Omer Yildiran

modelingaudiovisualtiming

Models of Bimodal Duration Estimation

Model Coding:

Created: June 18, 2025 6:09 PM
Tags: article note

1- Unimodal

1.1 Measurement

imageHo

m=N(s,σ2)m=N(s,\sigma^2) 
# unimodal measurements
def unimodalMeasurements(sigma, S):
    # P(x|s) # generate measurements from a normal distribution
    m = np.random.normal(S, sigma**2, 10000)  # true duration is S seconds
    return m

1.2 Likelihood

p(m∣s)=12πσexp⁡(−(m−s)22σ2)p(m|s)=\frac{1}{\sqrt{2\pi} \sigma}\exp(-\frac{(m-s)^2}{2\sigma^2}) 
def gaussianPDF(m,s,sigma):
    return (1/(np.sqrt(2*np.pi)*sigma))*np.exp(-((x-s)**2)/(2*(sigma**2)))

# likelihood function
def likelihood(S, sigma):
    # P(m|s) # likelihood of measurements given the true duration
    m=np.linspace(S - 4*sigma, S + 4*sigma, 500)
    p_x=gaussianPDF(m,S,sigma)
    return x, p_x

1.2.1 Plot Likelihoods analytically

def plotLikelihood(S,sigma):
    x, p_x = likelihood(S, sigma)
    plt.plot(x, p_x, label='Likelihood Function')
    plt.xlabel('Measurement $m$')
    plt.ylabel('Probability Density')
    plt.title('Analytical Likelihood $P(m|s)$')
    plt.legend()
    
def plotMeasurements(sigma, S):
    m = unimodalMeasurements(sigma, S)
    plt.hist(m, bins=50, density=True, alpha=0.5, label='Measurements Histogram')
    plt.xlabel('Measurement $m$')
    plt.ylabel('Density')
    plt.title('Unimodal Measurements Histogram')
    plt.legend()

![image.png](/assets/images/modelsOfbimodal/image 1.png)

![image.png](/assets/images/modelsOfbimodal/image 2.png)

2 Bimodal

graph TD
  A(C) --> B1(C=1)
  A --> b2(C=2)
  B1-->b11(S)
  b11-->b111(m_a)
  b11-->b112(m_v)
  b2-->b21(S_a)
  b2-->b22(S_v)
  b21-->b211(m_a)
  b22-->b212(m_v)

2.1 Fusion (C=1)

2.1.1 Fusion of one interval

S^av,a=S^av,v=σav,a−2ma+σav,v−2mvσav,a−2+σav,v−2:=waSa+wvSv\hat{S}_{av,a}=\hat{S}_{av,v}= \frac{\sigma_{av,a}^{-2} m_a+\sigma_{av,v}^{-2} m_v}{\sigma_{av,a}^{-2} + \sigma_{av,v}^{-2}}\\ := w_aS_a+w_vS_v Ja=1σav,a2Jv=1σav,v2σav2=1J1+J2J_a=\frac{1}{\sigma_{av,a}^{2}} \\ J_v=\frac{1}{\sigma_{av,v}^{2}}\\ \sigma_{av}^2=\frac{1}{J_1+J_2} p(S∣ma,mv)∼p(S)p(ma∣S)p(mv∣S)p(S|m_a,m_v)\sim p(S)p(m_a|S)p(m_v|S)\\ p(S∣ma,mv)∼N(S^av,σav2)p(S|m_a,m_v)\sim N(\hat S_{av},\sigma_{av}^2)\\ p(S∣ma,mv)=12πσavexp⁡(−(S−S^av)22σav2)p(S|m_a,m_v)= \frac{1}{\sqrt{2\pi} \sigma_{av}}\exp(-\frac{(S-\hat S_{av})^2}{2\sigma_{av}^2}) 
def fusionAV(sigmaAV_A,sigmaAV_V, S_a, visualConflict):
    m_a=unimodalMeasurements(sigmaAV_A, S_a)
    S_v=S_a+visualConflict
  m_v = unimodalMeasurements(sigmaAV_V,S_v )  # visual measurement
  # compute the precisons inverse of variances
  J_AV_A= sigmaAV_A**-2 # auditory precision
  J_AV_V=sigmaAV_V**-2 # visual precision
  # compute the fused estimate using reliability weighted averaging
  hat_S_AV= (J_AV_A*m_a+J_AV_V*m_v)/(J_AV_V+J_AV_A)
    mu_Shat = w_a * S_a + w_v * S_v  # fused mean
  sigma_S_AV_hat = np.sqrt(1 / (J_a + J_v))  # fused standard deviation

  return hat_S_AV , sigma_S_AV_hat # belief about fused stimulus value

![image.png](/assets/images/modelsOfbimodal/image 3.png)

2.1.2 Fusion of 2-IFC Duration Difference

Δt−s\Delta_{t-s} is the duration difference between t (test interval) and s (standard interval)

ma,tm_{a,t} = measurement for auditory test duration.

ma,tm_{a,t} = measurement for auditory standard duration.

cc = conflict duration incorporated to the visual standard stimulus.

Δt−s=wa(ma,t−ma,s)+wv(mv,t−mv,s)=waΔSa+wvΔSv\Delta_{t-s}=w_a({m_{a,t}} -m_{a,s})+ w_v ({m_{v,t}} -m_{v,s})\\=w_a\Delta S_a +w_v \Delta S_v ma,t−ma,s∼N(Δsa,2σa2)mv,t−mv,s∼N(Δsv,2σv2) m_{a,t} - m_{a,s} \sim N(\Delta s_a, 2\sigma^2_a)\\ m_{v,t} - m_{v,s} \sim N(\Delta s_v, 2\sigma^2_v)

The difference between two interval within single trial:

Δt−s=wa(ma,t−ma,s)+wv(mv,t−(mv,s))∼N(waΔsa+wvΔsv,2σav2) \Delta_{t-s} = w_a(m_{a,t} - m_{a,s}) + w_v(m_{v,t} - (m_{v,s}))\\ \sim N(w_a\Delta s_a + w_v\Delta s_v, 2\sigma^2_{av}) Δt−s∼N(S^t−S^s,2σav2)\Delta_{t-s} \sim N(\hat S_t - \hat S_s,2\sigma_{av}^2)

Expected value of duration difference

Test Interval:

E[mv,t]=E[ma,t]=Sv,t=Sa,tE[m_{v,t}]=E[m_{a,t}]=S_{v,t}=S_{a,t}

Standard interval:

E[ma,s]=Sv,s=Sa,s+cE[m_{a,s}]=S_{v,s}=S_{a,s}+c

Duration difference:

E[Δt−s]=wa(sa,t−sa,s)+wv(sv,t−sv,s)E[\Delta_{t-s}] = w_a(s_{a,t} - s_{a,s}) + w_v(s_{v,t} - s_{v,s})

substituting:

Sv,s=Sa,s+cS_{v,s}=S_{a,s}+c E[Δt−s]=wa(sa,t−sa,s)+wv[sa,t−(sa,s+c)]=wa(sa,t−sa,s)+wv(sa,t−sa,s−c)=(wa+wv)(sa,t−sa,s)−wvc=(sa,t−sa,s)−wvc\begin{aligned} E[\Delta_{t-s}] &= w_a(s_{a,t} - s_{a,s}) + w_v[s_{a,t} - (s_{a,s} + c)] \\ &= w_a(s_{a,t} - s_{a,s}) + w_v(s_{a,t} - s_{a,s} - c) \\ &= (w_a + w_v)(s_{a,t} - s_{a,s}) - w_v c \\ &= (s_{a,t} - s_{a,s}) - w_v c \end{aligned}

Notice that sum of weights equal to 1 as we assume fusion.

Final predict:

Δt−s=N((Sa,t−Sa,s)−wvc,2σav2)\Delta_{t-s}=N((S_{a,t}-S_{a,s})-w_vc,2\sigma_{av}^2)

wvcw_vc is predicted bias:

  • if c>0 standard visual is longer), the standard is perceived as longer, so test needs to be even longer to be matched.
  • The PSE shifts by wvcw_vc

Decision Rule:

P("test longer")=Φ((Sa,t−Sa,s)−wvc2σav)P(\text{"test longer"}) = \Phi\left(\frac{(S_{a,t} - S_{a,s}) - w_v c}{\sqrt{2}\sigma_{av}}\right)