# ABC approximation Bias

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?

• I do not think this Taylor approximation to the likelihood error is particularly helpful for ABC applications. Mar 4 at 19:19
• @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. Mar 4 at 19:27
• What matters is the approximation of the posterior, not of the likelihood, imho. Mar 4 at 20:09