How to create a cause specific hazard function for a nonparametric Bayesian model for survival analysis?

I am reading an article on a nonparametric Bayesian model for survival analysis with competing risks, which can be used for jointly assessing a patient’s risk of multiple (competing) adverse outcomes.

Let $D=\{X_i,T_i,k_i\}_{i=1}^n, X_i\in\chi$, a d-dimensional vector of covariates associated with subject $i$, $T_i\in\mathbb{R}_+$ is time until an event occur.

• $k_i\in K$ the type of event that occurred, $K=\{\varnothing,1,\dots,K\}$ being a set of K mutually exclusive, competing events that could occur to subject $i$, and where $\varnothing$ is right censoring

$T$ is a multivariate random variable, $T=(T^1,\dots,T^k)$ where $T^k,k\in K$ is the net survival time of a subject with respect to event $k$. They assume that $T$ is drawn from a conditional density (which I don't understand on which condition they base themselves upon). function that depends on the subjects covariates. For every subject $i$, we only observe the occurrence time for the earliest event, that is to say, $T_i=\min (T_i^1,\dots,T_i^k), k_i=\arg\min_j T_i^j$

The cause specific hazard function $\lambda_k(t,X)$ represents the instantaneous risk of event $k$ and is formally defined as :

$$\lambda_k=\lim_{dt\rightarrow0}\frac{1}{dt}P(t\le T^k<t+dt,\underbrace{k|T^k\ge t,X}_{\mbox{?}})$$

I don't get the $?$ part.

• First because it seems that it is already taken into account in $P(t\le T^k\dots$ part.

• Second as far as I am not sure about what we are talking about here : we only take events $k$ such that the net survival time of a subject with respect to these event is greater than $t$ for a given d-dimensional vector of covariates associated with subject $i$ ? That is to say the events that may occur at $t$ and kill someone ? Is it the instantaneous probability of death at $t$, conditional on survival until $t$ ?

The hazard function $\lambda_k(t)$ describes the instantaneous potential per unit time for the $k$-th event to occur, given that the individual remained event-free up to time $t$. You can think of it as an approximate conditional probability (it is not bounded from above) that the $k$-th event occurs in the small time interval $[t; t + dt[$, given that the individual remained event-free up to time $t$. Conditioning on $T^k \geq t$ is necessary to ensure that the individual is still event-free at time $t$. In particular, we are not interested in individuals that experienced the event prior to $t$. If the condition is dropped, the hazard function would also need to capture the potential of experiencing the event in the interval $[t; t + dt[$ if an individual already experienced the event prior to $t$, which would be a confusing definition, in particular when the event is death. If that would be the case, we would consider recurrent events (e.g. heart attacks), which are a separate topic in survival analysis.