# How to define the Lee-Carter model and its constrains in Bayesian setting

I'm running the Bayesian version of Lee-Carter model on jags, using rjags R package. Given a matrix of data $M$ such that $M_{x,t}=\log m_x(t)$ where $m_x(t)$ is defined as the central death rate at age $x$ in calendar year $t$, the Lee-Carter model states that: $$\log m_x(t) = \alpha_x + \beta_x\kappa_t+\epsilon_{x,t}$$

where $\epsilon_{x,t} \sim \mathcal{N}(0,\sigma^2_\epsilon)$. In a frequentist analysis, in order to estimate the parameters, two constraints are done:

$$\sum_x \beta_x = 1\quad \text{and}\quad \sum_t \kappa_t = 0$$

and they allow a specific interpretation of parameters. My question is: when I run this model in JAGS, are these constraints necessary in order to obtain estimations? Or they just allow to interpretate in the same way the parameters?

• I edited your title to something more informative, feel free to edit it again if you find it unsuitable. – Tim Jun 11 '18 at 12:02
• Well maybe my question was not very direct to the problem. I meant if the algorithm performance could be affected by the constraints used in frequentist approach. – zick094 Jun 11 '18 at 12:08

I'm not an expert on Lee-Carter model, but from what I gasp, $\alpha_x$ is the mean per age group, $\beta_x$ is the effect per age group and $\kappa_t$ is the effect per time. As noted in the Understanding the Lee-Carter Mortality Forecasting Method paper by Girosi and King (I adapted the notation):

The constraint $\sum_t \kappa_t = 0$ immediately implies that the parameter $\alpha_x$ is simply the empirical average over time of the age profile in age group $x$ . . . We therefore rewrite the model in terms of the mean centered log-mortality rate . . .

What leads to the following model

$$\log \tilde{m_{xt}} = \beta_x \kappa_t +\varepsilon_{xt}$$

where $\log \tilde{m_{xt}}$ is the centered $\log m_{xt}$, what looses the constraint on $\kappa_t$. As about $\beta_x$, you can use Dirichlet distribution for this parameter, as it already satisfies the constraint and is a common choice for a "distribution over probabilities".

• Thank you for the answer. Well I used Dirichlet distribution but it is not very appropriate since it does not allows negative values, while in some applications it's possible to see that for some ages $\beta_x$ is negative. Anyway my question was if I don't use these constraints, could this affect the estmation in JAGS? I mean not in absolute values term, but in performance of algorithm. – zick094 Jun 11 '18 at 12:06
• @zick094 as far as I understand, the constraints are only for interpretability. The model does not seem to differ from a regression model with interaction terms and the regression models do not have any such constraints, so no, it won't affect anything. – Tim Jun 11 '18 at 12:09
• This answers to my question. The problem was that while in regression we have observable variables, here everything is latent. – zick094 Jun 11 '18 at 12:11