Using certain formula with conditional probability in discrete and continuous case

Suppose I have some recurrent formula for probability:

$$f_k(t)=e_k(x_t)\sum_{i}f_i(t-1)p_{ik}. \tag{1}$$

Where $1 \leq i,k \leq N$ denote finite number of states, time $t \leq T$ is discrete and $x_t$ is some discrete observation value at time $t$.

The conditional probabilities $p_{ik}=P(\pi_t=k \mid \pi_{t-1}=i)$ and $e_k(x_t)=P(\text{observation}=x_t \mid \pi_{t}=k)$ doesn't depend on $t$ (I just used time in the notation. These are transition probabilities of time-homogeneous Markov chain and probability of associated observation given the state of the chain).

$f_k(t)$ and $\sum_{k=1}^N f_k(t)$ are indeed probabilities ($<1$), I'm actually interested in $\sum_{k=1}^N f_k(T)$.

Now I want to generalize this formula to the continuous distributions.

Suppose that I have $T$ observations as before, but the state space and observations space are $\mathbb{R}$. I have two conditional probability density functions $p_{Y \mid Y'}(y \mid y' )$ and $e(x \mid y)$. Transitions (and observations) occur at some discrete times $t=1,...,T$.

I intend to compute the following:

$$f_t(y) = e(x \mid y)\int_{\mathbb{R}}f_{t-1}(y') p_{Y \mid Y'}(y \mid y' ) dy'$$

I think here $f$ tends to be something similar to density, integration over $\mathbb{R}$ should give me the probability I want:

$$\int_{\mathbb{R}}f_T(y)dy \leq 1.$$

However, direct computation gives me very big result $1e+226$. The apparent reason is: the density $p_{Y \mid Y'}(y \mid y' )$ has very big values and is integrated over $y'$, not $y$.

What can be wrong with this idea and what can I do to apply this approach to continuous state space?

• math.stackexchange.com/questions/151599/… When the state space become continuous, usually you called markov process, and you may want to take a look at the Kolmogorov Equations en.wikipedia.org/wiki/Kolmogorov_equations – BGM Jun 2 '16 at 3:14
• @BGM Thank you! This looks similar to Kolmogorov equations, but it is too complicated for me to determine whether it is related to them or not. The problem is that this algorithm ("Forward algorithm" for hidden markov model) is defined only for discrete models in literature. This is my related question on stats.stackexchange. I fear that I will receive a little help this way, because most of people are unfamiliar with it, but this my question involve just ordinary probability theory at the moment. (see next comment >>) – Slowpoke Jun 2 '16 at 3:46
• @BGM Just from a probabilistic point of view: Can conditional probabilities in a such formula as (1) be replaced with densities and sum with integration $p_{ik} \rightarrow p(x \mid x')dx'$? Or are there some pitfalls switching to densities this way? – Slowpoke Jun 2 '16 at 3:50