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In Approximate Bayesian Computation, we approximate the (true) likelihood of our model, $f(x_{obs}|\theta)$, with the following integral

$$f_{ABC}(y_{obs}|\theta)=\int K_{h}(x-x_{obs})f(x|\theta)dx $$

with the use of a Taylor expansion, we can express the ABC likelihood approximately as

$$f_{ABC}(y_{obs}|\theta)\approx f(x_{obs}|\theta)+\frac{1}{2}h^{2}Var[K_{h}(x-x_{obs})]f^{''}(x_{obs}|\theta),$$

where $h$ is the scale of the kernel $K_{h}(\cdot)$ and $f^{''}(x_{obs}|\theta)$ the second derivative of the likelihood.

So, the bias of the true likelihood approximation can be expressed as

$$f_{ABC}(x_{obs}|\theta)-f(x_{obs}|\theta)=\frac{1}{2}h^{2}Var[K_{h}(x-x_{obs})]f^{''}(x_{obs}|\theta)$$

In the case where the $f(x_{obs}|\theta)$ is a continuous function (it is the result of a density), it is meaningful to calculate the derivates. However, in the case where the likelihood is discrete, i.e $\frac{df(x|\theta)}{dx}$ has no meaning.

Thus, would it be equivalent to express bias as $h^{2}Var[K_{h}(x-x_{obs})]$, because $\frac{1}{2}f^{''}(x_{obs}|\theta)$ is a constant, so it will not affect the shape of bias over a grid of $h$ values?

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  • $\begingroup$ I do not think this Taylor approximation to the likelihood error is particularly helpful for ABC applications. $\endgroup$
    – Xi'an
    Mar 4 at 19:19
  • $\begingroup$ @Xi'an But is there a closed-form formula that can assess the bias of the likelihood approximation? This Taylor Expansion is used, as I understand for continuous functions and not for discrete. $\endgroup$
    – Fiodor1234
    Mar 4 at 19:27
  • $\begingroup$ What matters is the approximation of the posterior, not of the likelihood, imho. $\endgroup$
    – Xi'an
    Mar 4 at 20:09

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