Skip to main content
Cross Validated

Return to Question

added 33 characters in body
Source Link
Dimitris Rizopoulos
Dimitris Rizopoulos
  • 21.5k
  • 2
  • 25
  • 51

In the paper Deep Survival Analysis (Ranganath, 2016), the author specifies a generative model for deep survival analysis.

Say $\mathbf{x}$ denotes the set of covariates, $\mathbf{\beta}$ the parameters for the data with some prior $p(\mathbf{\beta})$, $k$ a fixed scalar, and $n$ be the index of an observation. The generative model is defined as

\begin{gather*} b \sim \mathcal{N}(0, \sigma_b)\\ a \sim \mathcal{N}(0, \sigma_W)\\ z_n \sim \text{DEF}(\mathbf{W})\\ \mathbf{x}_n \sim p(\cdot | \mathbf{\beta}, z_n)\\ t_n \sim Weibull(log(1+\exp(z_n^Ta+b)),k) \end{gather*}\begin{gather*} b \sim \mathcal{N}(0, \sigma_b)\\ a \sim \mathcal{N}(0, \sigma_W)\\ z_n \sim \text{DEF}(\mathbf{W})\\ \mathbf{x}_n \sim p(\cdot \mid \mathbf{\beta}, z_n)\\ t_n \sim \mbox{Weibull}(\log\{1+\exp(z_n^Ta+b)\}, k) \end{gather*}

The latent variable $z_i$ comes from a DEF which generates the observed covariates and the time to failure. Given covariates $\mathbf{x}$, the model makes predictions via the posterior predictive distribution:

\begin{gather*} p(t|\mathbf{x}) = \displaystyle\int_zp(t|z)p(z|\mathbf{x})dz. \end{gather*}\begin{gather*} p(t \mid \mathbf{x}) = \displaystyle\int_zp(t \mid z)p(z \mid \mathbf{x}) \; dz. \end{gather*}

My question now is the following:

  1. How can this specific posterior predictive distribution make these predictions computationally? Do you first generate your most probable latent variables $z_n$ given your data $\mathbf{x}$ and do you then try to find the most probable $t$, given your most probable latent variables $z_n$?

In the paper Deep Survival Analysis (Ranganath, 2016), the author specifies a generative model for deep survival analysis.

Say $\mathbf{x}$ denotes the set of covariates, $\mathbf{\beta}$ the parameters for the data with some prior $p(\mathbf{\beta})$, $k$ a fixed scalar, and $n$ be the index of an observation. The generative model is defined as

\begin{gather*} b \sim \mathcal{N}(0, \sigma_b)\\ a \sim \mathcal{N}(0, \sigma_W)\\ z_n \sim \text{DEF}(\mathbf{W})\\ \mathbf{x}_n \sim p(\cdot | \mathbf{\beta}, z_n)\\ t_n \sim Weibull(log(1+\exp(z_n^Ta+b)),k) \end{gather*}

The latent variable $z_i$ comes from a DEF which generates the observed covariates and the time to failure. Given covariates $\mathbf{x}$, the model makes predictions via the posterior predictive distribution:

\begin{gather*} p(t|\mathbf{x}) = \displaystyle\int_zp(t|z)p(z|\mathbf{x})dz. \end{gather*}

My question now is the following:

  1. How can this specific posterior predictive distribution make these predictions computationally? Do you first generate your most probable latent variables $z_n$ given your data $\mathbf{x}$ and do you then try to find the most probable $t$, given your most probable latent variables $z_n$?

In the paper Deep Survival Analysis (Ranganath, 2016), the author specifies a generative model for deep survival analysis.

Say $\mathbf{x}$ denotes the set of covariates, $\mathbf{\beta}$ the parameters for the data with some prior $p(\mathbf{\beta})$, $k$ a fixed scalar, and $n$ be the index of an observation. The generative model is defined as

\begin{gather*} b \sim \mathcal{N}(0, \sigma_b)\\ a \sim \mathcal{N}(0, \sigma_W)\\ z_n \sim \text{DEF}(\mathbf{W})\\ \mathbf{x}_n \sim p(\cdot \mid \mathbf{\beta}, z_n)\\ t_n \sim \mbox{Weibull}(\log\{1+\exp(z_n^Ta+b)\}, k) \end{gather*}

The latent variable $z_i$ comes from a DEF which generates the observed covariates and the time to failure. Given covariates $\mathbf{x}$, the model makes predictions via the posterior predictive distribution:

\begin{gather*} p(t \mid \mathbf{x}) = \displaystyle\int_zp(t \mid z)p(z \mid \mathbf{x}) \; dz. \end{gather*}

My question now is the following:

  1. How can this specific posterior predictive distribution make these predictions computationally? Do you first generate your most probable latent variables $z_n$ given your data $\mathbf{x}$ and do you then try to find the most probable $t$, given your most probable latent variables $z_n$?
Source Link
hendiadys
hendiadys
  • 1
  • 1

How to make predictions with the posterior predictive distribution?

In the paper Deep Survival Analysis (Ranganath, 2016), the author specifies a generative model for deep survival analysis.

Say $\mathbf{x}$ denotes the set of covariates, $\mathbf{\beta}$ the parameters for the data with some prior $p(\mathbf{\beta})$, $k$ a fixed scalar, and $n$ be the index of an observation. The generative model is defined as

\begin{gather*} b \sim \mathcal{N}(0, \sigma_b)\\ a \sim \mathcal{N}(0, \sigma_W)\\ z_n \sim \text{DEF}(\mathbf{W})\\ \mathbf{x}_n \sim p(\cdot | \mathbf{\beta}, z_n)\\ t_n \sim Weibull(log(1+\exp(z_n^Ta+b)),k) \end{gather*}

The latent variable $z_i$ comes from a DEF which generates the observed covariates and the time to failure. Given covariates $\mathbf{x}$, the model makes predictions via the posterior predictive distribution:

\begin{gather*} p(t|\mathbf{x}) = \displaystyle\int_zp(t|z)p(z|\mathbf{x})dz. \end{gather*}

My question now is the following:

  1. How can this specific posterior predictive distribution make these predictions computationally? Do you first generate your most probable latent variables $z_n$ given your data $\mathbf{x}$ and do you then try to find the most probable $t$, given your most probable latent variables $z_n$?