There are two other possibilities not mentioned in the above answers.
Consider the time series $x_{t+1}=\beta{x}_t+\epsilon_{t+1}$. Assume $t\in\{81,82\}$ are missing. The question is why are they missing.
Consider three possible cases. The first is that it is a holiday or a similar day where there was no activity. The second is omission as a recording error. The third is hiding the data as it may be embarrassing to someone.
For the first two, it is simple. You alter your likelihood function in this one case. Noting for this example that $$x_{81}=\beta{x}_{80}+\epsilon_{81}$$ and $$x_{82}=\beta^2{x}_{80}+\beta\epsilon_{81}+\epsilon_{82}$$ and $$x_{83}=\beta^3x_{80}+\beta^2\epsilon_{81}+\beta\epsilon_{82}+\epsilon_{83}.$$
To make the example simple, let us assume that $\epsilon_t\sim\mathcal{N}(0,\sigma^2),\forall{t}$.
The simple case, where a holiday intervened is simple. You would note that no error could have happened on those days, so $\epsilon_{81}=\epsilon_{82}=0$.
You would update the posterior probability of $(\beta;\sigma^2)$ by simply cubing $\beta$ in the likelihood function. You can do this because $$\pi(\beta,\sigma^2|x_1\dots{x}_{80};x_{83})$$ follows from $$\pi(\beta,\sigma^2|x_1\dots{x}_{80})$$ via Bayes theorem where $$\pi(\beta,\sigma^2|x_1\dots{x}_{80})=\frac{\prod_{t=1}^{80}f(x_t|\beta;\sigma^2)\pi(\beta;\sigma^2)}{\int_0^\infty\int_{-\infty}^\infty\prod_{t=1}^{80}f(x_t|\beta;\sigma^2)\pi(\beta;\sigma^2)\mathrm{d}\sigma^2\mathrm{d}\beta}$$ and $$\pi(\beta,\sigma^2|x_1\dots{x}_{80};x_{83})=\frac{f(x_{83}|\beta;\sigma^2)\pi(\beta,\sigma^2|x_1\dots{x}_{80})}{\int_0^\infty\int_{-\infty}^\infty{f}(x_{83}|\beta;\sigma^2)\pi(\beta,\sigma^2|x_1\dots{x}_{80})\mathrm{d}\sigma^2\mathrm{d}\beta}.$$
In the case of an omission, there are two choices. The first is to treat it like a holiday because the alternative is to estimate the posterior of $\{\beta,\sigma^2,\epsilon_{81},\epsilon_{82}\}.$ It will end up with an answer that is no different than omitting them unless you have other variables which are correlated with $x_t$. In the case of at least one associated variable, you can improve the estimation of $\beta$ and $\sigma^2$ by estimating the missing variables as if they were parameters and then marginalizing them out via Bayes rule.
In the third case, where the omission is purposeful and assuming there is no correlated data, you should estimate the missing variable my inserting a prior density for them that estimates the size of the hidden variables.