Front page

abc

likelihood-free Bayesian inference

  • common optimization and sampling methods (e.g. MCMC) require the (unnormalized) likelihood
  • can happen: numerical evaluation infeasible
  • ... but still possible to simulate data $y\sim\pi(y|\theta)$

mini-intro abc

  • approximate Bayesian computation enables Bayesian inference for $$\pi(\theta|y_\text{obs}) \propto \pi(y_\text{obs}|\theta)\pi(\theta)$$ even if the likelihood cannot be evaluated
  • until $N$ acceptances:
    1. sample parameters $\theta\sim\pi(\theta)$
    2. simulate data $y\sim\pi(y|\theta)$
    3. accept if $d(s(y), s(y_\text{obs}))\leq\varepsilon$
    with summary statistics $s$, distance function $d$
  • often combined with an SMC scheme, $\varepsilon\rightarrow\varepsilon_t$, $\pi(\theta)\rightarrow g_t(\theta)$, $d\rightarrow d_t$, $t=1,\ldots,n_t$

the problem:

fitting heterogeneous data

ABC results can vary a lot depending on the employed methods

how to choose good summary statistics and distance functions?

vol. 1

robust adaptive distances

background: adaptive distances

  • a common distance metric in ABC is a weighted Euclidean distance $$d(y,y_\text{obs}) = \left(\sum_{i_y}(r_{i_y} \cdot (y_{i_y} - y_{{i_y},\text{obs}}))^2\right)^{1/2}$$
  • e.g. $r_{i_y} = \operatorname{Var}[\{y_{i,i_y}\}_{i\geq 1}]^{-1/2}$ to normalize scales
  • Prangle 2017: in ABC-SMC, updating weights $r^t$ iteratively to adjust to the changed proposal structure improves performance substantially

problems

  • highly sensitive to outliers
  • can perform badly for high-dimensional problems

solutions

  • robust Manhattan norm with bounded variance
  • active online outlier detection and down-weighting by bias assessment, $$r_{i_y} = \mathbb{E}[(y_{i_y} - y_{{i_y},\text{obs}})^2]^{-1} = (\text{Var}[\{y^i_{i_y}\}_{i}] + \text{Bias}[\{y^i_{i_y}\}_{i},y_{{i_y},\text{obs}}]^2)^{-1/2}$$ (or robust variants based on MAD and MADO)
  • apply bias corection if only few coordinates have large bias
  • yields a widely applicable efficient and robust distance metric

see our preprint Schälte et al., bioRxiv 2021

theorem (convergence of ABC posterior)

Consider prior $\pi(\theta)$ and likelihood $\pi(y|\theta)$ densities, with posterior $\pi(\theta|y_\text{obs})\propto \pi(y_\text{obs}|\theta)\pi(\theta)$, and measurable distance metrics $d_t$. Assume that for the acceptance regions $A_t = \{y:d_t(y,y_\text{obs})\leq\varepsilon_t\}$ holds $\lim_{t\rightarrow \infty}|A_t| = 0$, and that the $A_t$ have bounded eccentricity at $y_\text{obs}$. Then, for functions $\xi:\mathbb{R}^{n_\theta}\rightarrow\mathbb{R}$ with $\mathbb{E}_{\pi(\theta)}[|\xi|]<\infty$ holds $$ \mathbb{E}_{\pi_{\text{ABC},\varepsilon_t(\theta|y_\text{obs})}}[\xi] \xrightarrow{t\rightarrow\infty}\mathbb{E}_{\pi(\theta|y_\text{obs})}[\xi]\quad\text{for almost-all}~y_\text{obs}. $$

Proof: Yep.

results

accurate results on various problem types, on outlier-free and outlier-corrupted data
good data fits and reliable outlier down-weighting

application example M6: tumor growth

based on Jagiella et al., Cell Systems 2017 Tumor gif
  • cells modeled as interacting stochastic agents, dynamics of extracellular substances by PDEs
  • simulation time on the order of seconds
  • 7 parameters
better parameter estimates compared to established methods, also on outlier-free data
good data fits and reliable outlier down-weighting
vol. 2

informative distances and summary statistics

background: regression-based summary statistics

background: regression-based summary statistics

  • Fearnhead et al. 2012: Good statistics are $s(y) = \mathbb{E}[\theta|y]$
  • use a linear approximation $\mathbb{E}[\theta|y] \approx s(y) = Ay + b$
  • learn model $s: y\mapsto\theta$ from calibration samples, with (augmented) data as features, and parameters as targets
  • alternative regression models: DNN (Jiang et al. 2017), GP (Borowska et al. 2020)

problems

  • scale-normalized distances alone do not account for informativeness
  • identification of a high-density region for training
  • regression-based summary statistics rely on accuracy of the employed model
  • the same problems motivating adaptive distances, shifted to "parameter" space, apply
  • parameter non-identifiability problematic for regression models e.g. if $y \sim \theta^2$, $\theta\sim U[-1, 1]$

solutions

  • integrate summary statistics learning in ABC-SMC workflow
  • combine regression-based summary statistics with scale-normalized weights
  • regression-based sensitivity distance weights as an alternative way of incorporating informativeness
  • employ higher-order moments as regression targets to tackle non-identifiability

regression-based sensitivity distance weights

idea: employ regression model $s:y\mapsto\theta$ not to construct summary statistics, but to define sensitivity weights \begin{equation}\label{eq:info_weight} q_{i_y} = \sum_{i_\theta=1}^{n_\theta} \frac{\left|S_{i_yi_\theta}\right|}{ \sum_{j_y=1}^{n_y}\left|S_{j_yi_\theta}\right|}, \end{equation} as the sum of the absolute sensitivities of all parameters with respect to model output $i_y$, normalized per parameter, where \begin{equation}\label{eq:info_S} S = \nabla_y s(y_\text{obs})\in\mathbb{R}^{n_y \times n_\theta} \end{equation}

theorem (optimal summary statistics)

[...] Given $\lambda:\mathbb{R}^{n_\theta}\rightarrow\mathbb{R}^{n_\lambda}$ such that $\mathbb{E}_{\pi(\theta)}[|\lambda(\theta)|]<\infty$, define summary statistics as the conditional expectation $$s(y) := \mathbb{E}[\lambda(\Theta)|Y=y] = \int \lambda(\theta)\pi(\theta|y)d\theta.$$ Then, it holds $\left\lVert{\mathbb{E}_{\pi_{\text{ABC},\varepsilon}}[\lambda(\Theta)|s(y_\text{obs})] - s(y_\text{obs})}\right\rVert \leq \varepsilon$, and therefore \begin{equation}\label{eq:sreg_conv} \lim_{\varepsilon\rightarrow 0}\mathbb{E}_{\pi_{\text{ABC},\varepsilon}}[\lambda(\Theta)|s(y_\text{obs} )] = \mathbb{E}[\lambda(\Theta)|Y=y_\text{obs}]. \end{equation}

Proof: Yep.

In practice: Train regression model $s: y \mapsto \lambda(\theta) = (\theta^1,\ldots,\theta^k)$.

implementation



  from pyabc import *

  distance: Distance = AdaptivePNormDistance(
      sumstat=ModelSelectionPredictorSumstat(
          predictors=[
              LinearPredictor(),
              GPPredictor(kernel=['RBF', 'WhiteKernel']),
              MLPPredictor(hidden_layer_sizes=(50, 50, 50)),
          ],
      ),
      scale_function=rmse,
      pre=[lambda x: x, lambda x: x**2],
      par_trafo=[lambda y: y, lambda y: y**2],
  )
pyabc.rtfd.io/en/develop/examples/informative.html

🦁 a boss model

  • $y_1\sim\mathcal{N}(\theta_1,0.1^2)$ is informative of $\theta_1$, with a relatively wide corresponding prior $\theta_1\sim U[-7, 7]$,
  • $y_2\sim\mathcal{N}(\theta_2,100^2)$ is informative of $\theta_2$, with corresponding prior $\theta_2\sim U[-700, 700]$,
  • $y_3\sim\mathcal{N}(\theta_3, 4 \cdot 100^2)^{\otimes 4}$ is a four-dimensional vector informative of $\theta_3$, with corresponding prior $\theta_3\sim U[-700, 700]$,
  • $y_4\sim\mathcal{N}(\theta_4^2, 0.1^2)$ is informative of $\theta_4$, with corresponding symmetric prior $\theta_4\sim U[-1, 1]$, however is quadratic in the parameter, resulting in a bimodal posterior distribution for $y_{\text{obs},4}\neq 0$,
  • $y_5\sim\mathcal{N}(0, 10)^{\otimes 10}$ is an uninformative 10-dimensional vector.
only combination of scale normalization, informativeness assessment, and regression target augmentation permits accurate inference
sensitivity analysis permits further insights
especially sensitivity weights widely, robustly applicable; restriction to high-density region better; scale normalization better
sensitivity-weighting improves estimates on application problem substantially
sensitivity weights re-prioritize data points, in particular identifying uninformative ones
applicable to outlier-corrupted data

discussion

discussion

summary

  • accounting for both scale and informativeness often useful
  • provided two approaches using regression-based summary statistics and distance weights, with scale normalization and addressing parameter non-identifiability
  • robustly applicable also to outlier-corrupted data

outlook

  • model selection and alternative regression models
  • training specification and assessment
  • maybe a goal: fully integrated reliable workflow
  • ...

acknowledgments

Thanks to: Jan Hasenauer, Emad Alamoudi, Elba Raimúndez, the whole lab, ...

thx! qs?