Find $\theta^\star: P(Data \big|\, \theta) \to \max$, where $\theta=(N_p, m_{pq}, s_p)$

• Two alleles in considered locus: $A$ and $a$
• X(t) — the rel. frequency of allele $A$ in populations at generation $t$
• Wright-Fisher model: $X(t)$ — Markov chain
• If $N \to \infty$ then $X(t)$ $\to$ Markov process
• Consider transition probability density $p(\tau, x, y)$ to go from state $x$ to state $y$ in time step of $\tau$

• Consider transition probability density $p(\tau, x, y)$ to go from state $x$ to state $y$ in time step of $\tau$

• then $p(\tau, x, y)$ satisfies Kolmogorov forward equation:

  $$\frac{\partial}{\partial \tau}p(\tau, x, y) = \frac{1}{2}\sum_{p=1}^P \frac{\partial^2}{{\partial y_p}^2} \left[b_{pp}(y)p(\tau, x, y)\right] - \sum_{p=1}^P \frac{\partial}{\partial y_p} \left[a_p(y) p(\tau, x, y)\right]$$

• and $p(\tau, x, y)$ satisfies Kolmogorov backward equation:

  $$\frac{\partial}{\partial \tau}p(\tau, x, y) = \frac{1}{2}\sum_{p=1}^P b_{pp}(x) \frac{\partial^2}{\partial {x_p}^2}\left[p(\tau, x, y)\right] + \sum_{p=1}^P a_p(x) \frac{\partial}{\partial x_p} \left[p(\tau, x, y)\right]$$

  where $a_p(x)$ and $b_{pp}(x)$ — infinitesimal mean and variance

The pair $(X(t), C(t))$ is a hidden Markov model if:

• $X(t)$ is a Markov process that is considered to be hidden
• $C(t)$ is observable process whose outcomes are "influenced" by the outcomes of $X$ in a known way

• Outcome $C(t_k)$ is "influenced" exclusively by the outcome $X(t_k)$:

  $$P(C_k = c \,|\, X_1=x_1, \ldots, X_k=x_k) = P(C_k = c \,|\, X_k=x_k)$$

$P(C(t_k) = c \, |\, X(t_k)=x_k)$ is called emission probability

Goal: learn about $X(t)$ by observing $C(t)$

The $\alpha_k$ satisfy the forward recursion:

Transition probability density $p(\tau, x, y)$ satisfies Kolmogorov forward equation:

Hence, $\alpha_k$ satisfies Kolmogorov forward equation:

The $\beta_k$ satisfy the backward recursion:

Transition probability density $p(\tau, x, y)$ satisfies Kolmogorov backward equation:

Hence, $\beta_k$ satisfies Kolmogorov backward equation:

We consider population sizes $2N_p$, migration rates $m_{pq} \in [0,1]$ and fitness coefficients:

If $X(t)=x$, then mean $X(t+1)$ can be expressed:

We allow the two alleles $A$ and $a$ to mutate into each other:

Then infinitesimal mean and variance:

In order to compute $P\left(\{c_k\}_{k=0}^K \big| \theta\right)$ in condition of hidden Wright-Fisher model with parameters $\theta=[N_p, m_{pq}, s_p]_{p,q=1,..,P}$:

• Build infinitesimal mean $a_p(x)$ and variance $b_{pp}(x)$ according to $\theta$
• For each $k=1,\ldots,K-1$ do:

  • Obtain $\alpha_k(t_{k+1} - t_k,y)$ by solving forward equation:

    $$\frac{\partial}{\partial \tau}\alpha_k(\tau, y) = \frac{1}{2}\sum_{p=1}^P \frac{\partial^2}{{\partial y_p}^2} \left[b_{pp}(y)\alpha_k(\tau, y)\right] - \sum_{p=1}^P \frac{\partial}{\partial y_p} \left[a_p(y) \alpha_k(\tau, y)\right]$$ $$\text{subject to: }\alpha_k(0, y) = \alpha_{k-1}(t_{k} - t_{k-1}, y) \mathcal{B}(c_{k} | y)$$

• Evaluate:

  $$P\left[C(t_0) = c_0,\ldots, C(t_K)=c_K \, \big| \, \theta\right] = \int \alpha_{K}(0,y) \,dy$$

In order to compute $P\left(\{c_k\}_{k=0}^K \big| \theta\right)$ in condition of hidden Wright-Fisher model with parameters $\theta=[N_p, m_{pq}, s_p]_{p,q=1,..,P}$:

• Build infinitesimal mean $a_p(x)$ and variance $b_{pp}(x)$ according to $\theta$
• For each $k=K-1,\ldots,0$ do:

  • Obtain $\beta_k(t_{k+1} - t_k,x)$ by solving backward equation:

    $$\frac{\partial}{\partial \tau}\beta_k(\tau, x) = \frac{1}{2}\sum_{p=1}^P b_{pp}(x) \frac{\partial^2}{\partial {x_p}^2}\left[\beta_k(\tau, x)\right] + \sum_{p=1}^P a_p(x) \frac{\partial}{\partial x_p} \left[\beta_k(\tau, x)\right]$$ $$\text{subject to: }\beta_k(0, x) = \beta_{k+1}(t_{k+2} - t_{k+1}, x)\mathcal{B}(c_{k+1} | x)$$

• Evaluate:

  $$P\left[C(t_0) = c_0,\ldots, C(t_K)=c_K \, \big| \, \theta\right] = \int \beta_{0}(\Delta t_0,x)\mathcal{B}(c_{0} | x) \,dx$$

We propose reflecting boundary conditions for both forward and backward equations

• Forward equation

  $$\frac{\partial}{\partial \tau}\alpha(\tau, y) = \frac{1}{2}\sum_{p=1}^P \frac{\partial^2}{{\partial y_p}^2} \left[b_{pp}(y)\alpha(\tau, y)\right] - \sum_{p=1}^P \frac{\partial}{\partial y_p} \left[a_p(y) \alpha(\tau, y)\right]$$ $$\text{subject to: } J_p(\tau,y) = \frac{1}{2}\frac{\partial}{\partial y_p} \left[b_{pp}(y)\alpha(\tau, y)\right] - a_p(y) \alpha(\tau, y)=0, \quad \text{for } y \in \{0, 1\}^P$$

• Backward equation

  $$\frac{\partial}{\partial \tau}\beta_k(\tau, x) = \frac{1}{2}\sum_{p=1}^P b_{pp}(x) \frac{\partial^2}{\partial {x_p}^2}\left[\beta_k(\tau, x)\right] + \sum_{p=1}^P a_p(x) \frac{\partial}{\partial x_p} \left[\beta_k(\tau, x)\right]$$ $$\text{subject to: } \frac{\partial}{\partial x_p} \left[\beta(\tau, x)\right] =0, \quad \text{for } x \in \{0, 1\}^P$$

• Regular: process can both enter and leave from the boundary
  • Reflecting: the process cannot stay on the boundary?
• Absorbing: if the particle reach the boundary it is removed from the system
• Periodic: two boundaries are identified with each other, if process enter one boundary it can pop out from another one
• Entrance: process cannot get to boundary, but it can start from it
• Natural: the process can neither get to nor start at boundary
• Exit: starting close to the boundary it is impossible to reach intermediate state close to the boundary?

Boundary conditions for Fokker Plank equation on $[a, b]^d$

• Reflecting:

  $$ J(t, x) = 0, \ \text{for } x \in \{a, b\}^d$$
• Absorbing:

  $$p(t, x) = 0,\ \text{for } x \in \{a, b\}^d$$
• Periodic:

  $$\lim_{x\to b-} p(t, x) = \lim_{x\to a+} p(t, x) \quad \lim_{x\to b-} J(t, x) = \lim_{x\to a+} J(t, x)$$
• Entrance:

  $$A(t, a) > 0$$
• Natural:

  $$A(t, a) = 0$$
• Exit:

  $$A(t, a) < 0$$

Boundary conditions for Fokker Plank equation in population genetics

Conditional probability density $\phi(t, x)$ that allele freq. is $x$ at time $t$ given initial freq.

Proposed "absorbing" barriers (3.16, 3.17, 3.18 and 3.19):

where $f(t,x) = \frac{1}{2N}\phi(t, x)$

Boundary conditions for Fokker Plank equation in population genetics

$\phi(t, x)$ — "density" of derived mutations (no mutation in A(x))

Using "absorbing" barriers from Kimura 1964:

However, code is different? Mutation is injected at freq $x_1$ of the solution

Boundary conditions for backward equation with migrations

The behaviour on $x=0$ depends on the value of $\mu_{a \to A}$:

1) Probability that allele will survive at time $\tau$ if its frequency at time $t$ was equal to $x$

2) Backward probability (conditioned to selection coef. and allele age)