks it is hard to make much of it. This is a fairly large dataset we are analyzing and its hard to view an uncertainty interval; in this case its plotting the median and 95% confidence interval in grey and black and the observed counts in green. fido also has a simpler function that summarizes the posterior predictive check.
ppc_summary(posterior)
#> Proportions of Observations within 95% Credible Interval: 0.9898776Here we see that the model appears to be fitting well (at least based on the posterior predictive check) and that only about 1.5% of observations fall outside of the 95% posterior predictive density (this is good).
Some readers will look at the above ppc plots and think
“looks like over-fitting”. However, note that there are two ways of
using ppc. One is to predict the counts based on the
samples of \(\eta\) (Eta; as we did
above); the other is to predict “from scratch” that is to predict
starting form the posterior samples of \(\Lambda\) (Lambda) then sampling \(\eta\) and only then sampling \(Y\). This later functionality can be
accessed by also passing the parameters from_scratch=TRUE
to the ppc function. Note: these two posterior predictive
checks have different meanings, one is not better than the other.
ppc(posterior, from_scratch=TRUE) +ggplot2::coord_cartesian(ylim=c(0, 30000))
ppc_summary(posterior, from_scratch=TRUE)
#> Proportions of Observations within 95% Credible Interval: 0.9721633Now we are going to finally look at the posterior distribution of our regression parameters, but because there are so many we will focus on just those that have a 95% credible interval not including zero (i.e., those that the model is fairly certain are non-zero). We are also going to ignore the intercept term and just look at parameters associated with age and disease status.
posterior_summary <- summary(posterior, pars="Lambda")$Lambda
focus <- posterior_summary[sign(posterior_summary$p2.5) == sign(posterior_summary$p97.5),]
focus <- unique(focus$coord)
plot(posterior, par="Lambda", focus.coord = focus, focus.cov = rownames(X)[2:4])
#> Scale for 'colour' is already present. Adding another scale for 'colour', which will
#> replace the existing scale.
The first, and most obvious ting to notice is that the covariate
age has pretty much no effect at all, whatever effect it
may have is incredibly weak. So we are going to remove age from the plot
and just look at those coordinates with non-zero effect for diagnosis
CD
posterior_summary <- filter(posterior_summary, covariate=="diagnosisCD")
focus <- posterior_summary[sign(posterior_summary$p2.5) == sign(posterior_summary$p97.5),]
focus <- unique(focus$coord)
tax_table(dat)[taxa_names(dat)[which(names_coords(posterior) %in% focus)]]
#> Taxonomy Table: [12 taxa by 7 taxonomic ranks]:
#> Kingdom Phylum Class Order
#> 74305 "Bacteria" "Proteobacteria" "Epsilonproteobacteria" "Campylobacterales"
#> 4449236 "Bacteria" "Proteobacteria" "Betaproteobacteria" "Burkholderiales"
#> 1105919 "Bacteria" "Proteobacteria" "Betaproteobacteria" "Burkholderiales"
#> 4477696 "Bacteria" "Proteobacteria" "Gammaproteobacteria" "Pasteurellales"
#> 4448331 "Bacteria" "Proteobacteria" "Gammaproteobacteria" "Enterobacteriales"
#> 4154872 "Bacteria" "Bacteroidetes" "Flavobacteriia" "Flavobacteriales"
#> 4452538 "Bacteria" "Fusobacteria" "Fusobacteriia" "Fusobacteriales"
#> 341322 "Bacteria" "Firmicutes" "Bacilli" "Turicibacterales"
#> 1015143 "Bacteria" "Firmicutes" "Bacilli" "Gemellales"
#> 176318 "Bacteria" "Firmicutes" "Clostridia" "Clostridiales"
#> 1788466 "Bacteria" "Firmicutes" "Clostridia" "Clostridiales"
#> 1896700 "Bacteria" "Firmicutes" "Clostridia" "Clostridiales"
#> Family Genus Species
#> 74305 "Helicobacteraceae" NA NA
#> 4449236 "Alcaligenaceae" NA NA
#> 1105919 "Oxalobacteraceae" NA NA
#> 4477696 "Pasteurellaceae" NA NA
#> 4448331 "Enterobacteriaceae" NA NA
#> 4154872 "[Weeksellaceae]" NA NA
#> 4452538 "Fusobacteriaceae" NA NA
#> 341322 "Turicibacteraceae" NA NA
#> 1015143 "Gemellaceae" NA NA
#> 176318 "Christensenellaceae" NA NA
#> 1788466 "Lachnospiraceae" NA NA
#> 1896700 "Peptostreptococcaceae" NA NA
plot(posterior, par="Lambda", focus.coord = focus, focus.cov = rownames(X)[2])
#> Scale for 'colour' is already present. Adding another scale for 'colour', which will
#> replace the existing scale.
More Technical Details
A few notes on model inference and parameter collapsing
Along with some algorithmic speed-ups enabled by the C++ Eigen library fido uses conjugate priors for the regression component of the model allowing the last three lines of the model to be collapsed into 1 line. After this the last three lines of the model can be re-expanded using fully conjugate sampling schemes that do not require optimization or MCMC (only matrix operations).
Here are the details: The collapsed model is given by \[ \begin{align} Y_j & \sim \text{Multinomial}\left(\pi_j, n_j\right) \\ \pi_j & = \phi^{-1}(\eta_j) \\ \eta_j &\sim T_{(D-1)\times N}(\upsilon, \Theta X, \Xi, I_N + X^T \Gamma X) \end{align} \] where \(A=(I_N + X^T \Gamma, X)^{-1}\) and \(T_{(D-1)\times N}\) refers to the Matrix T-distribution the \((D-1)\times N\) matrix \(\eta\) with log density given by \[\log T_{(D-1)\times N}(\eta | \upsilon, \Theta X, \Xi, A) \propto -\frac{\upsilon+N-D-2}{2}\log | I_{D-1}+\Xi^{-1}(\eta-\Theta X)A(\eta-\Theta X)^T |.\] Rather than using MCMC to sample \(\eta\) fido uses MAP estimation. Additionally, fido allows quantification of uncertainty in MAP estimates using a Laplace approximation. We found that in practice this MAP based Laplace approximation produced comparable results to a full MCMC sampler but with tremendous improvements in compute time.
Once samples of \(\eta\) are produced using the Laplace approximation closed form solutions for the conditional density of \(\Lambda\) and \(\Sigma\) given \(\eta\) are used to “uncollapse” the collapsed model and produce posterior samples from the target model. This uncollapsing is fast and given by the following matrix equations:
\[ \begin{align} \upsilon_N &= \upsilon+N \\ \Gamma_N &= (XX^T+\Gamma^{-1})^{-1} \\ \Theta_N &= (\eta X^T+\Theta\Gamma^{-1})\Gamma_N \\ \Xi_N &= \Xi + (\eta - \Theta_N X)(\eta - \Theta_N X)^T + (\Theta_N - \Theta)\Gamma(\Theta_N- \Theta)^T \\ p(\Sigma | \eta, X) &= W^{-1}(\Xi_N, \upsilon_N)\\ p(\Lambda | \Sigma, \eta, X) &= MN_{(D-1)\times Q}(\Lambda_N, \Sigma, \Gamma_N). \end{align} \] If Laplace approximation is too slow, unstable (see below) or simply not needed, the default behavior of pibble is to preform the above matrix calculations and produce a single point estimate of \(\Sigma\) and \(\Lambda\) based on the posterior means of \(p(\Sigma | \eta, X)\) and \((\Lambda | \Sigma, \eta, X)\).