Probability Reasoning over Time

Date: 23.04.25 ~ 23.04.31

Writer: 9tailwolf

Introduction

In previous chapter, we learned about various proability reasoning. In this chapter, new condition time is added. we will learn about Probability Reasoning over Time.

Filtering

Filtering is a method that calculate recent state with belief state(evidences). Following is a fomular of filtering.

$$P(X_{t+1}\mid e_{1:t+1}) = P(X_{t+1}\mid e_{1:t},e_{t+1})$$ $$ = \alpha P(e_{t+1}\mid X_{t+1},e_{1:t})P(X_{t+1}\mid e_{1:t})$$ $$ = \alpha P(e_{t+1}\mid X_{t+1})P(X_{t+1}\mid e_{1:t})$$ $$ P(X_{t+1}\mid e_{1:t}) = \Sigma_{x_{t}}P(X_{t+1}\mid x_{t})P(x_{t}\mid e_{1:t})$$

Smoothing

Smoothing is a method that calculate past state with belief state(evidences).Following is a fomular of filtering.

$$P(X_{k}\mid e_{1:t}) = P(X_{k}\mid e_{1:k},e_{k+1\mid t})$$ $$ = \alpha P(X_{k}\mid e_{1:k})P(e_{k+1:t}\mid e_{1:k},X_{k})$$ $$ = \alpha P(X_{k}\mid e_{1:k})P(e_{k+1:t}\mid X_{k})$$

By above way, we can do smoothing. $P(X_{k}\mid e_{1:k})$ is a forward filtering. $P(e_{k+1:t}\mid X_{k})$ is a backward. below is a process of backward.

$$P(e_{k+1:t}\mid X_{k}) = \Sigma_{x_{k}}P(e_{k+1:t}\mid X_{k},x_{k+1})P(x_{k+1}\mid X_{k})$$ $$ = \Sigma_{x_{k}}P(e_{k+1},e_{k+2:t}\mid x_{k+1})P(x_{k+1}\mid X_{k})$$ $$ = \Sigma_{x_{k}}P(e_{k+1}\mid x_{k+1})P(e_{k+2:t}\mid x_{k+1})P(x_{k+1}\mid X_{k})$$

Algorithm

def forward_backward(ev,prior):
    fv = [0 for _ in range(time)]
    b = vector()
    sv = [0 for _ in range(time)]
    fv[0] = prior
    for i in range(1,t):
        fv[i] = forward(fv[i-1],ev)
    for i in range(t,0,-1):
        sv = normalize(fv[i] * b)
        b = backward(b,ev)
    return sv

Most Likely Explanation

Most likely Explanation is a method that calculte state that will happen most frequency. By below way, we can find solution.

$$\max_{x_{1}:x_{t}} P(x_{1},...x_{t}\mid e_{1:t})$$ $$= \alpha P(e_{t+1}\mid X_{t+1}) \max_{x_{t}}(P(X_{t+1}\mid x_{t} \max _{x_{1}:x_{t}} P(x_{1},...,x_{t}\mid e_{1:t}))$$